Efficient Semiparametric Marginal Estimation for the Partially Linear Additive Model for Longitudinal/Clustered Data

Abstract

We consider the efficient estimation of a regression parameter in a partially linear additive nonparametric regression model from repeated measures data when the covariates are multivariate. To date, while there is some literature in the scalar covariate case, the problem has not been addressed in the multivariate additive model case. Ours represents a first contribution in this direction. As part of this work, we first describe the behavior of nonparametric estimators for additive models with repeated measures when the underlying model is not additive. These results are critical when one considers variants of the basic additive model. We apply them to the partially linear additive repeated-measures model, deriving an explicit consistent estimator of the parametric component; if the errors are in addition Gaussian, the estimator is semiparametric efficient. We also apply our basic methods to a unique testing problem that arises in genetic epidemiology; in combination with a projection argument we develop an efficient and easily computed testing scheme. Simulations and an empirical example from nutritional epidemiology illustrate our methods.

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

“…, d. Recently, there are extensive literature on the estimation for model (1). For example, Carroll et al (2009) developed an efficient estimation of β in model (1) based on local linear smooth backfitting, Lian et al (2014) studied the marginal generalized additive partial linear models with diverging number of covariates by using generalized estimating equations (GEE), and Cheng et al (2014) proposed efficient estimation of the Euclidean parameters in generalized additive partial linear models. As far as we know, the GEE method does not possess robust property because it is in principle very similar to the weighted least squares method.…”

Variable selection in strong hierarchical semiparametric models for longitudinal data

“…When the working covariance matrix is misspecified, the resulting estimate is still consistent, but not efficient [20, 12]. In order to improve the estimation efficiency of the regression parameters and reduce the bias of the semiparametric estimate for longitudinal data, it’s essential to specify the working covariance matrix correctly [27, 7]. Here we first assume balanced data with n i = n for all i and propose the maximum likelihood estimation for V i , and then present the implementation with unbalanced data later.…”

“…To relax the assumptions on parametric forms, we consider the semiparametric additive partially linear model (APLM) in this paper, which combines the parsimony of parametric regression and flexibility of nonparametric regression, and thus provides a nice trade-off between model interpretability and flexibility. Estimation for APLMs has been receiving increasing attention, and there is a considerable amount of relevant studies (see, e.g., [7, 22, 2, 8]).…”

“…However, variable selection for APLMs with longitudinal data is rather challenging due to the semiparametric relationship and the within-subject correlation structure among the repeated observations of the same subjects, and thus has not received sufficient attention. Taking account of the within-subject correlation correctly is essential for obtaining efficient estimators for longitudinal data [27, 7]. …”

Marginal longitudinal semiparametric regression via penalized splines

“…This spawned a flurry of activity on the problem. Relevant references include: Lin and Carroll (2000, 2006), Welsh et al (2002), Wang (2003), Linton et al (2003), Lin et al (2004), Carroll et al (2004), Hu et al (2004), Chen and Jin (2005), Wang et al (2005) and Fan et al (2007), Sun et al (2007), Fan and Wu (2008) and Carroll et al (2009a,b). …”