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Longitudinal studies provide a potent means to analyse patterns over time in contrast to cross-sectional studies which are focused on a single time point. However, the complexity and costs involved in conducting longitudinal studies (e.g. drug trials) can result in smaller sample sizes. This then leads to the need to collate evidence across studies. Consequently, this thesis explores how Bayesian methodology provides a flexible framework for handling the problems that typically arise when conducting longitudinal meta-analysis. This thesis then gives a brief overview of longitudinal data and meta-analysis. The introduction section is followed by a concise overview of pharmacology and why it makes a good case study. Next is a review of Bayesian Hamiltonian Monte-Carlo inference (HMC) sampling. HMC was found to provide useful diagnostic tools and relatively fast inference times for conducting Bayesian longitudinal meta-analysis (BLMA). We then cover prior recommendations for BLMA. We found discrepancies between the confidence intervals of Bayesian and frequentist methods and how prior selection contributes to this difference. Next is a treatment of import practical considerations for undertaking BLMA. This section is followed by a two-level meta-analysis of a pharmaceutical data where we study the effect of the drug Naproxen on osteoarthritis pain. We found a significant effect of the drug on reported pain levels; however, this effect only existed in studies with high basal pain levels. We then conducted a three-level BLMA on a simulated dataset of a small number of trails where we found that Bayesian priors provide a useful means of inference regularisation for overcoming issues with random-effects. Finally, we conducted a BLMA on a simulated dataset with non-normal random-effects and found utility when conducting sensitivity analyses. In conclusion, Bayesian methodology provides a unified framework for conducting BLMA. Specifically, the framework allows for flexible stochastic modelling capabilities that help overcome inference issues when conducting BLMA on a small number of trails.

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Intensive longitudinal design involves sequential measurements and intervals between measurements are small (daily basis, or even hourly) and occasionally uneven. Due to short periods of time between measurements, serial correlations are more prominent in such design than in traditional longitudinal design. Various approaches have been suggested for the analysis of longitudinal studies. Mixed models constitute a popular class of models in this context, but other models have been proposed. Bayesian hierarchical Ornstein-Uhlenbeck models, which is derived from Ornstein-Uhlenbeck stochastic process, aims at modelling data with serial correlations and unequal time intervals between measurements – the very nature of intensive longitudinal data. Previous research (Oravecz & Tuerlinckx, 2011; Oravecz, Tuerlinckx & Vandekerckhove, 2016) in this fields have explored the practical application of the HOU model with Bayesian MCMC sampling method in continuous outcomes data and the comparison with traditional techniques such as mixed models. In this paper, we will discuss the background of Bayesian hierarchical Ornstein-Uhlenbeck models and extend its application in all common research setting by reconstructing the Bayesian hierarchical Ornstein-Uhlenbeck models framework so that it is applicable in both continuous and categorical outcomes, e.g. binary, ordinal, nominal response data, etc. In addition, the comparison between the new Bayesian hierarchical Ornstein-Uhlenbeck models framework and traditional techniques such as mixed models will be discussed. Like continuous outcomes case which was investigated in previous research, the present simulation study has shown that methods using mixed models with frequentist approach can provide a good estimate of the fixed effect parameters in hierarchical Ornstein–Uhlenbeck models with binary outcomes. However, they perform poorly in the variance estimation of the random effects. Practical application of hierarchical Ornstein–Uhlenbeck models with ordinal outcomes is also conducted on two core affect datasets (Kuppens, Oravecz, & Tuerlinckx, 2010; Vansteelandt & Verbeke, 2016) which results show there are prominent serial correlations on core affects but there is no strong evidence support individuals difference in autocorrelations level.