Nonparametric additive regression for repeatedly measured data

Abstract

We develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems. Our methodology easily copes with various settings, such as when some covariates are the same over repeated response measurements. We allow for a working covariance matrix for the regression errors, showing that our method is most efficient when the correct covariance matrix is used. The component functions achieve the known asymptotic variance lower bound for the scalar argument case. Smooth backfitting also leads directly to design-independent biases in the local linear case. Simulations show our estimator has smaller variance than the usual kernel estimator. This is also illustrated by an example from nutritional epidemiology.

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

“…In this section, we describe the local linear smooth backfitting estimator for repeated measures data (Carroll et al 2009), state its properties when an additive model holds, and then describe its properties when the additive model fails. The latter result is new.…”

“…The minimizer of (3) exists and is given as follows (Carroll et al 2009). Define the following estimators, for q = 1, … , D , r = 0,1,2, $${\stackrel{true^}{u}}_s^r(x)={\displaystyle {\sum }_{j=1}^J{b}_{\mathit{\text{jj}},\mathrm{\Sigma }}{n}^{-1}}{\displaystyle {\sum }_{i=1}^n{true(\frac{X_{\text{italiciqj}}-x}{h_q}true)}^r}{K}_{h_q}(x,X_{\text{italiciqj}})$$ and for q , p = 1, … , D ; r , s = 0,1, $$\begin{array}{cc}{\hat{\upsilon }}_q^{\text{italicrs}}(x,y)=\hfill & {\displaystyle {\sum }_{j=1}^J{{\displaystyle {\sum }_{k=1,\ne j}^J{b}_{\mathit{\text{jk}},\mathrm{\Sigma }}{n}^{-1}{\displaystyle {\sum }_{i=1}^n{true(\frac{X_{\text{italiciqj}}-x}{h_q}true)}^r}true(\frac{X_{\text{italiciqk}}-y}{h_q}true)}}^s{K}_{h_q}(x,X_{\text{italiciqj}})K_{h_q}(y,X_{\mathit{\text{iqk}}});}\hfill \\ {\hat{\upsilon }}_{\text{italicqp}}^{\text{italicrs}}(x,y)=\hfill & {\displaystyle {\sum }_{j=1}^J{{\displaystyle {\sum }_{k=1}^J{b}_{\mathit{\text{jk}},\mathrm{\Sigma }}{n}^{-1}{\displaystyle {\sum }_{i=1}^n{true(\frac{{}_{}}{}}^{}}}}^{}}\hfill \end{array}$$…”

“…If θ true were known, then the functions { m d ,true (·)} can be estimated efficiently by a local linear smooth backfitting algorithm (Carroll et al 2009) and reviewed in Section 2. We combine this methodology with a semiparametric profiling argument (Lin and Carroll 2006) to develop estimates of θ true and inference for it in Section 3.…”

Continuously additive models for nonlinear functional regression

“…Here we pursue a different kind of additivity that occurs in the time domain rather than in the spectral domain and develop nonlinear functional regression models that are not dependent on a preliminary functional principal component analysis yet are structurally stable and not weighed down by the curse of dimensionality (Hall et al 2009). Additive models (Friedman & Stuetzle 1981;Stone 1985) have been successfully used for many regression situations that involve continuous predictors and both continuous and generalized responses (Mammen & Park 2005;Yu et al 2008;Carroll et al 2008). …”

Untitled

“…Carroll et al (2009) had the same observation for a special case with repeated measurement data when g is the identity function.…”