Evaluation of Inequality Constrained Hypotheses Using an Akaike-Type Information Criterion

The Akaike information criterion (AIC) is one of the best known information criteria that can be used to evaluate hypotheses containing only equality restrictions on model parameters. The GORIC is a generalization of the AIC that can be utilized to evaluate hypotheses containing equality and/or inequality restrictions on model parameters, but only for normal linear models. This dissertation proposes a new information criterion, the GORICA, that mimics the performance of the GORIC on selecting the best hypothesis in a set of competing hypotheses for normal linear models. The GORICA can be used to evaluate (in)equality constrained hypotheses under a broad range of statistical models: generalized linear models, generalized linear mixed models, structural equation models, and contingency tables. The GORICA is an useful method in evaluating (in)equality constrained hypotheses, because the hypotheses under evaluation can contain either linear restrictions on model parameters or non-linear restrictions on model parameters. For example, the GORICA can be used to evaluate hypotheses containing (in)equality restrictions on odds ratios, which are formulated using non-linear functions of cell probabilities in the context of contingency tables. The GORICA evaluation of (in)equality constrained hypotheses is flexible in the sense that it only requires the estimates of model parameters used in the specification of the hypotheses under evaluation and their covariance matrix as input. These inputs can be obtained using a suitable estimation method such as maximum likelihood estimation, nonparametric bootstrapping, and Gibbs sampling.
Three main features of the GORICA evaluation of (in)equality constrained hypotheses need further investigation. First of all, although a simulation study shows that the GORICA mimics the GORIC on selecting the best hypothesis in a set of competing hypotheses, there is no normal proof that ensures this empirical evidence inside or outside of normal linear models. Secondly, the performance of the GORICA on selecting the best hypothesis depends on the technique used in estimating model parameters and their covariance matrix. Thus, future studies are required to compare the performance of the GORICA using these estimation techniques under different sets of conditions. Thirdly, the GORICA is a method that needs large sample sizes in evaluating hypotheses. Thus, researchers should be careful when using the GORICA to evaluate hypotheses for small samples, because its performance for small samples has not been fully investigated yet.
In sum, although the GORICA is a flexible and useful method that can be used to evaluate very complicated hypotheses under a broad range of statistical models, its limitations should be taken into account when using it to evaluate these hypotheses.