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 2^k frequencies, and nonlinear methods that use the full information in these 2^k 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 2^k 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. 1994 The British Psychological Society