### Dynamical Correlation for Multivariate Longitudinal Data

Joel A Dubin, Hans-Georg Müller
2005 Journal of the American Statistical Association
Nonparametric methodology for longitudinal data analysis is becoming increasingly popular. The analysis of multivariate longitudinal data, where data on several time courses are recorded per subject, has received considerably less attention, in spite of its importance for practical data analysis. In particular, there is a need for measures and estimates to capture dependency between the components of vector-valued longitudinal data. We propose and analyze a simple and effective nonparametric
more » ... ve nonparametric method to quantify the covariation of components of multivariate longitudinal observations, which are viewed as realizations of a random process. This includes the notion of a correlation between derivatives and time-shifted versions. The concept of dynamical correlation is based on a scalar product obtained from pairs of standardized smoothed curves. The proposed method can be utilized when observation times are irregular and not matching between subjects or between responses within subject. For higher-dimensional data, one may construct a dynamical correlation matrix which then serves as a starting point for standard multivariate analysis techniques such as principal components. Our methods are illustrated via simulations as well as with data on five acute phase blood proteins measured longitudinally from a study of hemodialysis patients. KEY WORDS: Acute phase proteins; Curve data; Dependency; Random effects model; Smoothing; Stochastic process. ∞ r=0 σ 2 r,k < ∞, and ε 0,k is uncorrelated with ε r,k , r ≥ 1. The functions η r , r = 0, 1, ..., are assumed to form an orthonormal basis of L 2 (dw), i.e., η i , η j = 0 for i = j and η i , η j = 1 for all i = j. The functions µ k , η r , 1 ≤ k ≤ p, 0 ≤ r < ∞, are furthermore assumed to be smooth, say twice continuously differentiable. We also assume that all random variables ε r,j , 0 ≤ r < ∞, 1 ≤ k ≤ p have a joint distribution.