Olav Laudy

Inequality Constrained Analysis of Variance: A Bayesian approach.

Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers 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 this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means.

Bayesian Evaluation of Inequality and Equality Constrained Hypotheses for Contingency Tables

In this article, a Bayesian model selection approach is introduced that can select the best of a set of inequality and equality constrained hypotheses for contingency tables. The hypotheses are presented in terms of cell probabilities allowing researchers to test (in)equality constrained hypotheses in a format that is directly related to the data. The proposed method is investigated by several simulation studies and shows good performance. Software that allows researchers to apply the Bayesian approach to their own data is also provided.

Bayesian methods for the analysis of inequality constrained contingency tables

A Bayesian methodology for the analysis of inequality constrained models for contingency tables is presented. The problem of interest lies in obtaining the estimates of functions of cell probabilities subject to inequality constraints, testing hypotheses and selection of the best model. Constraints on conditional cell probabilities and on local, global, continuation and cumulative odds ratios are discussed. A Gibbs sampler to obtain a discrete representation of the posterior distribution of the inequality constrained parameters is used. Using this discrete representation, the credibility regions of functions of cell probabilities can be constructed. Posterior model probabilities are used for model selection and hypotheses are tested using posterior predictive checks. The Bayesian methodology proposed is illustrated in two examples.

Applications of confirmatory latent class analysis in developmental psychology

In the field of developmental, psychology researchers may have several competing theories with respect to their research subject. In this paper an approach will be proposed that can be used to select the best of these theories. It will be shown that a theory can be translated in a constrained latent class model using inequality constraints. This can be done for several (possibly competing) theories. Subsequently, fit-measures can be used to determine which model (and thus which theory) is supported most by the data. The approach will be introduced using data with respect to self-reported child and adult antisocial behaviour. It will be further illustrated using data obtained using the figural intersection task.

Bayesian eggs and Bayesian omelettes: Reply to Stern (2005).

In this response to Sterns (2005) discussion of Klugkist, Laudy, and Hoijtink (2005), model inference based on posterior probabilities on the parameter space is discussed. Furthermore, the authors respond to Sterns example in which all possible orderings are included via a short discussion of exploratory versus theory-based modeling. Finally, the authors show that the Bayesian approach is flexible and can deal with many types of constraints. This is illustrated using a model with constraints on the differences between means.

Inequality Constrained Contingency Table Analysis

In recent years, there has been growing interest in statistical models incorporating inequality constraints on model parameters. This is because the omnibus hypotheses can be replaced by more specific inequality constrained hypotheses [2]. In the extensive review in [3], literature is discussed on order restricted statistical models for contingency tables. What becomes clear in this review is that many order restricted models can be estimated and tested – however, not without thorough technical knowledge of the matter. Even if the software is provided, still the (applied) researcher is required to know a great deal about parameterizations of log-linear models.