  We consider varying-coefficient models for mixed synchronous and asynchronous
longitudinal covariates, where asynchronicity refers to the misalignment of
longitudinal measurement times within an individual. We propose three different
methods of parameter estimation and inference. The first method is a one-step
approach that estimates non-parametric regression functions for synchronous and
asynchronous longitudinal covariates simultaneously. The second method is a
two-step approach in which synchronous longitudinal covariates are regressed
with the longitudinal response by centering the synchronous longitudinal
covariates first and, in the second step, the residuals from the first step are
regressed with asynchronous longitudinal covariates. The third method is the
same as the second method except that in the first step, we omit the
asynchronous longitudinal covariate and include a non-parametric intercept in
the regression analysis of synchronous longitudinal covariates and the
longitudinal response. We further construct simultaneous confidence bands for
the non-parametric regression functions to quantify the overall magnitude of
variation. Extensive simulation studies provide numerical support for the
theoretical findings. The practical utility of the methods is illustrated on a
dataset from the ADNI study.
