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Using an Approximate Chi‐Square Statistic to Test the Number of Dimensions Underlying the Responses to a Set of Items
Author(s) -
Gessaroli Marc E.,
Champlain André F.
Publication year - 1996
Publication title -
journal of educational measurement
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.917
H-Index - 47
eISSN - 1745-3984
pISSN - 0022-0655
DOI - 10.1111/j.1745-3984.1996.tb00487.x
Subject(s) - statistic , mathematics , statistics , test statistic , chi square test , sample size determination , type i and type ii errors , dimension (graph theory) , statistical hypothesis testing , combinatorics
An approximate χ 2 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 χ 2 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 χ 2 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 χ 2 in addressing current measurement issues are discussed.