Extending Methodology for Analyzing Multivariate Longitudinal Data

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Longitudinal data comprise a response that is measured repeatedly over time, for a number of individual units, in a study. This includes measurement of soil acidity, at various fields across a large geographic area, repeatedly over time; it includes a humidity index measured over time for a large group of cities; or it includes neuronal activity, measured at a set number of time points, for a group of individuals whose brains are being studied. When there exist two or more distinct responses each measured over time (e.g., temperature, barometric pressure, and humidity), these can be defined as multivariate longitudinal data. In this proposal, we will be extending existing methodology for analyzing multivariate longitudinal data, under the typical setting where multiple responses are each collected at a common set of discrete time points within each given individual or unit, though the time points may be distinct between units. Three extensions are proposed. The first involves modeling relatively densely collected multivariate longitudinal data that follow cyclical patterns. One example comes from different hormonal cycles from humans, each measured daily over a 365-day period. Another example follows different measures of environmental air quality daily at various sites across a province, state or country. Two important components here will be filling in gaps when there are missing cycles, as well as prediction of future cycles at the individual level. The second extension involves the situation when states observed over time are measured imperfectly and where some topic-specific meaningful hidden states can be defined. This occurs in an environmental air quality setting, where two or three imperfectly measured responses are collected, each over time, but where an underlying process or hidden state(s) of overall air quality is driving the observed responses. Among other goals, the current proposal will focus on the prediction of future hidden states given history of both past hidden and observed states. This effort here will require a substantial computing component, due to (i) the dimensionality of the problem, (ii) the proper modeling of the hidden states, and (iii) one of the main goals being prediction of future states. The final extension applies to the setting where two or more responses are collected over time, such as two different proteins for mouse models, and there is interest in determining how correlated the longitudinal processes are over time, including consideration of any lagged effects. This builds on some earlier research, and the goal for the current proposal is how to properly and robustly quantify uncertainty in this context. As implied above, the relevance of this research work will apply across various scientific domains, including in statistics, and potentially environmental, soil, biological, and atmospheric sciences, among others. The findings should prove useful for researchers in Canada and beyond, due to the ever-increasing need to appropriately analyze high-dimensional data, such as the multivariate longitudinal data considered here.