Computes the M2 (Maydeu-Olivares & Joe, 2006) statistic when all data are dichotomous,
the collapsed M2* statistic (collapsing over univariate and bivariate response categories;
see Cai and Hansen, 2013), and the hybrid C2 statistic which only collapses only the bivariate
moments (Cai and Monro, 2014). The C2 variant is mainly useful when polytomous response models
do not have sufficient degrees of freedom to compute M2*. This function
also computes associated fit indices that are based on
fitting the null model. Supports single and multiple-group models.
If the latent trait density was approximated (e.g., Davidian curves, Empirical histograms, etc)
then passing use_dentype_estimate = TRUE will use the internally saved quadrature and
density components (where applicable).
Usage
M2(
obj,
type = "M2*",
calcNull = TRUE,
quadpts = NULL,
theta_lim = c(-6, 6),
CI = 0.9,
residmat = FALSE,
QMC = FALSE,
suppress = 1,
...
)Arguments
- obj
an estimated model object from the mirt package
- type
type of fit statistic to compute. Options are "M2", "M2*" for the univariate and bivariate collapsed version of the M2 statistic ("M2" currently limited to dichotomous response data only), and "C2" for a hybrid between M2 and M2* where only the bivariate moments are collapsed
- calcNull
logical; calculate statistics for the null model as well? Allows for statistics such as the limited information TLI and CFI. Only valid when items all have a suitable null model (e.g., those created via
createItemwill not)- quadpts
number of quadrature points to use during estimation. If
NULL, a suitable value will be chosen based on the rubric found infscores- theta_lim
lower and upper range to evaluate latent trait integral for each dimension
- CI
numeric value from 0 to 1 indicating the range of the confidence interval for RMSEA. Default returns the 90% interval
- residmat
logical; return the residual matrix used to compute the SRMSR statistic? Only the lower triangle of the residual correlation matrix will be returned (the upper triangle is filled with NA's)
- QMC
logical; use quasi-Monte Carlo integration? Useful for higher dimensional models. If
quadptsnot specified, 5000 nodes are used by default- suppress
a numeric value indicating which parameter residual dependency combinations to flag as being too high. Absolute values for the standardized residuals greater than this value will be returned, while all values less than this value will be set to NA. Must be used in conjunction with the argument
residmat = TRUE- ...
additional arguments to pass
Value
Returns a data.frame object with the M2-type statistic, along with the degrees of freedom,
p-value, RMSEA (with 90% confidence interval), SRMSR for each group,
and optionally the TLI and CFI model fit statistics if calcNull = TRUE.
References
Cai, L. & Hansen, M. (2013). Limited-information goodness-of-fit testing of hierarchical item factor models. British Journal of Mathematical and Statistical Psychology, 66, 245-276.
Cai, L. & Monro, S. (2014). A new statistic for evaluating item response theory models for ordinal data. National Center for Research on Evaluation, Standards, & Student Testing. Technical Report.
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
Maydeu-Olivares, A. & Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713-732.
Author
Phil Chalmers rphilip.chalmers@gmail.com
Examples
# \donttest{
dat <- as.matrix(expand.table(LSAT7))
(mod1 <- mirt(dat, 1))
#>
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
#>
#> Call:
#> mirt(data = dat, model = 1)
#>
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 28 EM iterations.
#> mirt version: 1.44.4
#> M-step optimizer: BFGS
#> EM acceleration: Ramsay
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian
#>
#> Log-likelihood = -2658.805
#> Estimated parameters: 10
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN
M2(mod1)
#> M2 df p RMSEA RMSEA_5 RMSEA_95 SRMSR
#> stats 11.93769 5 0.03565165 0.0372683 0.008950922 0.06496573 0.03195919
#> TLI CFI
#> stats 0.9369332 0.9684666
resids <- M2(mod1, residmat=TRUE) #lower triangle of residual correlation matrix
resids
#> Item.1 Item.2 Item.3 Item.4 Item.5
#> Item.1 NA NA NA NA NA
#> Item.2 -0.02212010 NA NA NA NA
#> Item.3 -0.03265942 0.03335601 NA NA NA
#> Item.4 0.05155184 -0.01642572 -0.012476892 NA NA
#> Item.5 0.05443241 -0.03867705 -0.001860395 7.235425e-05 NA
summary(resids[lower.tri(resids)])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.038677 -0.020697 -0.007169 0.001519 0.025035 0.054432
# M2 with missing data present
dat[sample(1:prod(dim(dat)), 250)] <- NA
mod2 <- mirt(dat, 1)
#>
Iteration: 1, Log-Lik: -2547.291, Max-Change: 0.18194
Iteration: 2, Log-Lik: -2539.425, Max-Change: 0.13938
Iteration: 3, Log-Lik: -2536.809, Max-Change: 0.10708
Iteration: 4, Log-Lik: -2534.711, Max-Change: 0.05139
Iteration: 5, Log-Lik: -2534.472, Max-Change: 0.04305
Iteration: 6, Log-Lik: -2534.308, Max-Change: 0.03620
Iteration: 7, Log-Lik: -2533.924, Max-Change: 0.01092
Iteration: 8, Log-Lik: -2533.912, Max-Change: 0.00981
Iteration: 9, Log-Lik: -2533.904, Max-Change: 0.00868
Iteration: 10, Log-Lik: -2533.883, Max-Change: 0.00250
Iteration: 11, Log-Lik: -2533.882, Max-Change: 0.00195
Iteration: 12, Log-Lik: -2533.882, Max-Change: 0.00175
Iteration: 13, Log-Lik: -2533.881, Max-Change: 0.00121
Iteration: 14, Log-Lik: -2533.881, Max-Change: 0.00111
Iteration: 15, Log-Lik: -2533.881, Max-Change: 0.00050
Iteration: 16, Log-Lik: -2533.881, Max-Change: 0.00043
Iteration: 17, Log-Lik: -2533.881, Max-Change: 0.00184
Iteration: 18, Log-Lik: -2533.881, Max-Change: 0.00036
Iteration: 19, Log-Lik: -2533.881, Max-Change: 0.00026
Iteration: 20, Log-Lik: -2533.881, Max-Change: 0.00025
Iteration: 21, Log-Lik: -2533.881, Max-Change: 0.00026
Iteration: 22, Log-Lik: -2533.881, Max-Change: 0.00023
Iteration: 23, Log-Lik: -2533.881, Max-Change: 0.00023
Iteration: 24, Log-Lik: -2533.881, Max-Change: 0.00108
Iteration: 25, Log-Lik: -2533.881, Max-Change: 0.00033
Iteration: 26, Log-Lik: -2533.881, Max-Change: 0.00020
Iteration: 27, Log-Lik: -2533.881, Max-Change: 0.00015
Iteration: 28, Log-Lik: -2533.881, Max-Change: 0.00071
Iteration: 29, Log-Lik: -2533.881, Max-Change: 0.00055
Iteration: 30, Log-Lik: -2533.881, Max-Change: 0.00043
Iteration: 31, Log-Lik: -2533.881, Max-Change: 0.00033
Iteration: 32, Log-Lik: -2533.881, Max-Change: 0.00020
Iteration: 33, Log-Lik: -2533.881, Max-Change: 0.00012
Iteration: 34, Log-Lik: -2533.881, Max-Change: 0.00005
M2(mod2)
#> M2 df p RMSEA RMSEA_5 RMSEA_95 SRMSR TLI
#> stats 8.181306 5 0.1465225 0.02523684 0 0.05513589 0.03213259 0.9636641
#> CFI
#> stats 0.9818321
# C2 statistic (useful when polytomous IRT models have too few df)
pmod <- mirt(Science, 1)
#>
Iteration: 1, Log-Lik: -1629.361, Max-Change: 0.50660
Iteration: 2, Log-Lik: -1617.374, Max-Change: 0.25442
Iteration: 3, Log-Lik: -1612.894, Max-Change: 0.16991
Iteration: 4, Log-Lik: -1610.306, Max-Change: 0.10461
Iteration: 5, Log-Lik: -1609.814, Max-Change: 0.09162
Iteration: 6, Log-Lik: -1609.534, Max-Change: 0.07363
Iteration: 7, Log-Lik: -1609.030, Max-Change: 0.03677
Iteration: 8, Log-Lik: -1608.988, Max-Change: 0.03200
Iteration: 9, Log-Lik: -1608.958, Max-Change: 0.02754
Iteration: 10, Log-Lik: -1608.878, Max-Change: 0.01443
Iteration: 11, Log-Lik: -1608.875, Max-Change: 0.00847
Iteration: 12, Log-Lik: -1608.873, Max-Change: 0.00515
Iteration: 13, Log-Lik: -1608.872, Max-Change: 0.00550
Iteration: 14, Log-Lik: -1608.872, Max-Change: 0.00318
Iteration: 15, Log-Lik: -1608.871, Max-Change: 0.00462
Iteration: 16, Log-Lik: -1608.871, Max-Change: 0.00277
Iteration: 17, Log-Lik: -1608.870, Max-Change: 0.00145
Iteration: 18, Log-Lik: -1608.870, Max-Change: 0.00175
Iteration: 19, Log-Lik: -1608.870, Max-Change: 0.00126
Iteration: 20, Log-Lik: -1608.870, Max-Change: 0.00025
Iteration: 21, Log-Lik: -1608.870, Max-Change: 0.00285
Iteration: 22, Log-Lik: -1608.870, Max-Change: 0.00108
Iteration: 23, Log-Lik: -1608.870, Max-Change: 0.00022
Iteration: 24, Log-Lik: -1608.870, Max-Change: 0.00059
Iteration: 25, Log-Lik: -1608.870, Max-Change: 0.00014
Iteration: 26, Log-Lik: -1608.870, Max-Change: 0.00068
Iteration: 27, Log-Lik: -1608.870, Max-Change: 0.00065
Iteration: 28, Log-Lik: -1608.870, Max-Change: 0.00019
Iteration: 29, Log-Lik: -1608.870, Max-Change: 0.00061
Iteration: 30, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 31, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 32, Log-Lik: -1608.870, Max-Change: 0.00058
Iteration: 33, Log-Lik: -1608.870, Max-Change: 0.00055
Iteration: 34, Log-Lik: -1608.870, Max-Change: 0.00015
Iteration: 35, Log-Lik: -1608.870, Max-Change: 0.00052
Iteration: 36, Log-Lik: -1608.870, Max-Change: 0.00010
# This fails with too few df:
# M2(pmod)
# This, however, works:
M2(pmod, type = 'C2')
#> M2 df p RMSEA RMSEA_5 RMSEA_95 SRMSR
#> stats 19.17929 2 6.84337e-05 0.1482174 0.09234204 0.2116368 0.07257313
#> TLI CFI
#> stats 0.7300952 0.9100317
# }