Principal component models applied to confirmatory factor analysis
Testing structural equation models, in practice, may not always go smoothly, and the solution in the output may be an improper solution. The term improper solution refers to several possible problems with model estimation. Improper solution involves boundary estimates or exterior estimates, such as Heywood cases. Improper solution can be found in the output even without an error message. The dissertation achieves the following goals: develop a stable algorithm to generate proper estimates of parameters, and the stable algorithm would be robust to data set variability and converge with high probability; use statistical theory to construct confidence intervals for functions of parameters, under non-normality and equations were derived in this thesis for computing confidence intervals; and use statistical theory to construct hypotheses tests, such as the goodness-of-fit tests and model comparison tests to determine the number of factors, structure of Lambda and structure of Phi, especially under non-normality. Based on the large simulation results, it can be demonstrated that the inference procedures for the proposed model work well enough to be used in practice and that the proposed model has advantages over the conventional model, in terms of proportion of proper solutions; average success rates and coverage rates of upper one-sided nominal 95% confidence intervals, lower one-sided nominal 95% confidence intervals, and two-sided nominal 95% confidence intervals; and average of the ratios of widths of two-sided nominal 95% confidence intervals.