Extending Methodology for Analyzing Multivariate Longitudinal Data

纵向数据包括在一项研究中对多个个体单位随时间重复测量的响应。这包括测量土壤酸度，在一个大的地理区域的不同领域，随着时间的推移重复;它包括一个湿度指数随着时间的推移测量了一个大的城市组;或者它包括神经元活动，在一定数量的时间点测量，为一组人的大脑正在被研究。当存在两个或更多个不同的响应时，每个响应随时间测量（例如，温度、气压和湿度），这些可以被定义为多变量纵向数据。在本提案中，我们将扩展现有的方法来分析多变量纵向数据，在典型的设置下，在每个给定的个人或单位内的一组共同的离散时间点收集多个响应，尽管单位之间的时间点可能是不同的。 提出了三个扩展。第一个涉及建模相对密集收集的多元纵向数据，遵循周期性模式。一个例子来自人类不同的荷尔蒙周期，每个周期在365天内每天测量。另一个例子是在一个省、州或国家的不同地点每天对环境空气质量进行不同的测量。这里的两个重要组成部分将是填补缺失周期的空白，以及在个人层面上预测未来周期。 第二个扩展涉及的情况下，随着时间的推移观察到的状态测量不完美，可以定义一些特定主题的有意义的隐藏状态。这发生在环境空气质量设置中，其中收集两个或三个不完全测量的响应，每个响应随时间推移，但是其中整体空气质量的潜在过程或隐藏状态正在驱动观察到的响应。除其他目标外，目前的建议将集中在预测未来的隐藏状态，给出过去的隐藏状态和观察状态的历史。由于（i）问题的维度，（ii）隐藏状态的正确建模，以及（iii）预测未来状态的主要目标之一，这里的这项工作将需要大量的计算组件。 最后的扩展适用于随着时间的推移收集两个或更多个响应的设置，例如小鼠模型的两种不同蛋白质，并且有兴趣确定纵向过程随时间的相关性，包括考虑任何滞后效应。这是建立在一些早期研究的基础上，目前提案的目标是如何在这种情况下正确和稳健地量化不确定性。 如上所述，这项研究工作的相关性将适用于各个科学领域，包括统计学，以及潜在的环境，土壤，生物和大气科学等。这些发现应该证明对加拿大及其他地区的研究人员有用，因为越来越需要适当地分析高维数据，例如这里考虑的多变量纵向数据。

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.