| bfactor {mirt} | R Documentation |
Full-Information Item Bi-factor and Two-Tier Analysis
Description
bfactor fits a confirmatory maximum likelihood two-tier/bifactor/testlet model to
dichotomous and polytomous data under the item response theory paradigm.
The IRT models are fit using a dimensional reduction EM algorithm so that regardless
of the number of specific factors estimated the model only uses the number of
factors in the second-tier structure plus 1. For the bifactor model the maximum
number of dimensions is only 2 since the second-tier only consists of a
ubiquitous unidimensional factor. See mirt for appropriate methods to be used
on the objects returned from the estimation.
Usage
bfactor(
data,
model,
model2 = paste0("G = 1-", ncol(data)),
group = NULL,
quadpts = NULL,
invariance = "",
...
)
Arguments
data |
a |
model |
a numeric vector specifying which factor loads on which
item. For example, if for a 4 item test with two specific factors, the first
specific factor loads on the first two items and the second specific factor
on the last two, then the vector is |
model2 |
a two-tier model specification object defined by |
group |
a factor variable indicating group membership used for multiple group analyses |
quadpts |
number of quadrature nodes to use after accounting for the reduced number of dimensions.
Scheme is the same as the one used in |
invariance |
see |
... |
additional arguments to be passed to the estimation engine. See |
Details
bfactor follows the item factor analysis strategy explicated by
Gibbons and Hedeker (1992), Gibbons et al. (2007), and Cai (2010).
Nested models may be compared via an approximate
chi-squared difference test or by a reduction in AIC or BIC (accessible via
anova). See mirt for more details regarding the
IRT estimation approach used in this package.
The two-tier model has a specific block diagonal covariance structure between the primary and secondary latent traits. Namely, the secondary latent traits are assumed to be orthogonal to all traits and have a fixed variance of 1, while the primary traits can be organized to vary and covary with other primary traits in the model.
\Sigma_{two-tier} = \left(\begin{array}{cc} G & 0 \\ 0 & diag(S) \end{array} \right)
The bifactor model is a special case of the two-tier model when G above is a 1x1 matrix,
and therefore only 1 primary factor is being modeled. Evaluation of the numerical integrals
for the two-tier model requires only ncol(G) + 1 dimensions for integration since the
S second order (or 'specific') factors require only 1 integration grid due to the
dimension reduction technique.
Note: for multiple group two-tier analyses only the second-tier means and variances should be freed since the specific factors are not treated independently due to the dimension reduction technique.
Value
function returns an object of class SingleGroupClass
(SingleGroupClass-class) or MultipleGroupClass(MultipleGroupClass-class).
Author(s)
Phil Chalmers rphilip.chalmers@gmail.com
References
Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika, 75, 581-612.
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
Bradlow, E.T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153-168.
Gibbons, R. D., & Hedeker, D. R. (1992). Full-information Item Bi-Factor Analysis. Psychometrika, 57, 423-436.
Gibbons, R. D., Darrell, R. B., Hedeker, D., Weiss, D. J., Segawa, E., Bhaumik, D. K., Kupfer, D. J., Frank, E., Grochocinski, V. J., & Stover, A. (2007). Full-Information item bifactor analysis of graded response data. Applied Psychological Measurement, 31, 4-19.
Wainer, H., Bradlow, E.T., & Wang, X. (2007). Testlet response theory and its applications. New York, NY: Cambridge University Press.
See Also
Examples
## No test:
### load SAT12 and compute bifactor model with 3 specific factors
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))
specific <- c(2,3,2,3,3,2,1,2,1,1,1,3,1,3,1,2,1,1,3,3,1,1,3,1,3,3,1,3,2,3,1,2)
mod1 <- bfactor(data, specific)
summary(mod1)
## G S1 S2 S3 h2
## Item.1 0.4078 0.0000 0.2273 0.0000 0.21799
## Item.2 0.6195 0.0000 0.0000 0.3391 0.49881
## Item.3 0.5573 0.0000 -0.0744 0.0000 0.31612
## Item.4 0.2808 0.0000 0.0000 0.3101 0.17503
## Item.5 0.4779 0.0000 0.0000 0.2544 0.29311
## Item.6 0.5341 0.0000 0.2724 0.0000 0.35948
## Item.7 0.4741 0.4210 0.0000 0.0000 0.40199
## Item.8 0.3537 0.0000 0.2732 0.0000 0.19971
## Item.9 0.2181 0.5321 0.0000 0.0000 0.33068
## Item.10 0.4849 0.3792 0.0000 0.0000 0.37895
## Item.11 0.6442 0.3321 0.0000 0.0000 0.52536
## Item.12 0.0699 0.0000 0.0000 0.1592 0.03023
## Item.13 0.5219 0.2743 0.0000 0.0000 0.34764
## Item.14 0.4791 0.0000 0.0000 0.4563 0.43776
## Item.15 0.5985 0.2402 0.0000 0.0000 0.41590
## Item.16 0.3885 0.0000 0.2049 0.0000 0.19292
## Item.17 0.6636 0.1177 0.0000 0.0000 0.45423
## Item.18 0.7163 0.0775 0.0000 0.0000 0.51903
## Item.19 0.4520 0.0000 0.0000 0.0198 0.20466
## Item.20 0.6578 0.0000 0.0000 0.1792 0.46478
## Item.21 0.2806 0.3451 0.0000 0.0000 0.19787
## Item.22 0.7025 -0.0281 0.0000 0.0000 0.49425
## Item.23 0.3236 0.0000 0.0000 0.2657 0.17536
## Item.24 0.5848 0.1030 0.0000 0.0000 0.35266
## Item.25 0.3732 0.0000 0.0000 0.3297 0.24799
## Item.26 0.6430 0.0000 0.0000 0.2124 0.45854
## Item.27 0.7374 0.1554 0.0000 0.0000 0.56792
## Item.28 0.5256 0.0000 0.0000 0.0758 0.28199
## Item.29 0.4185 0.0000 0.7071 0.0000 0.67516
## Item.30 0.2455 0.0000 0.0000 -0.0959 0.06946
## Item.31 0.8333 -0.0872 0.0000 0.0000 0.70202
## Item.32 0.0780 0.0000 0.0165 0.0000 0.00635
##
## SS loadings: 8.435 1.03 0.748 0.781
## Proportion Var: 0.264 0.032 0.023 0.024
##
## Factor correlations:
##
## G S1 S2 S3
## G 1
## S1 0 1
## S2 0 0 1
## S3 0 0 0 1
itemplot(mod1, 18, drop.zeros = TRUE) #drop the zero slopes to allow plotting
### Try with fixed guessing parameters added
guess <- rep(.1,32)
mod2 <- bfactor(data, specific, guess = guess)
coef(mod2)
## $Item.1
## a1 a2 a3 a4 d g u
## par 1.225 0 0.624 0 -1.822 0.1 1
##
## $Item.2
## a1 a2 a3 a4 d g u
## par 1.721 0 0 0.954 0.171 0.1 1
##
## $Item.3
## a1 a2 a3 a4 d g u
## par 2.415 0 -0.459 0 -2.602 0.1 1
##
## $Item.4
## a1 a2 a3 a4 d g u
## par 0.745 0 0 0.695 -0.989 0.1 1
##
## $Item.5
## a1 a2 a3 a4 d g u
## par 1.048 0 0 0.603 0.419 0.1 1
##
## $Item.6
## a1 a2 a3 a4 d g u
## par 3.06 0 0.501 0 -5.002 0.1 1
##
## $Item.7
## a1 a2 a3 a4 d g u
## par 1.121 0.839 0 0 1.373 0.1 1
##
## $Item.8
## a1 a2 a3 a4 d g u
## par 1.956 0 1.443 0 -3.772 0.1 1
##
## $Item.9
## a1 a2 a3 a4 d g u
## par 0.512 1.236 0 0 2.484 0.1 1
##
## $Item.10
## a1 a2 a3 a4 d g u
## par 1.68 1.506 0 0 -1.031 0.1 1
##
## $Item.11
## a1 a2 a3 a4 d g u
## par 1.655 0.842 0 0 5.441 0.1 1
##
## $Item.12
## a1 a2 a3 a4 d g u
## par 0.129 0 0 0.364 -0.641 0.1 1
##
## $Item.13
## a1 a2 a3 a4 d g u
## par 1.183 0.477 0 0 0.679 0.1 1
##
## $Item.14
## a1 a2 a3 a4 d g u
## par 1.125 0 0 1.058 1.164 0.1 1
##
## $Item.15
## a1 a2 a3 a4 d g u
## par 1.435 0.317 0 0 1.863 0.1 1
##
## $Item.16
## a1 a2 a3 a4 d g u
## par 0.95 0 0.573 0 -0.783 0.1 1
##
## $Item.17
## a1 a2 a3 a4 d g u
## par 1.547 0.059 0 0 4.112 0.1 1
##
## $Item.18
## a1 a2 a3 a4 d g u
## par 2.731 0.094 0 0 -1.808 0.1 1
##
## $Item.19
## a1 a2 a3 a4 d g u
## par 0.918 0 0 0.101 -0.001 0.1 1
##
## $Item.20
## a1 a2 a3 a4 d g u
## par 1.456 0 0 0.593 2.501 0.1 1
##
## $Item.21
## a1 a2 a3 a4 d g u
## par 0.596 0.493 0 0 2.49 0.1 1
##
## $Item.22
## a1 a2 a3 a4 d g u
## par 1.554 -0.242 0 0 3.428 0.1 1
##
## $Item.23
## a1 a2 a3 a4 d g u
## par 0.908 0 0 0.766 -1.488 0.1 1
##
## $Item.24
## a1 a2 a3 a4 d g u
## par 1.379 0.001 0 0 1.132 0.1 1
##
## $Item.25
## a1 a2 a3 a4 d g u
## par 1.03 0 0 1.094 -1.164 0.1 1
##
## $Item.26
## a1 a2 a3 a4 d g u
## par 1.985 0 0 0.747 -0.663 0.1 1
##
## $Item.27
## a1 a2 a3 a4 d g u
## par 1.909 0.348 0 0 2.642 0.1 1
##
## $Item.28
## a1 a2 a3 a4 d g u
## par 1.213 0 0 0.142 -0.097 0.1 1
##
## $Item.29
## a1 a2 a3 a4 d g u
## par 1.938 0 2.339 0 -2.209 0.1 1
##
## $Item.30
## a1 a2 a3 a4 d g u
## par 0.479 0 0 -0.128 -0.527 0.1 1
##
## $Item.31
## a1 a2 a3 a4 d g u
## par 3.173 -0.82 0 0 3.316 0.1 1
##
## $Item.32
## a1 a2 a3 a4 d g u
## par 0.534 0 -0.053 0 -2.786 0.1 1
##
## $GroupPars
## MEAN_1 MEAN_2 MEAN_3 MEAN_4 COV_11 COV_21 COV_31 COV_41 COV_22 COV_32
## par 0 0 0 0 1 0 0 0 1 0
## COV_42 COV_33 COV_43 COV_44
## par 0 1 0 1
anova(mod1, mod2)
## AIC SABIC HQ BIC logLik X2 df p
## mod1 19062.10 19179.44 19226.42 19484.21 -9435.052
## mod2 19009.55 19126.88 19173.87 19431.65 -9408.775 52.553 0 NaN
## don't estimate specific factor for item 32
specific[32] <- NA
mod3 <- bfactor(data, specific)
anova(mod3, mod1)
## AIC SABIC HQ BIC logLik X2 df p
## mod3 19060.12 19176.23 19222.73 19477.83 -9435.062
## mod1 19062.10 19179.44 19226.42 19484.21 -9435.052 0.02 1 0.886
# same, but declared manually (not run)
#sv <- mod2values(mod1)
#sv$value[220] <- 0 #parameter 220 is the 32 items specific slope
#sv$est[220] <- FALSE
#mod3 <- bfactor(data, specific, pars = sv) #with excellent starting values
#########
# mixed itemtype example
# simulate data
a <- matrix(c(
1,0.5,NA,
1,0.5,NA,
1,0.5,NA,
1,0.5,NA,
1,0.5,NA,
1,0.5,NA,
1,0.5,NA,
1,NA,0.5,
1,NA,0.5,
1,NA,0.5,
1,NA,0.5,
1,NA,0.5,
1,NA,0.5,
1,NA,0.5),ncol=3,byrow=TRUE)
d <- matrix(c(
-1.0,NA,NA,
-1.5,NA,NA,
1.5,NA,NA,
0.0,NA,NA,
2.5,1.0,-1,
3.0,2.0,-0.5,
3.0,2.0,-0.5,
3.0,2.0,-0.5,
2.5,1.0,-1,
2.0,0.0,NA,
-1.0,NA,NA,
-1.5,NA,NA,
1.5,NA,NA,
0.0,NA,NA),ncol=3,byrow=TRUE)
items <- rep('2PL', 14)
items[5:10] <- 'graded'
sigma <- diag(3)
dataset <- simdata(a,d,2000,itemtype=items,sigma=sigma)
itemstats(dataset)
## $overall
## N mean_total.score sd_total.score ave.r sd.r alpha
## 2000 15.256 4.454 0.173 0.035 0.73
##
## $itemstats
## N mean sd total.r total.r_if_rm alpha_if_rm
## Item_1 2000 0.306 0.461 0.433 0.344 0.717
## Item_2 2000 0.236 0.424 0.402 0.318 0.720
## Item_3 2000 0.773 0.419 0.415 0.333 0.719
## Item_4 2000 0.514 0.500 0.441 0.344 0.716
## Item_5 2000 1.903 0.972 0.572 0.396 0.711
## Item_6 2000 2.178 0.861 0.538 0.379 0.711
## Item_7 2000 2.140 0.900 0.559 0.395 0.710
## Item_8 2000 2.156 0.872 0.568 0.412 0.707
## Item_9 2000 1.870 0.981 0.581 0.405 0.710
## Item_10 2000 1.340 0.743 0.506 0.366 0.712
## Item_11 2000 0.304 0.460 0.418 0.327 0.718
## Item_12 2000 0.230 0.421 0.390 0.306 0.721
## Item_13 2000 0.795 0.403 0.388 0.307 0.721
## Item_14 2000 0.511 0.500 0.429 0.330 0.718
##
## $proportions
## 0 1 2 3
## Item_1 0.695 0.306 NA NA
## Item_2 0.764 0.236 NA NA
## Item_3 0.227 0.773 NA NA
## Item_4 0.486 0.514 NA NA
## Item_5 0.106 0.208 0.364 0.322
## Item_6 0.068 0.092 0.432 0.407
## Item_7 0.080 0.104 0.412 0.404
## Item_8 0.070 0.106 0.425 0.400
## Item_9 0.118 0.198 0.378 0.306
## Item_10 0.164 0.333 0.503 NA
## Item_11 0.696 0.304 NA NA
## Item_12 0.770 0.230 NA NA
## Item_13 0.204 0.795 NA NA
## Item_14 0.488 0.511 NA NA
specific <- c(rep(1,7),rep(2,7))
simmod <- bfactor(dataset, specific)
coef(simmod)
## $Item_1
## a1 a2 a3 d g u
## par 1.184 0.325 0 -1.058 0 1
##
## $Item_2
## a1 a2 a3 d g u
## par 1.151 0.021 0 -1.471 0 1
##
## $Item_3
## a1 a2 a3 d g u
## par 1.225 0.484 0 1.608 0 1
##
## $Item_4
## a1 a2 a3 d g u
## par 1.031 0.257 0 0.069 0 1
##
## $Item_5
## a1 a2 a3 d1 d2 d3
## par 1.064 0 0 2.542 0.962 -0.915
##
## $Item_6
## a1 a2 a3 d1 d2 d3
## par 1.21 -0.496 0 3.272 2.142 -0.498
##
## $Item_7
## a1 a2 a3 d1 d2 d3
## par 1.156 0.341 0 2.998 1.883 -0.496
##
## $Item_8
## a1 a2 a3 d1 d2 d3
## par 0.965 0 0.832 3.224 1.985 -0.539
##
## $Item_9
## a1 a2 a3 d1 d2 d3
## par 0.904 0 0.745 2.47 0.977 -1.037
##
## $Item_10
## a1 a2 a3 d1 d2
## par 0.868 0 0.426 1.917 0.019
##
## $Item_11
## a1 a2 a3 d g u
## par 0.903 0 0.606 -1.02 0 1
##
## $Item_12
## a1 a2 a3 d g u
## par 0.909 0 0.64 -1.491 0 1
##
## $Item_13
## a1 a2 a3 d g u
## par 0.85 0 0.754 1.682 0 1
##
## $Item_14
## a1 a2 a3 d g u
## par 0.803 0 0.514 0.054 0 1
##
## $GroupPars
## MEAN_1 MEAN_2 MEAN_3 COV_11 COV_21 COV_31 COV_22 COV_32 COV_33
## par 0 0 0 1 0 0 1 0 1
#########
# General testlet response model (Wainer, 2007)
# simulate data
set.seed(1234)
a <- matrix(0, 12, 4)
a[,1] <- rlnorm(12, .2, .3)
ind <- 1
for(i in 1:3){
a[ind:(ind+3),i+1] <- a[ind:(ind+3),1]
ind <- ind+4
}
print(a)
## [,1] [,2] [,3] [,4]
## [1,] 0.8503394 0.8503394 0.000000 0.0000000
## [2,] 1.3274088 1.3274088 0.000000 0.0000000
## [3,] 1.6910208 1.6910208 0.000000 0.0000000
## [4,] 0.6042850 0.6042850 0.000000 0.0000000
## [5,] 1.3892130 0.0000000 1.389213 0.0000000
## [6,] 1.4216480 0.0000000 1.421648 0.0000000
## [7,] 1.0279618 0.0000000 1.027962 0.0000000
## [8,] 1.0366667 0.0000000 1.036667 0.0000000
## [9,] 1.0311394 0.0000000 0.000000 1.0311394
## [10,] 0.9351846 0.0000000 0.000000 0.9351846
## [11,] 1.0584888 0.0000000 0.000000 1.0584888
## [12,] 0.9052755 0.0000000 0.000000 0.9052755
d <- rnorm(12, 0, .5)
sigma <- diag(c(1, .5, 1, .5))
dataset <- simdata(a,d,2000,itemtype=rep('2PL', 12),sigma=sigma)
itemstats(dataset)
## $overall
## N mean_total.score sd_total.score ave.r sd.r alpha
## 2000 6 2.929 0.175 0.068 0.717
##
## $itemstats
## N mean sd total.r total.r_if_rm alpha_if_rm
## Item_1 2000 0.426 0.495 0.438 0.287 0.708
## Item_2 2000 0.502 0.500 0.560 0.425 0.689
## Item_3 2000 0.571 0.495 0.575 0.445 0.686
## Item_4 2000 0.502 0.500 0.383 0.224 0.716
## Item_5 2000 0.464 0.499 0.549 0.413 0.690
## Item_6 2000 0.436 0.496 0.561 0.428 0.688
## Item_7 2000 0.440 0.497 0.500 0.356 0.698
## Item_8 2000 0.693 0.462 0.474 0.339 0.701
## Item_9 2000 0.511 0.500 0.481 0.334 0.701
## Item_10 2000 0.456 0.498 0.465 0.316 0.704
## Item_11 2000 0.458 0.498 0.459 0.309 0.705
## Item_12 2000 0.540 0.498 0.475 0.327 0.702
##
## $proportions
## 0 1
## Item_1 0.575 0.426
## Item_2 0.498 0.502
## Item_3 0.430 0.571
## Item_4 0.498 0.502
## Item_5 0.536 0.464
## Item_6 0.564 0.436
## Item_7 0.559 0.440
## Item_8 0.308 0.693
## Item_9 0.488 0.511
## Item_10 0.543 0.456
## Item_11 0.541 0.458
## Item_12 0.460 0.540
# estimate by applying constraints and freeing the latent variances
specific <- c(rep(1,4),rep(2,4), rep(3,4))
model <- "G = 1-12
CONSTRAIN = (1, a1, a2), (2, a1, a2), (3, a1, a2), (4, a1, a2),
(5, a1, a3), (6, a1, a3), (7, a1, a3), (8, a1, a3),
(9, a1, a4), (10, a1, a4), (11, a1, a4), (12, a1, a4)
COV = S1*S1, S2*S2, S3*S3"
simmod <- bfactor(dataset, specific, model)
coef(simmod, simplify=TRUE)
## $items
## a1 a2 a3 a4 d g u
## Item_1 0.794 0.794 0.000 0.000 -0.359 0 1
## Item_2 1.544 1.544 0.000 0.000 0.011 0 1
## Item_3 1.762 1.762 0.000 0.000 0.479 0 1
## Item_4 0.544 0.544 0.000 0.000 0.011 0 1
## Item_5 1.386 0.000 1.386 0.000 -0.244 0 1
## Item_6 1.497 0.000 1.497 0.000 -0.449 0 1
## Item_7 0.853 0.000 0.853 0.000 -0.312 0 1
## Item_8 0.953 0.000 0.953 0.000 1.101 0 1
## Item_9 0.981 0.000 0.000 0.981 0.058 0 1
## Item_10 0.913 0.000 0.000 0.913 -0.217 0 1
## Item_11 0.868 0.000 0.000 0.868 -0.204 0 1
## Item_12 0.966 0.000 0.000 0.966 0.206 0 1
##
## $means
## G S1 S2 S3
## 0 0 0 0
##
## $cov
## G S1 S2 S3
## G 1 0.000 0.000 0.000
## S1 0 0.452 0.000 0.000
## S2 0 0.000 1.135 0.000
## S3 0 0.000 0.000 0.432
# Constrained testlet model (Bradlow, 1999)
model2 <- "G = 1-12
CONSTRAIN = (1, a1, a2), (2, a1, a2), (3, a1, a2), (4, a1, a2),
(5, a1, a3), (6, a1, a3), (7, a1, a3), (8, a1, a3),
(9, a1, a4), (10, a1, a4), (11, a1, a4), (12, a1, a4),
(GROUP, COV_22, COV_33, COV_44)
COV = S1*S1, S2*S2, S3*S3"
simmod2 <- bfactor(dataset, specific, model2)
coef(simmod2, simplify=TRUE)
## $items
## a1 a2 a3 a4 d g u
## Item_1 0.744 0.744 0.000 0.000 -0.360 0 1
## Item_2 1.453 1.453 0.000 0.000 0.010 0 1
## Item_3 1.664 1.664 0.000 0.000 0.482 0 1
## Item_4 0.509 0.509 0.000 0.000 0.011 0 1
## Item_5 1.541 0.000 1.541 0.000 -0.241 0 1
## Item_6 1.670 0.000 1.670 0.000 -0.445 0 1
## Item_7 0.968 0.000 0.968 0.000 -0.313 0 1
## Item_8 1.075 0.000 1.075 0.000 1.098 0 1
## Item_9 0.927 0.000 0.000 0.927 0.059 0 1
## Item_10 0.854 0.000 0.000 0.854 -0.218 0 1
## Item_11 0.813 0.000 0.000 0.813 -0.205 0 1
## Item_12 0.908 0.000 0.000 0.908 0.207 0 1
##
## $means
## G S1 S2 S3
## 0 0 0 0
##
## $cov
## G S1 S2 S3
## G 1 0.000 0.000 0.000
## S1 0 0.667 0.000 0.000
## S2 0 0.000 0.667 0.000
## S3 0 0.000 0.000 0.667
anova(simmod2, simmod)
## AIC SABIC HQ BIC logLik X2 df p
## simmod2 30256.59 30317.19 30308.00 30396.61 -15103.3
## simmod 30248.79 30314.24 30304.32 30400.02 -15097.4 11.795 2 0.003
#########
# Two-tier model
# simulate data
set.seed(1234)
a <- matrix(c(
0,1,0.5,NA,NA,
0,1,0.5,NA,NA,
0,1,0.5,NA,NA,
0,1,0.5,NA,NA,
0,1,0.5,NA,NA,
0,1,NA,0.5,NA,
0,1,NA,0.5,NA,
0,1,NA,0.5,NA,
1,0,NA,0.5,NA,
1,0,NA,0.5,NA,
1,0,NA,0.5,NA,
1,0,NA,NA,0.5,
1,0,NA,NA,0.5,
1,0,NA,NA,0.5,
1,0,NA,NA,0.5,
1,0,NA,NA,0.5),ncol=5,byrow=TRUE)
d <- matrix(rnorm(16))
items <- rep('2PL', 16)
sigma <- diag(5)
sigma[1,2] <- sigma[2,1] <- .4
dataset <- simdata(a,d,2000,itemtype=items,sigma=sigma)
itemstats(dataset)
## $overall
## N mean_total.score sd_total.score ave.r sd.r alpha
## 2000 7.086 3.077 0.108 0.058 0.662
##
## $itemstats
## N mean sd total.r total.r_if_rm alpha_if_rm
## Item_1 2000 0.288 0.453 0.378 0.241 0.650
## Item_2 2000 0.571 0.495 0.422 0.276 0.646
## Item_3 2000 0.705 0.456 0.381 0.245 0.650
## Item_4 2000 0.133 0.340 0.289 0.183 0.656
## Item_5 2000 0.601 0.490 0.393 0.246 0.650
## Item_6 2000 0.587 0.492 0.419 0.274 0.646
## Item_7 2000 0.379 0.485 0.444 0.304 0.642
## Item_8 2000 0.378 0.485 0.400 0.256 0.649
## Item_9 2000 0.386 0.487 0.392 0.246 0.650
## Item_10 2000 0.322 0.467 0.400 0.261 0.648
## Item_11 2000 0.402 0.490 0.455 0.315 0.640
## Item_12 2000 0.318 0.466 0.414 0.278 0.646
## Item_13 2000 0.368 0.482 0.423 0.281 0.645
## Item_14 2000 0.498 0.500 0.424 0.277 0.646
## Item_15 2000 0.669 0.471 0.394 0.254 0.649
## Item_16 2000 0.482 0.500 0.444 0.300 0.642
##
## $proportions
## 0 1
## Item_1 0.713 0.288
## Item_2 0.430 0.571
## Item_3 0.295 0.705
## Item_4 0.867 0.133
## Item_5 0.400 0.601
## Item_6 0.413 0.587
## Item_7 0.621 0.379
## Item_8 0.622 0.378
## Item_9 0.614 0.386
## Item_10 0.678 0.322
## Item_11 0.598 0.402
## Item_12 0.681 0.318
## Item_13 0.632 0.368
## Item_14 0.502 0.498
## Item_15 0.330 0.669
## Item_16 0.518 0.482
specific <- c(rep(1,5),rep(2,6),rep(3,5))
model <- '
G1 = 1-8
G2 = 9-16
COV = G1*G2'
# quadpts dropped for faster estimation, but not as precise
simmod <- bfactor(dataset, specific, model, quadpts = 9, TOL = 1e-3)
coef(simmod, simplify=TRUE)
## $items
## a1 a2 a3 a4 a5 d g u
## Item_1 0.965 0.000 0.385 0.000 0.000 -1.100 0 1
## Item_2 1.076 0.000 0.550 0.000 0.000 0.363 0 1
## Item_3 0.898 0.000 0.592 0.000 0.000 1.068 0 1
## Item_4 0.896 0.000 0.710 0.000 0.000 -2.293 0 1
## Item_5 0.892 0.000 0.848 0.000 0.000 0.526 0 1
## Item_6 1.013 0.000 0.000 0.413 0.000 0.435 0 1
## Item_7 1.162 0.000 0.000 0.451 0.000 -0.639 0 1
## Item_8 0.945 0.000 0.000 0.609 0.000 -0.623 0 1
## Item_9 0.000 0.831 0.000 0.371 0.000 -0.544 0 1
## Item_10 0.000 0.925 0.000 0.610 0.000 -0.926 0 1
## Item_11 0.000 1.142 0.000 0.495 0.000 -0.517 0 1
## Item_12 0.000 0.978 0.000 0.000 0.634 -0.964 0 1
## Item_13 0.000 1.108 0.000 0.000 0.437 -0.694 0 1
## Item_14 0.000 1.004 0.000 0.000 0.321 -0.012 0 1
## Item_15 0.000 0.916 0.000 0.000 0.758 0.897 0 1
## Item_16 0.000 1.020 0.000 0.000 0.650 -0.096 0 1
##
## $means
## G1 G2 S1 S2 S3
## 0 0 0 0 0
##
## $cov
## G1 G2 S1 S2 S3
## G1 1.000 0.412 0 0 0
## G2 0.412 1.000 0 0 0
## S1 0.000 0.000 1 0 0
## S2 0.000 0.000 0 1 0
## S3 0.000 0.000 0 0 1
summary(simmod)
## G1 G2 S1 S2 S3 h2
## Item_1 0.484 0.000 0.193 0.000 0.000 0.271
## Item_2 0.516 0.000 0.263 0.000 0.000 0.335
## Item_3 0.446 0.000 0.294 0.000 0.000 0.285
## Item_4 0.437 0.000 0.346 0.000 0.000 0.311
## Item_5 0.425 0.000 0.404 0.000 0.000 0.343
## Item_6 0.501 0.000 0.000 0.204 0.000 0.293
## Item_7 0.551 0.000 0.000 0.214 0.000 0.349
## Item_8 0.463 0.000 0.000 0.299 0.000 0.304
## Item_9 0.000 0.431 0.000 0.192 0.000 0.222
## Item_10 0.000 0.456 0.000 0.300 0.000 0.298
## Item_11 0.000 0.541 0.000 0.235 0.000 0.348
## Item_12 0.000 0.474 0.000 0.000 0.307 0.319
## Item_13 0.000 0.533 0.000 0.000 0.210 0.329
## Item_14 0.000 0.501 0.000 0.000 0.160 0.277
## Item_15 0.000 0.441 0.000 0.000 0.365 0.328
## Item_16 0.000 0.488 0.000 0.000 0.311 0.336
##
## SS loadings: 1.839 1.88 0.476 0.359 0.395
## Proportion Var: 0.115 0.118 0.03 0.022 0.025
##
## Factor correlations:
##
## G1 G2 S1 S2 S3
## G1 1.000
## G2 0.412 1
## S1 0.000 0 1
## S2 0.000 0 0 1
## S3 0.000 0 0 0 1
itemfit(simmod, QMC=TRUE)
## item S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
## 1 Item_1 7.103 9 0.000 0.626
## 2 Item_2 13.326 10 0.013 0.206
## 3 Item_3 8.332 9 0.000 0.501
## 4 Item_4 8.531 10 0.000 0.577
## 5 Item_5 7.170 10 0.000 0.709
## 6 Item_6 3.967 10 0.000 0.949
## 7 Item_7 8.350 10 0.000 0.595
## 8 Item_8 16.010 10 0.017 0.099
## 9 Item_9 17.529 10 0.019 0.063
## 10 Item_10 12.058 10 0.010 0.281
## 11 Item_11 13.567 10 0.013 0.194
## 12 Item_12 13.907 9 0.017 0.126
## 13 Item_13 11.144 10 0.008 0.346
## 14 Item_14 7.852 10 0.000 0.643
## 15 Item_15 14.142 9 0.017 0.117
## 16 Item_16 5.926 10 0.000 0.821
M2(simmod, QMC=TRUE)
## M2 df p RMSEA RMSEA_5 RMSEA_95 SRMSR TLI CFI
## stats 86.28163 87 0.5015988 0 0 0.01201603 0.01662365 1.000285 1
residuals(simmod, QMC=TRUE)
## LD matrix (lower triangle) and standardized values.
##
## Upper triangle summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.046 -0.011 -0.002 -0.001 0.012 0.041
##
## Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
## Item_1 NA -0.011 -0.015 -0.007 0.016 0.003 0.006 -0.002 0.004 -0.009
## Item_2 0.263 NA -0.002 0.005 -0.001 0.016 -0.022 0.021 -0.003 -0.006
## Item_3 0.441 0.008 NA 0.014 -0.001 0.029 -0.017 -0.011 -0.021 -0.028
## Item_4 0.086 0.054 0.376 NA -0.014 -0.023 0.021 -0.004 0.020 -0.040
## Item_5 0.514 0.004 0.003 0.386 NA -0.028 0.011 0.014 -0.021 0.013
## Item_6 0.015 0.483 1.630 1.038 1.588 NA -0.022 -0.009 0.031 -0.004
## Item_7 0.077 0.996 0.590 0.852 0.258 0.992 NA -0.007 -0.004 0.013
## Item_8 0.012 0.858 0.264 0.039 0.377 0.154 0.094 NA -0.020 0.008
## Item_9 0.033 0.017 0.863 0.803 0.890 1.974 0.037 0.808 NA 0.001
## Item_10 0.157 0.084 1.528 3.158 0.360 0.038 0.330 0.128 0.001 NA
## Item_11 0.510 0.125 2.195 0.231 0.004 0.215 1.049 0.004 0.031 0.754
## Item_12 2.017 0.253 1.865 0.388 0.005 0.417 0.074 0.090 0.443 0.042
## Item_13 0.470 2.122 0.125 0.271 0.881 0.264 0.310 4.304 0.009 0.059
## Item_14 0.101 1.546 0.165 0.006 0.296 0.004 1.672 0.765 3.341 0.066
## Item_15 0.822 0.257 0.011 0.442 0.443 0.113 0.526 0.297 2.306 0.044
## Item_16 0.097 0.627 1.486 0.127 0.445 0.011 0.732 0.061 0.007 0.944
## Item_11 Item_12 Item_13 Item_14 Item_15 Item_16
## Item_1 -0.016 -0.032 0.015 -0.007 0.020 -0.007
## Item_2 0.008 0.011 -0.033 -0.028 -0.011 0.018
## Item_3 0.033 0.031 0.008 -0.009 -0.002 0.027
## Item_4 -0.011 0.014 -0.012 -0.002 -0.015 -0.008
## Item_5 -0.001 0.002 -0.021 -0.012 0.015 0.015
## Item_6 0.010 0.014 -0.011 0.001 -0.008 0.002
## Item_7 0.023 0.006 0.012 0.029 -0.016 0.019
## Item_8 -0.001 0.007 -0.046 -0.020 0.012 0.006
## Item_9 0.004 -0.015 -0.002 0.041 -0.034 -0.002
## Item_10 -0.019 0.005 0.005 0.006 -0.005 0.022
## Item_11 NA 0.009 0.000 -0.025 0.039 -0.011
## Item_12 0.149 NA -0.008 0.013 -0.002 -0.009
## Item_13 0.000 0.140 NA 0.013 -0.012 0.003
## Item_14 1.269 0.338 0.336 NA -0.007 -0.023
## Item_15 3.061 0.010 0.284 0.086 NA 0.010
## Item_16 0.250 0.153 0.015 1.026 0.190 NA
## End(No test)
[Package mirt version 1.40 Index]