Testing in finite mixture models and some applications in health

This dissertation explores an application of finite mixture modelling to self-assessed health (SAH) survey data in the British Household Panel Survey (BHPS), and then considers tests for homogeneity in some examples of finite mixture models. In the application of finite mixture modelling to SAH survey data, the problem studied is how different question wording and response items in the survey question may affect SAH responses. While the usual methods in the literature implicitly assume that all respondents react to the change in response items in a certain manner, a latent class model is introduced that relaxes this assumption. Results show that this latent class model reduces misinformation that may be introduced using the usual methods in the literature. The estimated effect of question wording and response items can potentially be used to predict SAH responses to different SAH questions. The latent class model is one example of finite mixture models, and while the application of the latent class model in the SAH question seems a good fit to the data in various aspects, the problem of whether different latent classes exist in the first place needs to be further explored. In the setting of finite mixture modelling, this is known as testing for homogeneity. The rest of the thesis explores testing for homogeneity in two other examples: the zero-inflated Poisson (ZIP), and the two-component finite mixture model. Testing for homogeneity in finite mixture models is a well-studied statistical problem. While many other studies have focused on deriving the relevant non-standard null distributions of test statistics, a different approach is considered here. By considering alternative models that are close in some sense to the finite mixture models, simple tests can be constructed for which the null distributions of test statistics are known, and which may also have power when the true data generating processes are the finite mixture models. For testing against the ZIP, the alternative model constructed is one that shares similar characteristics to the ZIP and the hurdle Poisson (HP) models. For testing against the two-component finite mixture model, the construction of the alternative model is done by means of a Gram-Charlier expansion. Simulation results show that this approach performs well in terms of size and power for both the ZIP and the two-component finite mixture data generating processes.