Evaluating a theory-based hypothesis against its complement using an AIC-type information criterion with an application to facial burn injury

An information criterion (IC) like the Akaike IC (AIC), can be used to select the best hypothesis from a set of competing theory-based hypotheses. An IC developed to evaluate theory-based order-restricted hypotheses is the GORIC. Like for any IC, the values themselves are not interpretable but only comparable. To improve the interpretation regarding the strength, GORIC weights and related evidence ratios can be computed. However, if the unconstrained hypothesis (the default) is used as competing hypothesis, the evidence ratio is not affected by sample-size nor effect-size in case the hypothesis of interest is (also) in agreement with the data. In practice, this means that in such a case strong support for the order-restricted hypothesis is not reflected by a high evidence ratio. Therefore, we introduce the evaluation of an order-restricted hypothesis against its complement using the GORIC (weights). We show how to compute the GORIC value for the complement, which cannot be achieved by current methods. In a small simulation study, we show that the evidence ratio for the order-restricted hypothesis versus the complement increases for larger samples and/or effect-sizes, while the evidence ratio for the order-restricted hypothesis versus the unconstrained hypothesis remains bounded. An empirical example about facial burn injury illustrates our method and shows that using the complement as competing hypothesis results in much more support for the hypothesis of interest than using the unconstrained hypothesis as competing hypothesis.

Generalization of the order-restricted information criterion for multivariate normal linear models (pre-print )

The Akaike information criterion for model selection presupposes that the parameter space is not subject to order restrictions or inequality constraints.Anraku (1999) proposed a modified version of this criterion, called the order-restricted information criterion, for model selection in the one-way analysis of variance model when the population means are monotonic.We propose a generalization of this to the case when the population means may be restricted by a mixture of linear equality and inequality constraints.If the model has no inequality constraints, then the generalized order-restricted information criterion coincides with the Akaike information criterion.Thus, the former extends the applicability of the latter to model selection in multi-way analysis of variance models when some models may have inequality constraints while others may not. Simulation shows that the information criterion proposed in this paper performs well in selecting the correct model.

An Akaike-type information criterion for model selection under inequality constraints (pre-print)

The Akaike information criterion for model selection presupposes that the parameter space is not subject to order restrictions or inequality constraints. Anraku (1999) proposed a modified version of this criterion, called the order-restricted information criterion, for model selection in the one-way analysis of variance model when the population means are monotonic. We propose a generalization of this to the case when the population means may be restricted by a mixture of linear equality and inequality constraints. If the model has no inequality constraints, then the generalized order-restricted information criterion coincides with the Akaike information criterion. Thus, the former extends the applicability of the latter to model selection in multi-way analysis of variance models when some models may have inequality constraints while others may not. Simulation shows that the information criterion proposed in this paper performs well in selecting the correct model.

Evaluation of Inequality Constrained Hypotheses Using a Generalization of the AIC

In the social and behavioral sciences, it is often not interesting to evaluate the null hypothesis by means of a p-value. Researchers are often more interested in quantifying the evidence in the data (as opposed to using p-values) with respect to their own expectationsrepresented by equality and/or inequality constrained hypotheses (as opposed to the null hypothesis). This article proposes an Akaike-type information criterion (AIC; Akaike, 1973, 1974) called the generalized order-restricted information criterion approximation (GORICA) that evaluates (in)equality constrained hypotheses under a very broad range of statistical models. The results of five simulation studies provide empirical evidence showing that the performance of the GORICA on selecting the best hypothesis out of a set of (in)equality constrained hypotheses is convincing. To illustrate the use of the GORICA, the expectations of researchers are investigated in a logistic regression, multilevel regression, and structural equation model.