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Limited information goodness-of-fit testing in multidimensional contingency tables. (English) Zbl 1306.62477

Summary: We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order \(r\). They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when \(r\) is small and multidimensional tables are sparse, the proposed statistics have accurate empirical type I errors, unlike Pearson’s \(X^2\). For this model in nonsparse situations, the proposed statistics are also more powerful than \(X^2\). In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.

MSC:

62P15 Applications of statistics to psychology
62H15 Hypothesis testing in multivariate analysis
62H17 Contingency tables

Software:

LISREL; NOHARM
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