Testing log‐linear models with inequality constraints: a comparison of asymptotic, bootstrap, and posterior predictive p‐values

An important aspect of applied research is the assessment of the goodness‐of‐fit of an estimated statistical model. In the analysis of contingency tables, this usually involves determining the discrepancy between observed and estimated frequencies using the likelihood‐ratio statistic. In models with inequality constraints, however, the asymptotic distribution of this statistic depends on the unknown model parameters and, as a result, there no longer exists an unique p‐value. Bootstrap p‐values obtained by replacing the unknown parameters by their maximum likelihood estimates may also be inaccurate, especially if many of the imposed inequality constraints are violated in the available sample. We describe the various problems associated with the use of asymptotic and bootstrap p‐values and propose the use of Bayesian posterior predictive checks as a better alternative for assessing the fit of log‐linear models with inequality constraints.