Extending Methodology for Analyzing Multivariate Longitudinal Data

纵向数据包括在一项研究中，随着时间的推移，针对多个单个单位重复测量的响应。这包括在一大片地理区域的不同领域，随着时间的推移重复测量土壤酸度；它包括一大群城市随着时间的推移测量的湿度指数；或者它包括在设定的时间点测量的一组人的神经元活动，这些人的大脑正在接受研究。当存在两个或更多不同的响应时(例如，温度、气压和湿度)，这些响应可以被定义为多变量纵向数据。在这项提案中，我们将扩展现有的多变量纵向数据分析方法，在典型情况下，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答复，尽管单位之间的时间点可能不同。 提出了三种扩展方案。第一个涉及对相对密集收集的多变量纵向数据进行建模，这些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，每种荷尔蒙在365天内每天测量一次。另一个例子是，在一个省、州或国家的不同地点，每天的环境空气质量指标都不同。这里的两个重要组成部分将是填补缺失周期时的空白，以及在个人层面上预测未来周期。 第二个扩展涉及到这样一种情况，即随着时间的推移观察到的状态被不完美地测量出来，并且可以定义一些特定于主题的有意义的隐藏状态。这发生在环境空气质量环境中，收集了两到三个不完美的测量响应，每个都随着时间的推移而变化，但总体空气质量的潜在过程或隐藏状态(S)正在驱动观察到的响应。在其他目标中，目前的提议将侧重于根据过去隐藏状态和观测状态的历史预测未来隐藏状态。这项工作将需要大量的计算组件，因为(I)问题的维度，(Ii)隐藏状态的适当建模，以及(Iii)主要目标之一是对未来状态的预测。 最后的扩展适用于随着时间的推移收集两个或更多响应的设置，例如小鼠模型的两个不同的蛋白质，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的，当前提案的目标是如何适当和有力地量化这一背景下的不确定性。 如上所述，这项研究工作的相关性将适用于各种科学领域，包括统计学，以及潜在的环境、土壤、生物和大气科学等。这些发现应该会被证明对加拿大和其他地方的研究人员有用，因为越来越多的人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.