We propose a nonparametric bivariate varying coefficient generalized linear model to predict a mean response trajectory in the future given an individual's characteristics at present or an earlier time point in a longitudinal study. Given the measurement time of the predictors, the coefficients vary as functions of the future time over which the prediction of the mean response is concerned and illustrate the dynamic association between the future response and the earlier measured predictors. We use a nonparametric approach that takes advantage of features of both the kernel and the spline methods for estimation. The resulting coefficient estimator is asymptotically consistent under mild regularity conditions. We also develop a new bootstrap approach to construct simultaneous confidence bands for statistical inference about the coefficients and the predicted response trajectory based on the coverage rate of bootstrap estimates. We use the Framingham Heart Study to illustrate the methodology. The proposed procedure is applied to predict the probability trajectory of hypertension risk given individuals' health condition in early adulthood and to examine the impact of risk factors in early adulthood on a long-term risk of hypertension over several decades.
Keywords: bootstrap simultaneous confidence band; generalized estimating equations; kernel method; predictive trajectory; spline method; varying coefficients.
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