Flexible methods for mixed longitudinal responses measured at irregular time intervals

为了衡量两个纵向反应之间的关联，在传统统计不感兴趣或可能需要一些限制性假设(例如，使用皮尔逊相关系数)的情况下，使用灵活的方法是非常有用的。在这项建议中，我们打算开发灵活的方法来查看不同纵向测量的关联和时间排序，其中测量可以是混合类型(例如，一个是连续的，另一个是二进制的)，并记录在不均匀的网格上和彼此不同的时间点上。这一建议建立在为双变量连续纵向响应提出相关性度量的工作基础上(Dubin和Müler，2005)。在这项先前的工作中，该方法被应用于肾脏病蛋白质数据，在连续的尺度上测量，并且人体内的测量时间点相同。该方法采用基于曲线的方法，从而允许测量与滞后的关联的能力，即使当测量不是在个体内相同和统一的设计上测量时也是如此。我们将继续使用基于曲线的方法，与Dubin和Müler(2005)的方法相关的非参数模型，尽管我们也将提出半参数方法，其基础是相关数据的广义线性模型的框架，其提供了灵活性和容易地将该方法与标准软件相结合的能力之间的平衡。目前的建议是基于一个纵向的例子，也是在肾脏学中，目前还不知道一种反应(作为炎症标志的连续测量的蛋白质)的变化是在不同的反应之前还是之后，不同的反应是二元测量的感染。这些测量是混合类型的，并在不同和不均匀的场合进行测量。我们希望，拟议的方法将被证明在各种科学学科中都是有用的。

For measuring the association between two longitudinal responses, it is quite useful to have flexible methods in case traditional statistics are not of interest or may require somewhat restrictive assumptions (e.g., using Pearson correlation coefficient). In this proposal, we intend to develop flexible methods to look at both association and temporal sequencing of distinct longitudinal measures, where the measures may be of mixed type (e.g., one continuous, the other binary) and recorded on non-uniform grids and different time points from one another. This proposal builds off work which proposed a measure of correlation for bivariate continuous longitudinal responses (Dubin and Müller, 2005). In this prior work, the method was applied to nephrology protein data, measured on a continuous scale, and where the time points for measurements within a person were the same.  This method took a curve-based approach, thereby allowing for the ability to measure associations with lags, even when the measurements were not measured on an identical and uniform design within an individual.   We will continue to use a curve-based approach, a nonparametric model related to the approach from Dubin and Müller (2005), though we will also propose a semiparametric approach, grounded in the framework for generalized linear models for correlated data, which provides a balance between flexibility and the ability to easily incorporate the method with standard software. The current proposal is motivated by a longitudinal example, also in nephrology, where it is not known whether changes in one response, a continuous measured protein that is a marker of inflammation, precedes or follows a different response, a binary measured infection. These measurements are of mixed type, and are measured at different and non-uniform occasions. We hope that the proposed methodology will prove useful across various scientific disciplines.