Polynomial Spline Estimation and Inference for Varying Coefficient Models with Longitudinal Data

We consider nonparametric estimation of coefficient functions in a varying coefficient model of the form Yij = X T i (tij)β(tij)+ i(tij) based on longitudinal observations {(Yij ,Xi(tij), tij), i = 1, . . . , n, j = 1, . . . , ni}, where tij and ni are the time of the jth measurement and the number of repeated measurements for the ith subject, and Yij and Xi(tij) = (Xi0(tij), . . . ,XiL(tij)) T for L ≥ 0 are the ith subject’s observed outcome and covariates at tij . We approximate each coefficient function by a polynomial spline and employ the least squares method to do the estimation. An asymptotic theory for the resulting estimates is established, including consistency, rate of convergence and asymptotic distribution. The asymptotic distribution results are used as a guideline to construct approximate confidence intervals and confidence bands for components of β(t). We also propose a polynomial spline estimate of the covariance structure of (t), which is used to estimate the variance of the spline estimate β̂(t). A data example in epidemiology and a simulation study are used to demonstrate our methods.