Longitudinal studies rely on data collected at several occasions from a set of selected individuals. The purpose of these studies is to use a regression-type model to express a response variable as a function of explanatory variables, or covariates. In this thesis, we use marginal models for the analysis of such data, which, coupled with the method of estimating equations, provide estimators of the main regression parameter. When some of the responses are missing or there is error in the recorded covariates, the original estimating equation may be biased. We use techniques available in the literature to modify it and regain the unbiasedness property. We prove the asymptotic normality of the regression estimator obtained under these more realistic circumstances, and provide theoretical and numerical examples to illustrate this approach.

Over the past decade the widespread proliferation of mobile devices and wearable technology has significantly changed the landscape of epidemiological data gathering and evolved into a field known as Digital Epidemiology. One source of active digital data collection is online participatory syndromic surveillance systems. These systems actively engage the general public in reporting health-related information and provide timely information about disease trends within the community. This dissertation comprehensively addresses how researchers can effectively use this type of data to answer questions about Influenza-like Illness (ILI) disease burden in the general population. We assess the representativeness and reporting habits of volunteers for these systems and use this information to develop statistically rigorous methods that adjust for potential biases. Specifically, we evaluate how different missing data methods, such as complete case and multiple imputation models, affect estimates of ILI disease burden using both simulated data as well as data from the Australian system, Flutracking.net. We then extend these methods to data from the American system, Flu Near You, which has different patterns. Finally, we provide examples of how this data has been used to answer questions about ILI in the general community and promote better understanding of disease surveillance and data literacy among volunteers.

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Dynamic Causal Modeling (DCM) uses dynamical systems to represent the high-level neural processing strategy for a given cognitive task. The logical network topology of the model is specified by a combination of prior knowledge and statistical analysis of the neuro-imaging signals. Parameters of this a-priori model are then estimated and competing models are compared to determine the most likely model given experimental data. Inter-subject analysis using DCM is complicated by differences in model topology, which can vary across subjects due to errors in the first-level statistical analysis of fMRI data or variations in cognitive processing. This requires considerable judgment on the part of the experimenter to decide on the validity of assumptions used in the modeling and statistical analysis; in particular, the dropping of subjects with insufficient activity in a region of the model and ignoring activation not included in the model. This manual data filtering is required so that the fMRI model's network size is consistent across subjects. This thesis proposes a solution to this problem by treating missing regions in the first-level analysis as missing data, and performing estimation of the time course associated with any missing region using one of four candidate methods: zero-filling, average-filling, noise-filling using a fixed stochastic process, or one estimated using expectation-maximization. The effect of this estimation scheme was analyzed by treating it as a preprocessing step to DCM and observing the resulting effects on model evidence. Simulation studies show that estimation using expectation-maximization yields the highest classification accuracy using a simple loss function and highest model evidence, relative to other methods. This result held for various data set sizes and varying numbers of model choice. In real data, application to Go/No-Go and Simon tasks allowed computation of signals from the missing nodes and the consequent computation of model evidence in all subjects compared to 62 and 48 percent respectively if no preprocessing was performed. These results demonstrate the face validity of the preprocessing scheme and open the possibility of using single-subject DCM as an individual cognitive phenotyping tool. / Ph. D.