Inside cells, axonal and dendritic transport by motor proteins is a process that is responsible for supplying cargo, such as vesicles and organelles, to support neuronal function. Motor proteins achieve transport through a cycle of chemical and mechanical processes. Particle tracking experiments are used to study this intracellular cargo transport by recording multi-dimensional, discrete cargo position trajectories over time. However, due to experimental limitations, much of the mechanochemical process cannot be directly observed, making mathematical modeling and statistical inference an essential tool for identifying the underlying mechanisms. The cargo movement during transport is modeled using a switching stochastic differential equation framework that involves classification into one of three proposed hidden regimes. Each regime is characterized by different levels of velocity and stochasticity. The equations are presented as a state-space model with Markovian properties. Through a stochastic expectation-maximization algorithm, statistical inference can be made based on the observed trajectory. Regime predictions and particle location predictions are calculated through an auxiliary particle filter and particle smoother. Based on these predictions, parameters are estimated through maximum likelihood. Diagnostics are proposed that can assess model performance and therefore also be a form of model selection criteria. Model selection is used to find the most accurate regime models and the optimal number of regimes for a certain motor-cargo system. A method for incorporating a second positional dimension is also introduced. These methods are tested on both simulated data and different types of experimental data.

Longitudinal studies contain correlated data due to the repeated measurements on the same subject. The changing values of the time-dependent covariates and their association with the outcomes presents another source of correlation. Most methods used to analyze longitudinal data average the effects of time-dependent covariates on outcomes over time and provide a single regression coefficient per time-dependent covariate. This denies researchers the opportunity to follow the changing impact of time-dependent covariates on the outcomes. This dissertation addresses such issue through the use of partitioned regression coefficients in three different papers.

In the first paper, an alternative approach to the partitioned Generalized Method of Moments logistic regression model for longitudinal binary outcomes is presented. This method relies on Bayes estimators and is utilized when the partitioned Generalized Method of Moments model provides numerically unstable estimates of the regression coefficients. It is used to model obesity status in the Add Health study and cognitive impairment diagnosis in the National Alzheimer’s Coordination Center database.

The second paper develops a model that allows the joint modeling of two or more binary outcomes that provide an overall measure of a subject’s trait over time. The simultaneous modelling of all outcomes provides a complete picture of the overall measure of interest. This approach accounts for the correlation among and between the outcomes across time and the changing effects of time-dependent covariates on the outcomes. The model is used to analyze four outcomes measuring overall the quality of life in the Chinese Longitudinal Healthy Longevity Study.

The third paper presents an approach that allows for estimation of cross-sectional and lagged effects of the covariates on the outcome as well as the feedback of the response on future covariates. This is done in two-parts, in part-1, the effects of time-dependent covariates on the outcomes are estimated, then, in part-2, the outcome influences on future values of the covariates are measured. These model parameters are obtained through a Generalized Method of Moments procedure that uses valid moment conditions between the outcome and the covariates. Child morbidity in the Philippines and obesity status in the Add Health data are analyzed.