VALIDITY OF THE CHI-SQUARE TEST IN DICHOTOMOUS VARIABLE FACTOR-ANALYSIS WHEN EXPECTED FREQUENCIES ARE SMALL

This paper presents a comparison of results from two methods for estimating and testing a model for the factor analysis of dichotomous variables. For k manifest dichotomous variables, the data can be cross-classified to form a vector of 2k frequencies, and nonlinear methods that use the full information in these 2k frequencies are available for factor analysis. In addition, another method that uses only the limited information in the first-, and second-order marginal frequencies is available for the same model. As k becomes larger, substantial differences between the full-information and limited-information methods become apparent in results from the test of fit. For large k, Type I and Type II error rates may be higher in the full-information approach, because as the vector of 2k frequencies becomes sparse, the chi-square approximation for the distribution of the goodness-of-fit test statistic becomes poorer. In this paper, Monte Carlo experiments are used under a variety of conditions to compare the methods for rate of Type I errors when the model matches the simulated data and for the rate of Type II errors when the model does not match the simulated data.