A Poor Person’s Posterior Predictive Checking of Structural Equation Models

Abstract

Posterior predictive model checking (PPMC) is a Bayesian model checking method that compares the observed data to (plausible) future observations from the posterior predictive distribution. We propose an alternative to PPMC in the context of structural equation modeling, which we term the poor person’s PPMC (PP-PPMC), for the situation wherein one cannot afford (or is unwilling) to draw samples from the full posterior. Using only by-products of likelihood-based estimation (maximum likelihood estimate and information matrix), the PP-PPMC offers a natural method to handle parameter uncertainty in model fit assessment. In particular, a coupling relationship between the classical p values from the model fit chi-square test and the predictive p values from the PP-PPMC method is carefully examined, suggesting that PP-PPMC might offer an alternative, principled approach for model fit assessment. We also illustrate the flexibility of the PP-PPMC approach by applying it to case-influence diagnostics.

Keywords:

• case influence
• posterior normality
• posterior predictive model checking

ACKNOWLEDGMENT

The authors wish to thank Michael Seltzer for insightful comments.

Notes

¹ For simplicity we do not use the unbiased sample covariance matrix estimate with (N - 1) as the divisor. All discussions assume large N so any difference will be negligible.

Additional information

Funding

This research was supported in part by an Institute of Education Sciences statistical methodology grant (R305D140046) and a grant from the Research Council of the University of Oklahoma Norman Campus. The views expressed here belong to the authors and do not reflect the views or policies of the funding agencies or grantees.