mixedmirt fits MIRT models using FIML estimation to dichotomous and polytomous
IRT models conditional on fixed and random effect of person and item level covariates.
This can also be understood as 'explanatory IRT' if only fixed effects are modeled, or
multilevel/mixed IRT if random and fixed effects are included. The method uses the MH-RM
algorithm exclusively. Additionally, computation of the log-likelihood can be sped up by
using parallel estimation via mirtCluster.
Usage
mixedmirt(
data,
covdata = NULL,
model = 1,
fixed = ~1,
random = NULL,
itemtype = "Rasch",
lr.fixed = ~1,
lr.random = NULL,
itemdesign = NULL,
constrain = NULL,
pars = NULL,
return.design = FALSE,
SE = TRUE,
internal_constraints = TRUE,
technical = list(SEtol = 1e-04),
...
)Arguments
- data
a
matrixordata.framethat consists of numerically ordered data, organized in the form of integers, with missing data coded asNA- covdata
a
data.framethat consists of thenrow(data)byK'person level' fixed and random predictors. If missing data are present in this object then the observations fromcovdataanddatawill be removed row-wise via therowSums(is.na(covdata)) > 0- model
an object returned from, or a string to be passed to,
mirt.model()to declare how the IRT model is to be estimated. Seemirt.modelandmirtfor more detail- fixed
a right sided R formula for specifying the fixed effect (aka 'explanatory') predictors from
covdataanditemdesign. To estimate the intercepts for each item the keyworditemsis reserved and automatically added to theitemdesigninput. If any polytomous items are being model theitemsare argument is not valid since all intercept parameters are freely estimated and identified with the parameterizations found inmirt, and the first column in the fixed design matrix (commonly the intercept or a reference group) is omitted- random
a right sided formula or list of formulas containing crossed random effects of the form
v1 + ... v_n | G, whereGis the grouping variable andv_nare random numeric predictors within each group. If no intercept value is specified then by default the correlations between thev's andGare estimated, but can be suppressed by including the~ -1 + ...or 0 constant.Gmay contain interaction terms, such asgroup:itemsto include cross or person-level interactions effects- itemtype
same as itemtype in
mirt, except when thefixedorrandominputs are used does not support the following item types:c('PC2PL', 'PC3PL', '2PLNRM', '3PLNRM', '3PLuNRM', '4PLNRM')- lr.fixed
an R formula (or list of formulas) to specify regression effects in the latent variables from the variables in
covdata. This is used to construct models such as the so-called 'latent regression model' to explain person-level ability/trait differences. If a named list of formulas is supplied (where the names correspond to the latent trait names inmodel) then specific regression effects can be estimated for each factor. Supplying a single formula will estimate the regression parameters for all latent traits by default.- lr.random
a list of random effect terms for modeling variability in the latent trait scores, where the syntax uses the same style as in the
randomargument. Useful for building so-called 'multilevel IRT' models which are non-Rasch (multilevel Rasch models do not technically require these because they can be built using thefixedandrandominputs alone)- itemdesign
a
data.frameobject used to create a design matrix for the items, where eachnrow(itemdesign) == nitemsand the number of columns is equal to the number of fixed effect predictors (i.e., item intercepts). By default anitemsvariable is reserved for modeling the item intercept parameters- constrain
a list indicating parameter equality constrains. See
mirtfor more detail- pars
used for parameter starting values. See
mirtfor more detail- return.design
logical; return the design matrices before they have (potentially) been reassigned?
- SE
logical; compute the standard errors by approximating the information matrix using the MHRM algorithm? Default is TRUE
- internal_constraints
logical; use the internally defined constraints for constraining effects across persons and items? Default is TRUE. Setting this to FALSE runs the risk of under-identification
- technical
the technical list passed to the MH-RM estimation engine, with the SEtol default increased to .0001. Additionally, the argument
RANDSTARTis available to indicate at which iteration (during the burn-in stage) the additional random effect variables should begin to be approximated (i.e., elements inlr.randomandrandom). The default forRANDSTARTis to start at iteration 100, and when random effects are included the default number of burn-in iterations is increased from 150 to 200. Seemirtfor further details- ...
additional arguments to be passed to the MH-RM estimation engine. See
mirtfor more details and examples
Value
function returns an object of class MixedClass
(MixedClass-class).
Details
For dichotomous response models, mixedmirt follows the general form
$$P(x = 1|\theta, \psi) = g + \frac{(u - g)}{1 + exp(-1 * [\theta a + X \beta + Z \delta])}$$
where X is a design matrix with associated \(\beta\) fixed effect intercept coefficients,
and Z is a design matrix with associated \(\delta\) random effects for the intercepts.
For simplicity and easier interpretation, the unique item intercept values typically found in
\(X \beta\)
are extracted and reassigned within mirt's 'intercept' parameters (e.g., 'd').
To observe how the design matrices are structured prior to reassignment and estimation pass
the argument return.design = TRUE.
Polytomous IRT models follow a similar format except the item intercepts are automatically
estimated internally, rendering the items argument in the fixed formula redundant and
therefore must be omitted from the specification. If there are a mixture of dichotomous and
polytomous items the intercepts for the dichotomous models are also estimated for consistency.
The decomposition of the \(\theta\) parameters is also possible to form
latent regression and multilevel IRT models by using the lr.fixed and lr.random
inputs. These effects decompose \(\theta\) such that
$$\theta = V \Gamma + W \zeta + \epsilon$$
where V and W are fixed and random effects design matrices for the associated coefficients.
To simulate expected a posteriori predictions for the random effect terms
use the randef function.
References
Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06
Chalmers, R. P. (2015). Extended Mixed-Effects Item Response Models with the MH-RM Algorithm. Journal of Educational Measurement, 52, 200-222. doi:10.1111/jedm.12072
Author
Phil Chalmers rphilip.chalmers@gmail.com
Examples
# \donttest{
# make some data
set.seed(1234)
N <- 750
a <- matrix(rlnorm(10,.3,1),10,1)
d <- matrix(rnorm(10), 10)
Theta <- matrix(sort(rnorm(N)))
pseudoIQ <- Theta * 5 + 100 + rnorm(N, 0 , 5)
pseudoIQ <- (pseudoIQ - mean(pseudoIQ))/10 #rescale variable for numerical stability
group <- factor(rep(c('G1','G2','G3'), each = N/3))
data <- simdata(a,d,N, itemtype = rep('2PL',10), Theta=Theta)
covdata <- data.frame(group, pseudoIQ)
itemstats(data)
#> $overall
#> N mean_total.score sd_total.score ave.r sd.r alpha SEM.alpha
#> 750 4.655 2.346 0.166 0.133 0.671 1.345
#>
#> $itemstats
#> N mean sd total.r total.r_if_rm alpha_if_rm
#> Item_1 750 0.363 0.481 0.368 0.172 0.678
#> Item_2 750 0.335 0.472 0.631 0.485 0.617
#> Item_3 750 0.428 0.495 0.711 0.580 0.594
#> Item_4 750 0.512 0.500 0.234 0.021 0.708
#> Item_5 750 0.628 0.484 0.638 0.490 0.615
#> Item_6 750 0.472 0.500 0.669 0.523 0.607
#> Item_7 750 0.385 0.487 0.471 0.286 0.657
#> Item_8 750 0.301 0.459 0.483 0.312 0.651
#> Item_9 750 0.319 0.466 0.481 0.307 0.652
#> Item_10 750 0.912 0.283 0.295 0.180 0.670
#>
#> $proportions
#> 0 1
#> Item_1 0.637 0.363
#> Item_2 0.665 0.335
#> Item_3 0.572 0.428
#> Item_4 0.488 0.512
#> Item_5 0.372 0.628
#> Item_6 0.528 0.472
#> Item_7 0.615 0.385
#> Item_8 0.699 0.301
#> Item_9 0.681 0.319
#> Item_10 0.088 0.912
#>
# use parallel computing
if(interactive()) mirtCluster()
# specify IRT model
model <- 'Theta = 1-10'
# model with no person predictors
mod0 <- mirt(data, model, itemtype = 'Rasch')
#>
# group as a fixed effect predictor (aka, uniform dif)
mod1 <- mixedmirt(data, covdata, model, fixed = ~ 0 + group + items)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1804, Max-Change = 0.1582, Max-Change = 0.1359, Max-Change = 0.1172, Max-Change = 0.1000, Max-Change = 0.0848, Max-Change = 0.0698, Max-Change = 0.0570, Max-Change = 0.0459, Max-Change = 0.0370, Max-Change = 0.0293, Max-Change = 0.0246, Max-Change = 0.0201, Max-Change = 0.0183, Max-Change = 0.0156, Max-Change = 0.0103, Max-Change = 0.0106, Max-Change = 0.0098, Max-Change = 0.0093, Max-Change = 0.0055, Max-Change = 0.0062, Max-Change = 0.0108, Max-Change = 0.0065, Max-Change = 0.0076, Max-Change = 0.0137, Max-Change = 0.0067, Max-Change = 0.0036, Max-Change = 0.0023, Max-Change = 0.0095, Max-Change = 0.0108, Max-Change = 0.0141, Max-Change = 0.0083, Max-Change = 0.0064, Max-Change = 0.0056, Max-Change = 0.0151, Max-Change = 0.0066, Max-Change = 0.0049, Max-Change = 0.0097, Max-Change = 0.0052, Max-Change = 0.0064, Max-Change = 0.0080, Max-Change = 0.0028, Max-Change = 0.0076, Max-Change = 0.0055, Max-Change = 0.0063, Max-Change = 0.0047, Max-Change = 0.0077, Max-Change = 0.0048, Max-Change = 0.0053, Max-Change = 0.0095, Max-Change = 0.0085, Max-Change = 0.0055, Max-Change = 0.0095, Max-Change = 0.0115, Max-Change = 0.0062, Max-Change = 0.0076, Max-Change = 0.0043, Max-Change = 0.0053, Max-Change = 0.0035, Max-Change = 0.0064, Max-Change = 0.0078, Max-Change = 0.0051, Max-Change = 0.0046, Max-Change = 0.0057, Max-Change = 0.0020, Max-Change = 0.0050, Max-Change = 0.0058, Max-Change = 0.0027, Max-Change = 0.0093, Max-Change = 0.0032, Max-Change = 0.0042, Max-Change = 0.0052, Max-Change = 0.0036, Max-Change = 0.0018, Max-Change = 0.0045, Max-Change = 0.0024, Max-Change = 0.0042, Max-Change = 0.0082, Max-Change = 0.0086, Max-Change = 0.0041, Max-Change = 0.0053, Max-Change = 0.0045, Max-Change = 0.0170, Max-Change = 0.0099, Max-Change = 0.0101, Max-Change = 0.0036, Max-Change = 0.0036, Max-Change = 0.0038, Max-Change = 0.0076, Max-Change = 0.0062, Max-Change = 0.0065, Max-Change = 0.0065, Max-Change = 0.0056, Max-Change = 0.0076, Max-Change = 0.0084, Max-Change = 0.0055, Max-Change = 0.0083, Max-Change = 0.0023, Max-Change = 0.0065, Max-Change = 0.0067, Max-Change = 0.0066, Max-Change = 0.0082, Max-Change = 0.0057, Max-Change = 0.0032, Max-Change = 0.0022, Max-Change = 0.0035, Max-Change = 0.0064, Max-Change = 0.0047, Max-Change = 0.0093, Max-Change = 0.0073, Max-Change = 0.0025, Max-Change = 0.0040, Max-Change = 0.0090, Max-Change = 0.0089, Max-Change = 0.0055, Max-Change = 0.0073, Max-Change = 0.0068, Max-Change = 0.0070, Max-Change = 0.0012, Max-Change = 0.0116, Max-Change = 0.0067, Max-Change = 0.0072, Max-Change = 0.0057, Max-Change = 0.0018, Max-Change = 0.0095, Max-Change = 0.0058, Max-Change = 0.0035, Max-Change = 0.0024, Max-Change = 0.0059, Max-Change = 0.0059, Max-Change = 0.0058, Max-Change = 0.0080, Max-Change = 0.0033, Max-Change = 0.0074, Max-Change = 0.0073, Max-Change = 0.0078, Max-Change = 0.0066, Max-Change = 0.0064, Max-Change = 0.0035, Max-Change = 0.0048, Max-Change = 0.0064, Max-Change = 0.0050, Max-Change = 0.0039, Max-Change = 0.0038, Max-Change = 0.0065, Max-Change = 0.0068, Max-Change = 0.0092, Max-Change = 0.0043, Max-Change = 0.0068, Max-Change = 0.0037, Max-Change = 0.0050, Max-Change = 0.0054, Max-Change = 0.0032, Max-Change = 0.0052, Max-Change = 0.0028, Max-Change = 0.0067, Max-Change = 0.0052, Max-Change = 0.0035, Max-Change = 0.0057, Max-Change = 0.0049, Max-Change = 0.0056, Max-Change = 0.0030, Max-Change = 0.0007, Max-Change = 0.0016, Max-Change = 0.0075, Max-Change = 0.0040, Max-Change = 0.0058, Max-Change = 0.0023, Max-Change = 0.0035, Max-Change = 0.0011, Max-Change = 0.0046, Max-Change = 0.0061, Max-Change = 0.0018, Max-Change = 0.0095, Max-Change = 0.0076, Max-Change = 0.0073, Max-Change = 0.0022, Max-Change = 0.0035, Max-Change = 0.0048, Max-Change = 0.0053, Max-Change = 0.0102, Max-Change = 0.0054, Max-Change = 0.0048, Max-Change = 0.0028, Max-Change = 0.0055, Max-Change = 0.0081, Max-Change = 0.0010, Max-Change = 0.0050, Max-Change = 0.0072, Max-Change = 0.0047, Max-Change = 0.0100, Max-Change = 0.0074, Max-Change = 0.0054, Max-Change = 0.0040, Max-Change = 0.0051, Max-Change = 0.0031, Max-Change = 0.0062, Max-Change = 0.0077, Max-Change = 0.0071, Max-Change = 0.0033, Max-Change = 0.0062, Max-Change = 0.0026, Max-Change = 0.0040, Max-Change = 0.0086, Max-Change = 0.0046, Max-Change = 0.0067, Max-Change = 0.0048, Max-Change = 0.0070, Max-Change = 0.0062, Max-Change = 0.0078, Max-Change = 0.0077, Max-Change = 0.0041, Max-Change = 0.0018, Max-Change = 0.0074, Max-Change = 0.0023, Max-Change = 0.0036, Max-Change = 0.0037, Max-Change = 0.0045, Max-Change = 0.0059, Max-Change = 0.0057, Max-Change = 0.0042, Max-Change = 0.0059, Max-Change = 0.0054, Max-Change = 0.0096, Max-Change = 0.0061, Max-Change = 0.0041, Max-Change = 0.0048, Max-Change = 0.0029, Max-Change = 0.0087, Max-Change = 0.0017, Max-Change = 0.0022, Max-Change = 0.0036, Max-Change = 0.0031, Max-Change = 0.0031, Max-Change = 0.0025, Max-Change = 0.0085, Max-Change = 0.0045, Max-Change = 0.0064, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0048, gam = 0.1057, Max-Change = 0.0032, gam = 0.0780, Max-Change = 0.0010, gam = 0.0629, Max-Change = 0.0022, gam = 0.0532, Max-Change = 0.0012, gam = 0.0464, Max-Change = 0.0022, gam = 0.0413, Max-Change = 0.0008, gam = 0.0374, Max-Change = 0.0012, gam = 0.0342, Max-Change = 0.0016, gam = 0.0316, Max-Change = 0.0006, gam = 0.0294, Max-Change = 0.0006, gam = 0.0276, Max-Change = 0.0006
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
anova(mod0, mod1)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod0 8799.543 8815.435 8819.126 8850.364 -4388.772
#> mod1 8111.326 8130.107 8134.469 8171.387 -4042.663 692.217 2 0
summary(mod1)
#>
#> Call:
#> mixedmirt(data = data, covdata = covdata, model = model, fixed = ~0 +
#> group + items)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> groupG1 -1.858 0.100 -18.495
#> groupG2 -0.748 0.094 -7.979
#> groupG3 0.515 0.094 5.493
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> Theta
#> Theta 0.118
#>
coef(mod1)
#> $Item_1
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 0 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA NA NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA NA NA NA
#>
#> $Item_2
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 -0.152 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA -0.388 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 0.084 NA NA
#>
#> $Item_3
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 0.340 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA 0.109 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 0.572 NA NA
#>
#> $Item_4
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 0.763 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA 0.532 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 0.994 NA NA
#>
#> $Item_5
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 1.353 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA 1.119 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 1.588 NA NA
#>
#> $Item_6
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 0.563 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA 0.332 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 0.793 NA NA
#>
#> $Item_7
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 0.120 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA -0.113 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 0.353 NA NA
#>
#> $Item_8
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 -0.339 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA -0.578 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA -0.101 NA NA
#>
#> $Item_9
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 -0.241 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA -0.478 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA -0.004 NA NA
#>
#> $Item_10
#> groupG1 groupG2 groupG3 a1 d g u
#> par -1.858 -0.748 0.515 1 3.432 0 1
#> CI_2.5 -2.055 -0.931 0.331 NA 3.117 NA NA
#> CI_97.5 -1.661 -0.564 0.699 NA 3.747 NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 0.118
#> CI_2.5 NA 0.072
#> CI_97.5 NA 0.164
#>
if (FALSE) { # \dontrun{
# same model as above in lme4
wide <- data.frame(id=1:nrow(data),data,covdata)
long <- reshape2::melt(wide, id.vars = c('id', 'group', 'pseudoIQ'))
library(lme4)
lmod0 <- glmer(value ~ 0 + variable + (1|id), long, family = binomial)
lmod1 <- glmer(value ~ 0 + group + variable + (1|id), long, family = binomial)
anova(lmod0, lmod1)
} # }
# model using 2PL items instead of Rasch
mod1b <- mixedmirt(data, covdata, model, fixed = ~ 0 + group + items, itemtype = '2PL')
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1860, Max-Change = 0.1696, Max-Change = 0.1434, Max-Change = 0.1281, Max-Change = 0.1024, Max-Change = 0.0980, Max-Change = 0.0816, Max-Change = 0.0806, Max-Change = 0.0636, Max-Change = 0.0644, Max-Change = 0.0457, Max-Change = 0.0484, Max-Change = 0.0573, Max-Change = 0.0544, Max-Change = 0.0444, Max-Change = 0.0289, Max-Change = 0.0242, Max-Change = 0.0313, Max-Change = 0.0332, Max-Change = 0.0317, Max-Change = 0.0343, Max-Change = 0.0348, Max-Change = 0.0514, Max-Change = 0.0715, Max-Change = 0.0273, Max-Change = 0.0443, Max-Change = 0.0254, Max-Change = 0.0261, Max-Change = 0.0364, Max-Change = 0.0200, Max-Change = 0.0392, Max-Change = 0.0491, Max-Change = 0.0365, Max-Change = 0.0491, Max-Change = 0.0339, Max-Change = 0.0376, Max-Change = 0.0239, Max-Change = 0.0464, Max-Change = 0.0513, Max-Change = 0.0378, Max-Change = 0.0793, Max-Change = 0.0370, Max-Change = 0.0447, Max-Change = 0.0359, Max-Change = 0.0696, Max-Change = 0.0790, Max-Change = 0.0264, Max-Change = 0.0346, Max-Change = 0.0233, Max-Change = 0.0334, Max-Change = 0.0345, Max-Change = 0.0441, Max-Change = 0.0441, Max-Change = 0.0380, Max-Change = 0.0255, Max-Change = 0.0391, Max-Change = 0.0394, Max-Change = 0.0293, Max-Change = 0.0720, Max-Change = 0.0462, Max-Change = 0.0417, Max-Change = 0.0285, Max-Change = 0.0245, Max-Change = 0.0603, Max-Change = 0.0343, Max-Change = 0.0166, Max-Change = 0.0561, Max-Change = 0.0314, Max-Change = 0.0257, Max-Change = 0.0314, Max-Change = 0.0342, Max-Change = 0.0598, Max-Change = 0.0402, Max-Change = 0.0349, Max-Change = 0.0471, Max-Change = 0.0189, Max-Change = 0.0324, Max-Change = 0.0257, Max-Change = 0.0750, Max-Change = 0.0510, Max-Change = 0.0721, Max-Change = 0.0339, Max-Change = 0.0547, Max-Change = 0.0469, Max-Change = 0.0607, Max-Change = 0.0375, Max-Change = 0.0417, Max-Change = 0.0378, Max-Change = 0.0610, Max-Change = 0.0517, Max-Change = 0.0244, Max-Change = 0.0349, Max-Change = 0.0239, Max-Change = 0.0503, Max-Change = 0.0232, Max-Change = 0.0854, Max-Change = 0.0282, Max-Change = 0.0397, Max-Change = 0.0406, Max-Change = 0.0820, Max-Change = 0.0267, Max-Change = 0.0839, Max-Change = 0.0307, Max-Change = 0.0237, Max-Change = 0.0285, Max-Change = 0.1084, Max-Change = 0.1160, Max-Change = 0.0390, Max-Change = 0.0355, Max-Change = 0.0703, Max-Change = 0.0391, Max-Change = 0.0954, Max-Change = 0.0553, Max-Change = 0.0308, Max-Change = 0.0652, Max-Change = 0.0169, Max-Change = 0.0664, Max-Change = 0.1173, Max-Change = 0.0311, Max-Change = 0.0509, Max-Change = 0.1200, Max-Change = 0.0904, Max-Change = 0.1736, Max-Change = 0.0696, Max-Change = 0.0348, Max-Change = 0.0707, Max-Change = 0.0487, Max-Change = 0.0574, Max-Change = 0.0342, Max-Change = 0.0850, Max-Change = 0.0477, Max-Change = 0.0656, Max-Change = 0.0731, Max-Change = 0.0842, Max-Change = 0.1089, Max-Change = 0.0266, Max-Change = 0.1092, Max-Change = 0.0731, Max-Change = 0.0200, Max-Change = 0.0487, Max-Change = 0.0573, Max-Change = 0.1227, Max-Change = 0.0633, Max-Change = 0.0499, Max-Change = 0.0307, Max-Change = 0.1556, Max-Change = 0.0620, Max-Change = 0.0445, Max-Change = 0.1713, Max-Change = 0.0844, Max-Change = 0.0719, Max-Change = 0.0258, Max-Change = 0.0742, Max-Change = 0.0565, Max-Change = 0.0339, Max-Change = 0.0182, Max-Change = 0.0373, Max-Change = 0.1227, Max-Change = 0.0404, Max-Change = 0.0894, Max-Change = 0.0172, Max-Change = 0.1133, Max-Change = 0.0502, Max-Change = 0.0338, Max-Change = 0.0758, Max-Change = 0.0635, Max-Change = 0.0280, Max-Change = 0.0781, Max-Change = 0.0398, Max-Change = 0.0805, Max-Change = 0.1266, Max-Change = 0.1815, Max-Change = 0.0498, Max-Change = 0.0387, Max-Change = 0.0311, Max-Change = 0.0463, Max-Change = 0.1127, Max-Change = 0.0519, Max-Change = 0.1120, Max-Change = 0.2000, Max-Change = 0.0178, Max-Change = 0.0509, Max-Change = 0.1411, Max-Change = 0.1258, Max-Change = 0.0486, Max-Change = 0.0713, Max-Change = 0.0751, Max-Change = 0.1150, Max-Change = 0.0370, Max-Change = 0.1355, Max-Change = 0.2000, Max-Change = 0.1107, Max-Change = 0.0404, Max-Change = 0.1141, Max-Change = 0.1374, Max-Change = 0.1567, Max-Change = 0.0502, Max-Change = 0.0305, Max-Change = 0.0899, Max-Change = 0.0245, Max-Change = 0.0828, Max-Change = 0.1181, Max-Change = 0.0241, Max-Change = 0.0989, Max-Change = 0.1326, Max-Change = 0.0390, Max-Change = 0.1871, Max-Change = 0.0252, Max-Change = 0.0234, Max-Change = 0.0791, Max-Change = 0.1058, Max-Change = 0.0712, Max-Change = 0.1591, Max-Change = 0.0433, Max-Change = 0.0432, Max-Change = 0.0488, Max-Change = 0.1257, Max-Change = 0.2000, Max-Change = 0.1857, Max-Change = 0.0790, Max-Change = 0.0391, Max-Change = 0.0559, Max-Change = 0.2000, Max-Change = 0.1776, Max-Change = 0.0181, Max-Change = 0.0494, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1295, Max-Change = 0.1152, Max-Change = 0.2000, Max-Change = 0.1530, Max-Change = 0.0186, Max-Change = 0.1586, Max-Change = 0.0536, Max-Change = 0.0350, Max-Change = 0.0466, Max-Change = 0.0195, Max-Change = 0.0606, Max-Change = 0.1190, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.1087, gam = 0.1057, Max-Change = 0.0780, gam = 0.0780, Max-Change = 0.0283, gam = 0.0629, Max-Change = 0.0549, gam = 0.0532, Max-Change = 0.0181, gam = 0.0464, Max-Change = 0.0171, gam = 0.0413, Max-Change = 0.0038, gam = 0.0374, Max-Change = 0.0220, gam = 0.0342, Max-Change = 0.0112, gam = 0.0316, Max-Change = 0.0208, gam = 0.0294, Max-Change = 0.0122, gam = 0.0276, Max-Change = 0.0115, gam = 0.0260, Max-Change = 0.0075, gam = 0.0246, Max-Change = 0.0027, gam = 0.0233, Max-Change = 0.0056, gam = 0.0222, Max-Change = 0.0078, gam = 0.0212, Max-Change = 0.0072, gam = 0.0203, Max-Change = 0.0164, gam = 0.0195, Max-Change = 0.0044, gam = 0.0188, Max-Change = 0.0097, gam = 0.0181, Max-Change = 0.0156, gam = 0.0175, Max-Change = 0.0020, gam = 0.0169, Max-Change = 0.0043, gam = 0.0164, Max-Change = 0.0047, gam = 0.0159, Max-Change = 0.0035, gam = 0.0154, Max-Change = 0.0101, gam = 0.0150, Max-Change = 0.0149, gam = 0.0146, Max-Change = 0.0065, gam = 0.0142, Max-Change = 0.0078, gam = 0.0139, Max-Change = 0.0128, gam = 0.0135, Max-Change = 0.0101, gam = 0.0132, Max-Change = 0.0038, gam = 0.0129, Max-Change = 0.0029, gam = 0.0126, Max-Change = 0.0119, gam = 0.0124, Max-Change = 0.0033, gam = 0.0121, Max-Change = 0.0080, gam = 0.0119, Max-Change = 0.0169, gam = 0.0116, Max-Change = 0.0034, gam = 0.0114, Max-Change = 0.0056, gam = 0.0112, Max-Change = 0.0060, gam = 0.0110, Max-Change = 0.0023, gam = 0.0108, Max-Change = 0.0043, gam = 0.0106, Max-Change = 0.0020, gam = 0.0104, Max-Change = 0.0057, gam = 0.0102, Max-Change = 0.0074, gam = 0.0101, Max-Change = 0.0063, gam = 0.0099, Max-Change = 0.0016, gam = 0.0098, Max-Change = 0.0024, gam = 0.0096, Max-Change = 0.0020, gam = 0.0095, Max-Change = 0.0067, gam = 0.0093, Max-Change = 0.0017, gam = 0.0092, Max-Change = 0.0083, gam = 0.0091, Max-Change = 0.0015, gam = 0.0089, Max-Change = 0.0056, gam = 0.0088, Max-Change = 0.0027, gam = 0.0087, Max-Change = 0.0033, gam = 0.0086, Max-Change = 0.0011, gam = 0.0085, Max-Change = 0.0055, gam = 0.0084, Max-Change = 0.0023, gam = 0.0082, Max-Change = 0.0023, gam = 0.0081, Max-Change = 0.0026, gam = 0.0080, Max-Change = 0.0037, gam = 0.0080, Max-Change = 0.0017, gam = 0.0079, Max-Change = 0.0048, gam = 0.0078, Max-Change = 0.0043, gam = 0.0077, Max-Change = 0.0037, gam = 0.0076, Max-Change = 0.0094, gam = 0.0075, Max-Change = 0.0067, gam = 0.0074, Max-Change = 0.0068, gam = 0.0073, Max-Change = 0.0047, gam = 0.0073, Max-Change = 0.0012, gam = 0.0072, Max-Change = 0.0018, gam = 0.0071, Max-Change = 0.0062, gam = 0.0070, Max-Change = 0.0020, gam = 0.0070, Max-Change = 0.0033, gam = 0.0069, Max-Change = 0.0012, gam = 0.0068, Max-Change = 0.0033, gam = 0.0068, Max-Change = 0.0062, gam = 0.0067, Max-Change = 0.0011, gam = 0.0066, Max-Change = 0.0029, gam = 0.0066, Max-Change = 0.0041, gam = 0.0065, Max-Change = 0.0012, gam = 0.0065, Max-Change = 0.0057, gam = 0.0064, Max-Change = 0.0021, gam = 0.0064, Max-Change = 0.0071, gam = 0.0063, Max-Change = 0.0066, gam = 0.0062, Max-Change = 0.0013, gam = 0.0062, Max-Change = 0.0006, gam = 0.0061, Max-Change = 0.0019, gam = 0.0061, Max-Change = 0.0009, gam = 0.0060, Max-Change = 0.0007, gam = 0.0060, Max-Change = 0.0083, gam = 0.0059, Max-Change = 0.0034, gam = 0.0059, Max-Change = 0.0028, gam = 0.0058, Max-Change = 0.0063, gam = 0.0058, Max-Change = 0.0073, gam = 0.0058, Max-Change = 0.0037, gam = 0.0057, Max-Change = 0.0018, gam = 0.0057, Max-Change = 0.0017, gam = 0.0056, Max-Change = 0.0037, gam = 0.0056, Max-Change = 0.0027, gam = 0.0055, Max-Change = 0.0012, gam = 0.0055, Max-Change = 0.0006, gam = 0.0055, Max-Change = 0.0031, gam = 0.0054, Max-Change = 0.0048, gam = 0.0054, Max-Change = 0.0019, gam = 0.0053, Max-Change = 0.0032, gam = 0.0053, Max-Change = 0.0005, gam = 0.0053, Max-Change = 0.0049, gam = 0.0052, Max-Change = 0.0029, gam = 0.0052, Max-Change = 0.0023, gam = 0.0052, Max-Change = 0.0056, gam = 0.0051, Max-Change = 0.0060, gam = 0.0051, Max-Change = 0.0031, gam = 0.0051, Max-Change = 0.0050, gam = 0.0050, Max-Change = 0.0049, gam = 0.0050, Max-Change = 0.0009, gam = 0.0050, Max-Change = 0.0036, gam = 0.0049, Max-Change = 0.0055, gam = 0.0049, Max-Change = 0.0032, gam = 0.0049, Max-Change = 0.0043, gam = 0.0048, Max-Change = 0.0007, gam = 0.0048, Max-Change = 0.0024, gam = 0.0048, Max-Change = 0.0019, gam = 0.0048, Max-Change = 0.0025, gam = 0.0047, Max-Change = 0.0025, gam = 0.0047, Max-Change = 0.0007, gam = 0.0047, Max-Change = 0.0005, gam = 0.0046, Max-Change = 0.0008
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
anova(mod1, mod1b) #better with 2PL models using all criteria (as expected, given simdata pars)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod1 8111.326 8130.107 8134.469 8171.387 -4042.663
#> mod1b 7975.487 8007.270 8014.651 8077.128 -3965.743 153.84 9 0
# continuous predictor with group
mod2 <- mixedmirt(data, covdata, model, fixed = ~ 0 + group + items + pseudoIQ)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1807, Max-Change = 0.1593, Max-Change = 0.1350, Max-Change = 0.1155, Max-Change = 0.1001, Max-Change = 0.0861, Max-Change = 0.0702, Max-Change = 0.0585, Max-Change = 0.0465, Max-Change = 0.0362, Max-Change = 0.0292, Max-Change = 0.0245, Max-Change = 0.0166, Max-Change = 0.0178, Max-Change = 0.0144, Max-Change = 0.0085, Max-Change = 0.0130, Max-Change = 0.0096, Max-Change = 0.0103, Max-Change = 0.0061, Max-Change = 0.0049, Max-Change = 0.0084, Max-Change = 0.0113, Max-Change = 0.0152, Max-Change = 0.0156, Max-Change = 0.0068, Max-Change = 0.0083, Max-Change = 0.0068, Max-Change = 0.0079, Max-Change = 0.0075, Max-Change = 0.0149, Max-Change = 0.0082, Max-Change = 0.0035, Max-Change = 0.0157, Max-Change = 0.0115, Max-Change = 0.0088, Max-Change = 0.0055, Max-Change = 0.0128, Max-Change = 0.0083, Max-Change = 0.0046, Max-Change = 0.0066, Max-Change = 0.0040, Max-Change = 0.0086, Max-Change = 0.0020, Max-Change = 0.0069, Max-Change = 0.0051, Max-Change = 0.0081, Max-Change = 0.0034, Max-Change = 0.0156, Max-Change = 0.0074, Max-Change = 0.0066, Max-Change = 0.0084, Max-Change = 0.0088, Max-Change = 0.0151, Max-Change = 0.0069, Max-Change = 0.0178, Max-Change = 0.0042, Max-Change = 0.0129, Max-Change = 0.0094, Max-Change = 0.0068, Max-Change = 0.0076, Max-Change = 0.0032, Max-Change = 0.0093, Max-Change = 0.0067, Max-Change = 0.0026, Max-Change = 0.0060, Max-Change = 0.0056, Max-Change = 0.0064, Max-Change = 0.0079, Max-Change = 0.0038, Max-Change = 0.0052, Max-Change = 0.0105, Max-Change = 0.0046, Max-Change = 0.0047, Max-Change = 0.0059, Max-Change = 0.0055, Max-Change = 0.0067, Max-Change = 0.0072, Max-Change = 0.0068, Max-Change = 0.0074, Max-Change = 0.0033, Max-Change = 0.0105, Max-Change = 0.0133, Max-Change = 0.0075, Max-Change = 0.0087, Max-Change = 0.0052, Max-Change = 0.0040, Max-Change = 0.0056, Max-Change = 0.0094, Max-Change = 0.0095, Max-Change = 0.0116, Max-Change = 0.0043, Max-Change = 0.0044, Max-Change = 0.0038, Max-Change = 0.0096, Max-Change = 0.0055, Max-Change = 0.0047, Max-Change = 0.0035, Max-Change = 0.0089, Max-Change = 0.0055, Max-Change = 0.0059, Max-Change = 0.0088, Max-Change = 0.0103, Max-Change = 0.0040, Max-Change = 0.0033, Max-Change = 0.0102, Max-Change = 0.0063, Max-Change = 0.0033, Max-Change = 0.0072, Max-Change = 0.0095, Max-Change = 0.0037, Max-Change = 0.0092, Max-Change = 0.0103, Max-Change = 0.0058, Max-Change = 0.0068, Max-Change = 0.0071, Max-Change = 0.0047, Max-Change = 0.0138, Max-Change = 0.0077, Max-Change = 0.0070, Max-Change = 0.0086, Max-Change = 0.0091, Max-Change = 0.0099, Max-Change = 0.0031, Max-Change = 0.0124, Max-Change = 0.0060, Max-Change = 0.0074, Max-Change = 0.0038, Max-Change = 0.0101, Max-Change = 0.0044, Max-Change = 0.0073, Max-Change = 0.0048, Max-Change = 0.0057, Max-Change = 0.0058, Max-Change = 0.0079, Max-Change = 0.0054, Max-Change = 0.0106, Max-Change = 0.0074, Max-Change = 0.0023, Max-Change = 0.0120, Max-Change = 0.0061, Max-Change = 0.0063, Max-Change = 0.0069, Max-Change = 0.0075, Max-Change = 0.0097, Max-Change = 0.0032, Max-Change = 0.0077, Max-Change = 0.0058, Max-Change = 0.0041, Max-Change = 0.0022, Max-Change = 0.0034, Max-Change = 0.0067, Max-Change = 0.0071, Max-Change = 0.0055, Max-Change = 0.0043, Max-Change = 0.0030, Max-Change = 0.0056, Max-Change = 0.0043, Max-Change = 0.0119, Max-Change = 0.0053, Max-Change = 0.0059, Max-Change = 0.0087, Max-Change = 0.0026, Max-Change = 0.0041, Max-Change = 0.0052, Max-Change = 0.0077, Max-Change = 0.0074, Max-Change = 0.0031, Max-Change = 0.0066, Max-Change = 0.0108, Max-Change = 0.0058, Max-Change = 0.0067, Max-Change = 0.0018, Max-Change = 0.0069, Max-Change = 0.0096, Max-Change = 0.0059, Max-Change = 0.0066, Max-Change = 0.0018, Max-Change = 0.0049, Max-Change = 0.0040, Max-Change = 0.0106, Max-Change = 0.0046, Max-Change = 0.0021, Max-Change = 0.0047, Max-Change = 0.0034, Max-Change = 0.0067, Max-Change = 0.0070, Max-Change = 0.0061, Max-Change = 0.0059, Max-Change = 0.0024, Max-Change = 0.0069, Max-Change = 0.0041, Max-Change = 0.0036, Max-Change = 0.0061, Max-Change = 0.0058, Max-Change = 0.0057, Max-Change = 0.0072, Max-Change = 0.0068, Max-Change = 0.0071, Max-Change = 0.0056, Max-Change = 0.0109, Max-Change = 0.0047, Max-Change = 0.0063, Max-Change = 0.0082, Max-Change = 0.0020, Max-Change = 0.0107, Max-Change = 0.0032, Max-Change = 0.0050, Max-Change = 0.0109, Max-Change = 0.0056, Max-Change = 0.0076, Max-Change = 0.0024, Max-Change = 0.0072, Max-Change = 0.0038, Max-Change = 0.0062, Max-Change = 0.0041, Max-Change = 0.0049, Max-Change = 0.0037, Max-Change = 0.0042, Max-Change = 0.0044, Max-Change = 0.0017, Max-Change = 0.0054, Max-Change = 0.0066, Max-Change = 0.0098, Max-Change = 0.0067, Max-Change = 0.0066, Max-Change = 0.0087, Max-Change = 0.0095, Max-Change = 0.0073, Max-Change = 0.0092, Max-Change = 0.0051, Max-Change = 0.0043, Max-Change = 0.0059, Max-Change = 0.0057, Max-Change = 0.0041, Max-Change = 0.0052, Max-Change = 0.0076, Max-Change = 0.0058, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0043, gam = 0.1057, Max-Change = 0.0022, gam = 0.0780, Max-Change = 0.0015, gam = 0.0629, Max-Change = 0.0023, gam = 0.0532, Max-Change = 0.0019, gam = 0.0464, Max-Change = 0.0010, gam = 0.0413, Max-Change = 0.0004, gam = 0.0374, Max-Change = 0.0012, gam = 0.0342, Max-Change = 0.0014, gam = 0.0316, Max-Change = 0.0007, gam = 0.0294, Max-Change = 0.0007, gam = 0.0276, Max-Change = 0.0006
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod2)
#>
#> Call:
#> mixedmirt(data = data, covdata = covdata, model = model, fixed = ~0 +
#> group + items + pseudoIQ)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> groupG1 -1.711 0.105 -16.317
#> groupG2 -0.750 0.093 -8.029
#> groupG3 0.366 0.099 3.714
#> pseudoIQ 0.268 0.058 4.594
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> Theta
#> Theta 0.105
#>
anova(mod1b, mod2)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod1b 7975.487 8007.270 8014.651 8077.128 -3965.743
#> mod2 8090.269 8110.494 8115.192 8154.950 -4031.134 -130.782 -8 NaN
# view fixed design matrix with and without unique item level intercepts
withint <- mixedmirt(data, covdata, model, fixed = ~ 0 + items + group, return.design = TRUE)
withoutint <- mixedmirt(data, covdata, model, fixed = ~ 0 + group, return.design = TRUE)
# notice that in result above, the intercepts 'items1 to items 10' were reassigned to 'd'
head(withint$X)
#> items1 items2 items3 items4 items5 items6 items7 items8 items9 items10
#> 1.1 1 0 0 0 0 0 0 0 0 0
#> 2.1 1 0 0 0 0 0 0 0 0 0
#> 3.1 1 0 0 0 0 0 0 0 0 0
#> 4.1 1 0 0 0 0 0 0 0 0 0
#> 5.1 1 0 0 0 0 0 0 0 0 0
#> 6.1 1 0 0 0 0 0 0 0 0 0
#> groupG2 groupG3
#> 1.1 0 0
#> 2.1 0 0
#> 3.1 0 0
#> 4.1 0 0
#> 5.1 0 0
#> 6.1 0 0
tail(withint$X)
#> items1 items2 items3 items4 items5 items6 items7 items8 items9 items10
#> 745.10 0 0 0 0 0 0 0 0 0 1
#> 746.10 0 0 0 0 0 0 0 0 0 1
#> 747.10 0 0 0 0 0 0 0 0 0 1
#> 748.10 0 0 0 0 0 0 0 0 0 1
#> 749.10 0 0 0 0 0 0 0 0 0 1
#> 750.10 0 0 0 0 0 0 0 0 0 1
#> groupG2 groupG3
#> 745.10 0 1
#> 746.10 0 1
#> 747.10 0 1
#> 748.10 0 1
#> 749.10 0 1
#> 750.10 0 1
head(withoutint$X) # no intercepts design here to be reassigned into item intercepts
#> groupG1 groupG2 groupG3
#> 1.1 1 0 0
#> 2.1 1 0 0
#> 3.1 1 0 0
#> 4.1 1 0 0
#> 5.1 1 0 0
#> 6.1 1 0 0
tail(withoutint$X)
#> groupG1 groupG2 groupG3
#> 745.10 0 0 1
#> 746.10 0 0 1
#> 747.10 0 0 1
#> 748.10 0 0 1
#> 749.10 0 0 1
#> 750.10 0 0 1
###################################################
### random effects
# make the number of groups much larger
covdata$group <- factor(rep(paste0('G',1:50), each = N/50))
# random groups
rmod1 <- mixedmirt(data, covdata, 1, fixed = ~ 0 + items, random = ~ 1|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1864, Max-Change = 0.1623, Max-Change = 0.1436, Max-Change = 0.1243, Max-Change = 0.1035, Max-Change = 0.0953, Max-Change = 0.0862, Max-Change = 0.0730, Max-Change = 0.0612, Max-Change = 0.0490, Max-Change = 0.0426, Max-Change = 0.0342, Max-Change = 0.0359, Max-Change = 0.0255, Max-Change = 0.0251, Max-Change = 0.0175, Max-Change = 0.0105, Max-Change = 0.0121, Max-Change = 0.0184, Max-Change = 0.0194, Max-Change = 0.0092, Max-Change = 0.0064, Max-Change = 0.0083, Max-Change = 0.0065, Max-Change = 0.0026, Max-Change = 0.0066, Max-Change = 0.0110, Max-Change = 0.0004, Max-Change = 0.0044, Max-Change = 0.0104, Max-Change = 0.0118, Max-Change = 0.0071, Max-Change = 0.0056, Max-Change = 0.0073, Max-Change = 0.0134, Max-Change = 0.0043, Max-Change = 0.0143, Max-Change = 0.0042, Max-Change = 0.0109, Max-Change = 0.0061, Max-Change = 0.0096, Max-Change = 0.0130, Max-Change = 0.0158, Max-Change = 0.0124, Max-Change = 0.0075, Max-Change = 0.0080, Max-Change = 0.0056, Max-Change = 0.0025, Max-Change = 0.0109, Max-Change = 0.0030, Max-Change = 0.0135, Max-Change = 0.0156, Max-Change = 0.0136, Max-Change = 0.0085, Max-Change = 0.0083, Max-Change = 0.0118, Max-Change = 0.0017, Max-Change = 0.0074, Max-Change = 0.0278, Max-Change = 0.0030, Max-Change = 0.0086, Max-Change = 0.0053, Max-Change = 0.0006, Max-Change = 0.0139, Max-Change = 0.0106, Max-Change = 0.0060, Max-Change = 0.0048, Max-Change = 0.0084, Max-Change = 0.0076, Max-Change = 0.0094, Max-Change = 0.0090, Max-Change = 0.0120, Max-Change = 0.0036, Max-Change = 0.0029, Max-Change = 0.0089, Max-Change = 0.0036, Max-Change = 0.0030, Max-Change = 0.0074, Max-Change = 0.0044, Max-Change = 0.0153, Max-Change = 0.0043, Max-Change = 0.0295, Max-Change = 0.0054, Max-Change = 0.0105, Max-Change = 0.0154, Max-Change = 0.0094, Max-Change = 0.0021, Max-Change = 0.0077, Max-Change = 0.0113, Max-Change = 0.0035, Max-Change = 0.0045, Max-Change = 0.0065, Max-Change = 0.2000, Max-Change = 0.1344, Max-Change = 0.0744, Max-Change = 0.0786, Max-Change = 0.0581, Max-Change = 0.0426, Max-Change = 0.0315, Max-Change = 0.0381, Max-Change = 0.0227, Max-Change = 0.0228, Max-Change = 0.0159, Max-Change = 0.0193, Max-Change = 0.0112, Max-Change = 0.0092, Max-Change = 0.0150, Max-Change = 0.0126, Max-Change = 0.0093, Max-Change = 0.0185, Max-Change = 0.0088, Max-Change = 0.0108, Max-Change = 0.0167, Max-Change = 0.0128, Max-Change = 0.0118, Max-Change = 0.0094, Max-Change = 0.0056, Max-Change = 0.0258, Max-Change = 0.0316, Max-Change = 0.0118, Max-Change = 0.0106, Max-Change = 0.0198, Max-Change = 0.0252, Max-Change = 0.0309, Max-Change = 0.0159, Max-Change = 0.0162, Max-Change = 0.0087, Max-Change = 0.0045, Max-Change = 0.0110, Max-Change = 0.0057, Max-Change = 0.0069, Max-Change = 0.0039, Max-Change = 0.0225, Max-Change = 0.0115, Max-Change = 0.0144, Max-Change = 0.0102, Max-Change = 0.0078, Max-Change = 0.0102, Max-Change = 0.0065, Max-Change = 0.0275, Max-Change = 0.0248, Max-Change = 0.0045, Max-Change = 0.0114, Max-Change = 0.0065, Max-Change = 0.0160, Max-Change = 0.0145, Max-Change = 0.0068, Max-Change = 0.0129, Max-Change = 0.0116, Max-Change = 0.0161, Max-Change = 0.0138, Max-Change = 0.0076, Max-Change = 0.0139, Max-Change = 0.0116, Max-Change = 0.0067, Max-Change = 0.0162, Max-Change = 0.0228, Max-Change = 0.0090, Max-Change = 0.0079, Max-Change = 0.0054, Max-Change = 0.0223, Max-Change = 0.0171, Max-Change = 0.0075, Max-Change = 0.0105, Max-Change = 0.0246, Max-Change = 0.0028, Max-Change = 0.0061, Max-Change = 0.0095, Max-Change = 0.0026, Max-Change = 0.0123, Max-Change = 0.0207, Max-Change = 0.0063, Max-Change = 0.0044, Max-Change = 0.0084, Max-Change = 0.0051, Max-Change = 0.0050, Max-Change = 0.0085, Max-Change = 0.0051, Max-Change = 0.0207, Max-Change = 0.0080, Max-Change = 0.0121, Max-Change = 0.0258, Max-Change = 0.0036, Max-Change = 0.0144, Max-Change = 0.0108, Max-Change = 0.0057, Max-Change = 0.0055, Max-Change = 0.0196, Max-Change = 0.0071, Max-Change = 0.0062, Max-Change = 0.0077, Max-Change = 0.0068, Max-Change = 0.0187, Max-Change = 0.0174, Max-Change = 0.0161, Max-Change = 0.0129, Max-Change = 0.0047, Max-Change = 0.0088, Max-Change = 0.0096, Max-Change = 0.0171, Max-Change = 0.0073, Max-Change = 0.0098, Max-Change = 0.0079, Max-Change = 0.0198, Max-Change = 0.0111, Max-Change = 0.0425, Max-Change = 0.0146, Max-Change = 0.0073, Max-Change = 0.0110, Max-Change = 0.0258, Max-Change = 0.0168, Max-Change = 0.0345, Max-Change = 0.0010, Max-Change = 0.0187, Max-Change = 0.0113, Max-Change = 0.0134, Max-Change = 0.0198, Max-Change = 0.0027, Max-Change = 0.0127, Max-Change = 0.0065, Max-Change = 0.0132, Max-Change = 0.0139, Max-Change = 0.0045, Max-Change = 0.0400, Max-Change = 0.0098, Max-Change = 0.0053, Max-Change = 0.0015, Max-Change = 0.0103, Max-Change = 0.0102, Max-Change = 0.0054, Max-Change = 0.0199, Max-Change = 0.0100, Max-Change = 0.0041, Max-Change = 0.0184, Max-Change = 0.0145, Max-Change = 0.0103, Max-Change = 0.0038, Max-Change = 0.0143, Max-Change = 0.0152, Max-Change = 0.0110, Max-Change = 0.0068, Max-Change = 0.0118, Max-Change = 0.0066, Max-Change = 0.0130, Max-Change = 0.0208, Max-Change = 0.0037, Max-Change = 0.0121, Max-Change = 0.0141, Max-Change = 0.0072, Max-Change = 0.0143, Max-Change = 0.0123, Max-Change = 0.0070, Max-Change = 0.0035, Max-Change = 0.0055, Max-Change = 0.0196, Max-Change = 0.0027, Max-Change = 0.0068, Max-Change = 0.0078, Max-Change = 0.0070, Max-Change = 0.0023, Max-Change = 0.0069, Max-Change = 0.0069, Max-Change = 0.0115, Max-Change = 0.0052, Max-Change = 0.0048, Max-Change = 0.0232, Max-Change = 0.0055, Max-Change = 0.0128, Max-Change = 0.0051, Max-Change = 0.0093, Max-Change = 0.0252, Max-Change = 0.0142, Max-Change = 0.0101, Max-Change = 0.0092, Max-Change = 0.0333, Max-Change = 0.0122, Max-Change = 0.0084, Max-Change = 0.0098, Max-Change = 0.0081, Max-Change = 0.0300, Max-Change = 0.0306, Max-Change = 0.0112, Max-Change = 0.0048, Max-Change = 0.0091, Max-Change = 0.0088, Max-Change = 0.0263, Max-Change = 0.0100, Max-Change = 0.0118, Max-Change = 0.0103, Max-Change = 0.0021, Max-Change = 0.0084, Max-Change = 0.0130, Max-Change = 0.0080, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0151, gam = 0.1057, Max-Change = 0.0072, gam = 0.0780, Max-Change = 0.0048, gam = 0.0629, Max-Change = 0.0014, gam = 0.0532, Max-Change = 0.0047, gam = 0.0464, Max-Change = 0.0016, gam = 0.0413, Max-Change = 0.0021, gam = 0.0374, Max-Change = 0.0007, gam = 0.0342, Max-Change = 0.0026, gam = 0.0316, Max-Change = 0.0035, gam = 0.0294, Max-Change = 0.0018, gam = 0.0276, Max-Change = 0.0009, gam = 0.0260, Max-Change = 0.0013, gam = 0.0246, Max-Change = 0.0007, gam = 0.0233, Max-Change = 0.0016, gam = 0.0222, Max-Change = 0.0012, gam = 0.0212, Max-Change = 0.0011, gam = 0.0203, Max-Change = 0.0024, gam = 0.0195, Max-Change = 0.0020, gam = 0.0188, Max-Change = 0.0010, gam = 0.0181, Max-Change = 0.0013, gam = 0.0175, Max-Change = 0.0003, gam = 0.0169, Max-Change = 0.0011, gam = 0.0164, Max-Change = 0.0008, gam = 0.0159, Max-Change = 0.0007, gam = 0.0154, Max-Change = 0.0008
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(rmod1)
#>
#> Call:
#> mixedmirt(data = data, covdata = covdata, model = 1, fixed = ~0 +
#> items, random = ~1 | group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 0.062
#>
#> $group
#> COV_group
#> COV_group 1.11
#>
coef(rmod1)
#> $Item_1
#> a1 d g u
#> par 1 -0.713 0 1
#> CI_2.5 NA -0.927 NA NA
#> CI_97.5 NA -0.500 NA NA
#>
#> $Item_2
#> a1 d g u
#> par 1 -0.867 0 1
#> CI_2.5 NA -1.081 NA NA
#> CI_97.5 NA -0.653 NA NA
#>
#> $Item_3
#> a1 d g u
#> par 1 -0.370 0 1
#> CI_2.5 NA -0.583 NA NA
#> CI_97.5 NA -0.156 NA NA
#>
#> $Item_4
#> a1 d g u
#> par 1 0.057 0 1
#> CI_2.5 NA -0.161 NA NA
#> CI_97.5 NA 0.276 NA NA
#>
#> $Item_5
#> a1 d g u
#> par 1 0.654 0 1
#> CI_2.5 NA 0.422 NA NA
#> CI_97.5 NA 0.886 NA NA
#>
#> $Item_6
#> a1 d g u
#> par 1 -0.145 0 1
#> CI_2.5 NA -0.361 NA NA
#> CI_97.5 NA 0.071 NA NA
#>
#> $Item_7
#> a1 d g u
#> par 1 -0.592 0 1
#> CI_2.5 NA -0.805 NA NA
#> CI_97.5 NA -0.379 NA NA
#>
#> $Item_8
#> a1 d g u
#> par 1 -1.057 0 1
#> CI_2.5 NA -1.273 NA NA
#> CI_97.5 NA -0.842 NA NA
#>
#> $Item_9
#> a1 d g u
#> par 1 -0.957 0 1
#> CI_2.5 NA -1.172 NA NA
#> CI_97.5 NA -0.743 NA NA
#>
#> $Item_10
#> a1 d g u
#> par 1 2.769 0 1
#> CI_2.5 NA 2.425 NA NA
#> CI_97.5 NA 3.113 NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 0.062
#> CI_2.5 NA 0.004
#> CI_97.5 NA 0.120
#>
#> $group
#> COV_group_group
#> par 1.112
#> CI_2.5 0.631
#> CI_97.5 1.592
#>
# random groups and random items
rmod2 <- mixedmirt(data, covdata, 1, random = list(~ 1|group, ~ 1|items))
#> , Max-Change = 0.0274, Max-Change = 0.0217, Max-Change = 0.0087, Max-Change = 0.0228, Max-Change = 0.0182, Max-Change = 0.0177, Max-Change = 0.0220, Max-Change = 0.0255, Max-Change = 0.0132, Max-Change = 0.0106, Max-Change = 0.0070, Max-Change = 0.0106, Max-Change = 0.0120, Max-Change = 0.0060, Max-Change = 0.0116, Max-Change = 0.0019, Max-Change = 0.0029, Max-Change = 0.0069, Max-Change = 0.0130, Max-Change = 0.0098, Max-Change = 0.0232, Max-Change = 0.0104, Max-Change = 0.0034, Max-Change = 0.0054, Max-Change = 0.0075, Max-Change = 0.0051, Max-Change = 0.0113, Max-Change = 0.0033, Max-Change = 0.0032, Max-Change = 0.0056, Max-Change = 0.0037, Max-Change = 0.0047, Max-Change = 0.0051, Max-Change = 0.0028, Max-Change = 0.0024, Max-Change = 0.0006, Max-Change = 0.0045, Max-Change = 0.0037, Max-Change = 0.0007, Max-Change = 0.0013, Max-Change = 0.0057, Max-Change = 0.0169, Max-Change = 0.0131, Max-Change = 0.0095, Max-Change = 0.0038, Max-Change = 0.0032, Max-Change = 0.0049, Max-Change = 0.0047, Max-Change = 0.0137, Max-Change = 0.0022, Max-Change = 0.0077, Max-Change = 0.0074, Max-Change = 0.0048, Max-Change = 0.0044, Max-Change = 0.0044, Max-Change = 0.0059, Max-Change = 0.0017, Max-Change = 0.0037, Max-Change = 0.0062, Max-Change = 0.0062, Max-Change = 0.0026, Max-Change = 0.0063, Max-Change = 0.0057, Max-Change = 0.0151, Max-Change = 0.0044, Max-Change = 0.0059, Max-Change = 0.0086, Max-Change = 0.0036, Max-Change = 0.0044, Max-Change = 0.0054, Max-Change = 0.0030, Max-Change = 0.0035, Max-Change = 0.0056, Max-Change = 0.0004, Max-Change = 0.0009, Max-Change = 0.0027, Max-Change = 0.0004, Max-Change = 0.0020, Max-Change = 0.0066, Max-Change = 0.0013, Max-Change = 0.0023, Max-Change = 0.0045, Max-Change = 0.0076, Max-Change = 0.0024, Max-Change = 0.0015, Max-Change = 0.0012, Max-Change = 0.0033, Max-Change = 0.0040, Max-Change = 0.0105, Max-Change = 0.0004, Max-Change = 0.0064, Max-Change = 0.0121, Max-Change = 0.0016, Max-Change = 0.0090, Max-Change = 0.0018, Max-Change = 0.0069, Max-Change = 0.0070, Max-Change = 0.0055, Max-Change = 0.0037, Max-Change = 0.0504, Max-Change = 0.1569, Max-Change = 0.0713, Max-Change = 0.0281, Max-Change = 0.0327, Max-Change = 0.0567, Max-Change = 0.0160, Max-Change = 0.0127, Max-Change = 0.0138, Max-Change = 0.0226, Max-Change = 0.0120, Max-Change = 0.0178, Max-Change = 0.0122, Max-Change = 0.0057, Max-Change = 0.0087, Max-Change = 0.0137, Max-Change = 0.0088, Max-Change = 0.0196, Max-Change = 0.0088, Max-Change = 0.0167, Max-Change = 0.0123, Max-Change = 0.0119, Max-Change = 0.0139, Max-Change = 0.0092, Max-Change = 0.0199, Max-Change = 0.0132, Max-Change = 0.0030, Max-Change = 0.0045, Max-Change = 0.0065, Max-Change = 0.0194, Max-Change = 0.0114, Max-Change = 0.0146, Max-Change = 0.0089, Max-Change = 0.0129, Max-Change = 0.0156, Max-Change = 0.0131, Max-Change = 0.0130, Max-Change = 0.0087, Max-Change = 0.0086, Max-Change = 0.0054, Max-Change = 0.0052, Max-Change = 0.0106, Max-Change = 0.0146, Max-Change = 0.0035, Max-Change = 0.0120, Max-Change = 0.0061, Max-Change = 0.0061, Max-Change = 0.0023, Max-Change = 0.0062, Max-Change = 0.0086, Max-Change = 0.0083, Max-Change = 0.0035, Max-Change = 0.0077, Max-Change = 0.0064, Max-Change = 0.0033, Max-Change = 0.0061, Max-Change = 0.0065, Max-Change = 0.0129, Max-Change = 0.0230, Max-Change = 0.0068, Max-Change = 0.0033, Max-Change = 0.0121, Max-Change = 0.0015, Max-Change = 0.0018, Max-Change = 0.0051, Max-Change = 0.0016, Max-Change = 0.0140, Max-Change = 0.0063, Max-Change = 0.0193, Max-Change = 0.0179, Max-Change = 0.0064, Max-Change = 0.0051, Max-Change = 0.0061, Max-Change = 0.0067, Max-Change = 0.0110, Max-Change = 0.0131, Max-Change = 0.0151, Max-Change = 0.0057, Max-Change = 0.0098, Max-Change = 0.0081, Max-Change = 0.0068, Max-Change = 0.0101, Max-Change = 0.0025, Max-Change = 0.0068, Max-Change = 0.0015, Max-Change = 0.0160, Max-Change = 0.0077, Max-Change = 0.0060, Max-Change = 0.0057, Max-Change = 0.0068, Max-Change = 0.0053, Max-Change = 0.0143, Max-Change = 0.0049, Max-Change = 0.0012, Max-Change = 0.0075, Max-Change = 0.0059, Max-Change = 0.0126, Max-Change = 0.0084, Max-Change = 0.0163, Max-Change = 0.0068, Max-Change = 0.0511, Max-Change = 0.0088, Max-Change = 0.0075, Max-Change = 0.0063, Max-Change = 0.0070, Max-Change = 0.0058, Max-Change = 0.0064, Max-Change = 0.0086, Max-Change = 0.0082, Max-Change = 0.0145, Max-Change = 0.0049, Max-Change = 0.0087, Max-Change = 0.0072, Max-Change = 0.0055, Max-Change = 0.0123, Max-Change = 0.0124, Max-Change = 0.0067, Max-Change = 0.0308, Max-Change = 0.0055, Max-Change = 0.0237, Max-Change = 0.0222, Max-Change = 0.0140, Max-Change = 0.0075, Max-Change = 0.0180, Max-Change = 0.0111, Max-Change = 0.0081, Max-Change = 0.0086, Max-Change = 0.0068, Max-Change = 0.0053, Max-Change = 0.0135, Max-Change = 0.0094, Max-Change = 0.0109, Max-Change = 0.0093, Max-Change = 0.0060, Max-Change = 0.0103, Max-Change = 0.0121, Max-Change = 0.0125, Max-Change = 0.0084, Max-Change = 0.0025, Max-Change = 0.0033, Max-Change = 0.0045, Max-Change = 0.0197, Max-Change = 0.0107, Max-Change = 0.0052, Max-Change = 0.0106, Max-Change = 0.0027, Max-Change = 0.0060, Max-Change = 0.0156, Max-Change = 0.0083, Max-Change = 0.0089, Max-Change = 0.0035, Max-Change = 0.0071, Max-Change = 0.0058, Max-Change = 0.0088, Max-Change = 0.0106, Max-Change = 0.0061, Max-Change = 0.0024, Max-Change = 0.0024, Max-Change = 0.0082, Max-Change = 0.0089, Max-Change = 0.0078, Max-Change = 0.0087, Max-Change = 0.0008, Max-Change = 0.0146, Max-Change = 0.0082, Max-Change = 0.0179, Max-Change = 0.0087, Max-Change = 0.0155, Max-Change = 0.0085, Max-Change = 0.0156, Max-Change = 0.0047, Max-Change = 0.0068, Max-Change = 0.0045, Max-Change = 0.0033, Max-Change = 0.0102, Max-Change = 0.0098, Max-Change = 0.0087, Max-Change = 0.0054, Max-Change = 0.0097, Max-Change = 0.0142, Max-Change = 0.0110, Max-Change = 0.0107, Max-Change = 0.0042, Max-Change = 0.0030, Max-Change = 0.0054, Max-Change = 0.0063, Max-Change = 0.0173, Max-Change = 0.0087, Max-Change = 0.0167, Max-Change = 0.0098, Max-Change = 0.0046, Max-Change = 0.0170, Max-Change = 0.0173, Max-Change = 0.0085, Max-Change = 0.0076, Max-Change = 0.0287, Max-Change = 0.0082, Max-Change = 0.0072, Max-Change = 0.0086, Max-Change = 0.0074, Max-Change = 0.0086, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0048, gam = 0.1057, Max-Change = 0.0035, gam = 0.0780, Max-Change = 0.0012, gam = 0.0629, Max-Change = 0.0029, gam = 0.0532, Max-Change = 0.0044, gam = 0.0464, Max-Change = 0.0044, gam = 0.0413, Max-Change = 0.0006, gam = 0.0374, Max-Change = 0.0018, gam = 0.0342, Max-Change = 0.0023, gam = 0.0316, Max-Change = 0.0016, gam = 0.0294, Max-Change = 0.0028, gam = 0.0276, Max-Change = 0.0004, gam = 0.0260, Max-Change = 0.0009, gam = 0.0246, Max-Change = 0.0004
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(rmod2)
#>
#> Call:
#> mixedmirt(data = data, covdata = covdata, model = 1, random = list(~1 |
#> group, ~1 | items))
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> (Intercept) -0.605 0.013 -47.455
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 0.0593
#>
#> $group
#> COV_group
#> COV_group 0.969
#>
#> $items
#> COV_items
#> COV_items 0.694
#>
eff <- randef(rmod2) #estimate random effects
# random slopes with fixed intercepts (suppressed correlation)
rmod3 <- mixedmirt(data, covdata, 1, fixed = ~ 0 + items, random = ~ -1 + pseudoIQ|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1864, Max-Change = 0.1623, Max-Change = 0.1436, Max-Change = 0.1243, Max-Change = 0.1035, Max-Change = 0.0953, Max-Change = 0.0862, Max-Change = 0.0730, Max-Change = 0.0612, Max-Change = 0.0490, Max-Change = 0.0426, Max-Change = 0.0342, Max-Change = 0.0359, Max-Change = 0.0255, Max-Change = 0.0251, Max-Change = 0.0175, Max-Change = 0.0105, Max-Change = 0.0121, Max-Change = 0.0184, Max-Change = 0.0194, Max-Change = 0.0092, Max-Change = 0.0064, Max-Change = 0.0083, Max-Change = 0.0065, Max-Change = 0.0026, Max-Change = 0.0066, Max-Change = 0.0110, Max-Change = 0.0004, Max-Change = 0.0044, Max-Change = 0.0104, Max-Change = 0.0118, Max-Change = 0.0071, Max-Change = 0.0056, Max-Change = 0.0073, Max-Change = 0.0134, Max-Change = 0.0043, Max-Change = 0.0143, Max-Change = 0.0042, Max-Change = 0.0109, Max-Change = 0.0061, Max-Change = 0.0096, Max-Change = 0.0130, Max-Change = 0.0158, Max-Change = 0.0124, Max-Change = 0.0075, Max-Change = 0.0080, Max-Change = 0.0056, Max-Change = 0.0025, Max-Change = 0.0109, Max-Change = 0.0030, Max-Change = 0.0135, Max-Change = 0.0156, Max-Change = 0.0136, Max-Change = 0.0085, Max-Change = 0.0083, Max-Change = 0.0118, Max-Change = 0.0017, Max-Change = 0.0074, Max-Change = 0.0278, Max-Change = 0.0030, Max-Change = 0.0086, Max-Change = 0.0053, Max-Change = 0.0006, Max-Change = 0.0139, Max-Change = 0.0106, Max-Change = 0.0060, Max-Change = 0.0048, Max-Change = 0.0084, Max-Change = 0.0076, Max-Change = 0.0094, Max-Change = 0.0090, Max-Change = 0.0120, Max-Change = 0.0036, Max-Change = 0.0029, Max-Change = 0.0089, Max-Change = 0.0036, Max-Change = 0.0030, Max-Change = 0.0074, Max-Change = 0.0044, Max-Change = 0.0153, Max-Change = 0.0043, Max-Change = 0.0295, Max-Change = 0.0054, Max-Change = 0.0105, Max-Change = 0.0154, Max-Change = 0.0094, Max-Change = 0.0021, Max-Change = 0.0077, Max-Change = 0.0113, Max-Change = 0.0035, Max-Change = 0.0045, Max-Change = 0.0065, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.0620, Max-Change = 0.0394, Max-Change = 0.0281, Max-Change = 0.0205, Max-Change = 0.0199, Max-Change = 0.0517, Max-Change = 0.0185, Max-Change = 0.0067, Max-Change = 0.0141, Max-Change = 0.0170, Max-Change = 0.0190, Max-Change = 0.0213, Max-Change = 0.0206, Max-Change = 0.0102, Max-Change = 0.0144, Max-Change = 0.0149, Max-Change = 0.0186, Max-Change = 0.0133, Max-Change = 0.0153, Max-Change = 0.0083, Max-Change = 0.0201, Max-Change = 0.0119, Max-Change = 0.0101, Max-Change = 0.0102, Max-Change = 0.0116, Max-Change = 0.0103, Max-Change = 0.0138, Max-Change = 0.0114, Max-Change = 0.0096, Max-Change = 0.0143, Max-Change = 0.0112, Max-Change = 0.0052, Max-Change = 0.0129, Max-Change = 0.0050, Max-Change = 0.0069, Max-Change = 0.0115, Max-Change = 0.0161, Max-Change = 0.0211, Max-Change = 0.0159, Max-Change = 0.0214, Max-Change = 0.0115, Max-Change = 0.0161, Max-Change = 0.0133, Max-Change = 0.0057, Max-Change = 0.0090, Max-Change = 0.0073, Max-Change = 0.0074, Max-Change = 0.0091, Max-Change = 0.0059, Max-Change = 0.0164, Max-Change = 0.0378, Max-Change = 0.0162, Max-Change = 0.0071, Max-Change = 0.0116, Max-Change = 0.0195, Max-Change = 0.0175, Max-Change = 0.0311, Max-Change = 0.0127, Max-Change = 0.0213, Max-Change = 0.0165, Max-Change = 0.0228, Max-Change = 0.0124, Max-Change = 0.0231, Max-Change = 0.0031, Max-Change = 0.0051, Max-Change = 0.0071, Max-Change = 0.0128, Max-Change = 0.0067, Max-Change = 0.0103, Max-Change = 0.0062, Max-Change = 0.0078, Max-Change = 0.0059, Max-Change = 0.0112, Max-Change = 0.0211, Max-Change = 0.0079, Max-Change = 0.0113, Max-Change = 0.0155, Max-Change = 0.0214, Max-Change = 0.0076, Max-Change = 0.0141, Max-Change = 0.0077, Max-Change = 0.0132, Max-Change = 0.0065, Max-Change = 0.0096, Max-Change = 0.0197, Max-Change = 0.0292, Max-Change = 0.0085, Max-Change = 0.0121, Max-Change = 0.0045, Max-Change = 0.0164, Max-Change = 0.0094, Max-Change = 0.0174, Max-Change = 0.0066, Max-Change = 0.0051, Max-Change = 0.0148, Max-Change = 0.0141, Max-Change = 0.0092, Max-Change = 0.0097, Max-Change = 0.0092, Max-Change = 0.0047, Max-Change = 0.0197, Max-Change = 0.0115, Max-Change = 0.0174, Max-Change = 0.0189, Max-Change = 0.0048, Max-Change = 0.0090, Max-Change = 0.0168, Max-Change = 0.0043, Max-Change = 0.0088, Max-Change = 0.0175, Max-Change = 0.0223, Max-Change = 0.0080, Max-Change = 0.0114, Max-Change = 0.0086, Max-Change = 0.0130, Max-Change = 0.0102, Max-Change = 0.0142, Max-Change = 0.0136, Max-Change = 0.0089, Max-Change = 0.0116, Max-Change = 0.0048, Max-Change = 0.0155, Max-Change = 0.0401, Max-Change = 0.0167, Max-Change = 0.0126, Max-Change = 0.0013, Max-Change = 0.0050, Max-Change = 0.0060, Max-Change = 0.0117, Max-Change = 0.0052, Max-Change = 0.0086, Max-Change = 0.0101, Max-Change = 0.0164, Max-Change = 0.0097, Max-Change = 0.0056, Max-Change = 0.0179, Max-Change = 0.0056, Max-Change = 0.0072, Max-Change = 0.0118, Max-Change = 0.0031, Max-Change = 0.0076, Max-Change = 0.0171, Max-Change = 0.0125, Max-Change = 0.0159, Max-Change = 0.0089, Max-Change = 0.0146, Max-Change = 0.0109, Max-Change = 0.0075, Max-Change = 0.0102, Max-Change = 0.0110, Max-Change = 0.0174, Max-Change = 0.0163, Max-Change = 0.0087, Max-Change = 0.0054, Max-Change = 0.0102, Max-Change = 0.0127, Max-Change = 0.0176, Max-Change = 0.0243, Max-Change = 0.0061, Max-Change = 0.0035, Max-Change = 0.0121, Max-Change = 0.0134, Max-Change = 0.0163, Max-Change = 0.0043, Max-Change = 0.0094, Max-Change = 0.0096, Max-Change = 0.0183, Max-Change = 0.0114, Max-Change = 0.0099, Max-Change = 0.0313, Max-Change = 0.0248, Max-Change = 0.0163, Max-Change = 0.0125, Max-Change = 0.0056, Max-Change = 0.0097, Max-Change = 0.0210, Max-Change = 0.0163, Max-Change = 0.0101, Max-Change = 0.0173, Max-Change = 0.0129, Max-Change = 0.0087, Max-Change = 0.0150, Max-Change = 0.0198, Max-Change = 0.0080, Max-Change = 0.0213, Max-Change = 0.0164, Max-Change = 0.0118, Max-Change = 0.0174, Max-Change = 0.0076, Max-Change = 0.0076, Max-Change = 0.0159, Max-Change = 0.0115, Max-Change = 0.0215, Max-Change = 0.0067, Max-Change = 0.0148, Max-Change = 0.0073, Max-Change = 0.0075, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0050, gam = 0.1057, Max-Change = 0.0160, gam = 0.0780, Max-Change = 0.0089, gam = 0.0629, Max-Change = 0.0076, gam = 0.0532, Max-Change = 0.0028, gam = 0.0464, Max-Change = 0.0033, gam = 0.0413, Max-Change = 0.0039, gam = 0.0374, Max-Change = 0.0038, gam = 0.0342, Max-Change = 0.0013, gam = 0.0316, Max-Change = 0.0024, gam = 0.0294, Max-Change = 0.0012, gam = 0.0276, Max-Change = 0.0031, gam = 0.0260, Max-Change = 0.0046, gam = 0.0246, Max-Change = 0.0009, gam = 0.0233, Max-Change = 0.0039, gam = 0.0222, Max-Change = 0.0026, gam = 0.0212, Max-Change = 0.0005, gam = 0.0203, Max-Change = 0.0010, gam = 0.0195, Max-Change = 0.0004
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(rmod3)
#>
#> Call:
#> mixedmirt(data = data, covdata = covdata, model = 1, fixed = ~0 +
#> items, random = ~-1 + pseudoIQ | group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 0.073
#>
#> $group
#> COV_group COV_pseudoIQ
#> COV_group 0.903 0.00
#> COV_pseudoIQ 0.000 0.16
#>
eff <- randef(rmod3)
str(eff)
#> List of 2
#> $ Theta: num [1:750, 1] -0.077264 0.04119 0.019919 0.000355 -0.083537 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr "F1"
#> $ group: num [1:50, 1:2] -1.65 -1.11 -1.36 -1.56 -1.51 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:50] "G1" "G2" "G3" "G4" ...
#> .. ..$ : chr [1:2] "group" "pseudoIQ"
###################################################
## LLTM, and 2PL version of LLTM
data(SAT12)
data <- key2binary(SAT12,
key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
model <- 'Theta = 1-32'
# for unconditional intercept comparison
mod <- mirt(data, model, itemtype='Rasch')
#>
coef(mod, simplify=TRUE)
#> $items
#> a1 d g u
#> Item.1 1 -1.077 0 1
#> Item.2 1 0.331 0 1
#> Item.3 1 -1.096 0 1
#> Item.4 1 -0.574 0 1
#> Item.5 1 0.581 0 1
#> Item.6 1 -1.912 0 1
#> Item.7 1 1.342 0 1
#> Item.8 1 -1.593 0 1
#> Item.9 1 2.324 0 1
#> Item.10 1 -0.362 0 1
#> Item.11 1 4.449 0 1
#> Item.12 1 -0.394 0 1
#> Item.13 1 0.791 0 1
#> Item.14 1 1.124 0 1
#> Item.15 1 1.724 0 1
#> Item.16 1 -0.402 0 1
#> Item.17 1 3.620 0 1
#> Item.18 1 -0.709 0 1
#> Item.19 1 0.237 0 1
#> Item.20 1 2.204 0 1
#> Item.21 1 2.684 0 1
#> Item.22 1 2.991 0 1
#> Item.23 1 -0.910 0 1
#> Item.24 1 1.153 0 1
#> Item.25 1 -0.590 0 1
#> Item.26 1 -0.179 0 1
#> Item.27 1 2.094 0 1
#> Item.28 1 0.150 0 1
#> Item.29 1 -0.769 0 1
#> Item.30 1 -0.274 0 1
#> Item.31 1 1.852 0 1
#> Item.32 1 -1.899 0 1
#>
#> $means
#> Theta
#> 0
#>
#> $cov
#> Theta
#> Theta 0.82
#>
# Suppose that the first 16 items were suspected to be easier than the last 16 items,
# and we wish to test this item structure hypothesis (more intercept designs are possible
# by including more columns).
itemdesign <- data.frame(itemorder = factor(c(rep('easier', 16), rep('harder', 16))))
rownames(itemdesign) <- colnames(data)
itemdesign
#> itemorder
#> Item.1 easier
#> Item.2 easier
#> Item.3 easier
#> Item.4 easier
#> Item.5 easier
#> Item.6 easier
#> Item.7 easier
#> Item.8 easier
#> Item.9 easier
#> Item.10 easier
#> Item.11 easier
#> Item.12 easier
#> Item.13 easier
#> Item.14 easier
#> Item.15 easier
#> Item.16 easier
#> Item.17 harder
#> Item.18 harder
#> Item.19 harder
#> Item.20 harder
#> Item.21 harder
#> Item.22 harder
#> Item.23 harder
#> Item.24 harder
#> Item.25 harder
#> Item.26 harder
#> Item.27 harder
#> Item.28 harder
#> Item.29 harder
#> Item.30 harder
#> Item.31 harder
#> Item.32 harder
# notice that the 'fixed = ~ ... + items' argument is omitted
LLTM <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, itemdesign = itemdesign,
SE = TRUE) # SE argument ensures that the information matrix is computed accurately
#> , Max-Change = 0.2000, Max-Change = 0.1211, Max-Change = 0.0434, Max-Change = 0.0359, Max-Change = 0.0327, Max-Change = 0.0388, Max-Change = 0.0141, Max-Change = 0.0267, Max-Change = 0.0141, Max-Change = 0.0104, Max-Change = 0.0137, Max-Change = 0.0148, Max-Change = 0.0114, Max-Change = 0.0116, Max-Change = 0.0112, Max-Change = 0.0118, Max-Change = 0.0093, Max-Change = 0.0096, Max-Change = 0.0059, Max-Change = 0.0056, Max-Change = 0.0114, Max-Change = 0.0124, Max-Change = 0.0062, Max-Change = 0.0059, Max-Change = 0.0072, Max-Change = 0.0089, Max-Change = 0.0046, Max-Change = 0.0061, Max-Change = 0.0051, Max-Change = 0.0022, Max-Change = 0.0050, Max-Change = 0.0068, Max-Change = 0.0016, Max-Change = 0.0028, Max-Change = 0.0049, Max-Change = 0.0067, Max-Change = 0.0019, Max-Change = 0.0038, Max-Change = 0.0054, Max-Change = 0.0082, Max-Change = 0.0038, Max-Change = 0.0019, Max-Change = 0.0055, Max-Change = 0.0044, Max-Change = 0.0065, Max-Change = 0.0020, Max-Change = 0.0034, Max-Change = 0.0029, Max-Change = 0.0062, Max-Change = 0.0021, Max-Change = 0.0029, Max-Change = 0.0023, Max-Change = 0.0030, Max-Change = 0.0008, Max-Change = 0.0130, Max-Change = 0.0023, Max-Change = 0.0062, Max-Change = 0.0028, Max-Change = 0.0057, Max-Change = 0.0035, Max-Change = 0.0029, Max-Change = 0.0017, Max-Change = 0.0020, Max-Change = 0.0012, Max-Change = 0.0021, Max-Change = 0.0028, Max-Change = 0.0012, Max-Change = 0.0020, Max-Change = 0.0019, Max-Change = 0.0036, Max-Change = 0.0029, Max-Change = 0.0029, Max-Change = 0.0051, Max-Change = 0.0044, Max-Change = 0.0053, Max-Change = 0.0021, Max-Change = 0.0016, Max-Change = 0.0037, Max-Change = 0.0021, Max-Change = 0.0044, Max-Change = 0.0073, Max-Change = 0.0023, Max-Change = 0.0070, Max-Change = 0.0025, Max-Change = 0.0043, Max-Change = 0.0035, Max-Change = 0.0026, Max-Change = 0.0011, Max-Change = 0.0024, Max-Change = 0.0019, Max-Change = 0.0024, Max-Change = 0.0031, Max-Change = 0.0037, Max-Change = 0.0065, Max-Change = 0.0016, Max-Change = 0.0041, Max-Change = 0.0015, Max-Change = 0.0028, Max-Change = 0.0038, Max-Change = 0.0031, Max-Change = 0.0016, Max-Change = 0.0069, Max-Change = 0.0031, Max-Change = 0.0026, Max-Change = 0.0034, Max-Change = 0.0028, Max-Change = 0.0035, Max-Change = 0.0050, Max-Change = 0.0047, Max-Change = 0.0036, Max-Change = 0.0028, Max-Change = 0.0040, Max-Change = 0.0049, Max-Change = 0.0024, Max-Change = 0.0062, Max-Change = 0.0044, Max-Change = 0.0023, Max-Change = 0.0040, Max-Change = 0.0037, Max-Change = 0.0031, Max-Change = 0.0024, Max-Change = 0.0037, Max-Change = 0.0040, Max-Change = 0.0030, Max-Change = 0.0023, Max-Change = 0.0020, Max-Change = 0.0022, Max-Change = 0.0064, Max-Change = 0.0008, Max-Change = 0.0028, Max-Change = 0.0063, Max-Change = 0.0073, Max-Change = 0.0020, Max-Change = 0.0055, Max-Change = 0.0041, Max-Change = 0.0032, Max-Change = 0.0033, Max-Change = 0.0030, Max-Change = 0.0027, Max-Change = 0.0036, Max-Change = 0.0049, Max-Change = 0.0020, Max-Change = 0.0024, Max-Change = 0.0041, Max-Change = 0.0026, Max-Change = 0.0031, Max-Change = 0.0018, Max-Change = 0.0002, Max-Change = 0.0031, Max-Change = 0.0031, Max-Change = 0.0054, Max-Change = 0.0037, Max-Change = 0.0008, Max-Change = 0.0040, Max-Change = 0.0019, Max-Change = 0.0045, Max-Change = 0.0029, Max-Change = 0.0032, Max-Change = 0.0029, Max-Change = 0.0013, Max-Change = 0.0032, Max-Change = 0.0034, Max-Change = 0.0036, Max-Change = 0.0046, Max-Change = 0.0044, Max-Change = 0.0075, Max-Change = 0.0045, Max-Change = 0.0043, Max-Change = 0.0046, Max-Change = 0.0012, Max-Change = 0.0027, Max-Change = 0.0060, Max-Change = 0.0018, Max-Change = 0.0031, Max-Change = 0.0019, Max-Change = 0.0085, Max-Change = 0.0035, Max-Change = 0.0013, Max-Change = 0.0054, Max-Change = 0.0023, Max-Change = 0.0016, Max-Change = 0.0031, Max-Change = 0.0021, Max-Change = 0.0027, Max-Change = 0.0059, Max-Change = 0.0024, Max-Change = 0.0027, Max-Change = 0.0041, Max-Change = 0.0032, Max-Change = 0.0014, Max-Change = 0.0021, Max-Change = 0.0075, Max-Change = 0.0044, Max-Change = 0.0061, Max-Change = 0.0026, Max-Change = 0.0014, Max-Change = 0.0076, Max-Change = 0.0022, Max-Change = 0.0014, Max-Change = 0.0052, Max-Change = 0.0035, Max-Change = 0.0014, Max-Change = 0.0029, Max-Change = 0.0028, Max-Change = 0.0006, Max-Change = 0.0050, Max-Change = 0.0027, Max-Change = 0.0054, Max-Change = 0.0038, Max-Change = 0.0051, Max-Change = 0.0040, Max-Change = 0.0028, Max-Change = 0.0055, Max-Change = 0.0050, Max-Change = 0.0020, Max-Change = 0.0037, Max-Change = 0.0067, Max-Change = 0.0031, Max-Change = 0.0011, Max-Change = 0.0024, Max-Change = 0.0075, Max-Change = 0.0025, Max-Change = 0.0044, Max-Change = 0.0043, Max-Change = 0.0024, Max-Change = 0.0034, Max-Change = 0.0008, Max-Change = 0.0025, Max-Change = 0.0021, Max-Change = 0.0031, Max-Change = 0.0038, Max-Change = 0.0012, Max-Change = 0.0064, Max-Change = 0.0045, Max-Change = 0.0014, Max-Change = 0.0030, Max-Change = 0.0028, Max-Change = 0.0024, Max-Change = 0.0055, Max-Change = 0.0039, Max-Change = 0.0063, Max-Change = 0.0022, Max-Change = 0.0013, Max-Change = 0.0015, Max-Change = 0.0027, Max-Change = 0.0006, Max-Change = 0.0032, Max-Change = 0.0020, Max-Change = 0.0071, Max-Change = 0.0010, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0050, gam = 0.1057, Max-Change = 0.0015, gam = 0.0780, Max-Change = 0.0014, gam = 0.0629, Max-Change = 0.0005, gam = 0.0532, Max-Change = 0.0005, gam = 0.0464, Max-Change = 0.0008
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(LLTM)
#>
#> Call:
#> mixedmirt(data = data, model = model, fixed = ~0 + itemorder,
#> itemdesign = itemdesign, SE = TRUE)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> itemordereasier 0.165 0.029 5.746
#> itemorderharder 0.456 0.029 15.757
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> Theta
#> Theta 0.359
#>
coef(LLTM)
#> $Item.1
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.2
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.3
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.4
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.5
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.6
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.7
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.8
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.9
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.10
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.11
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.12
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.13
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.14
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.15
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.16
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.17
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.18
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.19
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.20
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.21
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.22
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.23
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.24
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.25
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.26
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.27
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.28
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.29
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.30
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.31
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $Item.32
#> itemordereasier itemorderharder a1 d g u
#> par 0.165 0.456 1 0 0 1
#> CI_2.5 0.109 0.400 NA NA NA NA
#> CI_97.5 0.221 0.513 NA NA NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 0.359
#> CI_2.5 NA 0.300
#> CI_97.5 NA 0.417
#>
wald(LLTM)
#> itemordereasier.1.7.13.19.25.31.37.43.49.55.61.67.73.79.85.91.97.103.109.115.121.127.133.139.145.151.157.163.169.175.181.187
#> 0.165
#> itemorderharder.2.8.14.20.26.32.38.44.50.56.62.68.74.80.86.92.98.104.110.116.122.128.134.140.146.152.158.164.170.176.182.188
#> 0.456
#> COV_11.194
#> 0.359
L <- matrix(c(-1, 1, 0), 1)
wald(LLTM, L) #first half different from second
#> W df p
#> 1 92.08467 1 0
# compare to items with estimated slopes (2PL)
twoPL <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, itemtype = '2PL',
itemdesign = itemdesign)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1442, Max-Change = 0.1334, Max-Change = 0.1011, Max-Change = 0.1513, Max-Change = 0.1542, Max-Change = 0.1931, Max-Change = 0.1260, Max-Change = 0.1033, Max-Change = 0.0642, Max-Change = 0.0382, Max-Change = 0.0362, Max-Change = 0.0938, Max-Change = 0.1657, Max-Change = 0.0519, Max-Change = 0.0473, Max-Change = 0.0912, Max-Change = 0.1021, Max-Change = 0.0542, Max-Change = 0.0314, Max-Change = 0.0865, Max-Change = 0.1497, Max-Change = 0.0749, Max-Change = 0.0938, Max-Change = 0.0806, Max-Change = 0.0620, Max-Change = 0.1095, Max-Change = 0.0995, Max-Change = 0.0615, Max-Change = 0.0376, Max-Change = 0.0565, Max-Change = 0.1273, Max-Change = 0.0580, Max-Change = 0.0892, Max-Change = 0.1018, Max-Change = 0.1197, Max-Change = 0.0277, Max-Change = 0.0370, Max-Change = 0.0888, Max-Change = 0.1287, Max-Change = 0.0706, Max-Change = 0.0717, Max-Change = 0.1049, Max-Change = 0.0358, Max-Change = 0.0426, Max-Change = 0.0958, Max-Change = 0.0737, Max-Change = 0.0686, Max-Change = 0.0378, Max-Change = 0.0280, Max-Change = 0.0456, Max-Change = 0.0404, Max-Change = 0.0484, Max-Change = 0.0489, Max-Change = 0.0659, Max-Change = 0.0907, Max-Change = 0.0830, Max-Change = 0.0461, Max-Change = 0.0758, Max-Change = 0.0964, Max-Change = 0.0503, Max-Change = 0.1839, Max-Change = 0.0628, Max-Change = 0.0445, Max-Change = 0.0308, Max-Change = 0.1108, Max-Change = 0.0750, Max-Change = 0.0491, Max-Change = 0.1908, Max-Change = 0.0673, Max-Change = 0.1160, Max-Change = 0.0179, Max-Change = 0.0481, Max-Change = 0.1369, Max-Change = 0.0348, Max-Change = 0.0430, Max-Change = 0.0454, Max-Change = 0.0662, Max-Change = 0.1296, Max-Change = 0.0452, Max-Change = 0.0481, Max-Change = 0.1019, Max-Change = 0.0639, Max-Change = 0.0883, Max-Change = 0.0408, Max-Change = 0.1144, Max-Change = 0.0848, Max-Change = 0.0220, Max-Change = 0.0204, Max-Change = 0.1042, Max-Change = 0.0727, Max-Change = 0.0235, Max-Change = 0.1152, Max-Change = 0.1283, Max-Change = 0.2000, Max-Change = 0.1024, Max-Change = 0.0790, Max-Change = 0.0530, Max-Change = 0.1599, Max-Change = 0.0882, Max-Change = 0.0701, Max-Change = 0.0432, Max-Change = 0.0908, Max-Change = 0.0740, Max-Change = 0.0306, Max-Change = 0.1226, Max-Change = 0.0799, Max-Change = 0.0216, Max-Change = 0.0709, Max-Change = 0.0772, Max-Change = 0.0427, Max-Change = 0.0283, Max-Change = 0.1534, Max-Change = 0.0620, Max-Change = 0.1452, Max-Change = 0.0580, Max-Change = 0.1788, Max-Change = 0.0190, Max-Change = 0.0533, Max-Change = 0.0625, Max-Change = 0.0330, Max-Change = 0.0542, Max-Change = 0.1262, Max-Change = 0.0511, Max-Change = 0.0660, Max-Change = 0.0970, Max-Change = 0.0928, Max-Change = 0.0803, Max-Change = 0.0857, Max-Change = 0.0280, Max-Change = 0.1594, Max-Change = 0.0863, Max-Change = 0.0378, Max-Change = 0.0311, Max-Change = 0.0886, Max-Change = 0.0715, Max-Change = 0.1207, Max-Change = 0.0359, Max-Change = 0.0906, Max-Change = 0.1337, Max-Change = 0.0909, Max-Change = 0.0324, Max-Change = 0.1398, Max-Change = 0.1323, Max-Change = 0.0913, Max-Change = 0.0333, Max-Change = 0.0595, Max-Change = 0.0449, Max-Change = 0.1344, Max-Change = 0.2000, Max-Change = 0.0278, Max-Change = 0.0459, Max-Change = 0.1154, Max-Change = 0.0319, Max-Change = 0.0714, Max-Change = 0.1099, Max-Change = 0.0942, Max-Change = 0.0636, Max-Change = 0.0819, Max-Change = 0.0860, Max-Change = 0.0412, Max-Change = 0.2000, Max-Change = 0.0422, Max-Change = 0.1284, Max-Change = 0.0594, Max-Change = 0.0843, Max-Change = 0.0775, Max-Change = 0.0348, Max-Change = 0.1056, Max-Change = 0.0353, Max-Change = 0.0363, Max-Change = 0.1896, Max-Change = 0.0535, Max-Change = 0.0978, Max-Change = 0.0959, Max-Change = 0.0827, Max-Change = 0.0291, Max-Change = 0.1093, Max-Change = 0.0862, Max-Change = 0.1114, Max-Change = 0.0440, Max-Change = 0.0922, Max-Change = 0.0546, Max-Change = 0.0655, Max-Change = 0.1098, Max-Change = 0.0620, Max-Change = 0.0964, Max-Change = 0.1176, Max-Change = 0.0687, Max-Change = 0.0395, Max-Change = 0.1172, Max-Change = 0.1350, Max-Change = 0.1815, Max-Change = 0.0760, Max-Change = 0.0625, Max-Change = 0.0623, Max-Change = 0.1998, Max-Change = 0.0979, Max-Change = 0.1779, Max-Change = 0.0804, Max-Change = 0.1349, Max-Change = 0.0441, Max-Change = 0.0772, Max-Change = 0.1331, Max-Change = 0.1961, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0468, gam = 0.1057, Max-Change = 0.0182, gam = 0.0780, Max-Change = 0.0097, gam = 0.0629, Max-Change = 0.0159, gam = 0.0532, Max-Change = 0.0087, gam = 0.0464, Max-Change = 0.0160, gam = 0.0413, Max-Change = 0.0233, gam = 0.0374, Max-Change = 0.0305, gam = 0.0342, Max-Change = 0.0305, gam = 0.0316, Max-Change = 0.0061, gam = 0.0294, Max-Change = 0.0088, gam = 0.0276, Max-Change = 0.0146, gam = 0.0260, Max-Change = 0.0119, gam = 0.0246, Max-Change = 0.0078, gam = 0.0233, Max-Change = 0.0162, gam = 0.0222, Max-Change = 0.0076, gam = 0.0212, Max-Change = 0.0183, gam = 0.0203, Max-Change = 0.0075, gam = 0.0195, Max-Change = 0.0028, gam = 0.0188, Max-Change = 0.0026, gam = 0.0181, Max-Change = 0.0044, gam = 0.0175, Max-Change = 0.0026, gam = 0.0169, Max-Change = 0.0029, gam = 0.0164, Max-Change = 0.0043, gam = 0.0159, Max-Change = 0.0071, gam = 0.0154, Max-Change = 0.0045, gam = 0.0150, Max-Change = 0.0026, gam = 0.0146, Max-Change = 0.0028, gam = 0.0142, Max-Change = 0.0063, gam = 0.0139, Max-Change = 0.0102, gam = 0.0135, Max-Change = 0.0182, gam = 0.0132, Max-Change = 0.0051, gam = 0.0129, Max-Change = 0.0043, gam = 0.0126, Max-Change = 0.0050, gam = 0.0124, Max-Change = 0.0074, gam = 0.0121, Max-Change = 0.0018, gam = 0.0119, Max-Change = 0.0029, gam = 0.0116, Max-Change = 0.0115, gam = 0.0114, Max-Change = 0.0030, gam = 0.0112, Max-Change = 0.0032, gam = 0.0110, Max-Change = 0.0031, gam = 0.0108, Max-Change = 0.0034, gam = 0.0106, Max-Change = 0.0090, gam = 0.0104, Max-Change = 0.0032, gam = 0.0102, Max-Change = 0.0048, gam = 0.0101, Max-Change = 0.0014, gam = 0.0099, Max-Change = 0.0040, gam = 0.0098, Max-Change = 0.0042, gam = 0.0096, Max-Change = 0.0083, gam = 0.0095, Max-Change = 0.0041, gam = 0.0093, Max-Change = 0.0031, gam = 0.0092, Max-Change = 0.0018, gam = 0.0091, Max-Change = 0.0060, gam = 0.0089, Max-Change = 0.0014, gam = 0.0088, Max-Change = 0.0051, gam = 0.0087, Max-Change = 0.0093, gam = 0.0086, Max-Change = 0.0104, gam = 0.0085, Max-Change = 0.0017, gam = 0.0084, Max-Change = 0.0053, gam = 0.0082, Max-Change = 0.0017, gam = 0.0081, Max-Change = 0.0012, gam = 0.0080, Max-Change = 0.0032, gam = 0.0080, Max-Change = 0.0047, gam = 0.0079, Max-Change = 0.0016, gam = 0.0078, Max-Change = 0.0063, gam = 0.0077, Max-Change = 0.0021, gam = 0.0076, Max-Change = 0.0013, gam = 0.0075, Max-Change = 0.0016, gam = 0.0074, Max-Change = 0.0012, gam = 0.0073, Max-Change = 0.0017, gam = 0.0073, Max-Change = 0.0027, gam = 0.0072, Max-Change = 0.0019, gam = 0.0071, Max-Change = 0.0018, gam = 0.0070, Max-Change = 0.0038, gam = 0.0070, Max-Change = 0.0019, gam = 0.0069, Max-Change = 0.0021, gam = 0.0068, Max-Change = 0.0023, gam = 0.0068, Max-Change = 0.0064, gam = 0.0067, Max-Change = 0.0011, gam = 0.0066, Max-Change = 0.0022, gam = 0.0066, Max-Change = 0.0022, gam = 0.0065, Max-Change = 0.0008, gam = 0.0065, Max-Change = 0.0018, gam = 0.0064, Max-Change = 0.0019, gam = 0.0064, Max-Change = 0.0021, gam = 0.0063, Max-Change = 0.0035, gam = 0.0062, Max-Change = 0.0023, gam = 0.0062, Max-Change = 0.0014, gam = 0.0061, Max-Change = 0.0013, gam = 0.0061, Max-Change = 0.0028, gam = 0.0060, Max-Change = 0.0055, gam = 0.0060, Max-Change = 0.0030, gam = 0.0059, Max-Change = 0.0029, gam = 0.0059, Max-Change = 0.0025, gam = 0.0058, Max-Change = 0.0026, gam = 0.0058, Max-Change = 0.0021, gam = 0.0058, Max-Change = 0.0017, gam = 0.0057, Max-Change = 0.0016, gam = 0.0057, Max-Change = 0.0056, gam = 0.0056, Max-Change = 0.0021, gam = 0.0056, Max-Change = 0.0011, gam = 0.0055, Max-Change = 0.0018, gam = 0.0055, Max-Change = 0.0034, gam = 0.0055, Max-Change = 0.0019, gam = 0.0054, Max-Change = 0.0007, gam = 0.0054, Max-Change = 0.0020, gam = 0.0053, Max-Change = 0.0019, gam = 0.0053, Max-Change = 0.0032, gam = 0.0053, Max-Change = 0.0030, gam = 0.0052, Max-Change = 0.0024, gam = 0.0052, Max-Change = 0.0029, gam = 0.0052, Max-Change = 0.0014, gam = 0.0051, Max-Change = 0.0017, gam = 0.0051, Max-Change = 0.0045, gam = 0.0051, Max-Change = 0.0023, gam = 0.0050, Max-Change = 0.0024, gam = 0.0050, Max-Change = 0.0018, gam = 0.0050, Max-Change = 0.0020, gam = 0.0049, Max-Change = 0.0011, gam = 0.0049, Max-Change = 0.0010, gam = 0.0049, Max-Change = 0.0007, gam = 0.0048, Max-Change = 0.0018, gam = 0.0048, Max-Change = 0.0034, gam = 0.0048, Max-Change = 0.0037, gam = 0.0048, Max-Change = 0.0018, gam = 0.0047, Max-Change = 0.0018, gam = 0.0047, Max-Change = 0.0005, gam = 0.0047, Max-Change = 0.0036, gam = 0.0046, Max-Change = 0.0017, gam = 0.0046, Max-Change = 0.0010, gam = 0.0046, Max-Change = 0.0020, gam = 0.0046, Max-Change = 0.0016, gam = 0.0045, Max-Change = 0.0022, gam = 0.0045, Max-Change = 0.0008, gam = 0.0045, Max-Change = 0.0031, gam = 0.0045, Max-Change = 0.0011, gam = 0.0044, Max-Change = 0.0013, gam = 0.0044, Max-Change = 0.0016, gam = 0.0044, Max-Change = 0.0031, gam = 0.0044, Max-Change = 0.0027, gam = 0.0043, Max-Change = 0.0013, gam = 0.0043, Max-Change = 0.0033, gam = 0.0043, Max-Change = 0.0022, gam = 0.0043, Max-Change = 0.0024, gam = 0.0043, Max-Change = 0.0007, gam = 0.0042, Max-Change = 0.0015, gam = 0.0042, Max-Change = 0.0030, gam = 0.0042, Max-Change = 0.0014, gam = 0.0042, Max-Change = 0.0015, gam = 0.0041, Max-Change = 0.0012, gam = 0.0041, Max-Change = 0.0013, gam = 0.0041, Max-Change = 0.0032, gam = 0.0041, Max-Change = 0.0025, gam = 0.0041, Max-Change = 0.0008, gam = 0.0040, Max-Change = 0.0015, gam = 0.0040, Max-Change = 0.0012, gam = 0.0040, Max-Change = 0.0010, gam = 0.0040, Max-Change = 0.0017, gam = 0.0040, Max-Change = 0.0020, gam = 0.0040, Max-Change = 0.0017, gam = 0.0039, Max-Change = 0.0031, gam = 0.0039, Max-Change = 0.0007, gam = 0.0039, Max-Change = 0.0017, gam = 0.0039, Max-Change = 0.0013, gam = 0.0039, Max-Change = 0.0014, gam = 0.0038, Max-Change = 0.0016, gam = 0.0038, Max-Change = 0.0010, gam = 0.0038, Max-Change = 0.0015, gam = 0.0038, Max-Change = 0.0006, gam = 0.0038, Max-Change = 0.0020, gam = 0.0038, Max-Change = 0.0009, gam = 0.0037, Max-Change = 0.0018, gam = 0.0037, Max-Change = 0.0020, gam = 0.0037, Max-Change = 0.0009, gam = 0.0037, Max-Change = 0.0017, gam = 0.0037, Max-Change = 0.0011, gam = 0.0037, Max-Change = 0.0004, gam = 0.0036, Max-Change = 0.0011, gam = 0.0036, Max-Change = 0.0011, gam = 0.0036, Max-Change = 0.0013, gam = 0.0036, Max-Change = 0.0039, gam = 0.0036, Max-Change = 0.0023, gam = 0.0036, Max-Change = 0.0013, gam = 0.0036, Max-Change = 0.0013, gam = 0.0035, Max-Change = 0.0012, gam = 0.0035, Max-Change = 0.0028, gam = 0.0035, Max-Change = 0.0011, gam = 0.0035, Max-Change = 0.0021, gam = 0.0035, Max-Change = 0.0020, gam = 0.0035, Max-Change = 0.0018, gam = 0.0035, Max-Change = 0.0012, gam = 0.0034, Max-Change = 0.0007, gam = 0.0034, Max-Change = 0.0011, gam = 0.0034, Max-Change = 0.0011, gam = 0.0034, Max-Change = 0.0007, gam = 0.0034, Max-Change = 0.0034, gam = 0.0034, Max-Change = 0.0012, gam = 0.0034, Max-Change = 0.0008, gam = 0.0034, Max-Change = 0.0009, gam = 0.0033, Max-Change = 0.0007
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
# twoPL not mixing too well (AR should be between .2 and .5), decrease MHcand
twoPL <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, itemtype = '2PL',
itemdesign = itemdesign, technical = list(MHcand = 0.8))
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1580, Max-Change = 0.1493, Max-Change = 0.1590, Max-Change = 0.1453, Max-Change = 0.1497, Max-Change = 0.2000, Max-Change = 0.1382, Max-Change = 0.1473, Max-Change = 0.0963, Max-Change = 0.0840, Max-Change = 0.0516, Max-Change = 0.0416, Max-Change = 0.0567, Max-Change = 0.0582, Max-Change = 0.0715, Max-Change = 0.0502, Max-Change = 0.0950, Max-Change = 0.0412, Max-Change = 0.0287, Max-Change = 0.0941, Max-Change = 0.1107, Max-Change = 0.1413, Max-Change = 0.1043, Max-Change = 0.0567, Max-Change = 0.0544, Max-Change = 0.0555, Max-Change = 0.0547, Max-Change = 0.0563, Max-Change = 0.0415, Max-Change = 0.0593, Max-Change = 0.0620, Max-Change = 0.0563, Max-Change = 0.1559, Max-Change = 0.1268, Max-Change = 0.0883, Max-Change = 0.1939, Max-Change = 0.0395, Max-Change = 0.0775, Max-Change = 0.1430, Max-Change = 0.2000, Max-Change = 0.0591, Max-Change = 0.0400, Max-Change = 0.0300, Max-Change = 0.0472, Max-Change = 0.0765, Max-Change = 0.0602, Max-Change = 0.0876, Max-Change = 0.0224, Max-Change = 0.0339, Max-Change = 0.0400, Max-Change = 0.0394, Max-Change = 0.0533, Max-Change = 0.0396, Max-Change = 0.0462, Max-Change = 0.0959, Max-Change = 0.0746, Max-Change = 0.1065, Max-Change = 0.1064, Max-Change = 0.1416, Max-Change = 0.1050, Max-Change = 0.0815, Max-Change = 0.0823, Max-Change = 0.2000, Max-Change = 0.0448, Max-Change = 0.0489, Max-Change = 0.0875, Max-Change = 0.0658, Max-Change = 0.1561, Max-Change = 0.1379, Max-Change = 0.1195, Max-Change = 0.1394, Max-Change = 0.1107, Max-Change = 0.0919, Max-Change = 0.0363, Max-Change = 0.0454, Max-Change = 0.0601, Max-Change = 0.0507, Max-Change = 0.0918, Max-Change = 0.0841, Max-Change = 0.0292, Max-Change = 0.0758, Max-Change = 0.0761, Max-Change = 0.1493, Max-Change = 0.0472, Max-Change = 0.1030, Max-Change = 0.0278, Max-Change = 0.0412, Max-Change = 0.0247, Max-Change = 0.0416, Max-Change = 0.1399, Max-Change = 0.1462, Max-Change = 0.2000, Max-Change = 0.0908, Max-Change = 0.2000, Max-Change = 0.1294, Max-Change = 0.0560, Max-Change = 0.0362, Max-Change = 0.1113, Max-Change = 0.0892, Max-Change = 0.0927, Max-Change = 0.0413, Max-Change = 0.0558, Max-Change = 0.0763, Max-Change = 0.1423, Max-Change = 0.0551, Max-Change = 0.0409, Max-Change = 0.0732, Max-Change = 0.1227, Max-Change = 0.0615, Max-Change = 0.1213, Max-Change = 0.1253, Max-Change = 0.0463, Max-Change = 0.0600, Max-Change = 0.1780, Max-Change = 0.1414, Max-Change = 0.0771, Max-Change = 0.1539, Max-Change = 0.0594, Max-Change = 0.0397, Max-Change = 0.0837, Max-Change = 0.0690, Max-Change = 0.0433, Max-Change = 0.0624, Max-Change = 0.0244, Max-Change = 0.0493, Max-Change = 0.1388, Max-Change = 0.0777, Max-Change = 0.0834, Max-Change = 0.0300, Max-Change = 0.0289, Max-Change = 0.0614, Max-Change = 0.1249, Max-Change = 0.0689, Max-Change = 0.0364, Max-Change = 0.0754, Max-Change = 0.1376, Max-Change = 0.0851, Max-Change = 0.0522, Max-Change = 0.0723, Max-Change = 0.0945, Max-Change = 0.1353, Max-Change = 0.0725, Max-Change = 0.0754, Max-Change = 0.1172, Max-Change = 0.0907, Max-Change = 0.2000, Max-Change = 0.0900, Max-Change = 0.0957, Max-Change = 0.1489, Max-Change = 0.0403, Max-Change = 0.0805, Max-Change = 0.1271, Max-Change = 0.0559, Max-Change = 0.0699, Max-Change = 0.0948, Max-Change = 0.0515, Max-Change = 0.0648, Max-Change = 0.1273, Max-Change = 0.0964, Max-Change = 0.0342, Max-Change = 0.1370, Max-Change = 0.0237, Max-Change = 0.0425, Max-Change = 0.0494, Max-Change = 0.0508, Max-Change = 0.0394, Max-Change = 0.1035, Max-Change = 0.0739, Max-Change = 0.0707, Max-Change = 0.0908, Max-Change = 0.0850, Max-Change = 0.0558, Max-Change = 0.1103, Max-Change = 0.1808, Max-Change = 0.1738, Max-Change = 0.1432, Max-Change = 0.0501, Max-Change = 0.2000, Max-Change = 0.0585, Max-Change = 0.1046, Max-Change = 0.1443, Max-Change = 0.1918, Max-Change = 0.0686, Max-Change = 0.0578, Max-Change = 0.0658, Max-Change = 0.0790, Max-Change = 0.1219, Max-Change = 0.1074, Max-Change = 0.1143, Max-Change = 0.1944, Max-Change = 0.2000, Max-Change = 0.0279, Max-Change = 0.1737, Max-Change = 0.0589, Max-Change = 0.1507, Max-Change = 0.1349, Max-Change = 0.0647, Max-Change = 0.1399, Max-Change = 0.0462, Max-Change = 0.1044, Max-Change = 0.0908, Max-Change = 0.0564, Max-Change = 0.0726, Max-Change = 0.1225, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0746, gam = 0.1057, Max-Change = 0.0272, gam = 0.0780, Max-Change = 0.0412, gam = 0.0629, Max-Change = 0.0188, gam = 0.0532, Max-Change = 0.0088, gam = 0.0464, Max-Change = 0.0202, gam = 0.0413, Max-Change = 0.0106, gam = 0.0374, Max-Change = 0.0108, gam = 0.0342, Max-Change = 0.0181, gam = 0.0316, Max-Change = 0.0109, gam = 0.0294, Max-Change = 0.0220, gam = 0.0276, Max-Change = 0.0024, gam = 0.0260, Max-Change = 0.0170, gam = 0.0246, Max-Change = 0.0048, gam = 0.0233, Max-Change = 0.0220, gam = 0.0222, Max-Change = 0.0166, gam = 0.0212, Max-Change = 0.0132, gam = 0.0203, Max-Change = 0.0060, gam = 0.0195, Max-Change = 0.0056, gam = 0.0188, Max-Change = 0.0016, gam = 0.0181, Max-Change = 0.0061, gam = 0.0175, Max-Change = 0.0037, gam = 0.0169, Max-Change = 0.0085, gam = 0.0164, Max-Change = 0.0039, gam = 0.0159, Max-Change = 0.0031, gam = 0.0154, Max-Change = 0.0112, gam = 0.0150, Max-Change = 0.0045, gam = 0.0146, Max-Change = 0.0044, gam = 0.0142, Max-Change = 0.0059, gam = 0.0139, Max-Change = 0.0052, gam = 0.0135, Max-Change = 0.0117, gam = 0.0132, Max-Change = 0.0068, gam = 0.0129, Max-Change = 0.0021, gam = 0.0126, Max-Change = 0.0019, gam = 0.0124, Max-Change = 0.0023, gam = 0.0121, Max-Change = 0.0069, gam = 0.0119, Max-Change = 0.0021, gam = 0.0116, Max-Change = 0.0033, gam = 0.0114, Max-Change = 0.0018, gam = 0.0112, Max-Change = 0.0051, gam = 0.0110, Max-Change = 0.0019, gam = 0.0108, Max-Change = 0.0025, gam = 0.0106, Max-Change = 0.0068, gam = 0.0104, Max-Change = 0.0052, gam = 0.0102, Max-Change = 0.0087, gam = 0.0101, Max-Change = 0.0064, gam = 0.0099, Max-Change = 0.0088, gam = 0.0098, Max-Change = 0.0082, gam = 0.0096, Max-Change = 0.0075, gam = 0.0095, Max-Change = 0.0048, gam = 0.0093, Max-Change = 0.0023, gam = 0.0092, Max-Change = 0.0030, gam = 0.0091, Max-Change = 0.0031, gam = 0.0089, Max-Change = 0.0021, gam = 0.0088, Max-Change = 0.0045, gam = 0.0087, Max-Change = 0.0025, gam = 0.0086, Max-Change = 0.0039, gam = 0.0085, Max-Change = 0.0046, gam = 0.0084, Max-Change = 0.0100, gam = 0.0082, Max-Change = 0.0049, gam = 0.0081, Max-Change = 0.0032, gam = 0.0080, Max-Change = 0.0046, gam = 0.0080, Max-Change = 0.0049, gam = 0.0079, Max-Change = 0.0015, gam = 0.0078, Max-Change = 0.0015, gam = 0.0077, Max-Change = 0.0033, gam = 0.0076, Max-Change = 0.0011, gam = 0.0075, Max-Change = 0.0021, gam = 0.0074, Max-Change = 0.0011, gam = 0.0073, Max-Change = 0.0030, gam = 0.0073, Max-Change = 0.0022, gam = 0.0072, Max-Change = 0.0013, gam = 0.0071, Max-Change = 0.0025, gam = 0.0070, Max-Change = 0.0018, gam = 0.0070, Max-Change = 0.0041, gam = 0.0069, Max-Change = 0.0017, gam = 0.0068, Max-Change = 0.0009, gam = 0.0068, Max-Change = 0.0009, gam = 0.0067, Max-Change = 0.0033, gam = 0.0066, Max-Change = 0.0023, gam = 0.0066, Max-Change = 0.0013, gam = 0.0065, Max-Change = 0.0049, gam = 0.0065, Max-Change = 0.0013, gam = 0.0064, Max-Change = 0.0012, gam = 0.0064, Max-Change = 0.0048, gam = 0.0063, Max-Change = 0.0051, gam = 0.0062, Max-Change = 0.0041, gam = 0.0062, Max-Change = 0.0081, gam = 0.0061, Max-Change = 0.0026, gam = 0.0061, Max-Change = 0.0031, gam = 0.0060, Max-Change = 0.0029, gam = 0.0060, Max-Change = 0.0040, gam = 0.0059, Max-Change = 0.0034, gam = 0.0059, Max-Change = 0.0018, gam = 0.0058, Max-Change = 0.0015, gam = 0.0058, Max-Change = 0.0046, gam = 0.0058, Max-Change = 0.0012, gam = 0.0057, Max-Change = 0.0017, gam = 0.0057, Max-Change = 0.0007, gam = 0.0056, Max-Change = 0.0009, gam = 0.0056, Max-Change = 0.0025, gam = 0.0055, Max-Change = 0.0017, gam = 0.0055, Max-Change = 0.0020, gam = 0.0055, Max-Change = 0.0016, gam = 0.0054, Max-Change = 0.0013, gam = 0.0054, Max-Change = 0.0015, gam = 0.0053, Max-Change = 0.0018, gam = 0.0053, Max-Change = 0.0012, gam = 0.0053, Max-Change = 0.0023, gam = 0.0052, Max-Change = 0.0026, gam = 0.0052, Max-Change = 0.0011, gam = 0.0052, Max-Change = 0.0024, gam = 0.0051, Max-Change = 0.0017, gam = 0.0051, Max-Change = 0.0011, gam = 0.0051, Max-Change = 0.0007, gam = 0.0050, Max-Change = 0.0030, gam = 0.0050, Max-Change = 0.0017, gam = 0.0050, Max-Change = 0.0009, gam = 0.0049, Max-Change = 0.0018, gam = 0.0049, Max-Change = 0.0022, gam = 0.0049, Max-Change = 0.0015, gam = 0.0048, Max-Change = 0.0026, gam = 0.0048, Max-Change = 0.0009, gam = 0.0048, Max-Change = 0.0028, gam = 0.0048, Max-Change = 0.0064, gam = 0.0047, Max-Change = 0.0041, gam = 0.0047, Max-Change = 0.0020, gam = 0.0047, Max-Change = 0.0019, gam = 0.0046, Max-Change = 0.0017, gam = 0.0046, Max-Change = 0.0010, gam = 0.0046, Max-Change = 0.0030, gam = 0.0046, Max-Change = 0.0026, gam = 0.0045, Max-Change = 0.0017, gam = 0.0045, Max-Change = 0.0027, gam = 0.0045, Max-Change = 0.0016, gam = 0.0045, Max-Change = 0.0031, gam = 0.0044, Max-Change = 0.0011, gam = 0.0044, Max-Change = 0.0027, gam = 0.0044, Max-Change = 0.0007, gam = 0.0044, Max-Change = 0.0012, gam = 0.0043, Max-Change = 0.0018, gam = 0.0043, Max-Change = 0.0026, gam = 0.0043, Max-Change = 0.0032, gam = 0.0043, Max-Change = 0.0051, gam = 0.0043, Max-Change = 0.0007, gam = 0.0042, Max-Change = 0.0020, gam = 0.0042, Max-Change = 0.0013, gam = 0.0042, Max-Change = 0.0013, gam = 0.0042, Max-Change = 0.0018, gam = 0.0041, Max-Change = 0.0016, gam = 0.0041, Max-Change = 0.0010, gam = 0.0041, Max-Change = 0.0010, gam = 0.0041, Max-Change = 0.0012, gam = 0.0041, Max-Change = 0.0021, gam = 0.0040, Max-Change = 0.0011, gam = 0.0040, Max-Change = 0.0021, gam = 0.0040, Max-Change = 0.0007, gam = 0.0040, Max-Change = 0.0016, gam = 0.0040, Max-Change = 0.0016, gam = 0.0040, Max-Change = 0.0009, gam = 0.0039, Max-Change = 0.0005, gam = 0.0039, Max-Change = 0.0017, gam = 0.0039, Max-Change = 0.0007, gam = 0.0039, Max-Change = 0.0017, gam = 0.0039, Max-Change = 0.0009, gam = 0.0038, Max-Change = 0.0022, gam = 0.0038, Max-Change = 0.0009, gam = 0.0038, Max-Change = 0.0010, gam = 0.0038, Max-Change = 0.0026, gam = 0.0038, Max-Change = 0.0039, gam = 0.0038, Max-Change = 0.0021, gam = 0.0037, Max-Change = 0.0011, gam = 0.0037, Max-Change = 0.0017, gam = 0.0037, Max-Change = 0.0013, gam = 0.0037, Max-Change = 0.0016, gam = 0.0037, Max-Change = 0.0011, gam = 0.0037, Max-Change = 0.0024, gam = 0.0036, Max-Change = 0.0034, gam = 0.0036, Max-Change = 0.0008, gam = 0.0036, Max-Change = 0.0013, gam = 0.0036, Max-Change = 0.0030, gam = 0.0036, Max-Change = 0.0009, gam = 0.0036, Max-Change = 0.0008, gam = 0.0036, Max-Change = 0.0018, gam = 0.0035, Max-Change = 0.0021, gam = 0.0035, Max-Change = 0.0004, gam = 0.0035, Max-Change = 0.0009, gam = 0.0035, Max-Change = 0.0005
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
anova(twoPL, LLTM) #much better fit
#> AIC SABIC HQ BIC logLik X2 df p
#> twoPL 20484.33 20525.88 20542.52 20633.82 -10208.16
#> LLTM 25464.27 25467.94 25469.41 25477.47 -12729.14 -5041.948 -31 NaN
summary(twoPL)
#>
#> Call:
#> mixedmirt(data = data, model = model, fixed = ~0 + itemorder,
#> itemtype = "2PL", itemdesign = itemdesign, technical = list(MHcand = 0.8))
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> itemordereasier -1.669 0.087 -19.127
#> itemorderharder -1.644 0.095 -17.316
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> Theta
#> Theta 1
#>
coef(twoPL)
#> $Item.1
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 0.930 0 0 1
#> CI_2.5 -1.840 -1.831 0.696 NA NA NA
#> CI_97.5 -1.498 -1.458 1.165 NA NA NA
#>
#> $Item.2
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 2.526 0 0 1
#> CI_2.5 -1.840 -1.831 2.201 NA NA NA
#> CI_97.5 -1.498 -1.458 2.851 NA NA NA
#>
#> $Item.3
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.00 0 0 1
#> CI_2.5 -1.840 -1.831 0.77 NA NA NA
#> CI_97.5 -1.498 -1.458 1.23 NA NA NA
#>
#> $Item.4
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.288 0 0 1
#> CI_2.5 -1.840 -1.831 1.035 NA NA NA
#> CI_97.5 -1.498 -1.458 1.540 NA NA NA
#>
#> $Item.5
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 2.704 0 0 1
#> CI_2.5 -1.840 -1.831 2.359 NA NA NA
#> CI_97.5 -1.498 -1.458 3.049 NA NA NA
#>
#> $Item.6
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 0.42 0 0 1
#> CI_2.5 -1.840 -1.831 0.19 NA NA NA
#> CI_97.5 -1.498 -1.458 0.65 NA NA NA
#>
#> $Item.7
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 3.913 0 0 1
#> CI_2.5 -1.840 -1.831 3.461 NA NA NA
#> CI_97.5 -1.498 -1.458 4.365 NA NA NA
#>
#> $Item.8
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 0.513 0 0 1
#> CI_2.5 -1.840 -1.831 0.282 NA NA NA
#> CI_97.5 -1.498 -1.458 0.744 NA NA NA
#>
#> $Item.9
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 5.455 0 0 1
#> CI_2.5 -1.840 -1.831 4.840 NA NA NA
#> CI_97.5 -1.498 -1.458 6.071 NA NA NA
#>
#> $Item.10
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.628 0 0 1
#> CI_2.5 -1.840 -1.831 1.352 NA NA NA
#> CI_97.5 -1.498 -1.458 1.905 NA NA NA
#>
#> $Item.11
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 13.512 0 0 1
#> CI_2.5 -1.840 -1.831 10.680 NA NA NA
#> CI_97.5 -1.498 -1.458 16.344 NA NA NA
#>
#> $Item.12
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.245 0 0 1
#> CI_2.5 -1.840 -1.831 0.974 NA NA NA
#> CI_97.5 -1.498 -1.458 1.515 NA NA NA
#>
#> $Item.13
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 3.042 0 0 1
#> CI_2.5 -1.840 -1.831 2.668 NA NA NA
#> CI_97.5 -1.498 -1.458 3.416 NA NA NA
#>
#> $Item.14
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 3.488 0 0 1
#> CI_2.5 -1.840 -1.831 3.076 NA NA NA
#> CI_97.5 -1.498 -1.458 3.899 NA NA NA
#>
#> $Item.15
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 4.749 0 0 1
#> CI_2.5 -1.840 -1.831 4.225 NA NA NA
#> CI_97.5 -1.498 -1.458 5.272 NA NA NA
#>
#> $Item.16
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.469 0 0 1
#> CI_2.5 -1.840 -1.831 1.205 NA NA NA
#> CI_97.5 -1.498 -1.458 1.733 NA NA NA
#>
#> $Item.17
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 9.952 0 0 1
#> CI_2.5 -1.840 -1.831 8.451 NA NA NA
#> CI_97.5 -1.498 -1.458 11.454 NA NA NA
#>
#> $Item.18
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.408 0 0 1
#> CI_2.5 -1.840 -1.831 1.145 NA NA NA
#> CI_97.5 -1.498 -1.458 1.670 NA NA NA
#>
#> $Item.19
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 2.155 0 0 1
#> CI_2.5 -1.840 -1.831 1.825 NA NA NA
#> CI_97.5 -1.498 -1.458 2.486 NA NA NA
#>
#> $Item.20
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 5.787 0 0 1
#> CI_2.5 -1.840 -1.831 5.118 NA NA NA
#> CI_97.5 -1.498 -1.458 6.456 NA NA NA
#>
#> $Item.21
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 6.345 0 0 1
#> CI_2.5 -1.840 -1.831 5.595 NA NA NA
#> CI_97.5 -1.498 -1.458 7.094 NA NA NA
#>
#> $Item.22
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 7.826 0 0 1
#> CI_2.5 -1.840 -1.831 6.827 NA NA NA
#> CI_97.5 -1.498 -1.458 8.824 NA NA NA
#>
#> $Item.23
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 0.982 0 0 1
#> CI_2.5 -1.840 -1.831 0.732 NA NA NA
#> CI_97.5 -1.498 -1.458 1.231 NA NA NA
#>
#> $Item.24
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 3.622 0 0 1
#> CI_2.5 -1.840 -1.831 3.186 NA NA NA
#> CI_97.5 -1.498 -1.458 4.057 NA NA NA
#>
#> $Item.25
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.293 0 0 1
#> CI_2.5 -1.840 -1.831 1.021 NA NA NA
#> CI_97.5 -1.498 -1.458 1.565 NA NA NA
#>
#> $Item.26
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.907 0 0 1
#> CI_2.5 -1.840 -1.831 1.608 NA NA NA
#> CI_97.5 -1.498 -1.458 2.205 NA NA NA
#>
#> $Item.27
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 5.656 0 0 1
#> CI_2.5 -1.840 -1.831 5.018 NA NA NA
#> CI_97.5 -1.498 -1.458 6.293 NA NA NA
#>
#> $Item.28
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 2.120 0 0 1
#> CI_2.5 -1.840 -1.831 1.807 NA NA NA
#> CI_97.5 -1.498 -1.458 2.433 NA NA NA
#>
#> $Item.29
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.165 0 0 1
#> CI_2.5 -1.840 -1.831 0.910 NA NA NA
#> CI_97.5 -1.498 -1.458 1.420 NA NA NA
#>
#> $Item.30
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 1.442 0 0 1
#> CI_2.5 -1.840 -1.831 1.156 NA NA NA
#> CI_97.5 -1.498 -1.458 1.727 NA NA NA
#>
#> $Item.31
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 5.235 0 0 1
#> CI_2.5 -1.840 -1.831 4.637 NA NA NA
#> CI_97.5 -1.498 -1.458 5.832 NA NA NA
#>
#> $Item.32
#> itemordereasier itemorderharder a1 d g u
#> par -1.669 -1.644 0.097 0 0 1
#> CI_2.5 -1.840 -1.831 -0.163 NA NA NA
#> CI_97.5 -1.498 -1.458 0.357 NA NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 1
#> CI_2.5 NA NA
#> CI_97.5 NA NA
#>
wald(twoPL)
#> itemordereasier.1.7.13.19.25.31.37.43.49.55.61.67.73.79.85.91.97.103.109.115.121.127.133.139.145.151.157.163.169.175.181.187
#> -1.669
#> itemorderharder.2.8.14.20.26.32.38.44.50.56.62.68.74.80.86.92.98.104.110.116.122.128.134.140.146.152.158.164.170.176.182.188
#> -1.644
#> a1.3
#> 0.930
#> a1.9
#> 2.526
#> a1.15
#> 1.000
#> a1.21
#> 1.288
#> a1.27
#> 2.704
#> a1.33
#> 0.420
#> a1.39
#> 3.913
#> a1.45
#> 0.513
#> a1.51
#> 5.455
#> a1.57
#> 1.628
#> a1.63
#> 13.512
#> a1.69
#> 1.245
#> a1.75
#> 3.042
#> a1.81
#> 3.488
#> a1.87
#> 4.749
#> a1.93
#> 1.469
#> a1.99
#> 9.952
#> a1.105
#> 1.408
#> a1.111
#> 2.155
#> a1.117
#> 5.787
#> a1.123
#> 6.345
#> a1.129
#> 7.826
#> a1.135
#> 0.982
#> a1.141
#> 3.622
#> a1.147
#> 1.293
#> a1.153
#> 1.907
#> a1.159
#> 5.656
#> a1.165
#> 2.120
#> a1.171
#> 1.165
#> a1.177
#> 1.442
#> a1.183
#> 5.235
#> a1.189
#> 0.097
L <- matrix(0, 1, 34)
L[1, 1] <- 1
L[1, 2] <- -1
wald(twoPL, L) # n.s., which is the correct conclusion. Rasch approach gave wrong inference
#> W df p
#> 1 0.07451603 1 0.7848715
## LLTM with item error term
LLTMwithError <- mixedmirt(data, model = model, fixed = ~ 0 + itemorder, random = ~ 1|items,
itemdesign = itemdesign)
#> , Max-Change = 0.2000, Max-Change = 0.1211, Max-Change = 0.0434, Max-Change = 0.0359, Max-Change = 0.0327, Max-Change = 0.0388, Max-Change = 0.0141, Max-Change = 0.0267, Max-Change = 0.0141, Max-Change = 0.0104, Max-Change = 0.0137, Max-Change = 0.0148, Max-Change = 0.0114, Max-Change = 0.0116, Max-Change = 0.0112, Max-Change = 0.0118, Max-Change = 0.0093, Max-Change = 0.0096, Max-Change = 0.0059, Max-Change = 0.0056, Max-Change = 0.0114, Max-Change = 0.0124, Max-Change = 0.0062, Max-Change = 0.0059, Max-Change = 0.0072, Max-Change = 0.0089, Max-Change = 0.0046, Max-Change = 0.0061, Max-Change = 0.0051, Max-Change = 0.0022, Max-Change = 0.0050, Max-Change = 0.0068, Max-Change = 0.0016, Max-Change = 0.0028, Max-Change = 0.0049, Max-Change = 0.0067, Max-Change = 0.0019, Max-Change = 0.0038, Max-Change = 0.0054, Max-Change = 0.0082, Max-Change = 0.0038, Max-Change = 0.0019, Max-Change = 0.0055, Max-Change = 0.0044, Max-Change = 0.0065, Max-Change = 0.0020, Max-Change = 0.0034, Max-Change = 0.0029, Max-Change = 0.0062, Max-Change = 0.0021, Max-Change = 0.0029, Max-Change = 0.0023, Max-Change = 0.0030, Max-Change = 0.0008, Max-Change = 0.0130, Max-Change = 0.0023, Max-Change = 0.0062, Max-Change = 0.0028, Max-Change = 0.0057, Max-Change = 0.0035, Max-Change = 0.0029, Max-Change = 0.0017, Max-Change = 0.0020, Max-Change = 0.0012, Max-Change = 0.0021, Max-Change = 0.0028, Max-Change = 0.0012, Max-Change = 0.0020, Max-Change = 0.0019, Max-Change = 0.0036, Max-Change = 0.0029, Max-Change = 0.0029, Max-Change = 0.0051, Max-Change = 0.0044, Max-Change = 0.0053, Max-Change = 0.0021, Max-Change = 0.0016, Max-Change = 0.0037, Max-Change = 0.0021, Max-Change = 0.0044, Max-Change = 0.0073, Max-Change = 0.0023, Max-Change = 0.0070, Max-Change = 0.0025, Max-Change = 0.0043, Max-Change = 0.0035, Max-Change = 0.0026, Max-Change = 0.0011, Max-Change = 0.0024, Max-Change = 0.0019, Max-Change = 0.0024, Max-Change = 0.0031, Max-Change = 0.0037, Max-Change = 0.0065, Max-Change = 0.0016, Max-Change = 0.0041, Max-Change = 0.0015, Max-Change = 0.0028, Max-Change = 0.0038, Max-Change = 0.0162, Max-Change = 0.0703, Max-Change = 0.0721, Max-Change = 0.0756, Max-Change = 0.0767, Max-Change = 0.0753, Max-Change = 0.0733, Max-Change = 0.0733, Max-Change = 0.0746, Max-Change = 0.0813, Max-Change = 0.0766, Max-Change = 0.0805, Max-Change = 0.0598, Max-Change = 0.0810, Max-Change = 0.0682, Max-Change = 0.0495, Max-Change = 0.0376, Max-Change = 0.0505, Max-Change = 0.0389, Max-Change = 0.0431, Max-Change = 0.0425, Max-Change = 0.0365, Max-Change = 0.0017, Max-Change = 0.0223, Max-Change = 0.0212, Max-Change = 0.0175, Max-Change = 0.0118, Max-Change = 0.0094, Max-Change = 0.0110, Max-Change = 0.0050, Max-Change = 0.0071, Max-Change = 0.0469, Max-Change = 0.0090, Max-Change = 0.0240, Max-Change = 0.0292, Max-Change = 0.0250, Max-Change = 0.0413, Max-Change = 0.0280, Max-Change = 0.0109, Max-Change = 0.0112, Max-Change = 0.0110, Max-Change = 0.0070, Max-Change = 0.0045, Max-Change = 0.0100, Max-Change = 0.0423, Max-Change = 0.0076, Max-Change = 0.0067, Max-Change = 0.0157, Max-Change = 0.0166, Max-Change = 0.0028, Max-Change = 0.0170, Max-Change = 0.0066, Max-Change = 0.0326, Max-Change = 0.0109, Max-Change = 0.0608, Max-Change = 0.0353, Max-Change = 0.0181, Max-Change = 0.0166, Max-Change = 0.0089, Max-Change = 0.0236, Max-Change = 0.0192, Max-Change = 0.0196, Max-Change = 0.0130, Max-Change = 0.0084, Max-Change = 0.0269, Max-Change = 0.0053, Max-Change = 0.0115, Max-Change = 0.0035, Max-Change = 0.0131, Max-Change = 0.0098, Max-Change = 0.0172, Max-Change = 0.0192, Max-Change = 0.0064, Max-Change = 0.0090, Max-Change = 0.0053, Max-Change = 0.0098, Max-Change = 0.0136, Max-Change = 0.0036, Max-Change = 0.0086, Max-Change = 0.0073, Max-Change = 0.0156, Max-Change = 0.0275, Max-Change = 0.0075, Max-Change = 0.0196, Max-Change = 0.0265, Max-Change = 0.0065, Max-Change = 0.0024, Max-Change = 0.0069, Max-Change = 0.0211, Max-Change = 0.0112, Max-Change = 0.0036, Max-Change = 0.0203, Max-Change = 0.0169, Max-Change = 0.0201, Max-Change = 0.0116, Max-Change = 0.0366, Max-Change = 0.0321, Max-Change = 0.0214, Max-Change = 0.0076, Max-Change = 0.0036, Max-Change = 0.0295, Max-Change = 0.0102, Max-Change = 0.0080, Max-Change = 0.0075, Max-Change = 0.0052, Max-Change = 0.0053, Max-Change = 0.0056, Max-Change = 0.0135, Max-Change = 0.0038, Max-Change = 0.0131, Max-Change = 0.0118, Max-Change = 0.0025, Max-Change = 0.0028, Max-Change = 0.0016, Max-Change = 0.0049, Max-Change = 0.0037, Max-Change = 0.0056, Max-Change = 0.0049, Max-Change = 0.0053, Max-Change = 0.0093, Max-Change = 0.0041, Max-Change = 0.0051, Max-Change = 0.0041, Max-Change = 0.0045, Max-Change = 0.0218, Max-Change = 0.0083, Max-Change = 0.0033, Max-Change = 0.0025, Max-Change = 0.0034, Max-Change = 0.0044, Max-Change = 0.0031, Max-Change = 0.0012, Max-Change = 0.0034, Max-Change = 0.0059, Max-Change = 0.0066, Max-Change = 0.0054, Max-Change = 0.0035, Max-Change = 0.0013, Max-Change = 0.0056, Max-Change = 0.0026, Max-Change = 0.0028, Max-Change = 0.0015, Max-Change = 0.0034, Max-Change = 0.0090, Max-Change = 0.0061, Max-Change = 0.0072, Max-Change = 0.0051, Max-Change = 0.0036, Max-Change = 0.0027, Max-Change = 0.0047, Max-Change = 0.0005, Max-Change = 0.0095, Max-Change = 0.0015, Max-Change = 0.0021, Max-Change = 0.0031, Max-Change = 0.0042, Max-Change = 0.0020, Max-Change = 0.0016, Max-Change = 0.0061, Max-Change = 0.0032, Max-Change = 0.0063, Max-Change = 0.0035, Max-Change = 0.0016, Max-Change = 0.0042, Max-Change = 0.0032, Max-Change = 0.0039, Max-Change = 0.0085, Max-Change = 0.0131, Max-Change = 0.0041, Max-Change = 0.0046, Max-Change = 0.0041, Max-Change = 0.0125, Max-Change = 0.0036, Max-Change = 0.0048, Max-Change = 0.0058, Max-Change = 0.0059, Max-Change = 0.0079, Max-Change = 0.0025, Max-Change = 0.0065, Max-Change = 0.0044, Max-Change = 0.0013, Max-Change = 0.0040, Max-Change = 0.0020, Max-Change = 0.0077, Max-Change = 0.0023, Max-Change = 0.0028, Max-Change = 0.0068, Max-Change = 0.0068, Max-Change = 0.0090, Max-Change = 0.0024, Max-Change = 0.0017, Max-Change = 0.0051, Max-Change = 0.0049, Max-Change = 0.0061, Max-Change = 0.0052, Max-Change = 0.0062, Max-Change = 0.0030, Max-Change = 0.0112, Max-Change = 0.0029, Max-Change = 0.0061, Max-Change = 0.0036, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0037, gam = 0.1057, Max-Change = 0.0038, gam = 0.0780, Max-Change = 0.0033, gam = 0.0629, Max-Change = 0.0032, gam = 0.0532, Max-Change = 0.0016, gam = 0.0464, Max-Change = 0.0012, gam = 0.0413, Max-Change = 0.0016, gam = 0.0374, Max-Change = 0.0008, gam = 0.0342, Max-Change = 0.0003, gam = 0.0316, Max-Change = 0.0010
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(LLTMwithError)
#>
#> Call:
#> mixedmirt(data = data, model = model, fixed = ~0 + itemorder,
#> random = ~1 | items, itemdesign = itemdesign)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> itemordereasier 0.147 0.153 0.961
#> itemorderharder 0.567 0.138 4.107
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> Theta
#> Theta 0.77
#>
#> $items
#> COV_items
#> COV_items 2.36
#>
# large item level variance after itemorder is regressed; not a great predictor of item difficulty
coef(LLTMwithError)
#> $Item.1
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.2
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.3
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.4
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.5
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.6
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.7
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.8
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.9
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.10
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.11
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.12
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.13
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.14
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.15
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.16
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.17
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.18
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.19
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.20
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.21
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.22
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.23
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.24
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.25
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.26
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.27
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.28
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.29
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.30
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.31
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $Item.32
#> itemordereasier itemorderharder a1 d g u
#> par 0.147 0.567 1 0 0 1
#> CI_2.5 -0.153 0.296 NA NA NA NA
#> CI_97.5 0.446 0.838 NA NA NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 0.770
#> CI_2.5 NA 0.656
#> CI_97.5 NA 0.884
#>
#> $items
#> COV_items_items
#> par 2.363
#> CI_2.5 1.229
#> CI_97.5 3.497
#>
###################################################
### Polytomous example
# make an arbitrary group difference
covdat <- data.frame(group = rep(c('m', 'f'), nrow(Science)/2))
# partial credit model
mod <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1962, Max-Change = 0.1682, Max-Change = 0.1133, Max-Change = 0.0674, Max-Change = 0.0992, Max-Change = 0.0598, Max-Change = 0.0391, Max-Change = 0.0767, Max-Change = 0.0756, Max-Change = 0.0498, Max-Change = 0.0341, Max-Change = 0.0267, Max-Change = 0.0180, Max-Change = 0.0456, Max-Change = 0.0402, Max-Change = 0.0274, Max-Change = 0.0393, Max-Change = 0.0554, Max-Change = 0.0284, Max-Change = 0.0071, Max-Change = 0.0122, Max-Change = 0.0220, Max-Change = 0.0069, Max-Change = 0.0146, Max-Change = 0.0283, Max-Change = 0.0373, Max-Change = 0.0308, Max-Change = 0.0094, Max-Change = 0.0156, Max-Change = 0.0788, Max-Change = 0.0158, Max-Change = 0.0216, Max-Change = 0.0166, Max-Change = 0.0117, Max-Change = 0.0380, Max-Change = 0.0473, Max-Change = 0.0301, Max-Change = 0.0510, Max-Change = 0.0303, Max-Change = 0.0534, Max-Change = 0.0369, Max-Change = 0.0366, Max-Change = 0.0164, Max-Change = 0.0112, Max-Change = 0.0184, Max-Change = 0.0241, Max-Change = 0.0132, Max-Change = 0.0130, Max-Change = 0.0322, Max-Change = 0.0117, Max-Change = 0.0257, Max-Change = 0.0167, Max-Change = 0.0103, Max-Change = 0.0230, Max-Change = 0.0250, Max-Change = 0.0330, Max-Change = 0.0080, Max-Change = 0.0185, Max-Change = 0.0132, Max-Change = 0.0584, Max-Change = 0.0139, Max-Change = 0.0264, Max-Change = 0.0230, Max-Change = 0.0302, Max-Change = 0.0573, Max-Change = 0.0190, Max-Change = 0.0394, Max-Change = 0.0492, Max-Change = 0.1002, Max-Change = 0.0201, Max-Change = 0.0600, Max-Change = 0.0258, Max-Change = 0.0563, Max-Change = 0.0101, Max-Change = 0.0197, Max-Change = 0.0245, Max-Change = 0.0342, Max-Change = 0.0495, Max-Change = 0.0175, Max-Change = 0.0301, Max-Change = 0.0319, Max-Change = 0.0743, Max-Change = 0.0160, Max-Change = 0.0126, Max-Change = 0.0248, Max-Change = 0.0238, Max-Change = 0.0409, Max-Change = 0.0393, Max-Change = 0.0248, Max-Change = 0.0402, Max-Change = 0.0136, Max-Change = 0.0208, Max-Change = 0.0514, Max-Change = 0.0607, Max-Change = 0.0259, Max-Change = 0.0346, Max-Change = 0.0313, Max-Change = 0.0295, Max-Change = 0.0630, Max-Change = 0.0644, Max-Change = 0.0463, Max-Change = 0.0292, Max-Change = 0.0367, Max-Change = 0.0403, Max-Change = 0.0443, Max-Change = 0.0677, Max-Change = 0.0219, Max-Change = 0.0139, Max-Change = 0.0540, Max-Change = 0.0469, Max-Change = 0.0200, Max-Change = 0.0142, Max-Change = 0.0141, Max-Change = 0.0355, Max-Change = 0.0165, Max-Change = 0.0266, Max-Change = 0.0173, Max-Change = 0.0206, Max-Change = 0.0325, Max-Change = 0.0488, Max-Change = 0.0090, Max-Change = 0.0374, Max-Change = 0.0068, Max-Change = 0.0110, Max-Change = 0.0160, Max-Change = 0.0323, Max-Change = 0.0151, Max-Change = 0.0225, Max-Change = 0.0657, Max-Change = 0.0597, Max-Change = 0.0197, Max-Change = 0.0156, Max-Change = 0.0257, Max-Change = 0.0453, Max-Change = 0.0137, Max-Change = 0.0347, Max-Change = 0.0347, Max-Change = 0.0123, Max-Change = 0.0305, Max-Change = 0.0093, Max-Change = 0.0487, Max-Change = 0.0134, Max-Change = 0.0418, Max-Change = 0.0272, Max-Change = 0.0292, Max-Change = 0.0300, Max-Change = 0.0573, Max-Change = 0.0277, Max-Change = 0.0086, Max-Change = 0.0474, Max-Change = 0.0222, Max-Change = 0.0189, Max-Change = 0.0280, Max-Change = 0.0296, Max-Change = 0.0060, Max-Change = 0.0725, Max-Change = 0.0465, Max-Change = 0.0170, Max-Change = 0.0170, Max-Change = 0.0480, Max-Change = 0.0664, Max-Change = 0.0191, Max-Change = 0.0223, Max-Change = 0.0139, Max-Change = 0.0465, Max-Change = 0.0288, Max-Change = 0.0670, Max-Change = 0.0177, Max-Change = 0.0684, Max-Change = 0.0415, Max-Change = 0.0246, Max-Change = 0.0121, Max-Change = 0.0224, Max-Change = 0.0199, Max-Change = 0.0073, Max-Change = 0.0258, Max-Change = 0.0154, Max-Change = 0.0388, Max-Change = 0.0320, Max-Change = 0.0259, Max-Change = 0.0501, Max-Change = 0.1062, Max-Change = 0.0204, Max-Change = 0.0840, Max-Change = 0.0111, Max-Change = 0.0397, Max-Change = 0.0191, Max-Change = 0.0168, Max-Change = 0.0256, Max-Change = 0.0382, Max-Change = 0.0726, Max-Change = 0.0670, Max-Change = 0.0159, Max-Change = 0.0099, Max-Change = 0.0558, Max-Change = 0.0305, Max-Change = 0.0040, Max-Change = 0.0227, Max-Change = 0.0270, Max-Change = 0.0105, Max-Change = 0.0755, Max-Change = 0.0284, Max-Change = 0.0125, Max-Change = 0.0220, Max-Change = 0.0491, Max-Change = 0.0176, Max-Change = 0.0107, Max-Change = 0.0212, Max-Change = 0.0090, Max-Change = 0.0251, Max-Change = 0.0252, Max-Change = 0.0374, Max-Change = 0.0572, Max-Change = 0.0189, Max-Change = 0.0261, Max-Change = 0.0190, Max-Change = 0.0336, Max-Change = 0.0352, Max-Change = 0.0230, Max-Change = 0.0160, Max-Change = 0.0347, Max-Change = 0.0476, Max-Change = 0.0099, Max-Change = 0.0521, Max-Change = 0.0576, Max-Change = 0.0144, Max-Change = 0.0152, Max-Change = 0.0292, Max-Change = 0.0164, Max-Change = 0.0307, Max-Change = 0.0539, Max-Change = 0.0581, Max-Change = 0.0183, Max-Change = 0.0614, Max-Change = 0.0283, Max-Change = 0.0063, Max-Change = 0.0422, Max-Change = 0.0528, Max-Change = 0.0289, Max-Change = 0.0444, Max-Change = 0.0230, Max-Change = 0.0127, Max-Change = 0.0158, Max-Change = 0.0254, Max-Change = 0.0421, Max-Change = 0.0088, Max-Change = 0.0437, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0321, gam = 0.1057, Max-Change = 0.0075, gam = 0.0780, Max-Change = 0.0111, gam = 0.0629, Max-Change = 0.0052, gam = 0.0532, Max-Change = 0.0125, gam = 0.0464, Max-Change = 0.0073, gam = 0.0413, Max-Change = 0.0042, gam = 0.0374, Max-Change = 0.0100, gam = 0.0342, Max-Change = 0.0023, gam = 0.0316, Max-Change = 0.0089, gam = 0.0294, Max-Change = 0.0029, gam = 0.0276, Max-Change = 0.0057, gam = 0.0260, Max-Change = 0.0063, gam = 0.0246, Max-Change = 0.0033, gam = 0.0233, Max-Change = 0.0053, gam = 0.0222, Max-Change = 0.0039, gam = 0.0212, Max-Change = 0.0055, gam = 0.0203, Max-Change = 0.0013, gam = 0.0195, Max-Change = 0.0041, gam = 0.0188, Max-Change = 0.0022, gam = 0.0181, Max-Change = 0.0027, gam = 0.0175, Max-Change = 0.0050, gam = 0.0169, Max-Change = 0.0055, gam = 0.0164, Max-Change = 0.0043, gam = 0.0159, Max-Change = 0.0011, gam = 0.0154, Max-Change = 0.0016, gam = 0.0150, Max-Change = 0.0014, gam = 0.0146, Max-Change = 0.0014, gam = 0.0142, Max-Change = 0.0016, gam = 0.0139, Max-Change = 0.0026, gam = 0.0135, Max-Change = 0.0041, gam = 0.0132, Max-Change = 0.0024, gam = 0.0129, Max-Change = 0.0024, gam = 0.0126, Max-Change = 0.0012, gam = 0.0124, Max-Change = 0.0010, gam = 0.0121, Max-Change = 0.0047, gam = 0.0119, Max-Change = 0.0034, gam = 0.0116, Max-Change = 0.0035, gam = 0.0114, Max-Change = 0.0021, gam = 0.0112, Max-Change = 0.0016, gam = 0.0110, Max-Change = 0.0016, gam = 0.0108, Max-Change = 0.0022, gam = 0.0106, Max-Change = 0.0027, gam = 0.0104, Max-Change = 0.0019, gam = 0.0102, Max-Change = 0.0012, gam = 0.0101, Max-Change = 0.0024, gam = 0.0099, Max-Change = 0.0007, gam = 0.0098, Max-Change = 0.0020, gam = 0.0096, Max-Change = 0.0020, gam = 0.0095, Max-Change = 0.0023, gam = 0.0093, Max-Change = 0.0021, gam = 0.0092, Max-Change = 0.0010, gam = 0.0091, Max-Change = 0.0006, gam = 0.0089, Max-Change = 0.0012, gam = 0.0088, Max-Change = 0.0012, gam = 0.0087, Max-Change = 0.0010, gam = 0.0086, Max-Change = 0.0014, gam = 0.0085, Max-Change = 0.0008, gam = 0.0084, Max-Change = 0.0007, gam = 0.0082, Max-Change = 0.0017, gam = 0.0081, Max-Change = 0.0018, gam = 0.0080, Max-Change = 0.0010, gam = 0.0080, Max-Change = 0.0007, gam = 0.0079, Max-Change = 0.0024, gam = 0.0078, Max-Change = 0.0003, gam = 0.0077, Max-Change = 0.0009, gam = 0.0076, Max-Change = 0.0014, gam = 0.0075, Max-Change = 0.0013, gam = 0.0074, Max-Change = 0.0022, gam = 0.0073, Max-Change = 0.0009, gam = 0.0073, Max-Change = 0.0011, gam = 0.0072, Max-Change = 0.0027, gam = 0.0071, Max-Change = 0.0010, gam = 0.0070, Max-Change = 0.0007, gam = 0.0070, Max-Change = 0.0008, gam = 0.0069, Max-Change = 0.0006
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
coef(mod)
#> $Comfort
#> groupm a1 ak0 ak1 ak2 ak3 d0 d1 d2 d3
#> par -0.084 1 0 1 2 3 0 3.098 5.718 4.364
#> CI_2.5 -0.336 NA NA NA NA NA NA 2.086 4.677 3.257
#> CI_97.5 0.167 NA NA NA NA NA NA 4.110 6.759 5.472
#>
#> $Work
#> groupm a1 ak0 ak1 ak2 ak3 d0 d1 d2 d3
#> par -0.084 1 0 1 2 3 0 1.919 2.859 1.038
#> CI_2.5 -0.336 NA NA NA NA NA NA 1.455 2.311 0.352
#> CI_97.5 0.167 NA NA NA NA NA NA 2.382 3.406 1.723
#>
#> $Future
#> groupm a1 ak0 ak1 ak2 ak3 d0 d1 d2 d3
#> par -0.084 1 0 1 2 3 0 2.665 4.114 3.013
#> CI_2.5 -0.336 NA NA NA NA NA NA 2.026 3.406 2.214
#> CI_97.5 0.167 NA NA NA NA NA NA 3.304 4.822 3.813
#>
#> $Benefit
#> groupm a1 ak0 ak1 ak2 ak3 d0 d1 d2 d3
#> par -0.084 1 0 1 2 3 0 2.469 3.398 2.076
#> CI_2.5 -0.336 NA NA NA NA NA NA 1.932 2.779 1.349
#> CI_97.5 0.167 NA NA NA NA NA NA 3.006 4.016 2.803
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 0.985
#> CI_2.5 NA 0.707
#> CI_97.5 NA 1.264
#>
# gpcm to estimate slopes
mod2 <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group,
itemtype = 'gpcm')
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1776, Max-Change = 0.1068, Max-Change = 0.1180, Max-Change = 0.1121, Max-Change = 0.0996, Max-Change = 0.1305, Max-Change = 0.0338, Max-Change = 0.1041, Max-Change = 0.0588, Max-Change = 0.0460, Max-Change = 0.0885, Max-Change = 0.0954, Max-Change = 0.0450, Max-Change = 0.1303, Max-Change = 0.0976, Max-Change = 0.1155, Max-Change = 0.0919, Max-Change = 0.1057, Max-Change = 0.0780, Max-Change = 0.0979, Max-Change = 0.0675, Max-Change = 0.1021, Max-Change = 0.0427, Max-Change = 0.0814, Max-Change = 0.0625, Max-Change = 0.0662, Max-Change = 0.0919, Max-Change = 0.1059, Max-Change = 0.1302, Max-Change = 0.0629, Max-Change = 0.0813, Max-Change = 0.0782, Max-Change = 0.1577, Max-Change = 0.0302, Max-Change = 0.0778, Max-Change = 0.1032, Max-Change = 0.0458, Max-Change = 0.0519, Max-Change = 0.0532, Max-Change = 0.1538, Max-Change = 0.1558, Max-Change = 0.1330, Max-Change = 0.0485, Max-Change = 0.0976, Max-Change = 0.0576, Max-Change = 0.0236, Max-Change = 0.1744, Max-Change = 0.0174, Max-Change = 0.0716, Max-Change = 0.1091, Max-Change = 0.0652, Max-Change = 0.0722, Max-Change = 0.1248, Max-Change = 0.0696, Max-Change = 0.0928, Max-Change = 0.0199, Max-Change = 0.1060, Max-Change = 0.0868, Max-Change = 0.0901, Max-Change = 0.1216, Max-Change = 0.0595, Max-Change = 0.0360, Max-Change = 0.0834, Max-Change = 0.1512, Max-Change = 0.0561, Max-Change = 0.1249, Max-Change = 0.0744, Max-Change = 0.1373, Max-Change = 0.1111, Max-Change = 0.0304, Max-Change = 0.1097, Max-Change = 0.0966, Max-Change = 0.0830, Max-Change = 0.0232, Max-Change = 0.0628, Max-Change = 0.0403, Max-Change = 0.1226, Max-Change = 0.0310, Max-Change = 0.0457, Max-Change = 0.1221, Max-Change = 0.1108, Max-Change = 0.1636, Max-Change = 0.1611, Max-Change = 0.1112, Max-Change = 0.0778, Max-Change = 0.1025, Max-Change = 0.0633, Max-Change = 0.0344, Max-Change = 0.0296, Max-Change = 0.0352, Max-Change = 0.0954, Max-Change = 0.1280, Max-Change = 0.0501, Max-Change = 0.1303, Max-Change = 0.0549, Max-Change = 0.1357, Max-Change = 0.1346, Max-Change = 0.0546, Max-Change = 0.0694, Max-Change = 0.0904, Max-Change = 0.1042, Max-Change = 0.0233, Max-Change = 0.0545, Max-Change = 0.1438, Max-Change = 0.0567, Max-Change = 0.0701, Max-Change = 0.0924, Max-Change = 0.0772, Max-Change = 0.1517, Max-Change = 0.0887, Max-Change = 0.0719, Max-Change = 0.0368, Max-Change = 0.1098, Max-Change = 0.1057, Max-Change = 0.0267, Max-Change = 0.2000, Max-Change = 0.0747, Max-Change = 0.0943, Max-Change = 0.0283, Max-Change = 0.0432, Max-Change = 0.0995, Max-Change = 0.0564, Max-Change = 0.0250, Max-Change = 0.1342, Max-Change = 0.0971, Max-Change = 0.0706, Max-Change = 0.1048, Max-Change = 0.0733, Max-Change = 0.0640, Max-Change = 0.1025, Max-Change = 0.0336, Max-Change = 0.0938, Max-Change = 0.0412, Max-Change = 0.0341, Max-Change = 0.0649, Max-Change = 0.1457, Max-Change = 0.0507, Max-Change = 0.0572, Max-Change = 0.1275, Max-Change = 0.0559, Max-Change = 0.0260, Max-Change = 0.1585, Max-Change = 0.0572, Max-Change = 0.1437, Max-Change = 0.0925, Max-Change = 0.0352, Max-Change = 0.0378, Max-Change = 0.1587, Max-Change = 0.0881, Max-Change = 0.1472, Max-Change = 0.0677, Max-Change = 0.0775, Max-Change = 0.1794, Max-Change = 0.0888, Max-Change = 0.1212, Max-Change = 0.0466, Max-Change = 0.0483, Max-Change = 0.0559, Max-Change = 0.0440, Max-Change = 0.1637, Max-Change = 0.0334, Max-Change = 0.0264, Max-Change = 0.0531, Max-Change = 0.0262, Max-Change = 0.1146, Max-Change = 0.0283, Max-Change = 0.0489, Max-Change = 0.1149, Max-Change = 0.1332, Max-Change = 0.1170, Max-Change = 0.0390, Max-Change = 0.0484, Max-Change = 0.0567, Max-Change = 0.0678, Max-Change = 0.0505, Max-Change = 0.0526, Max-Change = 0.2000, Max-Change = 0.0915, Max-Change = 0.0471, Max-Change = 0.1616, Max-Change = 0.0679, Max-Change = 0.1139, Max-Change = 0.0587, Max-Change = 0.0723, Max-Change = 0.1630, Max-Change = 0.1455, Max-Change = 0.0390, Max-Change = 0.1358, Max-Change = 0.1216, Max-Change = 0.1024, Max-Change = 0.0847, Max-Change = 0.1856, Max-Change = 0.1123, Max-Change = 0.0340, Max-Change = 0.1251, Max-Change = 0.0451, Max-Change = 0.0858, Max-Change = 0.1119, Max-Change = 0.0819, Max-Change = 0.1402, Max-Change = 0.0619, Max-Change = 0.0700, Max-Change = 0.0546, Max-Change = 0.0847, Max-Change = 0.0283, Max-Change = 0.0607, Max-Change = 0.1731, Max-Change = 0.0976, Max-Change = 0.0916, Max-Change = 0.1646, Max-Change = 0.0688, Max-Change = 0.0275, Max-Change = 0.2000, Max-Change = 0.0603, Max-Change = 0.0934, Max-Change = 0.0944, Max-Change = 0.1044, Max-Change = 0.1739, Max-Change = 0.0828, Max-Change = 0.0794, Max-Change = 0.0359, Max-Change = 0.0265, Max-Change = 0.1483, Max-Change = 0.0836, Max-Change = 0.2000, Max-Change = 0.0339, Max-Change = 0.0317, Max-Change = 0.0652, Max-Change = 0.0273, Max-Change = 0.1724, Max-Change = 0.0454, Max-Change = 0.0369, Max-Change = 0.0521, Max-Change = 0.0760, Max-Change = 0.0256, Max-Change = 0.1141, Max-Change = 0.0382, Max-Change = 0.0813, Max-Change = 0.1710, Max-Change = 0.0684, Max-Change = 0.0900, Max-Change = 0.2000, Max-Change = 0.0473, Max-Change = 0.0458, Max-Change = 0.1378, Max-Change = 0.0302, Max-Change = 0.0392, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0665, gam = 0.1057, Max-Change = 0.0327, gam = 0.0780, Max-Change = 0.0223, gam = 0.0629, Max-Change = 0.0178, gam = 0.0532, Max-Change = 0.0253, gam = 0.0464, Max-Change = 0.0294, gam = 0.0413, Max-Change = 0.0359, gam = 0.0374, Max-Change = 0.0188, gam = 0.0342, Max-Change = 0.0164, gam = 0.0316, Max-Change = 0.0080, gam = 0.0294, Max-Change = 0.0069, gam = 0.0276, Max-Change = 0.0225, gam = 0.0260, Max-Change = 0.0079, gam = 0.0246, Max-Change = 0.0040, gam = 0.0233, Max-Change = 0.0105, gam = 0.0222, Max-Change = 0.0049, gam = 0.0212, Max-Change = 0.0051, gam = 0.0203, Max-Change = 0.0124, gam = 0.0195, Max-Change = 0.0096, gam = 0.0188, Max-Change = 0.0145, gam = 0.0181, Max-Change = 0.0066, gam = 0.0175, Max-Change = 0.0234, gam = 0.0169, Max-Change = 0.0129, gam = 0.0164, Max-Change = 0.0096, gam = 0.0159, Max-Change = 0.0039, gam = 0.0154, Max-Change = 0.0066, gam = 0.0150, Max-Change = 0.0084, gam = 0.0146, Max-Change = 0.0173, gam = 0.0142, Max-Change = 0.0095, gam = 0.0139, Max-Change = 0.0080, gam = 0.0135, Max-Change = 0.0065, gam = 0.0132, Max-Change = 0.0055, gam = 0.0129, Max-Change = 0.0156, gam = 0.0126, Max-Change = 0.0010, gam = 0.0124, Max-Change = 0.0063, gam = 0.0121, Max-Change = 0.0083, gam = 0.0119, Max-Change = 0.0064, gam = 0.0116, Max-Change = 0.0040, gam = 0.0114, Max-Change = 0.0010, gam = 0.0112, Max-Change = 0.0047, gam = 0.0110, Max-Change = 0.0012, gam = 0.0108, Max-Change = 0.0062, gam = 0.0106, Max-Change = 0.0039, gam = 0.0104, Max-Change = 0.0026, gam = 0.0102, Max-Change = 0.0022, gam = 0.0101, Max-Change = 0.0069, gam = 0.0099, Max-Change = 0.0022, gam = 0.0098, Max-Change = 0.0016, gam = 0.0096, Max-Change = 0.0025, gam = 0.0095, Max-Change = 0.0068, gam = 0.0093, Max-Change = 0.0016, gam = 0.0092, Max-Change = 0.0033, gam = 0.0091, Max-Change = 0.0024, gam = 0.0089, Max-Change = 0.0008, gam = 0.0088, Max-Change = 0.0018, gam = 0.0087, Max-Change = 0.0037, gam = 0.0086, Max-Change = 0.0033, gam = 0.0085, Max-Change = 0.0043, gam = 0.0084, Max-Change = 0.0015, gam = 0.0082, Max-Change = 0.0033, gam = 0.0081, Max-Change = 0.0017, gam = 0.0080, Max-Change = 0.0015, gam = 0.0080, Max-Change = 0.0023, gam = 0.0079, Max-Change = 0.0045, gam = 0.0078, Max-Change = 0.0084, gam = 0.0077, Max-Change = 0.0039, gam = 0.0076, Max-Change = 0.0057, gam = 0.0075, Max-Change = 0.0074, gam = 0.0074, Max-Change = 0.0028, gam = 0.0073, Max-Change = 0.0050, gam = 0.0073, Max-Change = 0.0041, gam = 0.0072, Max-Change = 0.0015, gam = 0.0071, Max-Change = 0.0017, gam = 0.0070, Max-Change = 0.0032, gam = 0.0070, Max-Change = 0.0028, gam = 0.0069, Max-Change = 0.0059, gam = 0.0068, Max-Change = 0.0013, gam = 0.0068, Max-Change = 0.0025, gam = 0.0067, Max-Change = 0.0022, gam = 0.0066, Max-Change = 0.0029, gam = 0.0066, Max-Change = 0.0010, gam = 0.0065, Max-Change = 0.0021, gam = 0.0065, Max-Change = 0.0022, gam = 0.0064, Max-Change = 0.0038, gam = 0.0064, Max-Change = 0.0062, gam = 0.0063, Max-Change = 0.0027, gam = 0.0062, Max-Change = 0.0034, gam = 0.0062, Max-Change = 0.0016, gam = 0.0061, Max-Change = 0.0009, gam = 0.0061, Max-Change = 0.0035, gam = 0.0060, Max-Change = 0.0053, gam = 0.0060, Max-Change = 0.0045, gam = 0.0059, Max-Change = 0.0027, gam = 0.0059, Max-Change = 0.0033, gam = 0.0058, Max-Change = 0.0043, gam = 0.0058, Max-Change = 0.0009, gam = 0.0058, Max-Change = 0.0033, gam = 0.0057, Max-Change = 0.0026, gam = 0.0057, Max-Change = 0.0025, gam = 0.0056, Max-Change = 0.0036, gam = 0.0056, Max-Change = 0.0012, gam = 0.0055, Max-Change = 0.0011, gam = 0.0055, Max-Change = 0.0044, gam = 0.0055, Max-Change = 0.0023, gam = 0.0054, Max-Change = 0.0014, gam = 0.0054, Max-Change = 0.0018, gam = 0.0053, Max-Change = 0.0028, gam = 0.0053, Max-Change = 0.0013, gam = 0.0053, Max-Change = 0.0060, gam = 0.0052, Max-Change = 0.0042, gam = 0.0052, Max-Change = 0.0029, gam = 0.0052, Max-Change = 0.0013, gam = 0.0051, Max-Change = 0.0012, gam = 0.0051, Max-Change = 0.0016, gam = 0.0051, Max-Change = 0.0022, gam = 0.0050, Max-Change = 0.0018, gam = 0.0050, Max-Change = 0.0008, gam = 0.0050, Max-Change = 0.0027, gam = 0.0049, Max-Change = 0.0021, gam = 0.0049, Max-Change = 0.0042, gam = 0.0049, Max-Change = 0.0008, gam = 0.0048, Max-Change = 0.0034, gam = 0.0048, Max-Change = 0.0020, gam = 0.0048, Max-Change = 0.0019, gam = 0.0048, Max-Change = 0.0020, gam = 0.0047, Max-Change = 0.0009, gam = 0.0047, Max-Change = 0.0027, gam = 0.0047, Max-Change = 0.0064, gam = 0.0046, Max-Change = 0.0032, gam = 0.0046, Max-Change = 0.0016, gam = 0.0046, Max-Change = 0.0038, gam = 0.0046, Max-Change = 0.0044, gam = 0.0045, Max-Change = 0.0015, gam = 0.0045, Max-Change = 0.0018, gam = 0.0045, Max-Change = 0.0019, gam = 0.0045, Max-Change = 0.0023, gam = 0.0044, Max-Change = 0.0010, gam = 0.0044, Max-Change = 0.0006, gam = 0.0044, Max-Change = 0.0012, gam = 0.0044, Max-Change = 0.0014, gam = 0.0043, Max-Change = 0.0025, gam = 0.0043, Max-Change = 0.0044, gam = 0.0043, Max-Change = 0.0040, gam = 0.0043, Max-Change = 0.0023, gam = 0.0043, Max-Change = 0.0011, gam = 0.0042, Max-Change = 0.0013, gam = 0.0042, Max-Change = 0.0020, gam = 0.0042, Max-Change = 0.0012, gam = 0.0042, Max-Change = 0.0027, gam = 0.0041, Max-Change = 0.0016, gam = 0.0041, Max-Change = 0.0035, gam = 0.0041, Max-Change = 0.0036, gam = 0.0041, Max-Change = 0.0044, gam = 0.0041, Max-Change = 0.0009, gam = 0.0040, Max-Change = 0.0007, gam = 0.0040, Max-Change = 0.0028, gam = 0.0040, Max-Change = 0.0029, gam = 0.0040, Max-Change = 0.0011, gam = 0.0040, Max-Change = 0.0026, gam = 0.0040, Max-Change = 0.0008, gam = 0.0039, Max-Change = 0.0011, gam = 0.0039, Max-Change = 0.0011, gam = 0.0039, Max-Change = 0.0017, gam = 0.0039, Max-Change = 0.0028, gam = 0.0039, Max-Change = 0.0012, gam = 0.0038, Max-Change = 0.0014, gam = 0.0038, Max-Change = 0.0020, gam = 0.0038, Max-Change = 0.0017, gam = 0.0038, Max-Change = 0.0020, gam = 0.0038, Max-Change = 0.0010, gam = 0.0038, Max-Change = 0.0028, gam = 0.0037, Max-Change = 0.0010, gam = 0.0037, Max-Change = 0.0015, gam = 0.0037, Max-Change = 0.0007, gam = 0.0037, Max-Change = 0.0008, gam = 0.0037, Max-Change = 0.0021, gam = 0.0037, Max-Change = 0.0017, gam = 0.0036, Max-Change = 0.0007, gam = 0.0036, Max-Change = 0.0020, gam = 0.0036, Max-Change = 0.0024, gam = 0.0036, Max-Change = 0.0015, gam = 0.0036, Max-Change = 0.0030, gam = 0.0036, Max-Change = 0.0018, gam = 0.0036, Max-Change = 0.0007, gam = 0.0035, Max-Change = 0.0012, gam = 0.0035, Max-Change = 0.0009, gam = 0.0035, Max-Change = 0.0007, gam = 0.0035, Max-Change = 0.0021, gam = 0.0035, Max-Change = 0.0040, gam = 0.0035, Max-Change = 0.0013, gam = 0.0035, Max-Change = 0.0032, gam = 0.0034, Max-Change = 0.0028, gam = 0.0034, Max-Change = 0.0010, gam = 0.0034, Max-Change = 0.0015, gam = 0.0034, Max-Change = 0.0009, gam = 0.0034, Max-Change = 0.0017, gam = 0.0034, Max-Change = 0.0024, gam = 0.0034, Max-Change = 0.0009, gam = 0.0034, Max-Change = 0.0003, gam = 0.0033, Max-Change = 0.0013, gam = 0.0033, Max-Change = 0.0014, gam = 0.0033, Max-Change = 0.0006, gam = 0.0033, Max-Change = 0.0013, gam = 0.0033, Max-Change = 0.0013, gam = 0.0033, Max-Change = 0.0011, gam = 0.0033, Max-Change = 0.0005, gam = 0.0033, Max-Change = 0.0046, gam = 0.0032, Max-Change = 0.0016, gam = 0.0032, Max-Change = 0.0018, gam = 0.0032, Max-Change = 0.0004, gam = 0.0032, Max-Change = 0.0020, gam = 0.0032, Max-Change = 0.0020, gam = 0.0032, Max-Change = 0.0007, gam = 0.0032, Max-Change = 0.0012, gam = 0.0032, Max-Change = 0.0015, gam = 0.0032, Max-Change = 0.0020, gam = 0.0031, Max-Change = 0.0005, gam = 0.0031, Max-Change = 0.0015, gam = 0.0031, Max-Change = 0.0017, gam = 0.0031, Max-Change = 0.0010, gam = 0.0031, Max-Change = 0.0008, gam = 0.0031, Max-Change = 0.0018, gam = 0.0031, Max-Change = 0.0024, gam = 0.0031, Max-Change = 0.0031, gam = 0.0031, Max-Change = 0.0023, gam = 0.0031, Max-Change = 0.0031, gam = 0.0030, Max-Change = 0.0004, gam = 0.0030, Max-Change = 0.0004, gam = 0.0030, Max-Change = 0.0009
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod2)
#>
#> Call:
#> mixedmirt(data = Science, covdata = covdat, model = 1, fixed = ~0 +
#> group, itemtype = "gpcm")
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> groupm -0.176 0.118 -1.494
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1
#>
anova(mod, mod2)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod 3264.979 3276.155 3287.014 3320.577 -1618.489
#> mod2 3258.379 3271.950 3285.136 3325.891 -1612.190 12.6 3 0.006
# graded model
mod3 <- mixedmirt(Science, covdat, model=1, fixed = ~ 0 + group,
itemtype = 'graded')
#> , Max-Change = 0.0936, Max-Change = 0.0628, Max-Change = 0.0682, Max-Change = 0.0779, Max-Change = 0.0530, Max-Change = 0.0384, Max-Change = 0.0643, Max-Change = 0.0358, Max-Change = 0.1088, Max-Change = 0.0556, Max-Change = 0.0754, Max-Change = 0.0696, Max-Change = 0.0500, Max-Change = 0.0316, Max-Change = 0.0371, Max-Change = 0.0448, Max-Change = 0.0569, Max-Change = 0.0753, Max-Change = 0.0459, Max-Change = 0.0565, Max-Change = 0.0385, Max-Change = 0.0177, Max-Change = 0.0258, Max-Change = 0.0425, Max-Change = 0.0579, Max-Change = 0.0233, Max-Change = 0.0269, Max-Change = 0.0206, Max-Change = 0.0494, Max-Change = 0.0379, Max-Change = 0.0439, Max-Change = 0.0775, Max-Change = 0.0386, Max-Change = 0.0435, Max-Change = 0.0212, Max-Change = 0.0944, Max-Change = 0.0190, Max-Change = 0.0412, Max-Change = 0.0670, Max-Change = 0.0318, Max-Change = 0.0405, Max-Change = 0.0383, Max-Change = 0.0675, Max-Change = 0.0429, Max-Change = 0.0710, Max-Change = 0.0385, Max-Change = 0.0516, Max-Change = 0.0490, Max-Change = 0.0381, Max-Change = 0.0909, Max-Change = 0.0201, Max-Change = 0.0393, Max-Change = 0.0424, Max-Change = 0.0310, Max-Change = 0.0379, Max-Change = 0.0371, Max-Change = 0.1221, Max-Change = 0.0474, Max-Change = 0.0486, Max-Change = 0.0651, Max-Change = 0.0363, Max-Change = 0.0310, Max-Change = 0.0917, Max-Change = 0.0539, Max-Change = 0.0282, Max-Change = 0.0437, Max-Change = 0.0413, Max-Change = 0.0446, Max-Change = 0.0343, Max-Change = 0.0360, Max-Change = 0.0701, Max-Change = 0.0552, Max-Change = 0.0295, Max-Change = 0.0505, Max-Change = 0.0599, Max-Change = 0.0442, Max-Change = 0.0321, Max-Change = 0.0408, Max-Change = 0.0260, Max-Change = 0.0358, Max-Change = 0.0379, Max-Change = 0.0366, Max-Change = 0.0425, Max-Change = 0.0911, Max-Change = 0.0482, Max-Change = 0.0729, Max-Change = 0.0495, Max-Change = 0.0480, Max-Change = 0.0313, Max-Change = 0.0560, Max-Change = 0.0323, Max-Change = 0.0339, Max-Change = 0.0423, Max-Change = 0.0947, Max-Change = 0.0268, Max-Change = 0.0233, Max-Change = 0.0482, Max-Change = 0.0317, Max-Change = 0.0484, Max-Change = 0.0427, Max-Change = 0.0749, Max-Change = 0.0558, Max-Change = 0.0468, Max-Change = 0.0896, Max-Change = 0.0333, Max-Change = 0.0351, Max-Change = 0.0371, Max-Change = 0.0444, Max-Change = 0.0392, Max-Change = 0.0493, Max-Change = 0.0454, Max-Change = 0.0264, Max-Change = 0.0272, Max-Change = 0.0301, Max-Change = 0.0299, Max-Change = 0.0344, Max-Change = 0.0460, Max-Change = 0.0483, Max-Change = 0.0884, Max-Change = 0.0503, Max-Change = 0.0337, Max-Change = 0.0218, Max-Change = 0.0212, Max-Change = 0.0804, Max-Change = 0.0338, Max-Change = 0.0272, Max-Change = 0.0332, Max-Change = 0.0395, Max-Change = 0.0609, Max-Change = 0.0310, Max-Change = 0.0271, Max-Change = 0.0554, Max-Change = 0.0289, Max-Change = 0.0460, Max-Change = 0.0677, Max-Change = 0.0711, Max-Change = 0.0312, Max-Change = 0.0287, Max-Change = 0.0526, Max-Change = 0.0342, Max-Change = 0.0345, Max-Change = 0.0297, Max-Change = 0.0384, Max-Change = 0.0285, Max-Change = 0.0479, Max-Change = 0.0354, Max-Change = 0.1089, Max-Change = 0.0675, Max-Change = 0.0314, Max-Change = 0.0452, Max-Change = 0.1232, Max-Change = 0.0390, Max-Change = 0.0892, Max-Change = 0.0303, Max-Change = 0.0327, Max-Change = 0.1439, Max-Change = 0.0321, Max-Change = 0.0940, Max-Change = 0.0379, Max-Change = 0.0813, Max-Change = 0.0346, Max-Change = 0.0404, Max-Change = 0.0554, Max-Change = 0.0336, Max-Change = 0.0281, Max-Change = 0.0589, Max-Change = 0.0316, Max-Change = 0.0912, Max-Change = 0.0500, Max-Change = 0.0364, Max-Change = 0.1087, Max-Change = 0.0267, Max-Change = 0.0234, Max-Change = 0.0413, Max-Change = 0.0177, Max-Change = 0.0276, Max-Change = 0.0715, Max-Change = 0.0415, Max-Change = 0.0250, Max-Change = 0.1006, Max-Change = 0.0210, Max-Change = 0.0568, Max-Change = 0.1370, Max-Change = 0.0315, Max-Change = 0.0599, Max-Change = 0.0186, Max-Change = 0.0953, Max-Change = 0.0973, Max-Change = 0.0536, Max-Change = 0.0967, Max-Change = 0.0323, Max-Change = 0.0438, Max-Change = 0.0825, Max-Change = 0.0453, Max-Change = 0.0471, Max-Change = 0.0624, Max-Change = 0.0290, Max-Change = 0.0384, Max-Change = 0.0372, Max-Change = 0.0810, Max-Change = 0.0457, Max-Change = 0.0415, Max-Change = 0.0329, Max-Change = 0.0675, Max-Change = 0.0283, Max-Change = 0.0571, Max-Change = 0.0277, Max-Change = 0.0567, Max-Change = 0.0597, Max-Change = 0.1081, Max-Change = 0.0365, Max-Change = 0.0409, Max-Change = 0.0624, Max-Change = 0.0567, Max-Change = 0.0347, Max-Change = 0.1347, Max-Change = 0.0570, Max-Change = 0.0683, Max-Change = 0.0604, Max-Change = 0.0916, Max-Change = 0.0895, Max-Change = 0.0384, Max-Change = 0.0310, Max-Change = 0.0302, Max-Change = 0.0528, Max-Change = 0.0244, Max-Change = 0.0991, Max-Change = 0.0630, Max-Change = 0.0419, Max-Change = 0.0702, Max-Change = 0.0346, Max-Change = 0.0397, Max-Change = 0.0542, Max-Change = 0.0295, Max-Change = 0.0251, Max-Change = 0.0376, Max-Change = 0.0436, Max-Change = 0.0139, Max-Change = 0.0323, Max-Change = 0.0293, Max-Change = 0.0452, Max-Change = 0.1383, Max-Change = 0.0361, Max-Change = 0.0899, Max-Change = 0.0155, Max-Change = 0.0330, Max-Change = 0.0329, Max-Change = 0.0665, Max-Change = 0.0744, Max-Change = 0.0201, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0371, gam = 0.1057, Max-Change = 0.0347, gam = 0.0780, Max-Change = 0.0089, gam = 0.0629, Max-Change = 0.0091, gam = 0.0532, Max-Change = 0.0179, gam = 0.0464, Max-Change = 0.0041, gam = 0.0413, Max-Change = 0.0156, gam = 0.0374, Max-Change = 0.0204, gam = 0.0342, Max-Change = 0.0039, gam = 0.0316, Max-Change = 0.0069, gam = 0.0294, Max-Change = 0.0038, gam = 0.0276, Max-Change = 0.0136, gam = 0.0260, Max-Change = 0.0049, gam = 0.0246, Max-Change = 0.0066, gam = 0.0233, Max-Change = 0.0074, gam = 0.0222, Max-Change = 0.0075, gam = 0.0212, Max-Change = 0.0039, gam = 0.0203, Max-Change = 0.0053, gam = 0.0195, Max-Change = 0.0072, gam = 0.0188, Max-Change = 0.0108, gam = 0.0181, Max-Change = 0.0049, gam = 0.0175, Max-Change = 0.0108, gam = 0.0169, Max-Change = 0.0109, gam = 0.0164, Max-Change = 0.0079, gam = 0.0159, Max-Change = 0.0041, gam = 0.0154, Max-Change = 0.0025, gam = 0.0150, Max-Change = 0.0063, gam = 0.0146, Max-Change = 0.0015, gam = 0.0142, Max-Change = 0.0017, gam = 0.0139, Max-Change = 0.0084, gam = 0.0135, Max-Change = 0.0026, gam = 0.0132, Max-Change = 0.0021, gam = 0.0129, Max-Change = 0.0017, gam = 0.0126, Max-Change = 0.0022, gam = 0.0124, Max-Change = 0.0036, gam = 0.0121, Max-Change = 0.0035, gam = 0.0119, Max-Change = 0.0030, gam = 0.0116, Max-Change = 0.0023, gam = 0.0114, Max-Change = 0.0019, gam = 0.0112, Max-Change = 0.0028, gam = 0.0110, Max-Change = 0.0014, gam = 0.0108, Max-Change = 0.0045, gam = 0.0106, Max-Change = 0.0017, gam = 0.0104, Max-Change = 0.0016, gam = 0.0102, Max-Change = 0.0021, gam = 0.0101, Max-Change = 0.0019, gam = 0.0099, Max-Change = 0.0030, gam = 0.0098, Max-Change = 0.0021, gam = 0.0096, Max-Change = 0.0016, gam = 0.0095, Max-Change = 0.0009, gam = 0.0093, Max-Change = 0.0015, gam = 0.0092, Max-Change = 0.0024, gam = 0.0091, Max-Change = 0.0024, gam = 0.0089, Max-Change = 0.0015, gam = 0.0088, Max-Change = 0.0032, gam = 0.0087, Max-Change = 0.0023, gam = 0.0086, Max-Change = 0.0046, gam = 0.0085, Max-Change = 0.0013, gam = 0.0084, Max-Change = 0.0015, gam = 0.0082, Max-Change = 0.0019, gam = 0.0081, Max-Change = 0.0014, gam = 0.0080, Max-Change = 0.0016, gam = 0.0080, Max-Change = 0.0030, gam = 0.0079, Max-Change = 0.0009, gam = 0.0078, Max-Change = 0.0022, gam = 0.0077, Max-Change = 0.0009, gam = 0.0076, Max-Change = 0.0027, gam = 0.0075, Max-Change = 0.0026, gam = 0.0074, Max-Change = 0.0018, gam = 0.0073, Max-Change = 0.0044, gam = 0.0073, Max-Change = 0.0013, gam = 0.0072, Max-Change = 0.0018, gam = 0.0071, Max-Change = 0.0013, gam = 0.0070, Max-Change = 0.0015, gam = 0.0070, Max-Change = 0.0030, gam = 0.0069, Max-Change = 0.0032, gam = 0.0068, Max-Change = 0.0006, gam = 0.0068, Max-Change = 0.0016, gam = 0.0067, Max-Change = 0.0018, gam = 0.0066, Max-Change = 0.0017, gam = 0.0066, Max-Change = 0.0020, gam = 0.0065, Max-Change = 0.0007, gam = 0.0065, Max-Change = 0.0008, gam = 0.0064, Max-Change = 0.0013, gam = 0.0064, Max-Change = 0.0023, gam = 0.0063, Max-Change = 0.0024, gam = 0.0062, Max-Change = 0.0020, gam = 0.0062, Max-Change = 0.0009, gam = 0.0061, Max-Change = 0.0008, gam = 0.0061, Max-Change = 0.0016, gam = 0.0060, Max-Change = 0.0011, gam = 0.0060, Max-Change = 0.0005, gam = 0.0059, Max-Change = 0.0013, gam = 0.0059, Max-Change = 0.0007, gam = 0.0058, Max-Change = 0.0009, gam = 0.0058, Max-Change = 0.0007
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
coef(mod3)
#> $Comfort
#> groupm a1 d1 d2 d3
#> par -0.248 1.000 4.951 2.735 -1.327
#> CI_2.5 -0.578 0.636 3.989 2.289 -1.682
#> CI_97.5 0.082 1.363 5.913 3.181 -0.971
#>
#> $Work
#> groupm a1 d1 d2 d3
#> par -0.248 1.212 3.034 1.020 -2.133
#> CI_2.5 -0.578 0.873 2.545 0.697 -2.554
#> CI_97.5 0.082 1.551 3.522 1.342 -1.712
#>
#> $Future
#> groupm a1 d1 d2 d3
#> par -0.248 2.581 5.761 2.524 -1.995
#> CI_2.5 -0.578 1.251 3.733 1.497 -2.807
#> CI_97.5 0.082 3.911 7.789 3.551 -1.182
#>
#> $Benefit
#> groupm a1 d1 d2 d3
#> par -0.248 1.075 3.456 1.110 -1.554
#> CI_2.5 -0.578 0.728 2.905 0.796 -1.925
#> CI_97.5 0.082 1.423 4.007 1.423 -1.183
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 1
#> CI_2.5 NA NA
#> CI_97.5 NA NA
#>
###################################################
# latent regression with Rasch and 2PL models
set.seed(1)
n <- 300
a <- matrix(1, 10)
d <- matrix(rnorm(10))
Theta <- matrix(c(rnorm(n, 0), rnorm(n, 1), rnorm(n, 2)))
covdata <- data.frame(group=rep(c('g1','g2','g3'), each=n))
dat <- simdata(a, d, N=n*3, Theta=Theta, itemtype = '2PL')
itemstats(dat)
#> $overall
#> N mean_total.score sd_total.score ave.r sd.r alpha SEM.alpha
#> 900 6.932 2.347 0.197 0.035 0.709 1.266
#>
#> $itemstats
#> N mean sd total.r total.r_if_rm alpha_if_rm
#> Item_1 900 0.570 0.495 0.604 0.443 0.673
#> Item_2 900 0.689 0.463 0.518 0.351 0.690
#> Item_3 900 0.526 0.500 0.531 0.352 0.691
#> Item_4 900 0.892 0.310 0.422 0.305 0.698
#> Item_5 900 0.748 0.435 0.522 0.367 0.687
#> Item_6 900 0.543 0.498 0.569 0.398 0.682
#> Item_7 900 0.759 0.428 0.530 0.379 0.685
#> Item_8 900 0.803 0.398 0.516 0.375 0.686
#> Item_9 900 0.777 0.417 0.489 0.337 0.692
#> Item_10 900 0.626 0.484 0.548 0.378 0.685
#>
#> $proportions
#> 0 1
#> Item_1 0.430 0.570
#> Item_2 0.311 0.689
#> Item_3 0.474 0.526
#> Item_4 0.108 0.892
#> Item_5 0.252 0.748
#> Item_6 0.457 0.543
#> Item_7 0.241 0.759
#> Item_8 0.197 0.803
#> Item_9 0.223 0.777
#> Item_10 0.374 0.626
#>
# had we known the latent abilities, we could have computed the regression coefs
summary(lm(Theta ~ covdata$group))
#>
#> Call:
#> lm(formula = Theta ~ covdata$group)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.9871 -0.6851 -0.0427 0.7170 3.8313
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.04434 0.05970 0.743 0.458
#> covdata$groupg2 0.93468 0.08443 11.071 <2e-16 ***
#> covdata$groupg3 1.88134 0.08443 22.284 <2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 1.034 on 897 degrees of freedom
#> Multiple R-squared: 0.3563, Adjusted R-squared: 0.3549
#> F-statistic: 248.3 on 2 and 897 DF, p-value: < 2.2e-16
#>
# but all we have is observed test data. Latent regression helps to recover these coefs
# Rasch model approach (and mirt equivalent)
rmod0 <- mirt(dat, 1, 'Rasch') # unconditional
#>
# these two models are equivalent
rmod1a <- mirt(dat, 1, 'Rasch', covdata = covdata, formula = ~ group)
#>
rmod1b <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items + group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1660, Max-Change = 0.1608, Max-Change = 0.1153, Max-Change = 0.1284, Max-Change = 0.1084, Max-Change = 0.0957, Max-Change = 0.0911, Max-Change = 0.0686, Max-Change = 0.0641, Max-Change = 0.0574, Max-Change = 0.0470, Max-Change = 0.0606, Max-Change = 0.0515, Max-Change = 0.0547, Max-Change = 0.0557, Max-Change = 0.0498, Max-Change = 0.0395, Max-Change = 0.0418, Max-Change = 0.0242, Max-Change = 0.0275, Max-Change = 0.0258, Max-Change = 0.0452, Max-Change = 0.0276, Max-Change = 0.0125, Max-Change = 0.0253, Max-Change = 0.0108, Max-Change = 0.0333, Max-Change = 0.0210, Max-Change = 0.0134, Max-Change = 0.0152, Max-Change = 0.0231, Max-Change = 0.0217, Max-Change = 0.0079, Max-Change = 0.0055, Max-Change = 0.0177, Max-Change = 0.0161, Max-Change = 0.0112, Max-Change = 0.0215, Max-Change = 0.0093, Max-Change = 0.0148, Max-Change = 0.0046, Max-Change = 0.0062, Max-Change = 0.0209, Max-Change = 0.0151, Max-Change = 0.0218, Max-Change = 0.0196, Max-Change = 0.0184, Max-Change = 0.0086, Max-Change = 0.0131, Max-Change = 0.0063, Max-Change = 0.0094, Max-Change = 0.0153, Max-Change = 0.0218, Max-Change = 0.0101, Max-Change = 0.0083, Max-Change = 0.0093, Max-Change = 0.0063, Max-Change = 0.0164, Max-Change = 0.0056, Max-Change = 0.0080, Max-Change = 0.0213, Max-Change = 0.0182, Max-Change = 0.0181, Max-Change = 0.0143, Max-Change = 0.0046, Max-Change = 0.0158, Max-Change = 0.0097, Max-Change = 0.0053, Max-Change = 0.0100, Max-Change = 0.0160, Max-Change = 0.0180, Max-Change = 0.0125, Max-Change = 0.0130, Max-Change = 0.0052, Max-Change = 0.0130, Max-Change = 0.0063, Max-Change = 0.0151, Max-Change = 0.0077, Max-Change = 0.0175, Max-Change = 0.0040, Max-Change = 0.0118, Max-Change = 0.0074, Max-Change = 0.0119, Max-Change = 0.0034, Max-Change = 0.0066, Max-Change = 0.0120, Max-Change = 0.0091, Max-Change = 0.0088, Max-Change = 0.0247, Max-Change = 0.0402, Max-Change = 0.0111, Max-Change = 0.0130, Max-Change = 0.0303, Max-Change = 0.0130, Max-Change = 0.0019, Max-Change = 0.0164, Max-Change = 0.0046, Max-Change = 0.0144, Max-Change = 0.0129, Max-Change = 0.0112, Max-Change = 0.0158, Max-Change = 0.0095, Max-Change = 0.0094, Max-Change = 0.0093, Max-Change = 0.0221, Max-Change = 0.0065, Max-Change = 0.0094, Max-Change = 0.0084, Max-Change = 0.0103, Max-Change = 0.0149, Max-Change = 0.0050, Max-Change = 0.0116, Max-Change = 0.0152, Max-Change = 0.0187, Max-Change = 0.0131, Max-Change = 0.0103, Max-Change = 0.0097, Max-Change = 0.0067, Max-Change = 0.0101, Max-Change = 0.0154, Max-Change = 0.0066, Max-Change = 0.0054, Max-Change = 0.0095, Max-Change = 0.0134, Max-Change = 0.0274, Max-Change = 0.0198, Max-Change = 0.0215, Max-Change = 0.0062, Max-Change = 0.0192, Max-Change = 0.0136, Max-Change = 0.0070, Max-Change = 0.0231, Max-Change = 0.0084, Max-Change = 0.0036, Max-Change = 0.0111, Max-Change = 0.0212, Max-Change = 0.0077, Max-Change = 0.0103, Max-Change = 0.0066, Max-Change = 0.0145, Max-Change = 0.0148, Max-Change = 0.0220, Max-Change = 0.0106, Max-Change = 0.0029, Max-Change = 0.0118, Max-Change = 0.0063, Max-Change = 0.0111, Max-Change = 0.0124, Max-Change = 0.0123, Max-Change = 0.0334, Max-Change = 0.0058, Max-Change = 0.0069, Max-Change = 0.0107, Max-Change = 0.0096, Max-Change = 0.0085, Max-Change = 0.0131, Max-Change = 0.0087, Max-Change = 0.0044, Max-Change = 0.0128, Max-Change = 0.0061, Max-Change = 0.0071, Max-Change = 0.0255, Max-Change = 0.0085, Max-Change = 0.0039, Max-Change = 0.0073, Max-Change = 0.0116, Max-Change = 0.0178, Max-Change = 0.0079, Max-Change = 0.0110, Max-Change = 0.0258, Max-Change = 0.0104, Max-Change = 0.0107, Max-Change = 0.0188, Max-Change = 0.0064, Max-Change = 0.0130, Max-Change = 0.0245, Max-Change = 0.0130, Max-Change = 0.0056, Max-Change = 0.0093, Max-Change = 0.0055, Max-Change = 0.0215, Max-Change = 0.0085, Max-Change = 0.0101, Max-Change = 0.0114, Max-Change = 0.0121, Max-Change = 0.0113, Max-Change = 0.0278, Max-Change = 0.0092, Max-Change = 0.0157, Max-Change = 0.0057, Max-Change = 0.0107, Max-Change = 0.0197, Max-Change = 0.0160, Max-Change = 0.0130, Max-Change = 0.0111, Max-Change = 0.0104, Max-Change = 0.0099, Max-Change = 0.0070, Max-Change = 0.0122, Max-Change = 0.0056, Max-Change = 0.0191, Max-Change = 0.0057, Max-Change = 0.0093, Max-Change = 0.0135, Max-Change = 0.0127, Max-Change = 0.0101, Max-Change = 0.0207, Max-Change = 0.0223, Max-Change = 0.0166, Max-Change = 0.0193, Max-Change = 0.0147, Max-Change = 0.0132, Max-Change = 0.0091, Max-Change = 0.0186, Max-Change = 0.0149, Max-Change = 0.0111, Max-Change = 0.0074, Max-Change = 0.0106, Max-Change = 0.0046, Max-Change = 0.0145, Max-Change = 0.0145, Max-Change = 0.0062, Max-Change = 0.0128, Max-Change = 0.0127, Max-Change = 0.0167, Max-Change = 0.0117, Max-Change = 0.0079, Max-Change = 0.0170, Max-Change = 0.0163, Max-Change = 0.0093, Max-Change = 0.0094, Max-Change = 0.0147, Max-Change = 0.0057, Max-Change = 0.0175, Max-Change = 0.0084, Max-Change = 0.0149, Max-Change = 0.0051, Max-Change = 0.0126, Max-Change = 0.0149, Max-Change = 0.0224, Max-Change = 0.0094, Max-Change = 0.0108, Max-Change = 0.0175, Max-Change = 0.0080, Max-Change = 0.0122, Max-Change = 0.0100, Max-Change = 0.0113, Max-Change = 0.0164, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0118, gam = 0.1057, Max-Change = 0.0087, gam = 0.0780, Max-Change = 0.0021, gam = 0.0629, Max-Change = 0.0056, gam = 0.0532, Max-Change = 0.0039, gam = 0.0464, Max-Change = 0.0022, gam = 0.0413, Max-Change = 0.0034, gam = 0.0374, Max-Change = 0.0035, gam = 0.0342, Max-Change = 0.0008, gam = 0.0316, Max-Change = 0.0007, gam = 0.0294, Max-Change = 0.0039, gam = 0.0276, Max-Change = 0.0024, gam = 0.0260, Max-Change = 0.0019, gam = 0.0246, Max-Change = 0.0009, gam = 0.0233, Max-Change = 0.0017, gam = 0.0222, Max-Change = 0.0011, gam = 0.0212, Max-Change = 0.0014, gam = 0.0203, Max-Change = 0.0008, gam = 0.0195, Max-Change = 0.0002, gam = 0.0188, Max-Change = 0.0021, gam = 0.0181, Max-Change = 0.0009, gam = 0.0175, Max-Change = 0.0015, gam = 0.0169, Max-Change = 0.0011, gam = 0.0164, Max-Change = 0.0010, gam = 0.0159, Max-Change = 0.0004, gam = 0.0154, Max-Change = 0.0009, gam = 0.0150, Max-Change = 0.0003
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
anova(rmod0, rmod1b)
#> AIC SABIC HQ BIC logLik X2 df p
#> rmod0 9698.483 9716.375 9718.663 9751.310 -4838.242
#> rmod1b 9472.610 9493.755 9496.459 9535.041 -4723.305 229.873 2 0
coef(rmod1a, simplify=TRUE)
#> $items
#> a1 d g u
#> Item_1 1 -0.466 0 1
#> Item_2 1 0.193 0 1
#> Item_3 1 -0.699 0 1
#> Item_4 1 1.795 0 1
#> Item_5 1 0.561 0 1
#> Item_6 1 -0.607 0 1
#> Item_7 1 0.636 0 1
#> Item_8 1 0.957 0 1
#> Item_9 1 0.759 0 1
#> Item_10 1 -0.167 0 1
#>
#> $means
#> F1
#> 0
#>
#> $cov
#> F1
#> F1 1.005
#>
#> $lr.betas
#> F1
#> (Intercept) 0.000
#> groupg2 0.797
#> groupg3 1.707
#>
summary(rmod1b)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items + group)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> groupg2 0.821 0.118 6.972
#> groupg3 1.683 0.111 15.140
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1
#>
# 2PL, requires different input to allow Theta variance to remain fixed
mod0 <- mirt(dat, 1) # unconditional
#>
mod1a <- mirt(dat, 1, covdata = covdata, formula = ~ group, itemtype = '2PL')
#>
mod1b <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items, lr.fixed = ~group, itemtype = '2PL')
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1937, Max-Change = 0.1590, Max-Change = 0.1455, Max-Change = 0.1217, Max-Change = 0.1043, Max-Change = 0.0948, Max-Change = 0.0799, Max-Change = 0.0691, Max-Change = 0.0633, Max-Change = 0.0481, Max-Change = 0.0417, Max-Change = 0.0393, Max-Change = 0.0264, Max-Change = 0.0549, Max-Change = 0.0198, Max-Change = 0.0260, Max-Change = 0.0205, Max-Change = 0.0201, Max-Change = 0.0274, Max-Change = 0.0250, Max-Change = 0.0381, Max-Change = 0.0146, Max-Change = 0.0227, Max-Change = 0.0456, Max-Change = 0.0278, Max-Change = 0.0151, Max-Change = 0.0267, Max-Change = 0.0274, Max-Change = 0.0234, Max-Change = 0.0229, Max-Change = 0.0192, Max-Change = 0.0254, Max-Change = 0.0336, Max-Change = 0.0151, Max-Change = 0.0182, Max-Change = 0.0177, Max-Change = 0.0190, Max-Change = 0.0151, Max-Change = 0.0118, Max-Change = 0.0229, Max-Change = 0.0186, Max-Change = 0.0140, Max-Change = 0.0197, Max-Change = 0.0176, Max-Change = 0.0128, Max-Change = 0.0154, Max-Change = 0.0254, Max-Change = 0.0187, Max-Change = 0.0143, Max-Change = 0.0170, Max-Change = 0.0234, Max-Change = 0.0362, Max-Change = 0.0341, Max-Change = 0.0320, Max-Change = 0.0249, Max-Change = 0.0158, Max-Change = 0.0274, Max-Change = 0.0170, Max-Change = 0.0274, Max-Change = 0.0203, Max-Change = 0.0240, Max-Change = 0.0293, Max-Change = 0.0185, Max-Change = 0.0166, Max-Change = 0.0328, Max-Change = 0.0220, Max-Change = 0.0272, Max-Change = 0.0278, Max-Change = 0.0197, Max-Change = 0.0209, Max-Change = 0.0114, Max-Change = 0.0254, Max-Change = 0.0274, Max-Change = 0.0209, Max-Change = 0.0179, Max-Change = 0.0176, Max-Change = 0.0122, Max-Change = 0.0209, Max-Change = 0.0207, Max-Change = 0.0190, Max-Change = 0.0240, Max-Change = 0.0207, Max-Change = 0.0182, Max-Change = 0.0190, Max-Change = 0.0265, Max-Change = 0.0289, Max-Change = 0.0183, Max-Change = 0.0164, Max-Change = 0.0246, Max-Change = 0.0345, Max-Change = 0.0196, Max-Change = 0.0266, Max-Change = 0.0163, Max-Change = 0.0237, Max-Change = 0.0174, Max-Change = 0.0131, Max-Change = 0.0152, Max-Change = 0.0165, Max-Change = 0.0232, Max-Change = 0.0191, Max-Change = 0.0154, Max-Change = 0.0209, Max-Change = 0.0144, Max-Change = 0.0282, Max-Change = 0.0247, Max-Change = 0.0117, Max-Change = 0.0103, Max-Change = 0.0224, Max-Change = 0.0156, Max-Change = 0.0179, Max-Change = 0.0159, Max-Change = 0.0229, Max-Change = 0.0217, Max-Change = 0.0193, Max-Change = 0.0191, Max-Change = 0.0189, Max-Change = 0.0414, Max-Change = 0.0325, Max-Change = 0.0230, Max-Change = 0.0280, Max-Change = 0.0241, Max-Change = 0.0183, Max-Change = 0.0318, Max-Change = 0.0178, Max-Change = 0.0209, Max-Change = 0.0166, Max-Change = 0.0181, Max-Change = 0.0169, Max-Change = 0.0252, Max-Change = 0.0146, Max-Change = 0.0267, Max-Change = 0.0209, Max-Change = 0.0270, Max-Change = 0.0175, Max-Change = 0.0263, Max-Change = 0.0149, Max-Change = 0.0140, Max-Change = 0.0228, Max-Change = 0.0180, Max-Change = 0.0383, Max-Change = 0.0207, Max-Change = 0.0251, Max-Change = 0.0118, Max-Change = 0.0148, Max-Change = 0.0181, Max-Change = 0.0310, Max-Change = 0.0256, Max-Change = 0.0140, Max-Change = 0.0130, Max-Change = 0.0195, Max-Change = 0.0199, Max-Change = 0.0365, Max-Change = 0.0221, Max-Change = 0.0294, Max-Change = 0.0219, Max-Change = 0.0207, Max-Change = 0.0159, Max-Change = 0.0211, Max-Change = 0.0287, Max-Change = 0.0139, Max-Change = 0.0225, Max-Change = 0.0134, Max-Change = 0.0103, Max-Change = 0.0168, Max-Change = 0.0301, Max-Change = 0.0250, Max-Change = 0.0232, Max-Change = 0.0402, Max-Change = 0.0227, Max-Change = 0.0194, Max-Change = 0.0263, Max-Change = 0.0164, Max-Change = 0.0164, Max-Change = 0.0213, Max-Change = 0.0324, Max-Change = 0.0187, Max-Change = 0.0201, Max-Change = 0.0249, Max-Change = 0.0327, Max-Change = 0.0269, Max-Change = 0.0392, Max-Change = 0.0311, Max-Change = 0.0137, Max-Change = 0.0276, Max-Change = 0.0189, Max-Change = 0.0218, Max-Change = 0.0157, Max-Change = 0.0336, Max-Change = 0.0302, Max-Change = 0.0216, Max-Change = 0.0214, Max-Change = 0.0190, Max-Change = 0.0213, Max-Change = 0.0266, Max-Change = 0.0149, Max-Change = 0.0213, Max-Change = 0.0228, Max-Change = 0.0215, Max-Change = 0.0333, Max-Change = 0.0090, Max-Change = 0.0261, Max-Change = 0.0156, Max-Change = 0.0148, Max-Change = 0.0208, Max-Change = 0.0142, Max-Change = 0.0172, Max-Change = 0.0320, Max-Change = 0.0210, Max-Change = 0.0249, Max-Change = 0.0289, Max-Change = 0.0183, Max-Change = 0.0131, Max-Change = 0.0257, Max-Change = 0.0169, Max-Change = 0.0257, Max-Change = 0.0114, Max-Change = 0.0163, Max-Change = 0.0158, Max-Change = 0.0116, Max-Change = 0.0185, Max-Change = 0.0238, Max-Change = 0.0268, Max-Change = 0.0185, Max-Change = 0.0219, Max-Change = 0.0204, Max-Change = 0.0216, Max-Change = 0.0216, Max-Change = 0.0324, Max-Change = 0.0184, Max-Change = 0.0172, Max-Change = 0.0248, Max-Change = 0.0177, Max-Change = 0.0218, Max-Change = 0.0126, Max-Change = 0.0175, Max-Change = 0.0239, Max-Change = 0.0197, Max-Change = 0.0267, Max-Change = 0.0247, Max-Change = 0.0338, Max-Change = 0.0156, Max-Change = 0.0186, Max-Change = 0.0153, Max-Change = 0.0425, Max-Change = 0.0265, Max-Change = 0.0145, Max-Change = 0.0254, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0225, gam = 0.1057, Max-Change = 0.0134, gam = 0.0780, Max-Change = 0.0108, gam = 0.0629, Max-Change = 0.0051, gam = 0.0532, Max-Change = 0.0074, gam = 0.0464, Max-Change = 0.0045, gam = 0.0413, Max-Change = 0.0046, gam = 0.0374, Max-Change = 0.0044, gam = 0.0342, Max-Change = 0.0056, gam = 0.0316, Max-Change = 0.0034, gam = 0.0294, Max-Change = 0.0035, gam = 0.0276, Max-Change = 0.0032, gam = 0.0260, Max-Change = 0.0024, gam = 0.0246, Max-Change = 0.0036, gam = 0.0233, Max-Change = 0.0029, gam = 0.0222, Max-Change = 0.0030, gam = 0.0212, Max-Change = 0.0034, gam = 0.0203, Max-Change = 0.0021, gam = 0.0195, Max-Change = 0.0019, gam = 0.0188, Max-Change = 0.0016, gam = 0.0181, Max-Change = 0.0019, gam = 0.0175, Max-Change = 0.0018, gam = 0.0169, Max-Change = 0.0011, gam = 0.0164, Max-Change = 0.0027, gam = 0.0159, Max-Change = 0.0013, gam = 0.0154, Max-Change = 0.0008, gam = 0.0150, Max-Change = 0.0019, gam = 0.0146, Max-Change = 0.0022, gam = 0.0142, Max-Change = 0.0011, gam = 0.0139, Max-Change = 0.0012, gam = 0.0135, Max-Change = 0.0015, gam = 0.0132, Max-Change = 0.0016, gam = 0.0129, Max-Change = 0.0012, gam = 0.0126, Max-Change = 0.0018, gam = 0.0124, Max-Change = 0.0013, gam = 0.0121, Max-Change = 0.0013, gam = 0.0119, Max-Change = 0.0008, gam = 0.0116, Max-Change = 0.0008, gam = 0.0114, Max-Change = 0.0011, gam = 0.0112, Max-Change = 0.0009, gam = 0.0110, Max-Change = 0.0022, gam = 0.0108, Max-Change = 0.0014, gam = 0.0106, Max-Change = 0.0014, gam = 0.0104, Max-Change = 0.0016, gam = 0.0102, Max-Change = 0.0011, gam = 0.0101, Max-Change = 0.0005, gam = 0.0099, Max-Change = 0.0009, gam = 0.0098, Max-Change = 0.0010
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
anova(mod0, mod1b)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod0 9706.089 9738.620 9742.780 9802.136 -4833.044
#> mod1b 9481.042 9516.826 9521.402 9586.694 -4718.521 229.047 2 0
coef(mod1a)$lr.betas
#> F1
#> (Intercept) 0.0000000
#> groupg2 0.7910307
#> groupg3 1.7064184
summary(mod1b)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items, itemtype = "2PL", lr.fixed = ~group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1
#>
#> --------------
#> LATENT REGRESSION FIXED EFFECTS:
#>
#> F1
#> (Intercept) 0.000
#> groupg2 0.767
#> groupg3 1.711
#>
#> Std.Error_F1 z_F1
#> (Intercept) NA NA
#> groupg2 NaN NaN
#> groupg3 NaN NaN
# specifying specific regression effects is accomplished by passing a list of formula
model <- 'F1 = 1-5
F2 = 6-10'
covdata$contvar <- rnorm(nrow(covdata))
mod2 <- mirt(dat, model, itemtype = 'Rasch', covdata=covdata,
formula = list(F1 = ~ group + contvar, F2 = ~ group))
#>
coef(mod2)[11:12]
#> $GroupPars
#> MEAN_1 MEAN_2 COV_11 COV_21 COV_22
#> par 0 0 1.030864 0 1.106502
#>
#> $lr.betas
#> F1 F2
#> (Intercept) 0.0000000 0.0000000
#> groupg2 0.7213049 0.9025158
#> groupg3 1.7596829 1.7089144
#> contvar 0.1063280 0.0000000
#>
mod2b <- mixedmirt(dat, covdata, model, fixed = ~ 0 + items,
lr.fixed = list(F1 = ~ group + contvar, F2 = ~ group))
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1871, Max-Change = 0.1647, Max-Change = 0.1423, Max-Change = 0.1203, Max-Change = 0.0967, Max-Change = 0.0829, Max-Change = 0.0648, Max-Change = 0.0558, Max-Change = 0.0477, Max-Change = 0.0468, Max-Change = 0.0372, Max-Change = 0.0342, Max-Change = 0.0246, Max-Change = 0.0216, Max-Change = 0.0279, Max-Change = 0.0167, Max-Change = 0.0355, Max-Change = 0.0187, Max-Change = 0.0122, Max-Change = 0.0108, Max-Change = 0.0141, Max-Change = 0.0334, Max-Change = 0.0181, Max-Change = 0.0162, Max-Change = 0.0246, Max-Change = 0.0111, Max-Change = 0.0114, Max-Change = 0.0119, Max-Change = 0.0093, Max-Change = 0.0158, Max-Change = 0.0147, Max-Change = 0.0201, Max-Change = 0.0130, Max-Change = 0.0171, Max-Change = 0.0203, Max-Change = 0.0195, Max-Change = 0.0162, Max-Change = 0.0119, Max-Change = 0.0120, Max-Change = 0.0208, Max-Change = 0.0135, Max-Change = 0.0095, Max-Change = 0.0202, Max-Change = 0.0151, Max-Change = 0.0107, Max-Change = 0.0116, Max-Change = 0.0133, Max-Change = 0.0137, Max-Change = 0.0116, Max-Change = 0.0162, Max-Change = 0.0113, Max-Change = 0.0107, Max-Change = 0.0146, Max-Change = 0.0209, Max-Change = 0.0135, Max-Change = 0.0114, Max-Change = 0.0157, Max-Change = 0.0159, Max-Change = 0.0132, Max-Change = 0.0168, Max-Change = 0.0140, Max-Change = 0.0166, Max-Change = 0.0170, Max-Change = 0.0209, Max-Change = 0.0151, Max-Change = 0.0139, Max-Change = 0.0210, Max-Change = 0.0140, Max-Change = 0.0153, Max-Change = 0.0105, Max-Change = 0.0085, Max-Change = 0.0149, Max-Change = 0.0192, Max-Change = 0.0137, Max-Change = 0.0160, Max-Change = 0.0170, Max-Change = 0.0152, Max-Change = 0.0204, Max-Change = 0.0174, Max-Change = 0.0096, Max-Change = 0.0135, Max-Change = 0.0142, Max-Change = 0.0087, Max-Change = 0.0140, Max-Change = 0.0154, Max-Change = 0.0326, Max-Change = 0.0249, Max-Change = 0.0149, Max-Change = 0.0147, Max-Change = 0.0134, Max-Change = 0.0144, Max-Change = 0.0217, Max-Change = 0.0182, Max-Change = 0.0115, Max-Change = 0.0150, Max-Change = 0.0131, Max-Change = 0.0070, Max-Change = 0.0186, Max-Change = 0.0098, Max-Change = 0.0087, Max-Change = 0.0055, Max-Change = 0.0131, Max-Change = 0.0158, Max-Change = 0.0168, Max-Change = 0.0146, Max-Change = 0.0216, Max-Change = 0.0226, Max-Change = 0.0107, Max-Change = 0.0101, Max-Change = 0.0067, Max-Change = 0.0144, Max-Change = 0.0214, Max-Change = 0.0129, Max-Change = 0.0250, Max-Change = 0.0111, Max-Change = 0.0116, Max-Change = 0.0134, Max-Change = 0.0176, Max-Change = 0.0120, Max-Change = 0.0131, Max-Change = 0.0125, Max-Change = 0.0235, Max-Change = 0.0172, Max-Change = 0.0211, Max-Change = 0.0095, Max-Change = 0.0185, Max-Change = 0.0074, Max-Change = 0.0236, Max-Change = 0.0237, Max-Change = 0.0143, Max-Change = 0.0158, Max-Change = 0.0113, Max-Change = 0.0153, Max-Change = 0.0207, Max-Change = 0.0210, Max-Change = 0.0154, Max-Change = 0.0110, Max-Change = 0.0063, Max-Change = 0.0102, Max-Change = 0.0113, Max-Change = 0.0184, Max-Change = 0.0116, Max-Change = 0.0167, Max-Change = 0.0145, Max-Change = 0.0172, Max-Change = 0.0136, Max-Change = 0.0168, Max-Change = 0.0192, Max-Change = 0.0193, Max-Change = 0.0206, Max-Change = 0.0150, Max-Change = 0.0080, Max-Change = 0.0182, Max-Change = 0.0141, Max-Change = 0.0119, Max-Change = 0.0204, Max-Change = 0.0170, Max-Change = 0.0180, Max-Change = 0.0128, Max-Change = 0.0349, Max-Change = 0.0169, Max-Change = 0.0154, Max-Change = 0.0060, Max-Change = 0.0167, Max-Change = 0.0235, Max-Change = 0.0171, Max-Change = 0.0233, Max-Change = 0.0151, Max-Change = 0.0189, Max-Change = 0.0143, Max-Change = 0.0237, Max-Change = 0.0289, Max-Change = 0.0089, Max-Change = 0.0172, Max-Change = 0.0106, Max-Change = 0.0115, Max-Change = 0.0157, Max-Change = 0.0116, Max-Change = 0.0133, Max-Change = 0.0162, Max-Change = 0.0226, Max-Change = 0.0144, Max-Change = 0.0337, Max-Change = 0.0176, Max-Change = 0.0161, Max-Change = 0.0145, Max-Change = 0.0160, Max-Change = 0.0206, Max-Change = 0.0174, Max-Change = 0.0121, Max-Change = 0.0162, Max-Change = 0.0104, Max-Change = 0.0117, Max-Change = 0.0156, Max-Change = 0.0265, Max-Change = 0.0140, Max-Change = 0.0122, Max-Change = 0.0135, Max-Change = 0.0094, Max-Change = 0.0094, Max-Change = 0.0122, Max-Change = 0.0118, Max-Change = 0.0137, Max-Change = 0.0084, Max-Change = 0.0082, Max-Change = 0.0197, Max-Change = 0.0092, Max-Change = 0.0153, Max-Change = 0.0191, Max-Change = 0.0099, Max-Change = 0.0096, Max-Change = 0.0153, Max-Change = 0.0138, Max-Change = 0.0115, Max-Change = 0.0079, Max-Change = 0.0235, Max-Change = 0.0152, Max-Change = 0.0118, Max-Change = 0.0141, Max-Change = 0.0140, Max-Change = 0.0206, Max-Change = 0.0139, Max-Change = 0.0196, Max-Change = 0.0080, Max-Change = 0.0097, Max-Change = 0.0152, Max-Change = 0.0129, Max-Change = 0.0160, Max-Change = 0.0121, Max-Change = 0.0088, Max-Change = 0.0185, Max-Change = 0.0117, Max-Change = 0.0226, Max-Change = 0.0131, Max-Change = 0.0223, Max-Change = 0.0179, Max-Change = 0.0183, Max-Change = 0.0104, Max-Change = 0.0151, Max-Change = 0.0167, Max-Change = 0.0156, Max-Change = 0.0088, Max-Change = 0.0114, Max-Change = 0.0211, Max-Change = 0.0113, Max-Change = 0.0115, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0254, gam = 0.1057, Max-Change = 0.0113, gam = 0.0780, Max-Change = 0.0050, gam = 0.0629, Max-Change = 0.0019, gam = 0.0532, Max-Change = 0.0039, gam = 0.0464, Max-Change = 0.0052, gam = 0.0413, Max-Change = 0.0037, gam = 0.0374, Max-Change = 0.0028, gam = 0.0342, Max-Change = 0.0026, gam = 0.0316, Max-Change = 0.0015, gam = 0.0294, Max-Change = 0.0023, gam = 0.0276, Max-Change = 0.0016, gam = 0.0260, Max-Change = 0.0020, gam = 0.0246, Max-Change = 0.0017, gam = 0.0233, Max-Change = 0.0011, gam = 0.0222, Max-Change = 0.0020, gam = 0.0212, Max-Change = 0.0019, gam = 0.0203, Max-Change = 0.0016, gam = 0.0195, Max-Change = 0.0014, gam = 0.0188, Max-Change = 0.0026, gam = 0.0181, Max-Change = 0.0019, gam = 0.0175, Max-Change = 0.0013, gam = 0.0169, Max-Change = 0.0018, gam = 0.0164, Max-Change = 0.0018, gam = 0.0159, Max-Change = 0.0021, gam = 0.0154, Max-Change = 0.0007, gam = 0.0150, Max-Change = 0.0011, gam = 0.0146, Max-Change = 0.0013, gam = 0.0142, Max-Change = 0.0019, gam = 0.0139, Max-Change = 0.0011, gam = 0.0135, Max-Change = 0.0005, gam = 0.0132, Max-Change = 0.0011, gam = 0.0129, Max-Change = 0.0011, gam = 0.0126, Max-Change = 0.0008, gam = 0.0124, Max-Change = 0.0007, gam = 0.0121, Max-Change = 0.0009
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod2b)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = model, fixed = ~0 +
#> items, lr.fixed = list(F1 = ~group + contvar, F2 = ~group))
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1 F2
#> F1 0.982 0.00
#> F2 0.000 1.09
#>
#> --------------
#> LATENT REGRESSION FIXED EFFECTS:
#>
#> F1 F2
#> (Intercept) 0.000 0.000
#> groupg2 0.695 0.817
#> groupg3 1.743 1.621
#> contvar 0.100 0.000
#>
#> Std.Error_F1 Std.Error_F2 z_F1 z_F2
#> (Intercept) NA NA NA NA
#> groupg2 NaN NaN NaN NaN
#> groupg3 NaN NaN NaN NaN
#> contvar 0.023 NA 4.364 NA
####################################################
## Simulated Multilevel Rasch Model
set.seed(1)
N <- 2000
a <- matrix(rep(1,10),10,1)
d <- matrix(rnorm(10))
cluster = 100
random_intercept = rnorm(cluster,0,1)
Theta = numeric()
for (i in 1:cluster)
Theta <- c(Theta, rnorm(N/cluster,0,1) + random_intercept[i])
group = factor(rep(paste0('G',1:cluster), each = N/cluster))
covdata <- data.frame(group)
dat <- simdata(a,d,N, itemtype = rep('2PL',10), Theta=matrix(Theta))
itemstats(dat)
#> $overall
#> N mean_total.score sd_total.score ave.r sd.r alpha SEM.alpha
#> 2000 5.414 2.749 0.25 0.019 0.769 1.321
#>
#> $itemstats
#> N mean sd total.r total.r_if_rm alpha_if_rm
#> Item_1 2000 0.424 0.494 0.573 0.433 0.750
#> Item_2 2000 0.560 0.497 0.602 0.467 0.745
#> Item_3 2000 0.373 0.484 0.547 0.405 0.754
#> Item_4 2000 0.781 0.414 0.524 0.402 0.754
#> Item_5 2000 0.577 0.494 0.585 0.447 0.748
#> Item_6 2000 0.378 0.485 0.571 0.434 0.750
#> Item_7 2000 0.598 0.491 0.568 0.428 0.751
#> Item_8 2000 0.652 0.476 0.568 0.432 0.750
#> Item_9 2000 0.620 0.486 0.577 0.440 0.749
#> Item_10 2000 0.453 0.498 0.584 0.445 0.748
#>
#> $proportions
#> 0 1
#> Item_1 0.577 0.424
#> Item_2 0.440 0.560
#> Item_3 0.627 0.373
#> Item_4 0.219 0.781
#> Item_5 0.424 0.577
#> Item_6 0.622 0.378
#> Item_7 0.402 0.598
#> Item_8 0.348 0.652
#> Item_9 0.380 0.620
#> Item_10 0.547 0.453
#>
# null model
mod1 <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items, random = ~ 1|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1825, Max-Change = 0.1571, Max-Change = 0.1285, Max-Change = 0.1118, Max-Change = 0.0969, Max-Change = 0.0772, Max-Change = 0.0708, Max-Change = 0.0561, Max-Change = 0.0421, Max-Change = 0.0388, Max-Change = 0.0371, Max-Change = 0.0292, Max-Change = 0.0279, Max-Change = 0.0280, Max-Change = 0.0298, Max-Change = 0.0281, Max-Change = 0.0155, Max-Change = 0.0170, Max-Change = 0.0196, Max-Change = 0.0178, Max-Change = 0.0237, Max-Change = 0.0187, Max-Change = 0.0213, Max-Change = 0.0099, Max-Change = 0.0244, Max-Change = 0.0166, Max-Change = 0.0162, Max-Change = 0.0050, Max-Change = 0.0021, Max-Change = 0.0102, Max-Change = 0.0067, Max-Change = 0.0116, Max-Change = 0.0072, Max-Change = 0.0132, Max-Change = 0.0120, Max-Change = 0.0046, Max-Change = 0.0212, Max-Change = 0.0191, Max-Change = 0.0018, Max-Change = 0.0017, Max-Change = 0.0211, Max-Change = 0.0152, Max-Change = 0.0106, Max-Change = 0.0155, Max-Change = 0.0061, Max-Change = 0.0041, Max-Change = 0.0070, Max-Change = 0.0047, Max-Change = 0.0054, Max-Change = 0.0031, Max-Change = 0.0151, Max-Change = 0.0071, Max-Change = 0.0130, Max-Change = 0.0058, Max-Change = 0.0035, Max-Change = 0.0044, Max-Change = 0.0068, Max-Change = 0.0106, Max-Change = 0.0058, Max-Change = 0.0065, Max-Change = 0.0030, Max-Change = 0.0078, Max-Change = 0.0160, Max-Change = 0.0089, Max-Change = 0.0077, Max-Change = 0.0106, Max-Change = 0.0092, Max-Change = 0.0155, Max-Change = 0.0181, Max-Change = 0.0104, Max-Change = 0.0151, Max-Change = 0.0145, Max-Change = 0.0030, Max-Change = 0.0058, Max-Change = 0.0041, Max-Change = 0.0064, Max-Change = 0.0187, Max-Change = 0.0024, Max-Change = 0.0019, Max-Change = 0.0066, Max-Change = 0.0035, Max-Change = 0.0057, Max-Change = 0.0059, Max-Change = 0.0063, Max-Change = 0.0050, Max-Change = 0.0150, Max-Change = 0.0118, Max-Change = 0.0072, Max-Change = 0.0049, Max-Change = 0.0037, Max-Change = 0.0012, Max-Change = 0.0023, Max-Change = 0.0076, Max-Change = 0.0033, Max-Change = 0.0071, Max-Change = 0.0045, Max-Change = 0.0169, Max-Change = 0.1604, Max-Change = 0.0823, Max-Change = 0.0655, Max-Change = 0.0506, Max-Change = 0.0506, Max-Change = 0.0465, Max-Change = 0.0366, Max-Change = 0.0268, Max-Change = 0.0292, Max-Change = 0.0274, Max-Change = 0.0089, Max-Change = 0.0328, Max-Change = 0.0200, Max-Change = 0.0162, Max-Change = 0.0139, Max-Change = 0.0238, Max-Change = 0.0122, Max-Change = 0.0054, Max-Change = 0.0121, Max-Change = 0.0102, Max-Change = 0.0059, Max-Change = 0.0210, Max-Change = 0.0241, Max-Change = 0.0091, Max-Change = 0.0027, Max-Change = 0.0034, Max-Change = 0.0108, Max-Change = 0.0070, Max-Change = 0.0126, Max-Change = 0.0176, Max-Change = 0.0136, Max-Change = 0.0162, Max-Change = 0.0077, Max-Change = 0.0107, Max-Change = 0.0056, Max-Change = 0.0137, Max-Change = 0.0097, Max-Change = 0.0057, Max-Change = 0.0045, Max-Change = 0.0058, Max-Change = 0.0074, Max-Change = 0.0012, Max-Change = 0.0096, Max-Change = 0.0073, Max-Change = 0.0058, Max-Change = 0.0048, Max-Change = 0.0091, Max-Change = 0.0057, Max-Change = 0.0058, Max-Change = 0.0106, Max-Change = 0.0070, Max-Change = 0.0027, Max-Change = 0.0044, Max-Change = 0.0036, Max-Change = 0.0032, Max-Change = 0.0063, Max-Change = 0.0053, Max-Change = 0.0049, Max-Change = 0.0114, Max-Change = 0.0080, Max-Change = 0.0054, Max-Change = 0.0143, Max-Change = 0.0183, Max-Change = 0.0110, Max-Change = 0.0025, Max-Change = 0.0047, Max-Change = 0.0071, Max-Change = 0.0039, Max-Change = 0.0101, Max-Change = 0.0093, Max-Change = 0.0067, Max-Change = 0.0014, Max-Change = 0.0041, Max-Change = 0.0047, Max-Change = 0.0073, Max-Change = 0.0108, Max-Change = 0.0033, Max-Change = 0.0120, Max-Change = 0.0109, Max-Change = 0.0149, Max-Change = 0.0138, Max-Change = 0.0082, Max-Change = 0.0119, Max-Change = 0.0063, Max-Change = 0.0074, Max-Change = 0.0123, Max-Change = 0.0063, Max-Change = 0.0044, Max-Change = 0.0047, Max-Change = 0.0098, Max-Change = 0.0102, Max-Change = 0.0207, Max-Change = 0.0144, Max-Change = 0.0101, Max-Change = 0.0039, Max-Change = 0.0079, Max-Change = 0.0092, Max-Change = 0.0020, Max-Change = 0.0040, Max-Change = 0.0080, Max-Change = 0.0104, Max-Change = 0.0072, Max-Change = 0.0118, Max-Change = 0.0109, Max-Change = 0.0041, Max-Change = 0.0042, Max-Change = 0.0096, Max-Change = 0.0043, Max-Change = 0.0046, Max-Change = 0.0031, Max-Change = 0.0090, Max-Change = 0.0019, Max-Change = 0.0042, Max-Change = 0.0068, Max-Change = 0.0022, Max-Change = 0.0023, Max-Change = 0.0132, Max-Change = 0.0032, Max-Change = 0.0025, Max-Change = 0.0070, Max-Change = 0.0065, Max-Change = 0.0040, Max-Change = 0.0055, Max-Change = 0.0056, Max-Change = 0.0031, Max-Change = 0.0085, Max-Change = 0.0067, Max-Change = 0.0029, Max-Change = 0.0080, Max-Change = 0.0046, Max-Change = 0.0137, Max-Change = 0.0070, Max-Change = 0.0052, Max-Change = 0.0119, Max-Change = 0.0087, Max-Change = 0.0039, Max-Change = 0.0069, Max-Change = 0.0050, Max-Change = 0.0104, Max-Change = 0.0009, Max-Change = 0.0057, Max-Change = 0.0095, Max-Change = 0.0100, Max-Change = 0.0024, Max-Change = 0.0044, Max-Change = 0.0044, Max-Change = 0.0057, Max-Change = 0.0026, Max-Change = 0.0058, Max-Change = 0.0074, Max-Change = 0.0071, Max-Change = 0.0028, Max-Change = 0.0094, Max-Change = 0.0088, Max-Change = 0.0034, Max-Change = 0.0017, Max-Change = 0.0107, Max-Change = 0.0048, Max-Change = 0.0058, Max-Change = 0.0038, Max-Change = 0.0046, Max-Change = 0.0038, Max-Change = 0.0072, Max-Change = 0.0089, Max-Change = 0.0011, Max-Change = 0.0016, Max-Change = 0.0039, Max-Change = 0.0060, Max-Change = 0.0053, Max-Change = 0.0025, Max-Change = 0.0028, Max-Change = 0.0076, Max-Change = 0.0029, Max-Change = 0.0118, Max-Change = 0.0030, Max-Change = 0.0035, Max-Change = 0.0043, Max-Change = 0.0055, Max-Change = 0.0024, Max-Change = 0.0072, Max-Change = 0.0038, Max-Change = 0.0030, Max-Change = 0.0019, Max-Change = 0.0023, Max-Change = 0.0024, Max-Change = 0.0070, Max-Change = 0.0108, Max-Change = 0.0044, Max-Change = 0.0071, Max-Change = 0.0100, Max-Change = 0.0029, Max-Change = 0.0073, Max-Change = 0.0056, Max-Change = 0.0049, Max-Change = 0.0081, Max-Change = 0.0053, Max-Change = 0.0076, Max-Change = 0.0016, Max-Change = 0.0056, Max-Change = 0.0062, Max-Change = 0.0020, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0020, gam = 0.1057, Max-Change = 0.0083, gam = 0.0780, Max-Change = 0.0013, gam = 0.0629, Max-Change = 0.0017, gam = 0.0532, Max-Change = 0.0011, gam = 0.0464, Max-Change = 0.0002, gam = 0.0413, Max-Change = 0.0012, gam = 0.0374, Max-Change = 0.0004, gam = 0.0342, Max-Change = 0.0012, gam = 0.0316, Max-Change = 0.0003, gam = 0.0294, Max-Change = 0.0004, gam = 0.0276, Max-Change = 0.0010, gam = 0.0260, Max-Change = 0.0006, gam = 0.0246, Max-Change = 0.0004, gam = 0.0233, Max-Change = 0.0003
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod1)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items, random = ~1 | group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1.02
#>
#> $group
#> COV_group
#> COV_group 0.853
#>
# include level 2 predictor for 'group' variance
covdata$group_pred <- rep(random_intercept, each = N/cluster)
mod2 <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items + group_pred, random = ~ 1|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1797, Max-Change = 0.1540, Max-Change = 0.1292, Max-Change = 0.1114, Max-Change = 0.0981, Max-Change = 0.0797, Max-Change = 0.0726, Max-Change = 0.0565, Max-Change = 0.0436, Max-Change = 0.0397, Max-Change = 0.0353, Max-Change = 0.0313, Max-Change = 0.0278, Max-Change = 0.0326, Max-Change = 0.0265, Max-Change = 0.0204, Max-Change = 0.0212, Max-Change = 0.0173, Max-Change = 0.0144, Max-Change = 0.0184, Max-Change = 0.0145, Max-Change = 0.0115, Max-Change = 0.0066, Max-Change = 0.0137, Max-Change = 0.0068, Max-Change = 0.0131, Max-Change = 0.0088, Max-Change = 0.0131, Max-Change = 0.0121, Max-Change = 0.0098, Max-Change = 0.0085, Max-Change = 0.0064, Max-Change = 0.0127, Max-Change = 0.0081, Max-Change = 0.0079, Max-Change = 0.0073, Max-Change = 0.0062, Max-Change = 0.0087, Max-Change = 0.0085, Max-Change = 0.0046, Max-Change = 0.0091, Max-Change = 0.0039, Max-Change = 0.0077, Max-Change = 0.0035, Max-Change = 0.0032, Max-Change = 0.0047, Max-Change = 0.0046, Max-Change = 0.0074, Max-Change = 0.0039, Max-Change = 0.0037, Max-Change = 0.0104, Max-Change = 0.0064, Max-Change = 0.0067, Max-Change = 0.0035, Max-Change = 0.0043, Max-Change = 0.0088, Max-Change = 0.0028, Max-Change = 0.0047, Max-Change = 0.0040, Max-Change = 0.0024, Max-Change = 0.0078, Max-Change = 0.0162, Max-Change = 0.0064, Max-Change = 0.0054, Max-Change = 0.0070, Max-Change = 0.0016, Max-Change = 0.0066, Max-Change = 0.0069, Max-Change = 0.0076, Max-Change = 0.0078, Max-Change = 0.0057, Max-Change = 0.0103, Max-Change = 0.0023, Max-Change = 0.0051, Max-Change = 0.0057, Max-Change = 0.0055, Max-Change = 0.0184, Max-Change = 0.0044, Max-Change = 0.0037, Max-Change = 0.0050, Max-Change = 0.0042, Max-Change = 0.0070, Max-Change = 0.0010, Max-Change = 0.0078, Max-Change = 0.0022, Max-Change = 0.0104, Max-Change = 0.0058, Max-Change = 0.0075, Max-Change = 0.0049, Max-Change = 0.0010, Max-Change = 0.0014, Max-Change = 0.0033, Max-Change = 0.0029, Max-Change = 0.0033, Max-Change = 0.0053, Max-Change = 0.0015, Max-Change = 0.0026, Max-Change = 0.0172, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1757, Max-Change = 0.1798, Max-Change = 0.1665, Max-Change = 0.0043, Max-Change = 0.0036, Max-Change = 0.0047, Max-Change = 0.0044, Max-Change = 0.0095, Max-Change = 0.0068, Max-Change = 0.0034, Max-Change = 0.0076, Max-Change = 0.0038, Max-Change = 0.0058, Max-Change = 0.0064, Max-Change = 0.0070, Max-Change = 0.0053, Max-Change = 0.0115, Max-Change = 0.0084, Max-Change = 0.0088, Max-Change = 0.0042, Max-Change = 0.0051, Max-Change = 0.0060, Max-Change = 0.0075, Max-Change = 0.0062, Max-Change = 0.0056, Max-Change = 0.0088, Max-Change = 0.0070, Max-Change = 0.0040, Max-Change = 0.0161, Max-Change = 0.0061, Max-Change = 0.0044, Max-Change = 0.0038, Max-Change = 0.0070, Max-Change = 0.0018, Max-Change = 0.0071, Max-Change = 0.0095, Max-Change = 0.0081, Max-Change = 0.0092, Max-Change = 0.0070, Max-Change = 0.0144, Max-Change = 0.0061, Max-Change = 0.0084, Max-Change = 0.0076, Max-Change = 0.0065, Max-Change = 0.0046, Max-Change = 0.0049, Max-Change = 0.0092, Max-Change = 0.0063, Max-Change = 0.0027, Max-Change = 0.0061, Max-Change = 0.0085, Max-Change = 0.0043, Max-Change = 0.0063, Max-Change = 0.0052, Max-Change = 0.0028, Max-Change = 0.0054, Max-Change = 0.0076, Max-Change = 0.0027, Max-Change = 0.0039, Max-Change = 0.0033, Max-Change = 0.0041, Max-Change = 0.0082, Max-Change = 0.0050, Max-Change = 0.0104, Max-Change = 0.0078, Max-Change = 0.0037, Max-Change = 0.0060, Max-Change = 0.0080, Max-Change = 0.0045, Max-Change = 0.0068, Max-Change = 0.0039, Max-Change = 0.0078, Max-Change = 0.0041, Max-Change = 0.0038, Max-Change = 0.0019, Max-Change = 0.0061, Max-Change = 0.0112, Max-Change = 0.0074, Max-Change = 0.0025, Max-Change = 0.0039, Max-Change = 0.0087, Max-Change = 0.0016, Max-Change = 0.0032, Max-Change = 0.0020, Max-Change = 0.0066, Max-Change = 0.0022, Max-Change = 0.0048, Max-Change = 0.0050, Max-Change = 0.0066, Max-Change = 0.0036, Max-Change = 0.0040, Max-Change = 0.0049, Max-Change = 0.0049, Max-Change = 0.0078, Max-Change = 0.0041, Max-Change = 0.0093, Max-Change = 0.0046, Max-Change = 0.0109, Max-Change = 0.0053, Max-Change = 0.0105, Max-Change = 0.0050, Max-Change = 0.0022, Max-Change = 0.0057, Max-Change = 0.0064, Max-Change = 0.0040, Max-Change = 0.0044, Max-Change = 0.0053, Max-Change = 0.0095, Max-Change = 0.0033, Max-Change = 0.0030, Max-Change = 0.0066, Max-Change = 0.0019, Max-Change = 0.0035, Max-Change = 0.0124, Max-Change = 0.0072, Max-Change = 0.0072, Max-Change = 0.0075, Max-Change = 0.0074, Max-Change = 0.0034, Max-Change = 0.0055, Max-Change = 0.0036, Max-Change = 0.0020, Max-Change = 0.0090, Max-Change = 0.0070, Max-Change = 0.0048, Max-Change = 0.0064, Max-Change = 0.0046, Max-Change = 0.0049, Max-Change = 0.0049, Max-Change = 0.0042, Max-Change = 0.0090, Max-Change = 0.0049, Max-Change = 0.0034, Max-Change = 0.0071, Max-Change = 0.0040, Max-Change = 0.0124, Max-Change = 0.0072, Max-Change = 0.0065, Max-Change = 0.0072, Max-Change = 0.0140, Max-Change = 0.0046, Max-Change = 0.0095, Max-Change = 0.0052, Max-Change = 0.0078, Max-Change = 0.0026, Max-Change = 0.0081, Max-Change = 0.0097, Max-Change = 0.0031, Max-Change = 0.0037, Max-Change = 0.0112, Max-Change = 0.0082, Max-Change = 0.0051, Max-Change = 0.0026, Max-Change = 0.0031, Max-Change = 0.0055, Max-Change = 0.0118, Max-Change = 0.0060, Max-Change = 0.0030, Max-Change = 0.0054, Max-Change = 0.0032, Max-Change = 0.0106, Max-Change = 0.0020, Max-Change = 0.0077, Max-Change = 0.0047, Max-Change = 0.0053, Max-Change = 0.0060, Max-Change = 0.0062, Max-Change = 0.0019, Max-Change = 0.0039, Max-Change = 0.0009, Max-Change = 0.0143, Max-Change = 0.0031, Max-Change = 0.0069, Max-Change = 0.0050, Max-Change = 0.0027, Max-Change = 0.0020, Max-Change = 0.0085, Max-Change = 0.0023, Max-Change = 0.0040, Max-Change = 0.0049, Max-Change = 0.0040, Max-Change = 0.0048, Max-Change = 0.0056, Max-Change = 0.0050, Max-Change = 0.0046, Max-Change = 0.0065, Max-Change = 0.0133, Max-Change = 0.0035, Max-Change = 0.0059, Max-Change = 0.0067, Max-Change = 0.0044, Max-Change = 0.0060, Max-Change = 0.0030, Max-Change = 0.0072, Max-Change = 0.0066, Max-Change = 0.0067, Max-Change = 0.0045, Max-Change = 0.0033, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0026, gam = 0.1057, Max-Change = 0.0077, gam = 0.0780, Max-Change = 0.0018, gam = 0.0629, Max-Change = 0.0029, gam = 0.0532, Max-Change = 0.0016, gam = 0.0464, Max-Change = 0.0007, gam = 0.0413, Max-Change = 0.0006, gam = 0.0374, Max-Change = 0.0004
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
# including group means predicts nearly all variability in 'group'
summary(mod2)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items + group_pred, random = ~1 | group)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> group_pred 1.017 0.05 20.282
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1.04
#>
#> $group
#> COV_group
#> COV_group 0.0335
#>
anova(mod1, mod2)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod1 23548.80 23577.89 23573.48 23616.01 -11762.40
#> mod2 22744.85 22776.36 22771.59 22817.66 -11359.43 805.95 1 0
# can also be fit for Rasch/non-Rasch models with the lr.random input
mod1b <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items, lr.random = ~ 1|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1825, Max-Change = 0.1571, Max-Change = 0.1285, Max-Change = 0.1118, Max-Change = 0.0969, Max-Change = 0.0772, Max-Change = 0.0708, Max-Change = 0.0561, Max-Change = 0.0421, Max-Change = 0.0388, Max-Change = 0.0371, Max-Change = 0.0292, Max-Change = 0.0279, Max-Change = 0.0280, Max-Change = 0.0298, Max-Change = 0.0281, Max-Change = 0.0155, Max-Change = 0.0170, Max-Change = 0.0196, Max-Change = 0.0178, Max-Change = 0.0237, Max-Change = 0.0187, Max-Change = 0.0213, Max-Change = 0.0099, Max-Change = 0.0244, Max-Change = 0.0166, Max-Change = 0.0162, Max-Change = 0.0050, Max-Change = 0.0021, Max-Change = 0.0102, Max-Change = 0.0067, Max-Change = 0.0116, Max-Change = 0.0072, Max-Change = 0.0132, Max-Change = 0.0120, Max-Change = 0.0046, Max-Change = 0.0212, Max-Change = 0.0191, Max-Change = 0.0018, Max-Change = 0.0017, Max-Change = 0.0211, Max-Change = 0.0152, Max-Change = 0.0106, Max-Change = 0.0155, Max-Change = 0.0061, Max-Change = 0.0041, Max-Change = 0.0070, Max-Change = 0.0047, Max-Change = 0.0054, Max-Change = 0.0031, Max-Change = 0.0151, Max-Change = 0.0071, Max-Change = 0.0130, Max-Change = 0.0058, Max-Change = 0.0035, Max-Change = 0.0044, Max-Change = 0.0068, Max-Change = 0.0106, Max-Change = 0.0058, Max-Change = 0.0065, Max-Change = 0.0030, Max-Change = 0.0078, Max-Change = 0.0160, Max-Change = 0.0089, Max-Change = 0.0077, Max-Change = 0.0106, Max-Change = 0.0092, Max-Change = 0.0155, Max-Change = 0.0181, Max-Change = 0.0104, Max-Change = 0.0151, Max-Change = 0.0145, Max-Change = 0.0030, Max-Change = 0.0058, Max-Change = 0.0041, Max-Change = 0.0064, Max-Change = 0.0187, Max-Change = 0.0024, Max-Change = 0.0019, Max-Change = 0.0066, Max-Change = 0.0035, Max-Change = 0.0057, Max-Change = 0.0059, Max-Change = 0.0063, Max-Change = 0.0050, Max-Change = 0.0150, Max-Change = 0.0118, Max-Change = 0.0072, Max-Change = 0.0049, Max-Change = 0.0037, Max-Change = 0.0012, Max-Change = 0.0023, Max-Change = 0.0076, Max-Change = 0.0033, Max-Change = 0.0071, Max-Change = 0.0045, Max-Change = 0.0169, Max-Change = 0.0704, Max-Change = 0.0689, Max-Change = 0.0489, Max-Change = 0.0433, Max-Change = 0.0410, Max-Change = 0.0380, Max-Change = 0.0347, Max-Change = 0.0321, Max-Change = 0.0280, Max-Change = 0.0249, Max-Change = 0.0214, Max-Change = 0.0182, Max-Change = 0.0159, Max-Change = 0.0134, Max-Change = 0.0112, Max-Change = 0.0144, Max-Change = 0.0072, Max-Change = 0.0063, Max-Change = 0.0066, Max-Change = 0.0080, Max-Change = 0.0093, Max-Change = 0.0062, Max-Change = 0.0128, Max-Change = 0.0042, Max-Change = 0.0035, Max-Change = 0.0051, Max-Change = 0.0045, Max-Change = 0.0125, Max-Change = 0.0124, Max-Change = 0.0186, Max-Change = 0.0070, Max-Change = 0.0146, Max-Change = 0.0084, Max-Change = 0.0090, Max-Change = 0.0087, Max-Change = 0.0026, Max-Change = 0.0069, Max-Change = 0.0104, Max-Change = 0.0086, Max-Change = 0.0023, Max-Change = 0.0116, Max-Change = 0.0048, Max-Change = 0.0024, Max-Change = 0.0099, Max-Change = 0.0040, Max-Change = 0.0122, Max-Change = 0.0177, Max-Change = 0.0101, Max-Change = 0.0027, Max-Change = 0.0057, Max-Change = 0.0282, Max-Change = 0.0083, Max-Change = 0.0079, Max-Change = 0.0087, Max-Change = 0.0150, Max-Change = 0.0057, Max-Change = 0.0038, Max-Change = 0.0046, Max-Change = 0.0086, Max-Change = 0.0077, Max-Change = 0.0059, Max-Change = 0.0037, Max-Change = 0.0041, Max-Change = 0.0053, Max-Change = 0.0033, Max-Change = 0.0042, Max-Change = 0.0113, Max-Change = 0.0037, Max-Change = 0.0240, Max-Change = 0.0113, Max-Change = 0.0101, Max-Change = 0.0107, Max-Change = 0.0018, Max-Change = 0.0060, Max-Change = 0.0068, Max-Change = 0.0036, Max-Change = 0.0208, Max-Change = 0.0056, Max-Change = 0.0050, Max-Change = 0.0046, Max-Change = 0.0040, Max-Change = 0.0077, Max-Change = 0.0057, Max-Change = 0.0027, Max-Change = 0.0062, Max-Change = 0.0106, Max-Change = 0.0084, Max-Change = 0.0156, Max-Change = 0.0045, Max-Change = 0.0033, Max-Change = 0.0047, Max-Change = 0.0060, Max-Change = 0.0102, Max-Change = 0.0108, Max-Change = 0.0071, Max-Change = 0.0064, Max-Change = 0.0054, Max-Change = 0.0063, Max-Change = 0.0056, Max-Change = 0.0072, Max-Change = 0.0031, Max-Change = 0.0037, Max-Change = 0.0040, Max-Change = 0.0152, Max-Change = 0.0039, Max-Change = 0.0064, Max-Change = 0.0041, Max-Change = 0.0056, Max-Change = 0.0082, Max-Change = 0.0027, Max-Change = 0.0019, Max-Change = 0.0034, Max-Change = 0.0036, Max-Change = 0.0074, Max-Change = 0.0074, Max-Change = 0.0067, Max-Change = 0.0098, Max-Change = 0.0066, Max-Change = 0.0039, Max-Change = 0.0180, Max-Change = 0.0060, Max-Change = 0.0005, Max-Change = 0.0055, Max-Change = 0.0100, Max-Change = 0.0078, Max-Change = 0.0070, Max-Change = 0.0094, Max-Change = 0.0027, Max-Change = 0.0117, Max-Change = 0.0056, Max-Change = 0.0116, Max-Change = 0.0065, Max-Change = 0.0048, Max-Change = 0.0084, Max-Change = 0.0158, Max-Change = 0.0072, Max-Change = 0.0078, Max-Change = 0.0027, Max-Change = 0.0103, Max-Change = 0.0035, Max-Change = 0.0060, Max-Change = 0.0029, Max-Change = 0.0074, Max-Change = 0.0067, Max-Change = 0.0049, Max-Change = 0.0053, Max-Change = 0.0152, Max-Change = 0.0037, Max-Change = 0.0036, Max-Change = 0.0120, Max-Change = 0.0011, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0146, gam = 0.1057, Max-Change = 0.0034, gam = 0.0780, Max-Change = 0.0012, gam = 0.0629, Max-Change = 0.0018, gam = 0.0532, Max-Change = 0.0005, gam = 0.0464, Max-Change = 0.0018, gam = 0.0413, Max-Change = 0.0019, gam = 0.0374, Max-Change = 0.0007, gam = 0.0342, Max-Change = 0.0006, gam = 0.0316, Max-Change = 0.0006
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod1b)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items, lr.random = ~1 | group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1.54
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $group
#> COV_group
#> COV_group 1.45
#>
mod2b <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items + group_pred, lr.random = ~ 1|group)
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1797, Max-Change = 0.1540, Max-Change = 0.1292, Max-Change = 0.1114, Max-Change = 0.0981, Max-Change = 0.0797, Max-Change = 0.0726, Max-Change = 0.0565, Max-Change = 0.0436, Max-Change = 0.0397, Max-Change = 0.0353, Max-Change = 0.0313, Max-Change = 0.0278, Max-Change = 0.0326, Max-Change = 0.0265, Max-Change = 0.0204, Max-Change = 0.0212, Max-Change = 0.0173, Max-Change = 0.0144, Max-Change = 0.0184, Max-Change = 0.0145, Max-Change = 0.0115, Max-Change = 0.0066, Max-Change = 0.0137, Max-Change = 0.0068, Max-Change = 0.0131, Max-Change = 0.0088, Max-Change = 0.0131, Max-Change = 0.0121, Max-Change = 0.0098, Max-Change = 0.0085, Max-Change = 0.0064, Max-Change = 0.0127, Max-Change = 0.0081, Max-Change = 0.0079, Max-Change = 0.0073, Max-Change = 0.0062, Max-Change = 0.0087, Max-Change = 0.0085, Max-Change = 0.0046, Max-Change = 0.0091, Max-Change = 0.0039, Max-Change = 0.0077, Max-Change = 0.0035, Max-Change = 0.0032, Max-Change = 0.0047, Max-Change = 0.0046, Max-Change = 0.0074, Max-Change = 0.0039, Max-Change = 0.0037, Max-Change = 0.0104, Max-Change = 0.0064, Max-Change = 0.0067, Max-Change = 0.0035, Max-Change = 0.0043, Max-Change = 0.0088, Max-Change = 0.0028, Max-Change = 0.0047, Max-Change = 0.0040, Max-Change = 0.0024, Max-Change = 0.0078, Max-Change = 0.0162, Max-Change = 0.0064, Max-Change = 0.0054, Max-Change = 0.0070, Max-Change = 0.0016, Max-Change = 0.0066, Max-Change = 0.0069, Max-Change = 0.0076, Max-Change = 0.0078, Max-Change = 0.0057, Max-Change = 0.0103, Max-Change = 0.0023, Max-Change = 0.0051, Max-Change = 0.0057, Max-Change = 0.0055, Max-Change = 0.0184, Max-Change = 0.0044, Max-Change = 0.0037, Max-Change = 0.0050, Max-Change = 0.0042, Max-Change = 0.0070, Max-Change = 0.0010, Max-Change = 0.0078, Max-Change = 0.0022, Max-Change = 0.0104, Max-Change = 0.0058, Max-Change = 0.0075, Max-Change = 0.0049, Max-Change = 0.0010, Max-Change = 0.0014, Max-Change = 0.0033, Max-Change = 0.0029, Max-Change = 0.0033, Max-Change = 0.0053, Max-Change = 0.0015, Max-Change = 0.0026, Max-Change = 0.0504, Max-Change = 0.0563, Max-Change = 0.0438, Max-Change = 0.0414, Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.0674, Max-Change = 0.0488, Max-Change = 0.1112, Max-Change = 0.0397, Max-Change = 0.0278, Max-Change = 0.0117, Max-Change = 0.0143, Max-Change = 0.0157, Max-Change = 0.0098, Max-Change = 0.0107, Max-Change = 0.0456, Max-Change = 0.0102, Max-Change = 0.0108, Max-Change = 0.0286, Max-Change = 0.0242, Max-Change = 0.0158, Max-Change = 0.0098, Max-Change = 0.0145, Max-Change = 0.0062, Max-Change = 0.0076, Max-Change = 0.0117, Max-Change = 0.0107, Max-Change = 0.0120, Max-Change = 0.0100, Max-Change = 0.0085, Max-Change = 0.0268, Max-Change = 0.0089, Max-Change = 0.0072, Max-Change = 0.0066, Max-Change = 0.0051, Max-Change = 0.0076, Max-Change = 0.0049, Max-Change = 0.0051, Max-Change = 0.0058, Max-Change = 0.0080, Max-Change = 0.0052, Max-Change = 0.0136, Max-Change = 0.0078, Max-Change = 0.0079, Max-Change = 0.0079, Max-Change = 0.0061, Max-Change = 0.0028, Max-Change = 0.0061, Max-Change = 0.0110, Max-Change = 0.0032, Max-Change = 0.0023, Max-Change = 0.0071, Max-Change = 0.0048, Max-Change = 0.0087, Max-Change = 0.0048, Max-Change = 0.0096, Max-Change = 0.0060, Max-Change = 0.0052, Max-Change = 0.0035, Max-Change = 0.0057, Max-Change = 0.0061, Max-Change = 0.0036, Max-Change = 0.0041, Max-Change = 0.0070, Max-Change = 0.0063, Max-Change = 0.0112, Max-Change = 0.0021, Max-Change = 0.0141, Max-Change = 0.0145, Max-Change = 0.0035, Max-Change = 0.0038, Max-Change = 0.0023, Max-Change = 0.0084, Max-Change = 0.0054, Max-Change = 0.0073, Max-Change = 0.0036, Max-Change = 0.0117, Max-Change = 0.0045, Max-Change = 0.0050, Max-Change = 0.0066, Max-Change = 0.0070, Max-Change = 0.0050, Max-Change = 0.0008, Max-Change = 0.0061, Max-Change = 0.0089, Max-Change = 0.0036, Max-Change = 0.0127, Max-Change = 0.0072, Max-Change = 0.0045, Max-Change = 0.0094, Max-Change = 0.0102, Max-Change = 0.0087, Max-Change = 0.0114, Max-Change = 0.0088, Max-Change = 0.0058, Max-Change = 0.0016, Max-Change = 0.0072, Max-Change = 0.0096, Max-Change = 0.0061, Max-Change = 0.0060, Max-Change = 0.0020, Max-Change = 0.0113, Max-Change = 0.0087, Max-Change = 0.0036, Max-Change = 0.0066, Max-Change = 0.0037, Max-Change = 0.0041, Max-Change = 0.0044, Max-Change = 0.0051, Max-Change = 0.0075, Max-Change = 0.0020, Max-Change = 0.0050, Max-Change = 0.0055, Max-Change = 0.0047, Max-Change = 0.0049, Max-Change = 0.0113, Max-Change = 0.0050, Max-Change = 0.0044, Max-Change = 0.0080, Max-Change = 0.0076, Max-Change = 0.0047, Max-Change = 0.0054, Max-Change = 0.0075, Max-Change = 0.0026, Max-Change = 0.0035, Max-Change = 0.0099, Max-Change = 0.0038, Max-Change = 0.0108, Max-Change = 0.0072, Max-Change = 0.0086, Max-Change = 0.0042, Max-Change = 0.0025, Max-Change = 0.0094, Max-Change = 0.0063, Max-Change = 0.0023, Max-Change = 0.0100, Max-Change = 0.0037, Max-Change = 0.0064, Max-Change = 0.0031, Max-Change = 0.0068, Max-Change = 0.0047, Max-Change = 0.0077, Max-Change = 0.0042, Max-Change = 0.0045, Max-Change = 0.0079, Max-Change = 0.0060, Max-Change = 0.0025, Max-Change = 0.0040, Max-Change = 0.0071, Max-Change = 0.0033, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0065, gam = 0.1057, Max-Change = 0.0048, gam = 0.0780, Max-Change = 0.0016, gam = 0.0629, Max-Change = 0.0011, gam = 0.0532, Max-Change = 0.0018, gam = 0.0464, Max-Change = 0.0011, gam = 0.0413, Max-Change = 0.0015, gam = 0.0374, Max-Change = 0.0010, gam = 0.0342, Max-Change = 0.0007, gam = 0.0316, Max-Change = 0.0013, gam = 0.0294, Max-Change = 0.0007, gam = 0.0276, Max-Change = 0.0015, gam = 0.0260, Max-Change = 0.0004, gam = 0.0246, Max-Change = 0.0005, gam = 0.0233, Max-Change = 0.0006
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod2b)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items + group_pred, lr.random = ~1 | group)
#>
#> --------------
#> FIXED EFFECTS:
#> Estimate Std.Error z.value
#> group_pred 1.036 0.044 23.766
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1.12
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $group
#> COV_group
#> COV_group 0.1
#>
anova(mod1b, mod2b)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod1b 23637.58 23664.24 23660.20 23699.19 -11807.79
#> mod2b 22749.34 22778.43 22774.02 22816.55 -11362.67 890.24 1 0
mod3 <- mixedmirt(dat, covdata, 1, fixed = ~ 0 + items, lr.random = ~ 1|group, itemtype = '2PL')
#> , Max-Change = 0.2000, Max-Change = 0.2000, Max-Change = 0.1738, Max-Change = 0.1500, Max-Change = 0.1263, Max-Change = 0.1114, Max-Change = 0.1014, Max-Change = 0.0822, Max-Change = 0.0827, Max-Change = 0.0623, Max-Change = 0.0533, Max-Change = 0.0476, Max-Change = 0.0405, Max-Change = 0.0350, Max-Change = 0.0288, Max-Change = 0.0341, Max-Change = 0.0200, Max-Change = 0.0251, Max-Change = 0.0278, Max-Change = 0.0296, Max-Change = 0.0227, Max-Change = 0.0251, Max-Change = 0.0128, Max-Change = 0.0190, Max-Change = 0.0346, Max-Change = 0.0151, Max-Change = 0.0156, Max-Change = 0.0152, Max-Change = 0.0234, Max-Change = 0.0312, Max-Change = 0.0126, Max-Change = 0.0161, Max-Change = 0.0104, Max-Change = 0.0176, Max-Change = 0.0121, Max-Change = 0.0184, Max-Change = 0.0109, Max-Change = 0.0192, Max-Change = 0.0312, Max-Change = 0.0234, Max-Change = 0.0202, Max-Change = 0.0160, Max-Change = 0.0187, Max-Change = 0.0125, Max-Change = 0.0146, Max-Change = 0.0210, Max-Change = 0.0128, Max-Change = 0.0280, Max-Change = 0.0095, Max-Change = 0.0179, Max-Change = 0.0180, Max-Change = 0.0198, Max-Change = 0.0162, Max-Change = 0.0227, Max-Change = 0.0190, Max-Change = 0.0175, Max-Change = 0.0161, Max-Change = 0.0179, Max-Change = 0.0180, Max-Change = 0.0196, Max-Change = 0.0331, Max-Change = 0.0150, Max-Change = 0.0124, Max-Change = 0.0152, Max-Change = 0.0151, Max-Change = 0.0181, Max-Change = 0.0116, Max-Change = 0.0185, Max-Change = 0.0169, Max-Change = 0.0155, Max-Change = 0.0132, Max-Change = 0.0220, Max-Change = 0.0219, Max-Change = 0.0216, Max-Change = 0.0183, Max-Change = 0.0150, Max-Change = 0.0161, Max-Change = 0.0175, Max-Change = 0.0191, Max-Change = 0.0234, Max-Change = 0.0195, Max-Change = 0.0121, Max-Change = 0.0264, Max-Change = 0.0156, Max-Change = 0.0221, Max-Change = 0.0141, Max-Change = 0.0168, Max-Change = 0.0245, Max-Change = 0.0215, Max-Change = 0.0192, Max-Change = 0.0181, Max-Change = 0.0134, Max-Change = 0.0205, Max-Change = 0.0155, Max-Change = 0.0164, Max-Change = 0.0185, Max-Change = 0.0150, Max-Change = 0.0166, Max-Change = 0.0210, Max-Change = 0.0473, Max-Change = 0.0461, Max-Change = 0.0412, Max-Change = 0.0413, Max-Change = 0.0340, Max-Change = 0.0316, Max-Change = 0.0291, Max-Change = 0.0308, Max-Change = 0.0263, Max-Change = 0.0238, Max-Change = 0.0303, Max-Change = 0.0196, Max-Change = 0.0270, Max-Change = 0.0153, Max-Change = 0.0199, Max-Change = 0.0317, Max-Change = 0.0225, Max-Change = 0.0130, Max-Change = 0.0198, Max-Change = 0.0186, Max-Change = 0.0191, Max-Change = 0.0128, Max-Change = 0.0198, Max-Change = 0.0147, Max-Change = 0.0193, Max-Change = 0.0131, Max-Change = 0.0147, Max-Change = 0.0083, Max-Change = 0.0112, Max-Change = 0.0137, Max-Change = 0.0113, Max-Change = 0.0099, Max-Change = 0.0167, Max-Change = 0.0127, Max-Change = 0.0201, Max-Change = 0.0106, Max-Change = 0.0136, Max-Change = 0.0101, Max-Change = 0.0236, Max-Change = 0.0105, Max-Change = 0.0137, Max-Change = 0.0112, Max-Change = 0.0255, Max-Change = 0.0077, Max-Change = 0.0084, Max-Change = 0.0166, Max-Change = 0.0123, Max-Change = 0.0106, Max-Change = 0.0126, Max-Change = 0.0116, Max-Change = 0.0607, Max-Change = 0.0115, Max-Change = 0.0106, Max-Change = 0.0147, Max-Change = 0.0134, Max-Change = 0.0115, Max-Change = 0.0167, Max-Change = 0.0107, Max-Change = 0.0197, Max-Change = 0.0114, Max-Change = 0.0129, Max-Change = 0.0109, Max-Change = 0.0091, Max-Change = 0.0131, Max-Change = 0.0050, Max-Change = 0.0086, Max-Change = 0.0109, Max-Change = 0.0122, Max-Change = 0.0118, Max-Change = 0.0093, Max-Change = 0.0092, Max-Change = 0.0115, Max-Change = 0.0121, Max-Change = 0.0123, Max-Change = 0.0128, Max-Change = 0.0101, Max-Change = 0.0093, Max-Change = 0.0131, Max-Change = 0.0152, Max-Change = 0.0073, Max-Change = 0.0066, Max-Change = 0.0100, Max-Change = 0.0141, Max-Change = 0.0091, Max-Change = 0.0114, Max-Change = 0.0183, Max-Change = 0.0095, Max-Change = 0.0086, Max-Change = 0.0208, Max-Change = 0.0086, Max-Change = 0.0149, Max-Change = 0.0069, Max-Change = 0.0095, Max-Change = 0.0091, Max-Change = 0.0139, Max-Change = 0.0117, Max-Change = 0.0094, Max-Change = 0.0104, Max-Change = 0.0130, Max-Change = 0.0095, Max-Change = 0.0111, Max-Change = 0.0144, Max-Change = 0.0103, Max-Change = 0.0111, Max-Change = 0.0168, Max-Change = 0.0082, Max-Change = 0.0091, Max-Change = 0.0100, Max-Change = 0.0122, Max-Change = 0.0162, Max-Change = 0.0082, Max-Change = 0.0097, Max-Change = 0.0077, Max-Change = 0.0198, Max-Change = 0.0207, Max-Change = 0.0113, Max-Change = 0.0158, Max-Change = 0.0121, Max-Change = 0.0188, Max-Change = 0.0107, Max-Change = 0.0114, Max-Change = 0.0156, Max-Change = 0.0122, Max-Change = 0.0133, Max-Change = 0.0126, Max-Change = 0.0077, Max-Change = 0.0111, Max-Change = 0.0134, Max-Change = 0.0164, Max-Change = 0.0138, Max-Change = 0.0115, Max-Change = 0.0137, Max-Change = 0.0124, Max-Change = 0.0099, Max-Change = 0.0106, Max-Change = 0.0096, Max-Change = 0.0151, Max-Change = 0.0135, Max-Change = 0.0120, Max-Change = 0.0106, Max-Change = 0.0117, Max-Change = 0.0133, Max-Change = 0.0083, Max-Change = 0.0150, Max-Change = 0.0153, Max-Change = 0.0076, Max-Change = 0.0113, Max-Change = 0.0138, Max-Change = 0.0088, Max-Change = 0.0066, Max-Change = 0.0110, gam = 0.0000, Max-Change = 0.0000, gam = 0.1778, Max-Change = 0.0088, gam = 0.1057, Max-Change = 0.0072, gam = 0.0780, Max-Change = 0.0034, gam = 0.0629, Max-Change = 0.0035, gam = 0.0532, Max-Change = 0.0031, gam = 0.0464, Max-Change = 0.0023, gam = 0.0413, Max-Change = 0.0020, gam = 0.0374, Max-Change = 0.0019, gam = 0.0342, Max-Change = 0.0024, gam = 0.0316, Max-Change = 0.0020, gam = 0.0294, Max-Change = 0.0011, gam = 0.0276, Max-Change = 0.0026, gam = 0.0260, Max-Change = 0.0019, gam = 0.0246, Max-Change = 0.0019, gam = 0.0233, Max-Change = 0.0010, gam = 0.0222, Max-Change = 0.0012, gam = 0.0212, Max-Change = 0.0015, gam = 0.0203, Max-Change = 0.0012, gam = 0.0195, Max-Change = 0.0008, gam = 0.0188, Max-Change = 0.0008, gam = 0.0181, Max-Change = 0.0008
#>
#> Calculating information matrix...
#>
#> Calculating log-likelihood...
summary(mod3)
#>
#> Call:
#> mixedmirt(data = dat, covdata = covdata, model = 1, fixed = ~0 +
#> items, itemtype = "2PL", lr.random = ~1 | group)
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $Theta
#> F1
#> F1 1
#>
#>
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#>
#> $group
#> COV_group
#> COV_group 1
#>
anova(mod1b, mod3)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod1b 23637.58 23664.24 23660.20 23699.19 -11807.79
#> mod3 23571.39 23619.87 23612.52 23683.41 -11765.69 84.189 9 0
head(cbind(randef(mod3)$group, random_intercept))
#> group random_intercept
#> G1 1.2440442 1.51178117
#> G2 -0.5307017 0.38984324
#> G3 -0.4682958 -0.62124058
#> G4 -2.3288002 -2.21469989
#> G5 0.7959928 1.12493092
#> G6 -0.8524128 -0.04493361
# }