Abstract

In multilevel research, the data structure in the population is hierarchical, and the sample data are viewed as a multistage sample from this hierarchical population. For instance in educational research, the population consists of schools and pupils within these schools. In this scenario, pupils are said to be nested within
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schools. In this dissertation we look at some issues in the analysis of hierarchical (multilevel) data using hierarchical linear models. The main focus is on hierarchical linear models with constraints on the model parameters. In Chapter 2, methods that can be used to test homogeneity in a random intercept model are presented. The chapter begins with a review of asymptotic, posterior predictive and plug-in p-values and it is shown how each of them can be computed. Four test quantities for testing the homogeneity of residual variances are presented. Thereafter two test quantities for testing the homogeneity of regression slopes are presented. Subsequently each test quantity is evaluated using asymptotic, posterior predictive and plug in p-values. A simulation study is used to investigate the frequency properties of these p-values. It is shown that discrepancy measures evaluated using either posterior predictive or plug-in p-values have both good ‘classical’ and ‘Bayesian’ frequency properties. This suggests that whenever homogeneity of residual variances and /or homogeneity of regression slopes are to be evaluated in the random intercept model, it is advisable to use discrepancy measures evaluated with either posterior predictive or plug-in p-values. Homogeneity in more complex hierarchical models can be investigated by natural extensions of what has been presented for the random intercept model. Researchers in most fields of study usually have several competing theories/expectations about the outcome(s) of their research. These theories may emanate from subject-area experts or from previous studies. In Chapter 3, we use classical statistics to investigate the tenability of the theories presented by the subject experts. Data from two projects in clinical psychology are presented and analysed. The analyses illustrate how the linear mixed model can be adapted and used to analyse structured repeated measurements. In Chapters 4 and 5, the statistical modelling takes into account the theories a researcher has with respect to the outcome of the investigation at hand. In Chapter 4, several issues are presented. First, we show that by putting constraints on the model parameters, competing theories can be translated into a set of competing models. A fully Bayesian approach is used to select the ‘best’ model from the set of competing models via the computation of posterior probabilities using a novel method which we denote the ‘method of encompassing priors’. It is shown that for inequality constrained hierarchical models, posterior probabilities computed using the method are virtually independent of the specification of the prior distribution. The model with the largest posterior probability is chosen as the one that conforms best to the data. Further, we illustrate how the parameters in inequality constrained hierarchical models can be estimated. An advantage of the Bayesian approach is that estimates of the model parameters can be obtained easily using samples from the posterior distribution and the approach allows several models to be compared. This is in contrast to classical hypothesis testing where one can only have two hypotheses (null and alternative) with a necessity for the null to be nested in the alternative. The last chapter (Chapter 5) deals with posterior predictive model selection in the context of inequality constrained hierarchical models. Model selection is accomplished through choosing a model that minimizes the mean squared distance between the observed data and a future observation vector. Basically this approach to model selection chooses a model that would best predict the observed data. We argue that posterior predictive model selection has several (potential) advantages over classical hypothesis testing
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