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Using an Approximate Chi-Square Statistic to Test the Number of Dimensions Underlying the Responses to a Set of Items

An approximate χ2 statistic based on McDonald's (1967) nonlinear factor analytic representation of item response theory was proposed and investigated with simulated data. The results were compared with Stout's T statistic (Nandakumar & Stout, 1993; Stout, 1987). Unidimensional and two-dimensional item response data were simulated under varying levels of sample size, test length, test reliability, and dimension dominance. The approximate χ2 statistic had good control over Type I errors when unidimensional data were generated and displayed very good power in identifying the two-dimensional data. The performance of the approximate χ2 was at least as good as Stout's T statistic in all conditions and was better than Stout's T statistic with smaller sample sizes and shorter tests. Further implications regarding the potential use of nonlinear factor analysis and the approximate χ2 in addressing current measurement issues are discussed.