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Longitudinal growth patterns are routinely seen in medical studies where developments of individuals on one or more outcome variables are followed over a period of time. Many current methods for modeling growth presuppose a parametric relationship between the outcome and time (e.g., linear, quadratic); however, these relationships may not accurately capture growth over time. Functional mixed effects (FME) models provide flexibility in handling longitudinal data with nonparametric temporal trends because they allow the data to determine the shape of the curve. Although FME methods are well-developed for continuous, normally distributed outcome measures, nonparametric methods for handling categorical outcomes are limited.
In this thesis, we propose a Bayesian hierarchical FME model to account for growth curves with non-Gaussian outcomes. In particular, we extend traditional FME models which assume normally distributed outcomes by modeling the probabilities associated with the binomially distributed outcomes and adding an additional level to the hierarchical model to correctly specify the outcomes as binomially distributed.
We then extend the proposed binomial FME model to the multinomial setting where the outcomes consist of more than two nominal categories. Current modeling approaches include modeling each category of a multinomial outcome separately via linear and nonlinear mixed effects models; yet, these approaches ignore the inherent correlation among the categories of the outcome. Our model captures this correlation through a sequence of conditional binomial FME models which results in one model simultaneously estimating probabilities in all categories.
Lastly, we extend our binomial FME model to address a common medical situation where multiple outcomes are measured on subjects over time and investigators are interested in simultaneously assessing the impact of all outcomes. We account for the relationship between outcomes by altering the correlation structure in the hierarchical model and simultaneously estimating the outcome curves.
Our methods are assessed via simulation studies and real data analyses where we investigate the ability of the models to accurately predict the underlying growth trajectory of individuals and populations. Our applications include analyses of speech development data in adults and children with cochlear implants and analyses on eye-tracking data used to assess word processing in cochlear implant patients.