Item-level model comparison using Wald, LR or LM tests

This function evaluates whether the saturated G-DINA model can be replaced by reduced CDMs without significant loss in model data fit for each item using the Wald test, likelihood ratio (LR) test or Lagrange multiplier (LM) test. For Wald test, see de la Torre (2011), de la Torre and Lee (2013), Ma, Iaconangelo and de la Torre (2016) and Ma & de la Torre (2018) for details. For LR test and a two-step LR approximation procedure, see Sorrel, de la Torre, Abad, and Olea (2017), Ma (2017) and Ma & de la Torre (2019). For LM test, which is only applicable for DINA, DINO and ACDM, see Sorrel, Abad, Olea, de la Torre, and Barrada (2017). This function also calculates the dissimilarity between the reduced models and the G-DINA model, which can be viewed as a measure of effect size (Ma, Iaconangelo & de la Torre, 2016).

modelcomp(
  GDINA.obj = NULL,
  method = "Wald",
  items = "all",
  p.adjust.methods = "holm",
  models = c("DINA", "DINO", "ACDM", "LLM", "RRUM"),
  decision.args = list(rule = "simpler", alpha.level = 0.05, adjusted = FALSE),
  DS = FALSE,
  Wald.args = list(SE.type = 2, varcov = NULL),
  LR.args = list(LR.approx = FALSE),
  LM.args = list(reducedMDINA = NULL, reducedMDINO = NULL, reducedMACDM = NULL, SE.type =
    2)
)

# S3 method for modelcomp
extract(
  object,
  what = c("stats", "pvalues", "adj.pvalues", "df", "DS", "selected.model"),
  digits = 4,
  ...
)

# S3 method for modelcomp
summary(object, ...)

Arguments

GDINA.obj: An estimated model object of class GDINA
method: method for item level model comparison; can be wald, LR or LM.
items: a vector of items to specify the items for model comparsion
p.adjust.methods: adjusted p-values for multiple hypothesis tests. This is conducted using p.adjust function in stats, and therefore all adjustment methods supported by p.adjust can be used, including "holm", "hochberg", "hommel", "bonferroni", "BH" and "BY". See p.adjust for more details. "holm" is the default, indicating the Holm method.
models: a vector specifying which reduced CDMs are possible reduced CDMs for each item. The default is "DINA","DINO","ACDM","LLM",and "RRUM".
decision.args: a list of options for determining the most appropriate models including (1) rule can be either "simpler" or "largestp". See details; (2) alpha.level for the nominal level of decision; and (3) adjusted can be either TRUE or FALSE indicating whether the decision is based on p value (adjusted = FALSE) or adjusted p values.
DS: whether dissimilarity index should be calculated? FALSE is the default.
Wald.args: a list of options for Wald test including (1) SE.type giving the type of covariance matrix for the Wald test; by default, it uses outer product of gradient based on incomplete information matrix; (2) varcov for user specified variance-covariance matrix. If supplied, it must be a list, giving the variance covariance matrix of success probability for each item or category. The default is NULL, in which case, the estimated variance-covariance matrix from the GDINA function is used.
LR.args: a list of options for LR test including for now only LR.approx, which is either TRUE or FALSE, indicating whether a two-step LR approximation is implemented or not.
LM.args: a list of options for LM test including reducedMDINA, reducedMDINO, and reducedMACDM for DINA, DINO and ACDM estimates from the GDINA function; SE.type specifies the type of covariance matrix.
object: object of class modelcomp for various S3 methods
what: argument for S3 method extract indicating what to extract; It can be "wald" for wald statistics, "wald.p" for associated p-values, "df" for degrees of freedom, and "DS" for dissimilarity between G-DINA and other CDMs.
digits: How many decimal places in each number? The default is 4.
...: additional arguments

Value

an object of class modelcomp. Elements that can be extracted using extract method include

stats: Wald or LR statistics
pvalues: p-values associated with the test statistics
adj.pvalues: adjusted p-values
df: degrees of freedom
DS: dissimilarity between G-DINA and other CDMs

Details

After the test statistics for each reduced CDM were calculated for each item, the reduced models with p values less than the pre-specified alpha level were rejected. If all reduced models were rejected for an item, the G-DINA model was used as the best model; if at least one reduced model was retained, two diferent rules can be implemented for selecting the best model specified in argument decision.args:

(1) when rule="simpler", which is the default,

If (a) the DINA or DINO model was one of the retained models, then the DINA or DINO model with the larger p value was selected as the best model; but if (b) both DINA and DINO were rejected, the reduced model with the largest p value was selected as the best model for this item. Note that when the p-values of several reduced models were greater than 0.05, the DINA and DINO models were preferred over the A-CDM, LLM, and R-RUM because of their simplicity. This procedure is originally proposed by Ma, Iaconangelo, and de la Torre (2016).

(2) When rule="largestp":

The reduced model with the largest p-values is selected as the most appropriate model.

Methods (by generic)

extract(modelcomp): extract various elements from modelcomp objects
summary(modelcomp): print summary information

References

de la Torre, J., & Lee, Y. S. (2013). Evaluating the wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50, 355-373.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection and attribute classification. Applied Psychological Measurement, 40, 200-217.

Ma, W. (2017). A Sequential Cognitive Diagnosis Model for Graded Response: Model Development, Q-Matrix Validation,and Model Comparison. Unpublished doctoral dissertation. New Brunswick, NJ: Rutgers University.

Ma, W., & de la Torre, J. (2019). Category-Level Model Selection for the Sequential G-DINA Model. Journal of Educational and Behavioral Statistics. 44, 61-82.

Ma, W., & de la Torre, J. (2020). GDINA: An R Package for Cognitive Diagnosis Modeling. Journal of Statistical Software, 93(14), 1-26.

Sorrel, M. A., Abad, F. J., Olea, J., de la Torre, J., & Barrada, J. R. (2017). Inferential Item-Fit Evaluation in Cognitive Diagnosis Modeling. Applied Psychological Measurement, 41, 614-631.

Sorrel, M. A., de la Torre, J., Abad, F. J., & Olea, J. (2017). Two-Step Likelihood Ratio Test for Item-Level Model Comparison in Cognitive Diagnosis Models. Methodology, 13, 39-47.

Author

Wenchao Ma, The University of Alabama, wenchao.ma@ua.edu
Miguel A. Sorrel, Universidad Autonoma de Madrid
Jimmy de la Torre, The University of Hong Kong

Examples

if (FALSE) {
dat <- sim10GDINA$simdat
Q <- sim10GDINA$simQ
# --- GDINA model ---#
fit <- GDINA(dat = dat, Q = Q, model = "GDINA")
fit

###################
#
# Wald test
#
###################

w <- modelcomp(fit)
w
# wald statistics
extract(w,"stats")
#p values
extract(w,"pvalues")
# selected models
extract(w,"selected.model")
##########################
#
# LR and Two-step LR test
#
##########################

lr <- modelcomp(fit,method = "LR")
lr
TwostepLR <- modelcomp(fit,items =c(6:10),method = "LR",LR.args = list(LR.approx = TRUE))
TwostepLR

##########################
#
# LM test
#
##########################

dina <- GDINA(dat = dat, Q = Q, model = "DINA")
dino <- GDINA(dat = dat, Q = Q, model = "DINO")
acdm <- GDINA(dat = dat, Q = Q, model = "ACDM")
lm <- modelcomp(method = "LM",LM.args=list(reducedMDINA = dina,
reducedMDINO = dino, reducedMACDM  = acdm))
lm


}