Nonparametric estimation of covariance structure in longitudinal data

In longitudinal studies, the effect of various treatments over time is usually of prime interest. However, observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure. A nonparametric estimate of the covariance structure is useful, either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions. The sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects. The variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity. In this paper, we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity. The value of the estimator as a diagnostic tool is demonstrated in two applications, one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of CD4 cell numbers in HIV seroconverters. The use of the estimator in more formal statistical inferences concerning the mean profiles requires further study.