Limited‐information goodness‐of‐fit testing of hierarchical item factor models

In applications of item response theory, assessment of model fit is a critical issue. Recently, limited‐information goodness‐of‐fit testing has received increased attention in the psychometrics literature. In contrast to full‐information test statistics such as Pearson’s X² or the likelihood ratio G², these limited‐information tests utilize lower‐order marginal tables rather than the full contingency table. A notable example is Maydeu‐Olivares and colleagues’M₂ family of statistics based on univariate and bivariate margins. When the contingency table is sparse, tests based on M₂ retain better Type I error rate control than the full‐information tests and can be more powerful. While in principle the M₂ statistic can be extended to test hierarchical multidimensional item factor models (e.g., bifactor and testlet models), the computation is non‐trivial. To obtain M₂, a researcher often has to obtain (many thousands of) marginal probabilities, derivatives, and weights. Each of these must be approximated with high‐dimensional numerical integration. We propose a dimension reduction method that can take advantage of the hierarchical factor structure so that the integrals can be approximated far more efficiently. We also propose a new test statistic that can be substantially better calibrated and more powerful than the original M₂ statistic when the test is long and the items are polytomous. We use simulations to demonstrate the performance of our new methods and illustrate their effectiveness with applications to real data.