Inequality constrained normal linear models

Abstract

This dissertation deals with normal linear models with inequality constraints among model parameters. It consists of an introduction and four chapters that are papers submitted for publication. The first chapter introduces the use of inequality constraints. Scientists often have one or more theories or expectations with respect to the outcome of their empirical research. To evaluate these theories they have to be translated into statistical models. When scientists talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In the second chapter, (frequentist) null hypothesis testing with inequality constrained alternatives is investigated. In the literature, tests for univariate normal models are available, however, the possibilities for multivariate constrained testing are limited. In this chapter a computational method to sample the null distribution of any test statistic in the context of both univariate and multivariate constrained alternative hypotheses is presented. The remainder of this thesis discusses Bayesian estimation and model selection in the context of (competing) inequality constrained normal linear models. It is very natural to include the inequality constraints in the prior distribution of the model parameters. Each theory constitutes it's own prior knowledge and therefore for each model the appropriate prior is defined. In Chapter 3, a motivation for Bayesian model selection as opposed to null hypothesis testing is provided. Subsequently, the Bayesian approach is explained in a non-technical way (the technical details can however be found in the two appendices). In this chapter, the methods that are used for the specification of the prior distributions and for the estimation of Bayes factors, are based on results from the fourth chapter. In Chapter 4, the idea of encompassing priors is introduced and examined. Since inequality constrained models are all nested in one unconstrained, encompassing model just one prior for this encompassing model is specified. The prior distributions for the constrained models follow from the so-called encompassing prior by truncation of the parameters space. Sensitivity to the prior distribution in this application is twofold. The differences between the theories are incorporated in the model specific priors by truncation of the parameter space. Sensitivity to the prior in this case is desirable. It provides information about the fit of each of the models, that is, of each of the theories. However, sensitivity to the specification of the encompassing prior is not desirable. In this chapter sensitivity to encompassing priors is examined and it is shown that for specific classes of models the selection is virtually objective, that is, independent of the encompassing prior. The encompassing prior also leads to a nice interpretation of Bayes factors. The Bayes factor for any constrained model with the encompassing model reduces to the ratio of two proportions, namely the proportion of respectively the encompassing prior and posterior, in agreement with the constraints. This enables easy and straightforward estimation of the Bayes factor and its standard error. It is also rather efficient because with only one sample from the encompassing prior and one sample from the encompassing posterior, Bayes factors for all pairs of models are obtained. The goal of the final chapter, is to show the potential of posterior model probabilities and model selection as an alternative for the use of p-values in traditional hypothesis testing.