Marginal longitudinal nonparametric regression: Locality and efficiency of spline and kernel methods

We consider nonparametric regression in a longitudinal marginal model of generalized estimating equation (GEE) type with a time-varying covariate in the situation where the number of observations per subject is finite and the number of subjects is large, In such models, the basic shape of the regression function is affected only by the covariate values and not otherwise by the ordering of the observations. Two methods of estimating the nonparametric function can be considered: kernel methods and spline methods, Recently, surprising evidence has emerged suggesting that for kernel methods previously proposed in the literature, it is generally asymptotically preferable to ignore the correlation structure in our marginal model and instead assume that the data are independent, that is, working independence in the GEE jargon. As seen through equivalent kernel results, in univariate independent data problems splines and kernels have similar behavior: smoothing splines are equivalent to kernel regression with a specific higher-order kernel, and hence smoothing splines are local. This equivalence suggests that in our marginal model, working independence might be preferable for spline methods, Our results suggest the opposite; via theoretical and numerical calculations, we provide evidence suggesting that for our marginal model, marginal smoothing and penalized regression splines are not local in their behavior. In contrast to the kernel results, our evidence that spline suggests that when using spline methods, it is worthwhile to account for the correlation structure. Our results also suggest that spline methods appear to be more efficient than the previously proposed kernel methods for our marginal model.