Kernel estimation in semiparametric mixed effect longitudinal modeling

Abstract

HIV damages the immune system by targeting the CD4 cells. Hence CD4 count data modeling is important in the analysis of HIV infection. This paper considers a semiparametric mixed effects model for the analysis of CD4 longitudinal data. The model is a natural extension to the linear mixed and semiparametric models that uses parametric linear model to present the covariate effects and an arbitrary nonparametric smooth function to model the time effect and account for the within subject correlation using random effects. We approximate the nonparametric function by the profile kernel method, and make use of the weighted least squares to estimate the regression coefficients. Under some regularity conditions, the asymptotic normality of the resulting estimator is established and the performance is compared with the backfitting method. Although, two estimators share the same asymptotic variance matrix for independent data, it is shown that, backfitting often produces larger bias and variance than the profile-kernel method, asymptotically. Consequently, the use of backfitting method is no longer advised in semiparametric mixed effect longitudinal model. For practical implementation and also improve efficiency, the estimation of the covariance function is accomplished using an iterative algorithm. Performance of the proposed methods are compared through a simulation study and the analysis of CD4 data.

Appendix

In this section, we provide the proof of the main results along with a brief introduction of the EM method, which is used for the convergency of Algorithm 1. In the following proofs, recall that ${\varvec{t}}$, $\varvec{X}$ and ${\varvec{Y}}$ denote the observations over all the subjects; that is ${\varvec{t}}=({\varvec{t}}_1^\top , \ldots , {\varvec{t}}_n^\top )^\top$, ${\varvec{t}}_i=(t_{i1}, \ldots , t_{in_i})$ and similarly for $\varvec{X}$ and ${\varvec{Y}}$. Also, ${\varvec{V}}$ and ${\varvec{V}}_{o}$ stand for the $N \times N$ assumed and true covariance matrices for all data, respectively.

1.1 Proof of Theorem 1

For the kernel estimator, based on expression (4.1),

$$\begin{aligned} \sqrt{n}\Big ({\widehat{{\varvec{\beta }}}}_{K}-{\varvec{\beta }}\Big ) =D_n^{-1}\lbrace \sqrt{n}C_n\rbrace +o_p(1), \end{aligned}$$

where

$$\begin{aligned} D_n =\frac{1}{n}\sum _{i=1}^n {\tilde{\varvec{X}}}_i^\top {\varvec{V}}_i^{-1} {\tilde{\varvec{X}}}_i \end{aligned}$$

and

$$\begin{aligned} C_n=\frac{1}{n}\sum _{i=1}^n {\tilde{\varvec{X}}}_i^\top {\varvec{V}}_i^{-1} \big [{\varvec{Y}}_i - \big (\varvec{X}^\top _i {\varvec{\beta }}+ {\widehat{{\varvec{g}}}}({\varvec{t}}_i)\big )\big ] . \end{aligned}$$

Denote,

$$\begin{aligned} D=\underset{n\longrightarrow \infty }{\lim }D_n=E\big \lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1}{\tilde{\varvec{X}}}\big \rbrace . \end{aligned}$$

Simple calculations show that $C_n$, can be expanded as $C_n = C_{1n}- C_{2n}, + o_p(1)$, where, denoting ${\varvec{\mu }}_i=\varvec{X}^\top _i {\varvec{\beta }}+ {\varvec{g}}({\varvec{t}}_i)$ and ${\varvec{W}}_{1i}={\varvec{V}}_i^{-1}{\tilde{\varvec{X}}}_i$,

$$\begin{aligned} C_{1n}=\frac{1}{n}\sum _{i=1}^n {\tilde{\varvec{X}}}_i^\top {\varvec{V}}_i^{-1} \big [{\varvec{Y}}_i - {\varvec{\mu }}_i\big ]=\frac{1}{n}\sum _{i=1}^n {\varvec{W}}_{1i}^{\top } \big [{\varvec{Y}}_i - {\varvec{\mu }}_i\big ] \end{aligned}$$

and

$$\begin{aligned} C_{2n}=\frac{1}{n}\sum _{i=1}^n {\tilde{\varvec{X}}}_i^\top {\varvec{V}}_i^{-1}\big [{\widehat{{\varvec{g}}}}({\varvec{t}}_i,{\varvec{\beta }}) -{\varvec{g}}({\varvec{t}}_i)]. \end{aligned}$$

Obtaining asymptotic distribution of $\sqrt{n}C_{1n}$ is simple. Now examine the distribution of $\sqrt{n}C_{2n}$. Following Lin and Carroll (2001), we have

$$\begin{aligned} C_{2n}=\frac{1}{n}\sum _{i=1}^n {\varvec{W}}_{2i}^{\top } \big [{\varvec{Y}}_i - {\varvec{\mu }}_i\big ]+\frac{h^2}{2}E\big \lbrace \varvec{X}^\top {\varvec{V}}^{-1}{\varvec{g}}^2({\varvec{t}}) \big \rbrace +o_p(1), \end{aligned}$$

where ${\varvec{W}}_{2i}=\lbrace {\varvec{W}}_{2i1}, \ldots ,{\varvec{W}}_{2im}\rbrace ^\top$ and

$$\begin{aligned} {\varvec{W}}_{2ij}=\dfrac{\lbrace \sum _{k=1}^{m}\sum _{l=1}^{m}E({\tilde{\varvec{X}}}_k{\varvec{V}}^{kl}|{\varvec{t}}_l=t_{ij})\rbrace f_j(t_{ij}))}{\sum _{l=1}^{m}f_l(t_{ij}))}. \end{aligned}$$

It follows that

$$\begin{aligned} \sqrt{n}\Big ({\widehat{{\varvec{\beta }}}}_{K}-{\varvec{\beta }}\Big ) =D^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n}({\varvec{W}}_{1i}-{\varvec{W}}_{2i})({\varvec{Y}}_i-{\varvec{\mu }}_i)+\sqrt{nh^4}{\varvec{b}}_K({\varvec{\beta }}, {\varvec{g}})/2+o_p(1), \end{aligned}$$

where the bias term ${\varvec{b}}_K({\varvec{\beta }}, {\varvec{g}})=D^{-1}E\lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1}{\varvec{g}}^{(2)}({\varvec{t}}) \rbrace$. Equivalently,

$$\begin{aligned} \sqrt{n}\lbrace {\widehat{{\varvec{\beta }}}}_K-{\varvec{\beta }}+h^2{\varvec{b}}_K({\varvec{\beta }}, {\varvec{g}})/2\rbrace \overset{D}{ \rightarrow } N_p(0, {\varvec{V}}_K) \end{aligned}$$

, where ${\varvec{V}}_K=D^{-1} E\big \lbrace ({\varvec{W}}_1-{\varvec{W}}_2)^\top {\varvec{V}}_o({\varvec{W}}_1-{\varvec{W}}_2)\big \rbrace D^{-1}$ with ${\varvec{V}}_o=var({\varvec{Y}}|\varvec{X},{\varvec{t}})$.

1.2 Proof of Theorem 3

The asymptotic variance ${\varvec{V}}_{BF}$ has its central component generated from $\varvec{X}^\top {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^*$, where ${\varvec{\varepsilon }}^*={\varvec{Z}}^\top {\varvec{b}}+{\varvec{\varepsilon }}$. Similarly, the central component in the asymptotic variance ${\varvec{V}}_K$ is from $\varvec{X}^\top ({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^*$, which is ${\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^*$ asymptotically. To compare ${\varvec{V}}_K$ and ${\varvec{V}}_{BF}$, it is thus sufficient to compare the variances of the two central terms.

We now show that, under the condition that ${\varvec{V}}$ and ${\varvec{V}}_o$ depend only on ${\varvec{t}}$,

$$\begin{aligned} \hbox {cov} \lbrace \varvec{X}^\top {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace \ge \hbox {cov} \lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace . \end{aligned}$$

For the backfitting estimator,

$$\begin{aligned} \hbox {cov} \lbrace \varvec{X}^\top {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace&=\hbox {E}\Big \lbrace \varvec{X}^\top {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} \varvec{X}| {\varvec{t}}\Big \rbrace \\&= \mu _{\varvec{X}}^\top ({\varvec{t}}) {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1}\mu _{\varvec{X}}({\varvec{t}}) \\&\quad +\hbox {tr} \big [{\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} \hbox {cov} (\varvec{X}|{\varvec{t}})\big ]. \end{aligned}$$

(A1)

In this expression, ${\varvec{\mu }}_X({\varvec{t}})$ is generally nonzero and the first term is positive semidefinite because ${\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1}$ is positive semidefinite. Also,

$$\begin{aligned} \hbox {cov} \lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace&=\hbox {E}\Big \lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o ({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} {\tilde{\varvec{X}}}| {\varvec{t}}\Big \rbrace \\&=\hbox {E}({\tilde{\varvec{X}}}|{\varvec{t}})^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} \hbox {E}({\tilde{\varvec{X}}}|{\varvec{t}}) \\&\quad + \hbox {tr} \big [{\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{V}}_o({\varvec{I}}-{\varvec{S}})^\top {\varvec{V}}^{-1} \hbox {cov} ({\tilde{\varvec{X}}}|{\varvec{t}})\big ]. \end{aligned}$$

(A2)

Note that

$$\begin{aligned} \hbox {E}({\tilde{\varvec{X}}}_i|{\varvec{t}}_i){=}\hbox {E}\Big \lbrace \varvec{X}_i{-}{\varvec{\mu }}_X(t_i)|{\varvec{t}}_i\Big \rbrace {=}0, \qquad \hbox {cov}({\tilde{\varvec{X}}}_i|{\varvec{t}}_i){=}\hbox {cov} \Big \lbrace \varvec{X}_i{-}{\varvec{\mu }}_X({\varvec{t}}_i)|{\varvec{t}}_i\Big \rbrace {=}\hbox {cov} (\varvec{X}_i|{\varvec{t}}_i) \end{aligned}$$

Therefore, the first term in (A2) is 0, and the second terms in (A1) and (A2) are identical. It follows that $\hbox {cov} \lbrace \varvec{X}^\top {\varvec{V}}^{-1}({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace \ge \hbox {cov} \lbrace {\tilde{\varvec{X}}}^\top {\varvec{V}}^{-1} ({\varvec{I}}-{\varvec{S}}){\varvec{\varepsilon }}^* \rbrace$, and consequently ${\varvec{V}}_{BF}\ge {\varvec{V}}_K$.

1.3 EM algorithm-related to Algorithm 1

The EM algorithm, which is composed of the expectation (E) step and the maximization (M) step at each iteration, has several appealing features such as simplicity of implementation and monotone increase of the likelihood at each iteration. Toward this end, we let ${\varvec{\Theta }}=({\varvec{\beta }}, {\varvec{D}}, {\varvec{\Sigma }}, \phi )$ be the set of entire model parameters. Treating the unobservable random effects ${\varvec{b}}_i$ as the missing data, we have the complete data $\lbrace ({\varvec{y}}_i, {\varvec{b}}_i), i = 1, \ldots ,n\rbrace$. Let the density function of $({\varvec{y}}_i, {\varvec{b}}_i)$ be $f({\varvec{y}}_i, {\varvec{b}}_i|{\varvec{\Theta }})$ with parameters ${\varvec{\Theta }}\in \Omega$ and let the density function of ${\varvec{y}}_i$ given by

$$\begin{aligned} f({\varvec{y}}_i|{\varvec{\Theta }})=\prod _{i=1}^{n}\int f({\varvec{y}}_i,{\varvec{b}}_i|{\varvec{\Theta }})d{\varvec{b}}_i. \end{aligned}$$

(7.1)

The parameters ${\varvec{\Theta }}$ are to be estimated by maximizing $f({\varvec{y}}_i|{\varvec{\Theta }})$ over ${\varvec{\Theta }}\in \Omega$. In many statistical problems, maximization of the complete-data specification $f({\varvec{y}}_i,{\varvec{b}}_i|{\varvec{\Theta }})$ is simpler than that of the incomplete-data specification $f({\varvec{y}}_i|{\varvec{\Theta }})$. A main feature of the EM algorithm is maximization of $f({\varvec{y}}_i,{\varvec{b}}_i|{\varvec{\Theta }})$ over ${\varvec{\Theta }}\in \Omega$. Then the log-likelihood function is

$$\begin{aligned} L({\varvec{\Theta }})=\log f({\varvec{y}}_i|{\varvec{\Theta }})= Q({\varvec{\Theta }}^{'}|{\varvec{\Theta }})-H({\varvec{\Theta }}^{'}|{\varvec{\Theta }}), \end{aligned}$$

(7.2)

where $Q({\varvec{\Theta }}^{'}|{\varvec{\Theta }})=\hbox {E} \{\log f({\varvec{y}}_i|{\varvec{b}}_i,{\varvec{\Theta }})\}$ and $H({\varvec{\Theta }}^{'}|{\varvec{\Theta }})= \hbox {E}\{\log f({\varvec{b}}_i|{\varvec{\Theta }})\}$ are assumed to exist for all pairs $({\varvec{\Theta }}^{'}, {\varvec{\Theta }})$. We now define the EM iteration ${\varvec{\Theta }}_{r}\rightarrow {\varvec{\Theta }}_{r+1} \in M({\varvec{\Theta }}_{r})$ as follows:

E-step:

Determine $Q({\varvec{\Theta }}|{\varvec{\Theta }}_{r})$.

M-step:

Choose ${\varvec{\Theta }}_{r+1}$ to be any value of ${\varvec{\Theta }}\in \Omega$ which maximizes $Q({\varvec{\Theta }}|{\varvec{\Theta }}_{r})$.

Note that M is a point-to-set map, i.e., $M({\varvec{\Theta }}_{r})$ is the set of ${\varvec{\Theta }}$ values which maximizes $Q({\varvec{\Theta }}|{\varvec{\Theta }}_{r})$. Many applications of EM are for the curved exponential family, for which the E-step and M-step take special forms. Sometimes it may not be numerically feasible to perform the M-step. So let defined a generalized EM algorithm (a GEM algorithm) to be an iterative scheme ${\varvec{\Theta }}_{r}\rightarrow {\varvec{\Theta }}_{r+1} \in M({\varvec{\Theta }}_{r})$, where ${\varvec{\Theta }}\rightarrow M({\varvec{\Theta }})$ is a point-to-setm ap, such that

$$\begin{aligned} Q({\varvec{\Theta }}^{'}|{\varvec{\Theta }})\ge Q({\varvec{\Theta }}|{\varvec{\Theta }}). \end{aligned}$$

(7.3)

EM is a special case of GEM. For any instance $\{{\varvec{\Theta }}_{r}\}$ of a GEM algorithm,

$$\begin{aligned} L({\varvec{\Theta }}_{r+1})\ge L({\varvec{\Theta }}_{r}), \end{aligned}$$

(7.4)

and

$$\begin{aligned} H({\varvec{\Theta }}|{\varvec{\Theta }})\ge H({\varvec{\Theta }}^{'}|{\varvec{\Theta }}). \end{aligned}$$

(7.5)

See Wu (1983) for more details.