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 expectations represented 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. Translational AbstractEvaluation of Inequality Constrained Hypotheses Using a Generalization of the AIC: Researchers are interested in evaluating equality and/or inequality constrained hypotheses in the context not only of normal linear models, but also of the families outside of normal linear models using a suitable information criterion. However, the available information criteria in the literature are not capable of evaluating (in)equality constrained hypotheses under such a broad range of statistical models. The main aim of this paper is to close this research gap by proposing a new information criterion named the GORICA which can be utilized to evaluate these hypotheses for generalized linear (mixed) models and structural equation models. The GORICA enables researchers to quantify the evidence in the data for two or more (in)equality constrained hypotheses. Like all the other information criteria, the GORICA has the log likelihood and penalty parts. The superiority of the GORICA over the other information criteria lies behind the use of a simple formula when calculating its log likelihood. We investigated the performance of the GORICA on choosing the true hypothesis out of a set of competing hypotheses using simulation studies for logistic regression, multilevel regression, and structural equation model. The findings in these simulation studies suggest that the GORICA has a convincing performance on choosing the true hypothesis. The use of the GORICA is illustrated for (real) data sets in line with these simulation studies.