Linear Mixed-Effects Model Using Penalized Spline Based on Data Transformation Methods

Abstract

In this paper, we discuss two different data transformation techniques for dealing with censored data: Kaplan-Meier weights and the k-nearest neighbor imputation method. The main objective of this paper is to find penalized spline estimates for the components of a linear mixed effect model with right-censored data. In the context of a mixed model setting, the estimation procedure is performed based on the modified or transformed dataset obtained via these transformation techniques. In order to compare the outcomes from a linear mixed model using these two approaches, a Monte Carlo simulation and two real data examples are presented. According to our results, the k-nearest neighbor imputation is very successful in dealing with censored observations.

Appendices

Appendix A1—Proof of Equation (12.12)

One can see from  (12.11), minimization criterion is given by

$$\begin{aligned} {\text {PLSW}}({\pmb {\beta }}, \mathbf{u})=\arg \min \left( (\mathbf{T}-\mathbf{X} {\pmb {\beta }}-\mathbf{Z} \mathbf{u})^\prime \mathbf{W}(\mathbf{T}-\mathbf{X} {\pmb {\beta }}-\mathbf{Z} \mathbf{u})+\lambda \mathbf{u}^\prime \mathbf{D} \mathbf{u}\right) . \end{aligned}$$

From that equation, the system given below is obtained as

$$ \begin{aligned} \left( \begin{array}{c} \widehat{{\pmb {\beta }}}_{K M W} \\ \widehat{\mathbf{u}}_{K M W} \end{array}\right)&=\left( \begin{array}{cc} \mathbf{X}^\prime \mathbf{W} \mathbf{X} &{} \mathbf{X}^\prime \mathbf{W} \mathbf{Z} \\ \mathbf{Z}^\prime \mathbf{W} \mathbf{X} &{} \mathbf{Z}^\prime \mathbf{W} \mathbf{Z} +\lambda \mathbf{D} \end{array}\right) ^{-1}\left( \begin{array}{c} \mathbf{X}^\prime \mathbf{W} \mathbf{T} \\ \mathbf{Z}^\prime \mathbf{W} \mathbf{T} \end{array}\right) \\&=\left[ (\mathbf{X} : \mathbf{Z})^\prime \mathbf{W}(\mathbf{X} : \mathbf{Z})+\lambda \mathbf{D}\right] ^{-1}(\mathbf{X} : \mathbf{Z})^\prime \mathbf{W} \mathbf{T}. \end{aligned} $$

Let $\mathbf{R}=(\mathbf{X} : \mathbf{Z})$. Accordingly,  (12.12) is obtained as follows:

$$ \left[ \widehat{{\pmb {\beta }}}^\prime _{K M W}, \widehat{\mathbf{u}}^\prime _{K M W}\right] ^\prime =\left( \mathbf{R}^\prime \mathbf{W} \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \mathbf{R}^\prime \mathbf{W} \mathbf{T}. $$

Appendix A2—Proof of Lemma 12.1

Let ${\pmb {\theta }}=({\pmb {\beta }}^\prime ,\mathbf{u}^\prime )^\prime$ and ${\pmb {\theta }}$ can be written as the function of OLS estimator as

$$ \begin{aligned} (\widehat{{\pmb {\beta }}}^\prime , \widehat{\mathbf{u}}^\prime )^\prime&=\left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}\mathbf{R}^\prime \mathbf{T} \\&=\left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \mathbf{R}^\prime \mathbf{R}\left( \mathbf{R}^\prime \mathbf{R}\right) ^{-\mathbf {1}} \mathbf{R}^\prime \mathbf{T} \\&=\left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \mathbf{R}^\prime \mathbf{R} {\pmb {\theta }}. \end{aligned} $$

Thus,

$$ \begin{aligned} \mathrm{Cov}(\widehat{{\pmb {\beta }}}, \widehat{\mathbf{u}})&=\mathrm{Cov}({\pmb {\theta }})=\left[ \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}\mathbf{R}^\prime \mathbf{R}\right] \mathrm{Cov}({\pmb {\theta }})\left[ \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}\mathbf{R}^\prime \mathbf{R}\right] ^\prime \\&=\left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}\left( \mathbf{R}^\prime \mathbf{R}\right) \mathrm{Cov}({\pmb {\theta }})\left( \mathbf{R}^\prime \mathbf{R}\right) \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \\&=\left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}\left( \mathbf{R}^\prime \mathbf{R}\right) \frac{\sigma ^2}{n} \left( \mathbf{R}^\prime \mathbf{R}\right) ^{-1}\left( \mathbf{R}^\prime \mathbf{R}\right) \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \\&=\frac{\sigma ^2}{n} \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1} \left( \mathbf{R}^\prime \mathbf{R} \right) \left( \mathbf{R}^\prime \mathbf{R}+\lambda \mathbf{D}\right) ^{-1}, \end{aligned} $$

where $\sigma ^2/n$ comes from the variance of $\widehat{\mathbf{u}}$, which is the property of the penalized spline method (see Claeskens et al. [6]). Thus,  (12.21) would be obtained. It can be also obtained if $\mathbf{R}^\prime \mathbf{R}$ is replaced by $\mathbf{R}^\prime \mathbf{W}\mathbf{R}$ for KMW method.