Analysis of generalized semiparametric mixed varying-coefficients models for longitudinal data

Abstract

The generalized semiparametric mixed varying-coefficient effects model for longitudinal data can accommodate a variety of link functions and flexibly model different types of covariate effects, including time-constant, time-varying, and covariate-varying effects. The time-varying effects are unspecified functions of time and the covariate-varying effects are nonparametric functions of a possibly time-dependent exposure variable. A semiparametric estimation procedure is developed that uses local linear smoothing and profile weighted least squares, which requires smoothing in the two different and yet connected domains of time and the time-dependent exposure variable. The asymptotic properties of the estimators of both nonparametric and parametric effects are investigated. In addition, hypothesis testing procedures are developed to examine the covariate effects. The finite-sample properties of the proposed estimators and testing procedures are examined through simulations, indicating satisfactory performances. The proposed methods are applied to analyze the ACTG 244 clinical trial to investigate the effects of antiretroviral treatment switching in HIV-infected patients before and after developing the T215Y antiretroviral drug resistance mutation.

1. INTRODUCTION

Varying-coefficient models for cross-sectional data have been extensively studied (e.g., Hastie & Tibshirani, 1993; Yin et al., 2008). Research on varying-coefficient models for longitudinal data has mostly focused on time-varying effect models; see Martinussen & Scheike (2001), Lin & Ying (2001), Wu & Liang (2004), Fan & Li (2004), Hu et al. (2004), Qu & Li (2006), Fan et al. (2007), & Sun et al. (2013), among others. Semiparametric time-varying coefficient models specify some covariate effects to be time-constant and allow others to vary with time. Although time-varying coefficient models have attractive properties, in many applications certain covariate effects (e.g., treatment effects) may vary with an exposure variable.

The AIDS Clinical Trial Group (ACTG) 244 trial illustrates an application of the semiparametric covariate-varying coefficient model. ACTG 244 adaptively randomized HIV-infected patients to different antiretroviral regimens based on the occurrence of a drug resistance mutation, and the antiretroviral treatment effects on longitudinal markers of HIV disease progression may vary with time since treatment randomization/switching.

We now use this example to illustrate the newly proposed methods. ACTG 244 enrolled HIV-infected patients taking zidovudine (ZDV) monotherapy and monitored their HIV in plasma every eight weeks for presence of the T215Y/F ZDV resistance mutation (Principi et al., 1994; Japour et al., 1995). Upon detection of the 215 mutation in a participant’s plasma, she or he was randomized to either continue ZDV or to switch to ZDV plus didanosine (ddI) or ZDV plus ddI plus nevirapine (NVP). The primary objective of ACTG 244 was to assess how an adaptive treatment randomization affected longitudinal measurements of CD4 cell counts and plasma HIV viral loads (Qi et al., 2017). The newly developed methods can be applied to examine the possible time-varying effects of adaptive treatment randomization or switching (Chen, 2013). As such, the treatment randomization/switching constitutes a covariate-varying covariate effect with the exposure modifying variable the time since treatment randomization/switching. The methods can also be applied to study the nonlinear interactions between dose intensity and other covariates (Yin et al., 2008). In addition, the proposed methods can be applied generally to clinical settings where patients switched therapies based on the monitoring of a biomarker (Gilks et al., 2006; Phillips et al., 2008).

For these applications, we investigate the generalized semiparametric model with mixed covariate-varying effects for longitudinal data with a general link function. The ability to select from different link functions provides a rich family of models for longitudinal data, including categorical response data where little work has been done. Furthermore, the proposed methods do not assume any specific model for the sampling times, which accommodates the among-participant heterogeneity of study visits without risking assumptions that may be false.

A major contribution here is the nonparametric modeling of the possibly time-dependent covariate-varying effects. Qi et al. (2017) recently studied these longitudinal models with covariate-varying covariate effects. However, these models specified a parametric form for the covariate-varying effects, therefore at risk of model misspecifications. The new approach lets data speak and provides greater flexibility and robustness in modeling both time-dependent and covariate-varying effects. This new approach requires a substantially different and more challenging theoretical development. Another major contribution is the development of the hypothesis testing procedures to examine the covariate-varying effects and their monotonicity, which, to the best of our knowledge, have not been studied before.

The remainder of the article is organized as follows. We introduce the generalized semiparametric varying-coefficient regression model in Section 2. The estimation methods with the computational algorithm and bandwidth selection are developed in Section 3. We establish the asymptotic results for both nonparametric and parametric estimation methods in Section 4. In Section 5, we propose two hypothesis testing procedures for the presence of time- and exposure-varying effects. The finite-sample performance of the proposed estimators is examined via simulations in Section 6. The proposed methods are applied to the ACTG 244 data in Section 7, and concluding remarks are given in Section 8.

2. STATISTICAL MODELS

Consider a study that enrolls a random sample of n subjects with the follow-up through time τ. For subject i, suppose observations of the response process Y_i(t) are sampled at time points $0\le T_{i1}<T_{i2}\cdots <T_{i{n}_i}\le \tau$, where n_i is the number of observations from subject i, and i = 1, · · ·, n. The sampling times are often irregular and depend on covariates. In addition, some subjects may drop out of the study early. Let $N_i\left(t\right)={\sum }_{j=1}^{n_i}I\left(T_{ij}\le t\right)$ be the number of observations taken on the ith subject by time t, where I(·) is the indicator function. Let C_i be the end of follow-up time or censoring time, whichever occurs first. The responses for subject i can only be observed at the time points before C_i. Thus N_i(t) can be written as $N_i^{\ast }\left(t\wedge C_i\right)$, where $N_i^{\ast }\left(t\right)$ is the counting process of sampling times. Let X_i(t) and U_i(t) be the possibly time-dependent covariates for the ith subject. Suppose U_i(t) has support $\mathcal{U}$ and that {Y_i(·), X_i(·), U_i(·), N_i(·); i = 1, · · ·, n} are independent identically distributed (iid) random processes. We assume noninformative censoring time C_i in that $E\left\{d{N}_i^{\ast }\left(t\right)|X_i\left(t\right),U_i\left(t\right),C_i\ge t\right\}=E\left\{d{N}_i^{\ast }\left(t\right)|X_i\left(t\right),U_i\left(t\right)\right\}$ and $E\left\{Y_i\left(t\right)|X_i\left(t\right),U_i\left(t\right),C_i\ge t\right\}=E\left\{Y_i\left(t\right)|X_i\left(t\right),U_i\left(t\right)\right\}$. Assume that $d{N}_i^{\ast }\left(t\right)$ is independent of Y_i(t), conditional on X_i(t), U_i(t), and C_i ≥ t. The censoring time C_i is allowed to depend on X_i(·) and U_i(·).

Suppose that $X_i\left(t\right)={\left(X_{1i}^T\left(t\right),X_{2i}^T\left(t\right),X_{3i}^T\left(t\right)\right)}^T$ consists of three parts, each of dimension p₁, p₂, and p₃, respectively, over the time interval [0, τ ]. Let U_i(t) be a covariate process that has the potential to modify the effects of X_3i(t). We propose the generalized semiparametric regression model with varying coefficients:

$${\mu }_i\left(t\right)=E\left\{Y_i\left(t\right)|X_i\left(t\right),U\left(t\right)\right\}=g^{-1}\left\{{\alpha }^T\left(t\right)X_{1i}\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)+{\gamma }^T\left(U_i\left(t\right)\right)X_{3i}\left(t\right)\right\},$$ (1)

for 0 ≤ t ≤ τ, where g(·) is a known link function, α(·) is a p₁-dimensional vector of unspecified functions, β is a p₂-dimensional vector of unknown parameters, and γ(U_i(t)) is a p₃-dimensional vector of functions. The superscript T represents the transpose of a vector or matrix. The first component of X(t) is set to be 1, which codes a nonparametric baseline function. The function γ(u) represents the effect of X_3i(t) at level u of the covariate U_i(t). Setting the first component of X_3i(t) to 1 also allows nonparametric modeling of the covariate U_i(t).

The model has wide applications, for example, the adaptive treatment randomization problem studied by Qi et al. (2017), the treatment switching problem studied by Chen et al. (2013), and the nonlinear interactions between dose intensity and other covariates studied by Yin et al. (2008). Our previous work studied the version of model (1) that specified the right-most term parametrically, ${\gamma }^T\left(U_i\left(t\right);\theta \right)X_{3i}\left(t\right)$, where $\gamma \left(u,\theta \right)$ is a p₃-dimensional vector of possibly nonlinear parametric functions defined on the range $\mathcal{U}$ of U_i(·). The theoretical development for estimating the nonparametric component γ(u) is significantly challenging, because the nonparametric functions α(t) and γ(u) have different domains, yet the smoothing for α(t) and γ(u) cannot be totally separated, because γ(u) is a function of the time-varying covariate process U_i(t).

3. ESTIMATION PROCEDURES

Assume that α(t) and γ(u) are smooth so that their first and second derivatives $\dot{\alpha }\left(t\right)$, $\dot{\gamma }\left(u\right)$, $\ddot{\alpha }\left(t\right)$, and $\ddot{\gamma }\left(u\right)$ exist. Consider the local linear approximation for α(t) in a neighbourhood ${\mathcal{N}}_{u_0}$ of t₀, $\alpha \left(t\right)=\alpha \left(t_0\right)+\dot{\alpha }\left(t_0\right)\left(t-t_0\right)+O({\left(t-t_0\right)}^2)$, and the local linear approximation for γ(u) in a neighbourhood ${\mathcal{N}}_{u_0}$ of u₀,$\gamma \left(u\right)=\gamma \left(u_0\right)+\dot{\gamma }\left(u_0\right)\left(u-u_0\right)+O\left({\left(u-u_0\right)}^2\right)$. Then for $t\in {\mathcal{N}}_{t_0}$ and $U_i\left(t\right)\in {\mathcal{N}}_{u_0}$, model (1) can be approximated by

$$\tilde{\mu }\left(t,t_0,u_0,{\vartheta }^{\ast }\left(t_0,u_0\right)|X_i,U_i\right)=\phi \{{\vartheta }^{\ast T}\left(t_0,u_0\right){\tilde{X}}_i^{\ast }\left(t,t_0,u_0\right)+\beta X_{2i}\left(t\right)\},$$ (2)

where ${\vartheta }^{\ast }\left(t_0,u_0\right)={\left({\alpha }^T\left(t_0\right),{\gamma }^T\left(u_0\right),{\dot{\alpha }}^T\left(t_0\right),{\dot{\gamma }}^T\left(u_0\right)\right)}^T$, ${\tilde{X}}_i^{\ast }\left(t,t_0,u_0\right)={\left(X_{1i}^T\left(t\right),X_{3i}^T\left(t\right),X_{1i}^T\left(t\right)\left(t-t_0\right),X_{3i}^T\left(t\right)\left(U_i\left(t\right)-u_0\right)\right)}^T$. Here and thereafter, $\phi (\cdot )=g^{-1}(\cdot )$ is the inverse function of g(·). By the approximation (2), at each t₀ and u₀, and for fixed β, we consider the estimating function:

$$U_{\vartheta }\left({\vartheta }^{\ast };t_0,u_0,\beta \right)=\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right)\left[Y_i\left(t\right)-\tilde{\mu }\left(t,t_0,u_0,{\vartheta }^{\ast }\left(t_0,u_0\right)|X_i,U_i\right)\right]{\tilde{X}}_i^{\ast }\left(t,t_0,u_0\right)K_{h,b}\left(t,U_i\left(t\right);t_0,u_0\right)d{N}_i\left(t\right),$$ (3)

where $K_{h,b}\left(t,U_i\left(t\right);t_0,u_0\right)=K_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)$, $K_h(\cdot )=K_1\left(\cdot /h\right)/h$, $K_b(\cdot )=K_2\left(\cdot /b\right)/b$, K₁(·) and K₂(·) are kernel functions, and h = h_n and b = b_n are bandwidth parameters. Here W_i(t) = W (t, X_i(t), U_i(t)) is a nonnegative weight process associated with different time periods, which may potentially be used to improve estimation efficiency.

Let ${\tilde{\vartheta }}^{\ast }\left(t_0,u_0,\beta \right)$ be the solution to $U_{\vartheta }\left({\vartheta }^{\ast };t_0,u_0,\beta \right)$ for each t₀ and u₀. The first p₁ + p₃ components of ${\tilde{\vartheta }}^{\ast }\left(t_0,u_0,\beta \right)$ are denoted by $\tilde{\vartheta }\left(t_0,u_0,\beta \right)$. Let ${\tilde{X}}_i\left(t\right)={\left({\left(X_{1i}\left(t\right)\right)}^T,{\left(X_{3i}\left(t\right)\right)}^T\right)}^T$. The profile least-squares estimator $\widehat{\beta }$ is the root to the estimating function:

$$U_{\beta }\left(\beta \right)=\sum _{i=1}^n{\int }_{t_1}^{t_2}{W}_i\left(t\right)\left[Y_i\left(t\right)-\phi \left\{\tilde{\vartheta }\left(t,U_i\left(t\right),\beta \right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\right\}\right]\{{(\frac{\partial \tilde{\vartheta }\left(t,U_i\left(t\right),\beta \right)}{\partial \beta })}^T{\tilde{X}}_i\left(t\right)+X_{2i}\left(t\right)\}d{N}_i\left(t\right),$$ (4)

where $\partial \tilde{\vartheta }\left(t,U_i\left(t\right),\beta \right)/\partial \beta$ is the first p₁ + p₃ rows of $\partial {\tilde{\vartheta }}^{\ast }\left(t,U_i\left(t\right),\beta \right)/\partial \beta$, whose expression can be derived based on the identity $U_{\vartheta }\left({\tilde{\vartheta }}^{\ast }\left(t,U_i\left(t\right),\beta \right);t,U_i\left(t\right),\beta \right)\equiv 0$ as follows:

$$\frac{\partial {\tilde{\vartheta }}^{\ast }\left(t,U_i\left(t\right),\beta \right)}{\partial \beta }=-{\left\{\frac{\partial U_{\vartheta }\left({\vartheta }^{\ast };t,U_i\left(t\right),\beta \right)}{\partial {\vartheta }^{\ast }}\right\}}^{-1}\frac{\partial U_{\vartheta }\left({\vartheta }^{\ast };t,U_i\left(t\right),\beta \right)}{\partial \beta }{|}_{\vartheta \ast =\tilde{\vartheta }\ast \left(y,Ui\left(t\right),\beta \right)}.$$

Here we take [t₁, t₂] as a subinterval of (0, τ ) to avoid the boundary problems.

Plugging in $\widehat{\beta }$, we obtain the estimator $\widehat{\vartheta }\left(t_0,u_0\right)=\tilde{\vartheta }(t_0,u_0,\widehat{\beta })$ for the nonparametric components $\vartheta \left(t_0,u_0\right)={\left({\alpha }^T\left(t_0\right),{\gamma }^T\left(u_0\right)\right)}^T$. Let $\widehat{\alpha }\left(t_0,u_0\right)$ include the first p₁ elements of $\widehat{\vartheta }\left(t_0,u_0\right)$ and let $\widehat{\gamma }\left(t_0,u_0\right)$ be the vector including the elements of $\widehat{\vartheta }\left(t_0,u_0\right)$ from p₁ + 1 to p₁ + p₃. Thus,$\widehat{\vartheta }\left(t_0,u_0\right)={({\widehat{\alpha }}^T\left(t_0,u_0\right),{\widehat{\gamma }}^T\left(t_0,u_0\right))}^T$. Estimation of α(t₀) by $\widehat{\alpha }(t_0,u_0)$ is inefficient because it only utilizes the local observations with $U_i\left(t\right)\in {\mathcal{N}}_{u_0}$ for $t\in {\mathcal{N}}_{t_0}$. A more efficient estimator for α(t₀) at t₀ can be obtained through aggregation without restricting $U_i\left(t\right)\in {\mathcal{N}}_{u_0}$. Similarly, a more efficient estimator of γ(u₀) can be obtained through aggregating over t such that U_i(t) = u₀. We propose the following estimators $\widehat{\alpha }\left(t_0\right)$ and $\widehat{\gamma }\left(u_0\right)$ for α(t₀) and γ(u₀), respectively:

$$\widehat{\alpha }\left(t_0\right)=n^{-1}\sum _{j=1}^n\widehat{\alpha }\left(t_0,U_j\left(t_0\right)\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}\widehat{\gamma }\left(u_0\right)=n_{u_0}^{-1}\sum _{t_{u_0}\in {\mathcal{V}}_{u_0}}\widehat{\gamma }\left(t_{u_0},u_0\right),$$ (5)

Where $t_{u_0,j}\in U_j^{-1}\left(u_0\right)=\left\{t:U_j\left(t\right)=u_0\right\}$, and $n_{u_0}$ is the number of points in the union ${\mathcal{V}}_{u_0}={\cup }_{j=1}^n\{U_j^{-1}\left(u_0\right)\}$. For the motivating example, U_j(t) = t − S_j; thus,$t_{u_0,j}=u_0+S_j$. If it is difficult to find $U_j^{-1}\left(u_0\right),\gamma \left(u_0\right)$ can also be estimated by $\widehat{\gamma }\left(u_0\right)={\int }_0^{\tau }\widehat{\gamma }\left(t,u_0\right)dt$.

Computational algorithm

The estimators $\widehat{\alpha }\left(t_0\right)$, $\widehat{\gamma }\left(u_0\right)$, and $\widehat{\beta }$ can be calculated through the iterative algorithm:

1. Set initial values ${\widehat{\vartheta }}^{\left\{0\right\}}\left(t,u\right)$ and ${\widehat{\beta }}^{\left\{0\right\}}$;

2. For each jump point of {N_i(·), i = 1, · · ·, n}, say t, and u = U_i(t), the mth step estimate ${\widehat{\vartheta }}^{\ast \left\{m\right\}}\left(t,u\right)={\widehat{\vartheta }}^{\ast }\left(t,u,{\widehat{\beta }}^{\left\{m-1\right\}}\right)$ is the root of the estimating function (3), satisfying $U_{\vartheta }\left({\widehat{\vartheta }}^{\ast \left\{m\right\}}\left(t,u\right),t,u,{\widehat{\beta }}^{\left\{m-1\right\}}\right)=0$, where ${\widehat{\beta }}^{\left\{m-1\right\}}$ is the estimate of β at the (m − 1)th step;

3. The mth step estimate ${\widehat{\beta }}^{\left\{m\right\}}$ is the solution of (4), obtained after replacing $\tilde{\vartheta }\left(t,U_i\left(t\right),\beta \right)$ with ${\widehat{\vartheta }}^{\left\{m\right\}}\left(t,U_i\left(t\right)\right)$, which is the first p₁ + p₃ components of ${\widehat{\vartheta }}^{\ast }{}^{\left\{m\right\}}\left(t,U_i\left(t\right)\right)$;

4. Repeating Steps 2 and 3, ${\widehat{\vartheta }}^{\left\{m\right\}}\left(t,u\right)$ and ${\widehat{\beta }}^{\left\{m\right\}}$ are updated at each iteration until convergence. The estimate $\widehat{\beta }$ is ${\widehat{\beta }}^{\left\{m\right\}}$ at convergence, and $\widehat{\vartheta }\left(t,u\right)={\left({\widehat{\alpha }}^T\left(t,u\right),{\widehat{\gamma }}^T\left(t,u\right)\right)}^T$ is ${\widehat{\vartheta }}^{\left\{m\right\}}\left(t,u\right)$ at convergence.

5. The aggregated estimates $\widehat{\alpha }\left(t_0\right)$ and $\widehat{\gamma }\left(u_0\right)$ are obtained using (5).

Bandwidth Selection

Instead of selecting bandwidths for estimators by the leave-one-out cross-validation method suggested by Rice & Silverman (1991), we choose suitable bandwidths via the K-fold cross-validation procedure (Tian et al., 2005). Supposing that subjects are randomly divided into K groups, (D₁, D₂, · · ·, D_K), the K-fold cross-validation optimal bandwidth is

$$\left(h_{opt,K}{b}_{opt,K}\right)=\arg \underset{h,b}{\min }\sum _{k=1}^{K}P{E}_k\left(h,b\right),$$ (6)

where the kth prediction error is given by

$$\begin{gathered} P{E}_k\left(h,b\right)=\sum _{i\in D_k}\{{\int }_{t_1}^{t_2}{W}_i\left(t\right)[Y_i\left(t\right)-g^{-1}\{{\widehat{\alpha }}_{\left(-k\right)}^T\left(t\right)X_{1i}\left(t\right)+{\widehat{\beta }}_{\left(-k\right)}^T{X}_{2i}\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{+{\widehat{\gamma }}_{\left(-k\right)}^T\left(U_i\left(t\right)\right)X_{3i}\left(t\right)\}]d{N}_i\left(t\right)\}}^2, \end{gathered}$$

or k = 1, · · ·, K, where ${\widehat{\alpha }}_{\left(-k\right)}\left(t\right)$, ${\widehat{\beta }}_{\left(-k\right)}$, and ${\widehat{\gamma }}_{\left(-k\right)}\left(u\right)$ are estimated using the data excluding subjects in D_k. This procedure can be repeated a few times, say 10, and take the average of the resulting optimal bandwidths to increase the stability.

4. ASYMPTOTIC PROPERTIES

Let α₀(t), β₀, and γ₀(u) be the true value of α(t), β, and γ(u) under model (1), respectively. Let ${\mu }_i\left(t\right)=\phi \left\{{\alpha }_0\left(t\right)X_{1i}\left(t\right)+{\beta }_0{X}_{2i}\left(t\right)+{\gamma }_0\left(U_i\left(t\right)\right)X_{3i}\left(t\right)\right\}$ and ${\dot{\mu }}_i\left(t\right)=\dot{\phi }\left\{{\alpha }_0\left(t\right)X_{1i}\left(t\right)+{\beta }_0{X}_{2i}\left(t\right)+{\gamma }_0\left(U_i\left(t\right)\right)X_{3i}\left(t\right)\right\}$. Let ${\lambda }_i\left(t\right)dt=E\left\{d{N}_i^{\ast }\left(t\right)|X_i\left(t\right),U_i\left(t\right)\right\}dt$ be the conditional mean rate of the sampling times. Let ${\xi }_i\left(t\right)=I\left(C_i\ge t\right)$ and $w_i\left(t\right)=w\left(t,X_i\left(t\right),U_i\left(t\right)\right)$, where w(t, x, u) is the deterministic limit of W (t, x, u) in probability as n → ∞. For each t, f_U (t, u) is the density function of U_i(t) at u. Define $e_{11}\left(t,u\right)=E\left[w_i\left(t\right){\dot{\mu }}_i\left(t\right){\left\{{\tilde{X}}_i\left(t\right)\right\}}^{\otimes 2}{\xi }_i\left(t\right){\lambda }_i\left(t\right)|U_i\left(t\right)=u\right]f_U\left(t,u\right)$ and $e_{12}\left(t,u\right)=E[w_i\left(t\right){\dot{\mu }}_i\left(t\right){\tilde{X}}_i\left(t\right){\left(X_{2i}\left(t\right)\right)}^T{\xi }_i\left(t\right){\lambda }_i\left(t\right))|U_i\left(t\right)=u]f_U\left(t,u\right)$, where $a^{\otimes 2}=a{a}^T$ for a vector a. Let $R_i\left(t\right)=X_{2i}\left(t\right)-{\left(e_{12}\left(t,U_i\left(t\right)\right)\right)}^T{\left(e_{11}\left(t,U_i\left(t\right)\right)\right)}^{-1}{\tilde{X}}_i\left(t\right)$ and ${ϵ}_i\left(t\right)=Y_i\left(t\right)-{\mu }_i\left(t\right)$.

In the following, we state the asymptotic results of the estimators $\widehat{\beta }$, $\widehat{\alpha }\left(t\right)$ and $\widehat{\gamma }\left(u\right)$, together with the estimators of their asymptotic covariance matrices.

Theorem 1. Assume Condition A in the Appendix holds. Then as n → ∞, $\sqrt{n}\left(\widehat{\beta }-{\beta }_0\right)$ converges in distribution to a mean-zero normal distribution $N\left(0,A_{\beta }^{-1}{\sum }_{\beta }{A}_{\beta }^{-1}\right)$, where $A_{\beta }=E[{\int }_0^{\tau }{w}_i\left(t\right){\dot{\mu }}_i\left(t\right)R_i{\left(t\right)}^{\otimes 2}d{N}_i\left(t\right)]$ and ${\Sigma }_{\beta }=E{\left({\int }_{t_1}^{t_2}{w}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)d{N}_i\left(t\right)\right)}^{\otimes 2}$.

The asymptotic variance of $\widehat{\beta }$ can be derived using the empirical counterparts of A_β and Σ_β. Specifically, we define ${\widehat{\mu }}_i\left(t\right)=\phi \{{\widehat{\vartheta }}^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\}$, ${\widehat{\dot{\mu }}}_i\left(t\right)=\dot{\phi }\{{\widehat{\vartheta }}^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\}$, and ${\widehat{ϵ}}_i\left(t\right)=Y_i\left(t\right)-{\widehat{\mu }}_i\left(t\right)$. Let ${\widehat{E}}_{11}\left(t_0,u_0\right)=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{K}_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)W_i\left(t\right){\widehat{\dot{\mu }}}_i\left(t\right){\left\{{\tilde{X}}_i\left(t\right)\right\}}^{\otimes 2}d{N}_i\left(t\right)$, ${\widehat{E}}_{12}\left(t_0,u_0\right)=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{K}_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)W_i\left(t\right)\widehat{\dot{\mu }}\left(t\right){\tilde{X}}_i\left(t\right){\left(X_{2i}\left(t\right)\right)}^{T}d{N}_i\left(t\right)$, and ${\widehat{R}}_{ï}\left(t\right)=X_{2i}\left(t\right)-{\left({\widehat{E}}_{12}\left(t,U_i\left(t\right)\right)\right)}^T{\left({\widehat{E}}_{11}\left(t,U_i\left(t\right)\right)\right)}^{-1}{\tilde{X}}_i\left(t\right)$.

A consistent estimator of the matrix A_β is given by ${\widehat{A}}_{\beta }=n^{-1}{\sum }_{i=1}^n{\int }_{t_1}^{t_2}{W}_i\left(t\right){\widehat{\dot{\mu }}}_i\left(t\right){\widehat{R}}_i{\left(t\right)}^{\otimes 2}d{N}_i\left(t\right)$ and a consistent estimator of Σ_β is given by ${\widehat{\Sigma }}_{\beta }=n^{-1}{\sum }_{i=1}^n{\left({\int }_{t_1}^{t_2}{W}_i\left(t\right){\widehat{ϵ}}_i\left(t\right){\widehat{R}}_i\left(t\right)d{N}_i\left(t\right)\right)}^{\otimes 2}$.

Theorem 2. Assume Condition A in the Appendix holds. Then

1. $\underset{t\in \left[t_1,t_2\right]}{\sup }\left|\widehat{\alpha }\left(t\right)-{\alpha }_0\left(t\right)\right|=o_p\left(1\right);$

2. $\sqrt{nh}\left(\widehat{\alpha }\left(t\right)-{\alpha }_0\left(t\right)-\frac{1}{2}{h}^2{v}_2\ddot{\alpha }\left(t\right)\right)\stackrel{\mathcal{D}}{\to }N\left(0,{\sum }_{\alpha }\left(t\right)\right)$ where $n\to \infty$ and $t\in \left[t_1,t_2\right]$ where ${\sum }_{\alpha }\left(t\right)={\lim }_{n\to \infty }hE{\left[{\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(s\right){\mathcal{I}}_1{e}_{11}^{-1}\left(t,U_i\left(s\right)\right){\tilde{X}}_i\left(s\right)K_h\left(s-t\right)d{N}_i\left(s\right)\right]}^{\otimes 2}$,$v_2=\int u^2{K}^2\left(u\right)du$ and ${\mathcal{I}}_1=\left\{{\mathcal{I}}_{jk}\right\}$ is a p₁ × (p₁+p₃)matrix with ${\mathcal{I}}_{jk}=1$ for j = 1, …, p₁ and k = j, and ${\mathcal{I}}_{jk}=0$ otherwise.

The covariance matrix can be consistently estimated by ${\widehat{\Sigma }}_{\alpha }\left(t\right)=n^{-1}h{\sum }_{i=1}^n{\widehat{g}}_i{\left(t\right)}^{\otimes 2}$, where

$${\widehat{g}}_i\left(t\right)={\int }_0^{\tau }{W}_i\left(s\right){\widehat{ϵ}}_{í}\left(s\right){\mathcal{I}}_1{\widehat{E}}_{11}^{-1}\left(t,U_i\left(s\right)\right){\tilde{X}}_i\left(s\right)K_h\left(s-t\right)d{N}_i\left(s\right)-n^{-1}\sum _{j=1}^n{\mathcal{I}}_1{\widehat{E}}_{11}^{-1}\left(t,U_j\left(t\right)\right){\widehat{E}}_{12}\left(t,U_j\left(t\right)\right){\widehat{A}}_{\beta }^{-1}{\int }_0^{\tau }{W}_i\left(s\right){\widehat{ϵ}}_i\left(s\right){\widehat{R}}_i\left(s\right)d{N}_i\left(s\right).$$

Theorem 3. Assume Condition A in the Appendix holds. Let [u₁, u₂] be a subset of $\mathcal{U}$. Then

1. $\underset{u\in \left[u_1,u_2\right]}{\sup }\Vert \widehat{\gamma }\left(u\right)-{\gamma }_0\left(u\right)\Vert =o_p\left(1\right);$

2. $\sqrt{nb}\left(\widehat{\gamma }\left(u\right)-{\gamma }_0\left(u\right)-\frac{1}{2}{b}^2{v}_2\ddot{\gamma }\left(u\right)\right)\stackrel{\mathcal{D}}{\to }N\left(0,{\sum }_{\gamma }\left(u\right)\right)$ as $n\to \infty$, for $u\in \left[u_1,u_2\right]$ where ${\sum }_{\gamma }\left(u\right)={\lim }_{n\to \infty }bE{\left[{\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(t\right){\mathcal{I}}_2{e}_{11}{\left(t,u\right)}^{-1}{\tilde{X}}_i\left(t\right)K_b\left(U_i\left(t\right)-u\right)d{N}_i\left(t\right)\right]}^{\otimes 2}$, and ${\mathcal{I}}_3=\left\{{\mathcal{I}}_{jk}\right\}$ is a p₃ × (p₁+p₃)matrix with ${\mathcal{I}}_{jk}=1$ for j = 1, …, p₃ and k = j + p₁, and ${\mathcal{I}}_{jk}=0$ otherwise.

The asymptotic covariance matrix can be consistently estimated by ${\widehat{\Sigma }}_{\gamma }\left(u\right)=n^{-1}b{\sum }_{i=1}^n{\widehat{g}}_i^{\ast }{\left(u\right)}^{\otimes 2}$, where

$${\widehat{g}}_i^{\ast }\left(u\right)={\int }_0^{\tau }{W}_i\left(t\right){\widehat{ϵ}}_i\left(t\right){\mathcal{I}}_3{\widehat{E}}_{11}^{-1}\left(t,u\right){\tilde{X}}_i\left(t\right)K_b\left(U_i\left(t\right)-u\right)d{N}_i\left(t\right)-{n_u}^{-1}\sum _{t_u\in {\mathcal{V}}_u}{\mathcal{I}}_3{\widehat{E}}_{11}^{-1}\left(t_u,u\right){\widehat{E}}_{12}\left(t_u,u\right){\widehat{A}}_{\beta }^{-1}{\int }_0^{\tau }{W}_i\left(t\right){\widehat{ϵ}}_i\left(t\right){\widehat{R}}_i\left(t\right)d{N}_i\left(t\right),$$

Where ${\mathcal{V}}_u={\cup }_{j=1}^n\left\{U_j^{-1}\left(u\right)\right\}$, $U_j^{-1}\left(u\right)=\left\{t:U_j\left(t\right)=u\right\}$ and n_u is the number of points in ${\mathcal{V}}_u$.

Let ${\Gamma }_0\left(u\right)={\int }_{u_1}^u{\gamma }_0\left(s\right)ds$ and $\widehat{\Gamma }\left(u\right)={\int }_{u_1}^u\widehat{\gamma }\left(s\right)ds$. The following theorem presents a weak convergence result for $G_n\left(u\right)=n^{1/2}\left(\widehat{\Gamma }\left(u\right)-{\Gamma }_0\left(u\right)\right)$ over $u\in \left[u_1,u_2\right]\subset \mathcal{U}$.

Theorem 4. Under Condition A in the Appendix, $G_n\left(u\right)=n^{-1/2}{\sum }_{i=1}^n{H}_i\left(u\right)+o_p\left(1\right)$ uniformly for $u\in \left[u_1,u_2\right]$, where

$$H_i\left(u\right)={\int }_{u_1}^u{\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(t\right){\mathcal{I}}_3{e}_{11}^{-1}\left(t,v\right){\tilde{X}}_i\left(t\right)K_b\left(U_i\left(t\right)-v\right)d{N}_i\left(t\right)dv-{\int }_{u_1}^u{n}_v^{-1}\sum _{t_v\in {\mathcal{V}}_v}{\mathcal{I}}_3{e}_{11}^{-1}\left(t_v,v\right)e_{12}\left(t_v,v\right)e_{12}\left(t_v,v\right)dv{A}_{\beta }^{-1}{\int }_{t_1}^{t_2}{w}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)d{N}_i\left(t\right).$$

The process G_n(u) converges weakly to a zero-mean Gaussian process on [u₁, u₂].

5. HYPOTHESIS TESTING FOR γ(u)

This section develops hypothesis testing procedures for $\gamma \left(u\right),\hspace{0.17em}u\in \mathcal{U}$. Let Γ(u) be the cumulative coefficient function. Let γ_k(u) be the kth component of γ(u) and Γ_k(u) be the kth component of Γ(u).

First, we develop testing procedures for the hypotheses $H_{10}:{\gamma }_k\left(u\right)=0$ for $u\in \left[u_1,u_2\right]$, versus $H_{1a}:{\gamma }_k\left(u\right)\ne 0$ for some u ∈ [u₁, u₂], or $H_{1m}:{\gamma }_k\left(u\right)\le 0$ with strict inequality for some u ∈ [u₁, u₂]. The null hypothesis H₁₀ implies that the effect γ_k(u) of the kth component of X₃(t) is zero at any the exposure level u ∈ [u₁, u₂]. The alternative hypothesis H_1a means that the effect is not zero for at least some the exposure level of u, while H_1m suggests a positive effect for some u.

Consider the test process $Q_1\left(u\right)=n^{1/2}\widehat{\Gamma }\left(u\right)$,$u\in \left[u_1,u_2\right]$. Then $Q_1\left(u\right)=G_n\left(u\right)+n^{1/2}{\Gamma }_0\left(u\right)$,$u\in \left[u_1,u_2\right]$. Let ${\Gamma }_{0\left(k\right)}\left(u\right)$ be the kth component of Γ₀(u) and Q_1(k)(u) the kth component of Q₁(u). Under H₁₀, Γ_0(k)(u) = 0 for u ∈ [u₁, u₂]. We propose the supremum-type and integrated squared difference-type test statistics:

$$S_{a1}=\underset{u\in \left[u_1,u_2\right]}{\sup }\left|Q_{1\left(k\right)}\left(u\right)\right|,\hspace{1em}\hspace{1em}{S}_{a2}={\int }_{\left[u_1,u_2\right]}{\left\{Q_{1\left(k\right)}\left(u\right)\right\}}^{2}du,$$

$$S_{m1}=\underset{u\in \left[u_1,u_2\right]}{\inf }{Q}_{1\left(k\right)}\left(u\right),\hspace{1em}\hspace{1em}{S}_{m2}={\int }_{\left[u_1,u_2\right]}{Q}_{1\left(k\right)}\left(u\right)du.$$

The test statistics S_a1 and S_a2 capture general departures H_1a, while the test statistics S_m1 and S_m2 are sensitive to the monotone departures H_1m.

By Theorem 4, G_n(u), u ∈ [u₁, u₂] converges weakly to a mean-zero Gaussian process with continuous sample paths on u ∈ [u₁, u₂]. Furthermore, the distribution of G_n(u), for u ∈ [u₁, u₂], can be approximated using the Gaussian multipliers resampling method based on $G_n^{\ast }\left(u\right)=n^{-1/2}{\sum }_{i=1}^n{\widehat{H}}_i\left(u\right){\varphi }_i$, where ${\varphi }_1,\cdots ,{\varphi }_n$ are independent standard normal random variables, and the ${\widehat{H}}_i\left(t\right)$ are obtained by replacing the unknown quantities in H_i(t) with their corresponding empirical counterparts.

Let G_n(k)(u) and $G_{n\left(k\right)}^{\ast }\left(u\right)$ be the kth components of G_n(u) and $G_n^{\ast }\left(u\right)$, respectively. Then the distribution of Q_1(k)(u), u ∈ [u₁, u₂], can be approximated by the conditional distribution of $G_{n\left(k\right)}^{\ast }\left(u\right)$, $u\in \left[u_1,u_2\right]$ given the observed data sequence. Hence, the distributions of S_a1, S_a2, S_m1, and S_m2 under H₁₀ can be approximated by the conditional distribution of $S_{a1}^{\ast }={\sup }_{u\in \left[u_1,u_2\right]}|G_{n\left(k\right)}^{\ast }\left(u\right)|$, $S_{a2}^{\ast }={\int }_{\left[u_1,u_2\right]}{\{G_{n\left(k\right)}^{\ast }\left(u\right)\}}^{2}du$, $S_{m1}^{\ast }={\inf }_{\left[u_1,u_2\right]}{G}_{n\left(k\right)}^{\ast }\left(u\right)$, and $S_{m2}^{\ast }={\int }_{\left[u_1,u_2\right]}{G}_{n\left(k\right)}^{\ast }\left(u\right)du$, given the observed data sequence, respectively.

Next, we develop testing procedures for the hypotheses H₂₀: γ_k(u) does not depend on u for u ∈ [u₁, u₂], versus H_2a: γ_k(u) changes over u ∈ [u₁, u₂], or H_2m: γ_k(u) increases with u ∈ [u₁, u₂]. The null hypothesis H₂₀ implies that the effect γ_k(u) does not change with the exposure levels of u. The tests can be applied to test the alternative hypothesis that γ_k(u) decreases with u simply by replacing X₃(t) with −X₃(t). Let $Q_2\left(u\right)={\left(u-u_1\right)}^{-1}{n}^{1/2}\{\widehat{\Gamma }\left(u\right)-\widehat{\Gamma }\left(u_1\right)\}-{\left(u_2-u_1\right)}^{-1}{n}^{1/2}\{\widehat{\Gamma }\left(u_2\right)-\widehat{\Gamma }\left(u_1\right)\}$. Then

$$Q_2\left(u\right)=R\left(u,G_n\right)+n^{1/2}R\left(u,\Gamma \right),$$ (7)

where R(u, Γ) = (u − u₁)⁻¹{Γ(u) − Γ(u₁)} − (u₂ − u₁)⁻¹{Γ(u₂) − Γ(u₁)} is a transformation of Γ(·). Let Q_2(k)(u) and R(u, Γ_k) be the kth component of Q₂(u) and R(u, Γ), respectively.

Under H₂₀, R(u, Γ_k) = 0, the equation (7) motivates the test statistics:

$$T_{a1}=\underset{u\in \left[u_1^{\ast },u_2\right]}{\sup }\left|Q_{2\left(k\right)}\left(u\right)\right|,\hspace{1em}\hspace{1em}{T}_{a2}={\int }_{\left[u_1^{\ast },u_2\right]}{\left\{Q_{2\left(k\right)}\left(u\right)\right\}}^{2}du,$$

$$T_{m1}=\underset{u\in \left[u_1^{\ast },u_2\right]}{\inf }{Q}_{2\left(k\right)}\left(u\right),\hspace{1em}\hspace{1em}{T}_{m2}={\int }_{\left[u_1^{\ast },u_2\right]}{Q}_{2\left(k\right)}\left(u\right)du,$$

where $u_1^{\ast }$ is a number (u₁, u₂) so that the denominator of Q_2(k)(u) does not vanish. The $u_1^{\ast }$ can be chosen close to u₁ to make use of available data and to ensure the tests to be consistent.

Using the Gaussian multiplier method, the distributions of T_a1, T_a2, T_m1, and T_m2 under H₂₀

can be approximated by the conditional distributions of $T_{a1}^{\ast }={\sup }_{u\in \left[u_1^{\ast },u_2\right]}|R(u,G_{n\left(k\right)}^{\ast })|$, $T_{a2}^{\ast }={\int }_{\left[u_1^{\ast },u_2\right]}{\{R(u,G_{n\left(k\right)}^{\ast })\}}^{2}du$, $T_{m1}^{\ast }={\inf }_{u\in \left[u_1^{\ast },u_2\right]}R(u,G_{n\left(k\right)}^{\ast })$, and $T_{m2}^{\ast }={\int }_{\left[u_1^{\ast },u_2\right]}R(u,G_{n\left(k\right)}^{\ast })du$, given the observed data sequence, respectively.

The tests T_a1 and T_a2 capture general departures H_2a, while the tests T_m1 and T_m2 are sensitive to the monotone departure H_2m. Note that the derivative dR(u, Γ)/du = (u − a)⁻¹[Γ(u) − (u − a)⁻¹Γ(u)] ≥ 0 under H_2m with strict inequality for at least some u ∈ [u₁, u₂]. This, plus the fact that R(u, Γ) is non-decreasing with R(b, Γ) = 0, lead to the results that the tests based on T_m1 and T_m2 are consistent against H_2m, and the tests based on T_a1 and T_a2 are consistent against H_2a.

6. SIMULATION STUDIES

We conducted a simulation study to assess the finite-sample performance of the proposed methods under the model with three popular link functions:

$$E\left\{Y_i\left(t\right)|X_i,S_i\right\}=g^{-1}\left\{{\alpha }_0\left(t\right)+{\alpha }_1\left(t\right)X_{1i}\left(t\right)+\beta X_{2i}+\gamma \left(t-S_i\right)X_{3i}\right\},$$ (8)

for 0 ≤ t ≤ τ with τ = 3.5, where the covariate X_1i(t) = 0.05t + 0.1X_2i + V_i(t) is time-dependent with V_i(t) a standard normal random variable, X_2i is a Bernoulli random variable with success probability of 0.5, X_3i is a uniform random variable on [−1, 1], and S_i is uniform on [0, 1]. We consider α₀(t) = 0.1 cos(t), ${\alpha }_1\left(t\right)=0.2\sqrt{t}$, β = 0.4, and $\gamma \left(u\right)={\theta }_1+{\theta }_2\exp \left(-0.6u\right)$, where different values of θ₁ and θ₂ are used. For the identity link and the logarithm link, the error ${ϵ}_i\left(t\right)=Y_i\left(t\right)-E\left\{Y_i\left(t\right)|X_i,S_i\right\}$ has a normal distribution with mean ϕ_i and variance 0.04, and ϕ_i is N (0, 0.04). For the logit link, Y_i(t) has the Bernoulli distribution with the probability of success of E{Y_i(t)|X_i, S_i}. The observation times follow a Poisson process with the proportional mean rate model h(t|X_i, S_i) = 2 exp(0.9X_2i). The censoring times C_i are generated from a uniform distribution on [2.5, 8]. There are approximately 6 observations per subject in the interval [0, τ ], and about 25% subjects are censored.

We take the Epanechnikov kernel K(u) = 0.75(1 − u²)I(|u| ≤ 1) and a unit weight function W_i(t) = 1. We let t₁ = h/2 and t₂ = τ − h/2 in (4) to avoid larger variations on the boundaries. We consider the following settings for the values of (θ₁, θ₂) under model (8):

1. For the identity link function, I₁₀: (θ₁, θ₂) = (0, 0), I₁₁: (θ₁, θ₂) = (−0.02, 0), I₁₂: (θ₁, θ₂) = (−0.04, 0), and I₁₃: (θ₁, θ₂) = (−0.06, 0); I₂₀: (θ₁, θ₂) = (−0.2, 0), I₂₁: (θ₁, θ₂) = (0, −0.08), I₂₂: (θ₁, θ₂) = (0, −0.12), and I₂₃: (θ₁, θ₂) = (0, −0.16).

2. For the logarithm link function, E₁₀: (θ₁, θ₂) = (0, 0), E₁₁: (θ₁, θ₂) = (−0.03, 0), E₁₂: (θ₁, θ₂) = (−0.04, 0), and E₁₃: (θ₁, θ₂) = (−0.05, 0); E₂₀: (θ₁, θ₂) = (−0.2, 0), E₂₁: (θ₁, θ₂) = (0, −0.06), E₂₂: (θ₁, θ₂) = (0, −0.08), and E₂₃: (θ₁, θ₂) = (0, −0.1).

3. For the logit link function, L₁₀: (θ₁, θ₂) = (0, 0), L₁₁: (θ₁, θ₂) = (−0.15, 0), L₁₂: (θ₁, θ₂) = (−0.2, 0), and L₁₃: (θ₁, θ₂) = (−0.25, 0); L₂₀: (θ₁, θ₂) = (−0.2, 0), L₂₁: (θ₁, θ₂) = (0, −0.6), L₂₂: (θ₁, θ₂) = (0, −0.9), and L₂₃: (θ₁, θ₂) = (0, −1.2).

The performance of $\widehat{\beta }$, $\widehat{\alpha }\left(t\right)$, and $\widehat{\gamma }\left(u\right)$ at fixed points t and u is measured through the bias (Bias), the sample standard error of the estimators (SEE), the sample mean of the estimated standard errors (ESE), and the 95% empirical coverage probability (CP). The overall performance of the estimator $\widehat{\alpha }\left(t\right)$ is evaluated by the square root of the integrated mean square error $RMS{E}_{\alpha }={\left\{\frac{1}{N\left(\tau -2h\right)}{\sum }_{j=1}^N{\int }_h^{\tau -h}{\left({\widehat{\alpha }}_j\left(t\right)-{\alpha }_0\left(t\right)\right)}^{2}dt\right\}}^{1/2}$, where N is the number of simulations and ${\widehat{\alpha }}_j\left(t\right)$ is the estimate of α(t) in the jth simulation, j = 1, · · ·, N. The summary measure RMSE_γ is defined likewise.

Using the cross-validation procedure based on (6), an initial 10-fold cross-validation bandwidth selection in 10 runs yields bandwidths between 0.2 and 0.5 with the average close to 0.3. A larger sample size yields to smaller bandwidths. Table 1 summarizes the Bias, SEE, ESE, and CP of estimating β and RMSEs of estimating α(t) and γ(t) under model (8) for I₂₁, E₂₁, and L₂₁ with sample sizes n = 200, 400, and 600, using the bandwidths (h, b) = (0.2, 0.2), (0.25, 0.25), and (0.3, 0.3). Each entry of the tables is calculated based on 1,000 repetitions. For all three link functions, Table 1 shows that the biases are small, the estimated errors are close to the empirical standard errors. The coverage probabilities show slight under-coverage for the bandwidths (h, b) = (0.2, 0.2) for n = 200, but closer to their 95% nominal level for n = 400 and 600. Larger bandwidths (h, b) = (0.25, 0.25) and (0.3, 0.3) have better performance in terms of the coverage probabilities. The estimation standard errors decrease as the sample size increases.

TABLE 1:

Summary of Bias, SEE, ESE, and CP for β, and RMSEs for α(t) and γ(u) under model (8) with the settings I₂₁, E₂₁, and L₂₁ for sample sizes n = 200, 400, and 600, using the unit weight function, and bandwidths (h, b) = (0.2, 0.2), (0.25, 0.25), and (0.3, 0.3), based on 1,000 simulations.

Setting	(θ1, θ2)	n	h	b	Bias	SEE	ESE	CP	RMSEα0	RMSEα1	RMSEγ
Identity link
I21	(0, −0.08)	200	0.2	0.2	0.0038	0.0592	0.0532	0.917	0.0421	0.0236	0.0405
			0.25	0.25	0.0040	0.0591	0.0539	0.922	0.0412	0.0213	0.0378
			0.3	0.3	0.0039	0.0591	0.0544	0.926	0.0408	0.0200	0.0363
		400	0.2	0.2	0.0006	0.0408	0.0384	0.936	0.0286	0.0162	0.0281
			0.25	0.25	0.0005	0.0408	0.0387	0.941	0.0280	0.0148	0.0264
			0.3	0.3	0.0005	0.0406	0.0389	0.938	0.0276	0.0140	0.0255
		600	0.2	0.2	0.0015	0.0326	0.0317	0.938	0.0236	0.0131	0.0230
			0.25	0.25	0.0016	0.0326	0.0319	0.939	0.0231	0.0120	0.0217
			0.3	0.3	0.0016	0.0326	0.0321	0.946	0.0229	0.0114	0.0209
Logarithm link
E21	(0, −0.06)	200	0.2	0.2	0.0031	0.0457	0.0412	0.911	0.0366	0.0189	0.0307
			0.25	0.25	0.0017	0.0443	0.0417	0.937	0.0352	0.0171	0.0283
			0.3	0.3	0.0017	0.0435	0.0421	0.946	0.0346	0.0160	0.0267
		400	0.2	0.2	0.0006	0.0313	0.0296	0.936	0.0247	0.0129	0.0211
			0.25	0.25	0.0005	0.0313	0.0298	0.939	0.0241	0.0117	0.0198
			0.3	0.3	0.0005	0.0311	0.0300	0.937	0.0237	0.0111	0.0190
		600	0.2	0.2	0.0012	0.0251	0.0244	0.933	0.0203	0.0103	0.0172
			0.25	0.25	0.0013	0.0251	0.0245	0.939	0.0199	0.0095	0.0162
			0.3	0.3	0.0012	0.0251	0.0246	0.937	0.0197	0.0090	0.0155
Logit link
L21	(0, −0.6)	200	0.2	0.2	0.0524	0.1936	0.1813	0.928	0.2221	0.2128	0.2770
			0.25	0.25	0.0354	0.1840	0.1794	0.947	0.1987	0.1787	0.2362
			0.3	0.3	0.0263	0.1806	0.1790	0.949	0.1871	0.1623	0.2142
		400	0.2	0.2	0.0188	0.1394	0.1244	0.916	0.1474	0.1309	0.1775
			0.25	0.25	0.0119	0.1366	0.1242	0.929	0.1366	0.1158	0.1572
			0.3	0.3	0.0082	0.1356	0.1247	0.929	0.1308	0.1073	0.1446
		600	0.2	0.2	0.0085	0.1028	0.1008	0.948	0.1146	0.1025	0.1418
			0.25	0.25	0.0037	0.1016	0.1011	0.948	0.1068	0.0922	0.1268
			0.3	0.3	0.0009	0.1019	0.1016	0.945	0.1026	0.0863	0.1175

Table 1 also shows the overall performance of the estimators for ${\widehat{\alpha }}_0\left(t\right)$, ${\widehat{\alpha }}_1\left(t\right)$ and $\widehat{\gamma }\left(u\right)$ in terms of RMSE_α0, RMSE_α1, and RMSE_γ. It is clear that different link functions yield different levels of RMSEs. But they decrease as the sample size increases. Figure 1 shows the pointwise results for α₀(t), α₁(t), and γ(u) under the settings I₂₁, E₂₁, and L₂₁ for n = 600 with (h, b) = (0.3, 0.3). We can see that the local estimators are close to the true values and the ESE provides a good approximation for the SSE of the point estimates. The empirical coverage probabilities are reasonably close to the correct nominal level.

FIGURE 1:

Plots of Bias, SEE, ESE, and CP of ${\widehat{\alpha }}_0\left(t\right)$, ${\widehat{\alpha }}_1\left(t\right)$ and $\widehat{\gamma }\left(u\right)$ under model (8) with the settings I₂₁, E₂₁, and L₂₁ for n = 600, using the unit weight function, and bandwidths (h, b) = (0.3, 0.3), based on 1,000 simulations. The left panel is for ${\widehat{\alpha }}_0\left(t\right)$, the middle panel is for ${\widehat{\alpha }}_1\left(t\right)$, and the right panel is for $\widehat{\gamma }\left(u\right)$. The solid line is for identity link function, the dashed line is for logarithm link function, and the dotted line is for logit link function.

Next, we examine the finite sample performance of the proposed tests for testing H₁₀ : γ(u) = 0 and H₂₀ : γ(u) does not depend on u. We take u₁ = 0.15, u₂ = 2.35, and $u_1^{\ast }=0.2$. For n = 400 and n = 600 with (h, b) = (0.3, 0.3), the empirical sizes and the powers of the test statistics S_a1, S_a2, S_m1, and S_m2 for testing H₁₀ are shown in Table 2 and those of the test statistics T_a1, T_a2, T_m1, and T_m2 for testing H₂₀ are shown in Table 3. Our simulations show that for sample sizes n = 400 and 600, the empirical sizes are close to their nominal level 0.05 for the three link functions. The powers of the tests increase as the sample size increases. The powers of the tests also increase as the alternative models are increasingly different from the null models. The powers of the supremum-type tests are comparable to the powers of the integrated tests.

TABLE 2:

Empirical sizes and powers of the test statistics S_a1, S_a2, Sm1, and S_m2 for testing H₁₀ at the nominal level 0.05 under model (8) for three different link functions with various settings for sample sizes n = 400 and 600, using the unit weight function, and bandwidths (h, b) = (0.3, 0.3), based on 1,000 Gaussian multiplier samples and 1,000 simulations.

			n = 400				n = 600
Setting	(θ1, θ2)	Test	Sa1	Sa2	Sm1	Sm2	Sa1	Sa2	Sm1	Sm2
			Identity link
I10	(0, 0)	Size	0.052	0.051	0.055	0.065	0.065	0.070	0.059	0.066
I11	(−0.02, 0)	Power	0.182	0.177	0.263	0.257	0.223	0.221	0.308	0.311
I12	(−0.04, 0)		0.499	0.481	0.632	0.599	0.653	0.635	0.762	0.742
I13	(−0.06, 0)		0.822	0.813	0.894	0.882	0.938	0.930	0.963	0.960
			Logarithm link
E10	(0, 0)	Size	0.054	0.052	0.057	0.066	0.066	0.070	0.058	0.065
E11	(−0.03, 0)	Power	0.507	0.495	0.632	0.621	0.657	0.647	0.774	0.758
E12	(−0.04, 0)		0.748	0.746	0.828	0.827	0.880	0.886	0.929	0.933
E13	(−0.05, 0)		0.906	0.900	0.956	0.951	0.971	0.971	0.986	0.986
			Logit link
L10	(0, 0)	Size	0.048	0.053	0.053	0.059	0.051	0.053	0.058	0.058
L11	(−0.15, 0)	Power	0.600	0.531	0.718	0.635	0.769	0.722	0.843	0.793
L12	(−0.2, 0)		0.840	0.780	0.904	0.839	0.940	0.902	0.968	0.934
L13	(−0.25, 0)		0.951	0.925	0.973	0.946	0.994	0.983	0.996	0.986

TABLE 3:

Empirical sizes and powers of the test statistics T_a1, T_a2, T_m1, and T_m2 for testing H₂₀ at the nominal level 0.05 under model (8) for three different link functions with various settings for sample sizes n = 400 and 600, using the unit weight function, and bandwidths (h, b) = (0.3, 0.3), based on 1,000 Gaussian multiplier samples and 1,000 simulations.

			n = 400				n = 600
Setting	(θ1, θ2)	Test	Ta1	Ta2	Tm1	Tm2	Ta1	Ta2	Tm1	Tm2
			Identity link
I20	(−0.2, 0)	Size	0.076	0.061	0.068	0.053	0.067	0.060	0.061	0.061
I21	(0, −0.08)	Power	0.365	0.421	0.479	0.550	0.469	0.546	0.591	0.709
I22	(0, −0.12)		0.615	0.701	0.720	0.844	0.774	0.862	0.859	0.939
I23	(0, −0.16)		0.846	0.921	0.912	0.966	0.957	0.984	0.985	0.995
			Logarithm link
E20	(−0.2, 0)	Size	0.079	0.067	0.074	0.060	0.073	0.054	0.057	0.061
E21	(0, −0.06)	Power	0.383	0.433	0.497	0.553	0.496	0.568	0.620	0.711
E22	(0, −0.08)		0.573	0.628	0.662	0.765	0.714	0.802	0.808	0.894
E23	(0, −0.1)		0.728	0.833	0.831	0.919	0.883	0.943	0.933	0.975
			Logit link
L20	(−0.2, 0)	Size	0.069	0.058	0.071	0.059	0.067	0.067	0.057	0.059
L21	(0, −0.6)	Power	0.337	0.424	0.471	0.563	0.442	0.551	0.565	0.717
L22	(0, −0.9)		0.636	0.743	0.743	0.852	0.782	0.881	0.872	0.931
L23	(0, −1.2)		0.838	0.912	0.924	0.957	0.941	0.983	0.974	0.996

More discussions and additional simulation results for different models and bandwidths are given in Appendix B in the Web-based Supplementary Material.

7. REAL DATA APPLICATION

We apply the developed methods to the ACTG 244 randomized trial summarized in the Introduction. We recently analyzed the same data set using the version of the model with the exposure-varying covariate effects specified parametrically (Qi et al., 2017); here application of the nonparametric method is more defensible, given the lack of knowledge about the appropriateness of the selected parametric form. ACTG 244 enrolled HIV-infected patients receiving zidovudine (ZDV) monotherapy, and monitored them every eight weeks for occurrence of the 215 (T215Y/F) ZDV resistance mutation. Upon detection of the 215 mutation, a subject was randomized to either continue ZDV add ddI, or add both ddI and NVP. Of 289 enrolled participants, 5 were randomized but went off study before taking any medication, 284 were dispensed ZDV, and 57 developed T215Y/F during follow-up, of whom 49 were randomized to ZDV (n = 17), ZDV+ddI (n = 15), or ZDV+ddI+NVP (n = 17). Subsequent to an independent review by the Data Safety Monitoring Board, all subjects were offered randomization to either ZDV+ddI or ZDV+ddI+NVP with six months of new follow-up. An additional objective is to assess the effect of this randomization among subjects who had not yet acquired the 215 mutation at the time of this randomization, again studying whether the treatment effect depends on the time of treatment randomization/switching. At the time of the interim review 137 subjects were still taking ZDV and were randomized to ZDV+ddI (n = 69) or ZDV+ddI+NVP (n = 68). The data set includes 182 white, 84 black, and 23 hispanic and other races.

7.1. Analysis of the effects of switching treatments after drug-resistant virus was detected

Let Y (t) be the square root of CD4 count t years post–study entry, Z₁ be Gender (1 if Female; 0 if Male), Z₂ be Age at study entry in years, and Z₃ and Z₄ be indicator variables coding race (Z₃ = 1 if white and 0 otherwise; Z₄ = 1 if black and 0 otherwise). Let S₁ be the time between the dates of study entry and the first randomization triggered by occurrence of the T215Y/F mutation, such that U₁(t) = t − S₁ is the follow-up time starting at the date of the first randomization. Let T_A1(t) be the indicator of being randomized to ZDV and t > S₁, T_A2(t) be the indicator of being randomized to ZDV+ddI and t > S₁, and T_A3(t) be the indicator of being randomized to ZDV+ddI+NVP and t > S₁. These indicator variables are zero prior to detection of the 215 mutation.

The analysis includes all n = 284 enrolled subjects dispensed ZDV monotherapy, right-censoring the 98 subjects who stopped taking ZDV prior to the interim review at the time of stopping ZDV. Moreover, for this analysis subjects who did not develop the 215 mutation and underwent the second randomization are right-censored at the time of the second randomization.

The following model is used for the analysis:

$$Y_i\left(t\right)={\alpha }_0\left(t\right)+{\beta }_1{Z}_{1i}+{\beta }_2{Z}_{2i}+{\beta }_3{Z}_{3i}+{\beta }_4{Z}_{4i}+{\gamma }_1\left(U_{1i}\left(t\right)\right)T_{A1i}\left(t\right)+{\gamma }_2\left(U_{1i}\left(t\right)\right)T_{A2i}\left(t\right)+{\gamma }_3\left(U_{1i}\left(t\right)\right)T_{A3i}\left(t\right)+{ϵ}_i\left(t\right),$$ (9)

for t ∈ [0, τ ], where τ = 2.5 years. The average of 10 runs of 10-fold cross-validation yields h = 1.3 and b = 1.1. We find that the results are not overly sensitive to the choice of bandwidths. Table 4 shows the estimates of the time-constant parameters (first block of the table) for [t₁, t₂] = [0, 2.5], and Figure 2 shows the estimates of α₀(t), γ₁(u), γ₂(u), and γ₃(u) with 95% pointwise confidence intervals using the unit weight function.

TABLE 4:

Estimated effects of adaptive treatment randomizations in the ACTG 244 trial using the unit weight function.

Effect	Parameter	Estimate	Standard deviation	95% Confidence limits		P-value
Treatment effects after the T215Y/F mutation under model (9)
(first randomization)
Gender	β1	−0.8604	0.8330	−2.4931	0.7723	0.3017
Age	β2	0.5479	0.2620	0.0343	1.0614	0.0365
Race	β3	5.1418	0.4468	4.2661	6.0175	< 0.001
	β4	4.7831	0.5084	3.7866	5.7795	< 0.001
Treatment effects before the T215Y/F mutation under model (10)
(second randomization)
Gender	β1	−0.4558	0.7455	−1.9170	1.0054	0.5409
Age	β2	0.0417	0.2947	−0.5358	0.6192	0.8874
Race	β3	0.0026	1.2067	−2.3624	2.3677	0.9983
	β4	−0.8498	1.2824	−3.3634	1.6637	0.5075

FIGURE 2:

Estimated effects of switching treatments after drug-resistant virus was detected using the unit weight function based on the ACTG 244 data. (a) is the estimated baseline function ${\widehat{\alpha }}_0\left(t\right)$ with 95% pointwise confidence intervals; (b), (c), and (d) are estimated ${\widehat{\gamma }}_k\left(u\right)$, k = 1, 2, 3 with 95% pointwise confidence intervals under model (9), using h = 1.3 and b = 1.1.

We conduct the hypothesis tests for testing H₁₀ : γ_k(u) = 0 against H_1a : γ_k(u) ≠ 0 and H_1m : γ_k(u) ≥ 0. For testing H₂₀ against H_2a and H_2m, the monotone alternatives for γ₁(u) is decreasing, and for γ₂(u) and γ₃(u) are increasing. In the test statistics, we use the unit weight function, bandwidths (h, b) = (1.3, 1.1), [u₁, u₂] = [0.1, 2.0], and $u_1^{\ast }=0.11$.

Table 5 reports on the observed P -values of the test statistics S_a1, S_a2, S_m1, and S_m2 for testing H₁₀ and the test statistics T_a1, T_a2, T_m1, and T_m2 for testing H₂₀, based on 1,000 Gaussian multiplier samples using the unit weight function.

TABLE 5:

Observed P -values of the test statistics S_a1, S_a2, S_m1, and S_m2 for testing H₁₀ and the test statistics T_a1, T_a2, T_m1, and T_m2 for testing H₂₀, based on 1,000 Gaussian multiplier samples using the unit weight function.

	H10				H20
Testing	Sa1	Sa2	Sm1	Sm2	Ta1	Ta2	Tm1	Tm2
Treatment effects after the T215Y/F mutation under model (9)
(first randomization)
γ1(·)	0.342	0.516	0.676	0.676	0.110	0.046	0.054	0.010
γ2(·)	0.850	0.792	0.768	0.616	0.470	0.554	0.238	0.324
γ3(·)	0.638	0.566	0.998	0.722	< 0.001	0.001	< 0.001	0.002
Treatment effects before the T215Y/F mutation under model (10)
(second randomization)
γ2(·)	0.538	0.662	0.278	0.408	< 0.001	< 0.001	< 0.001	< 0.001
γ3(·)	0.014	0.048	0.006	0.040	< 0.001	< 0.001	< 0.001	< 0.001

The first block of Table 5 shows that CD4 cell counts decreases significantly after drug-resistant virus was detected for those continuing on ZDV monotherapy. CD4 cell counts increases significantly even after drug-resistant virus was detected for those switching to the combination therapy ZDV+ddI+NVP. While the combination therapy ZDV+ddI does not increase CD4 cell counts, it has some positive effects for not letting CD4 cell counts continue to decrease. In conclusion, switching to the combination therapies based on the T215Y mutation has positive benefits as compared to continuing with ZDV monotherapy ZDV even after drug-resistant virus was detected.

7.2. Analysis of the effects of switching treatments before drug-resistant virus was detected

Now we assess the effect of the randomization to ZDV+ddI versus ZDV+ddI+NVP among the 137 participants who had not developed the 215 mutation by the time of the interim review and were randomized. Let S₂ be the time between the date of the interim review and the date of the second randomization, and let U₂(t) = t − S₂. Let T_B2(t) be the indicator of randomization to ZDV+ddI and t > S₂, and let T_B3(t) be the indicator of randomization to ZDV+ddI+NVP and t > S₂. Zeros for both variables T_B2(t) = 0 and T_B3(t) = 0 indicate that a subject is taking ZDV at time t before the interim review.

We analyze the data with the following model:

$$Y_i\left(t\right)={\alpha }_0\left(t\right)+{\beta }_1{Z}_{1i}+{\beta }_2{Z}_{2i}+{\beta }_3{Z}_{3i}+{\beta }_4{Z}_{4i}+{\gamma }_2\left(U_{2i}\left(t\right)\right)T_{B2i}\left(t\right)+{\gamma }_3\left(U_{2i}\left(t\right)\right)T_{B3i}\left(t\right)+{ϵ}_i\left(t\right),$$ (10)

for t ∈ [0, 2.5]. The range of the observed values for U_2i(t), t ∈ [0, 2.5], is [0, 0.70]. We choose h = 0.7 and b = 0.8 selected by 10-fold cross-validation.

The estimates of the parameters are shown using the unit weight function in the second block of Table 4, and the point and 95% confidence interval estimates of α₀(t), γ₂(u), and γ₃(u) are presented in Figure 3. We conduct the hypothesis tests for testing H₁₀ : γ_k(u) = 0 against H_1a : γ_k(u) ≠ 0 and H_1m : γ_k(u) ≥ 0. For testing H₂₀, we consider the alternative H_2m, where γ₂(u) and γ₃(u) are increasing with u. In the test statistics, we use the unit weight function, bandwidths (h, b) = (0.7, 0.8), [u₁, u₂] = [0.1, 0.7], and $u_1^{\ast }=0.11$.

FIGURE 3:

Estimated effects of switching treatments before drug-resistant virus was detected using the unit weight function based on the ACTG 244 data. (a) is the estimated baseline function ${\widehat{\alpha }}_0\left(t\right)$ with 95% pointwise confidence intervals; (b) and (c) are estimated ${\widehat{\gamma }}_k\left(u\right)$, k = 2, 3 with 95% pointwise confidence intervals under model (10), using h = 0.7 and b = 0.8.l

The observed P -values shown in the second block of Table 5 suggests that switching to the combination therapies ZDV+ddI and ZDV+ddI+NVP both improve CD4 counts for patients without having yet developed the T215Y drug resistance mutation.

8. CONCLUDING REMARKS

This article develops a generalized semiparametric varying-coefficient model for longitudinal data. Our approach flexibly models three types of covariate effects: constant effects, nonparametric time-varying effects, and covariate-varying effects, where the central contribution of this work is to extend the method of Qi et al. (2017) that modeled the covariate-varying effects parametrically to model these effects nonparametrically. The new model provides greater flexibility and robustness. The asymptotic results for the nonparametric and parametric estimators are established. We have developed hypothesis testing procedures for testing whether and how the covariate effects such as X_3i(t) are modified by U_i(t).

We have conducted an extensive simulation study to examine the performances of the proposed estimation and hypothesis testing procedures. Additional simulation results for different models and bandwidths are placed in Web Appendix B. The simulation study suggests that the performance of the estimation procedure depends on the model, the number of repeated measurements on each subject, as well as the sample size. The convergence rate for the identity link function is the fastest, and the logit link function has the slowest convergence rate among the models we consider. The required computation time for each model for sample size 400 is about 5 hours for models under the identity link function and about 20 hours for models under the logarithm link function and logit link function running on the institution’s High Performance Computing Cluster.

The efficient selection of the weight function is also a very challenging problem that would depend on the link function and require the knowledge of the distribution of the error processes. In Web Appendix C, we discuss a two-stage procedure in the framework of marginal approaches to improve estimation efficiency through weight-function selection. We conducted a simulation study to investigate the two-stage estimation procedure for model (1) under three different error processes and found that the two-stage estimation improves efficiency over the estimation using the unit weight function. The two-stage estimation procedure and the additional simulation results for model (8) are presented in Web Appendix C. We have analyzed the ACTG 244 data using the two-stage estimation procedure. The estimation results for the effects of treatment switching presented in Web Appendix D are similar to the results using unit weight function but slightly smaller standard errors.

All the program codes produced for the simulation study and data analysis, the data set along with a detailed README file for the step-by-step instructions, is available as a part of the Web-based Supplementary Materials.

9. APPENDIX: PROOFS OF THEOREMS

Condition A.

• (A.1)The censoring time C_i is noninformative in the sense that $E\left\{d{N}_i^{\ast }\left(t\right)|X_i\left(t\right),U_i\left(t\right),C_i\ge t\right\}=E\left\{d{N}_i^{\ast }\left(t\right)|X_i\left(t\right),U_i\left(t\right)\right\}$ and $E\left\{Y_i\left(t\right)|X_i\left(t\right),U_i\left(t\right),C_i\ge t\right\}=E\left\{Y_i\left(t\right)|X_i\left(t\right),U_i\left(t\right)\right\}$; $d{N}_i^{\ast }\left(t\right)$ is independent of Y_i(t) conditional on X_i(t), U_i(t), and C_i ≥ t; the censoring time C_i is allowed to depend on the left continuous covariate process X_i(·);
• (A.2)The processes Y_i(t), X_i(t), and λ_i(t), 0 ≤ t ≤ τ, are bounded, and their total variations are bounded by a constant; E|N_i(t₂) − N_i(t₁)|² ≤ L(t₂ − t₁) for 0 ≤ t₁ ≤ t₂ ≤ τ, where L > 0 is a constant; E|N_i(t + h) − N_i(t − h)|^2+v = O(h), for some v > 0;
• (A.3)The kernel function K(·) is symmetric with compact support on [−1, 1] and Lipschitz continuous; Bandwidths $h\asymp b$; h → 0; nh² → ∞ and nh⁵ is bounded;
• (A.4)The function g⁻¹(·) is monotone and twice differentiable;
• (A.5)α₀(t), γ₀(u), e₁₁(t), and e₁₂(t) are twice differentiable; (e₁₁(t))⁻¹ is bounded over 0 ≤ t ≤ τ ; the matrices A and Σ are positive definite;
• (A.6)The limit ${\lim }_{n\to \infty }hE{[{\int }_0^{\tau }{ϵ}_i\left(t\right){\mathcal{I}}_1{e}_{11}{\left(t_0,U_i\left(t\right)\right)}^{-1}{X}_i\left(t\right)K_h\left(t-t_0\right)d{N}_i\left(t\right)]}^{\otimes 2}$, and ${\lim }_{n\to \infty }bE{[{\int }_0^{\tau }{\widehat{ϵ}}_i\left(t\right){\mathcal{I}}_3{e}_{11}{\left(t,u_0\right)}^{-1}{X}_i\left(t\right)K_b\left(U_i\left(t\right)-u_0\right)d{N}_i\left(t\right)]}^{\otimes 2}$ exist and are finite.

Next, we present technical lemmas to be used in the proofs of the main theorems. The proofs of these lemmas are placed in Web Appendix A in the Web-based Supplementary Material.

The following notations are used:$H=\text{diag}\left\{I_{p_1+p_3},h{I}_{p_1},b{I}_{p_3}\right\}$, $e_{\beta ,11}\left(t,u\right)=E[w_i\left(t\right)\dot{\phi }\left\{{\vartheta }_{\beta }^T\left(t,u\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\right\}{\left\{{\tilde{X}}_i\left(t\right)\right\}}^{\otimes 2}{\xi }_i\left(t\right){\lambda }_i\left(t\right)]$, and $e_{\beta ,12}\left(t,u\right)=E[w_i\left(t\right)\dot{\phi }\left\{{\vartheta }_{\beta }^T\left(t,u\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\right\}{\tilde{X}}_i\left(t\right){\left(X_{2i}\left(t\right)\right)}^T{\xi }_i\left(t\right){\lambda }_i\left(t\right))]$. Let the random function $\psi :{ℝ}^4\to ℝ$ satisfy: ψ(t, y, x, u) is continuous on {(t, x, u)}, uniformly in $y\in ℝ;\hspace{0.17em}E{\left|\psi \right|}^s<\infty$ for s > 2. Let ψ_i(t) = ψ(t, Y_i(t), X_i(t), U_i(t)). The kernel-weighted average for two-dimensional smoothers is defined as:

$${\Psi }_n\left(t_0,u_0\right)=n^{-1}\sum _{i=1}^n{\int }_{t_1}^{t_2}{\psi }_i\left(t\right)K_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)d{N}_i\left(t\right).$$

Lemma 1. Under Condition A, we have ${\sup }_{t_0\in \left[t_1,t_2\right],u_0\in \left[u_1,u_2\right]}\left|{\Psi }_n\left(t_0,u_0\right)-E\left\{{\psi }_i\left(t_0\right){\xi }_i\left(t_0\right){\lambda }_i\left(t_0\right)|U\left(t_0\right)=u_0\right\}{f}_U\left(t_0,u_0\right)\right|=O_p\left(\frac{\log n}{\sqrt{nhb}}\right)+O\left(h^2+b^2\right)$.

Lemma 2. Let Θ and $\mathcal{U}$ be compact sets in R^p and R^q, let Φ_n(θ, u) be random functions, and let Φ(θ, u) be a fixed function of $\left(\theta ,u\right)\in \Theta \times \mathcal{U}$. Let δ(u) be a fixed function of $u\in \mathcal{U}$ taking values in Θ. Assume that $\underset{\theta ,u}{\sup }\Vert {\Phi }_n\left(\theta ,u\right)-\Phi \left(\theta ,u\right)\Vert \stackrel{\mathcal{P}}{\to }0$ as n → ∞ and that for every ε > 0, we have $\underset{\Vert \theta -{\delta }_0\left(u\right)\Vert >\epsilon }{\inf }\Vert \Phi \left(\theta ,u\right)\Vert >0=\Vert \Phi \left({\delta }_0\left(u\right),u\right)\Vert$ for $u\in \mathcal{U}$. Then for any sequence of estimators $\widehat{\delta }\left(u\right)$, with ${\Phi }_n(\widehat{\delta }\left(u\right),u)=o_p\left(1\right)$ uniformly in $u\in \mathcal{U}$, we have that as n → ∞, $\widehat{\delta }\left(u\right)\stackrel{\mathcal{P}}{\to }{\delta }_0\left(u\right)$ uniformly in $u\in \mathcal{U}$.

Lemma 3. Under Condition A, we have that as n → ∞, $H{\tilde{\vartheta }}^{\ast }\left(t,u,\beta \right)\stackrel{\mathcal{P}}{\to }{({\vartheta }_{\beta }^{\ast T}\left(t,u\right),0^T)}^T$,

$$H\frac{\partial {\tilde{\vartheta }}^{\ast }\left(t,u,\beta \right)}{\partial \beta }\stackrel{\mathcal{P}}{\to }{\left({\left\{-e_{\beta ,11}^{-1}\left(t,u\right)e_{\beta ,12}\left(t,u\right)\right\}}^T,\text{0}\right)}^T$$

uniformly in t ∈ [t₁, t₂], u ∈ [u₁, u₂] and β in a neighbourhood of β₀.

Lemma 4. Under Condition A, as $h\asymp b$, nh² → ∞ and nh⁶ = O_p(1), we have

$$\begin{array}{l}\sqrt{nhb}\left\{\tilde{\vartheta }\left(t_0,u_0,{\beta }_0\right)-{\vartheta }_0\left(t_0,u_0\right)-\frac{1}{2}{h}^2{v}_2{e}_{11}^{-1}\left(t_0,u_0\right)b_{\alpha }\left(t_0,u_0\right)-\frac{1}{2}{b}^2{v}_2{e}_{11}^{-1}\left(t_0,u_0\right)b_{\gamma }\left(t_0,u_0\right)\right\}\\ ={\left(e_{11}\left(t_0,u_0\right)\right)}^{-1}\sqrt{nhb}{\text{A}}_n\left(t_0,u_0\right)+o_p\left(1\right),\end{array}$$ (A.1)

uniformly in t₀ ∈ [t₁, t₂] and u₀ ∈ [u₁, u₂], where

$$b_{\alpha }\left(t_0,u_0\right)=E\left\{w_i\left(t_0\right){\dot{\mu }}_i\left(t_0\right){\tilde{X}}_i\left(t_0\right)X_{1i}^T\left(t_0\right)|U_i\left(t_0\right)=u_0\right\}{f}_U\left(t_0,u_0\right)\ddot{\alpha }\left(t_0\right),$$

$$b_{\gamma }\left(t_0,u_0\right)=E\left\{w_i\left(t_0\right){\dot{\mu }}_i\left(t_0\right){\tilde{X}}_i\left(t_0\right)X_{3i}^T\left(t_0\right)|U_i\left(t_0\right)=u_0\right\}{f}_U\left(t_0,u_0\right)\ddot{\gamma }\left(u_0\right),$$

$${\text{A}}_n\left(t_0,u_0\right)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){ϵ}_i\left(t\right){\tilde{X}}_i\left(t\right)K_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)d{N}_i\left(t\right).$$

Furthermore, $\sqrt{nhb}{\text{A}}_n\left(t_0,u_0\right)=O_p\left(1\right)$, uniformly in t ∈ [t₁, t₂] and u ∈ [u₁, u₂].

Proof of Theorem 1.

By Lemma 1, Lemma 3, and application of the Glivenko-Cantelli theorem to the estimating function (4), we have that as n → ∞,

$$\begin{gathered} n^{-1}{U}_{\beta }\left(\beta \right) \\ \stackrel{\mathcal{P}}{\to }E{\int }_0^{\tau }{W}_i\left(t\right)\left[Y_i\left(t\right)-\phi \left\{{\vartheta }_{\beta }^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\right\}\right]\times \left\{X_{2i}\left(t\right)-{\left(e_{\beta ,12}\left(t,U_i\left(t\right)\right)\right)}^T{\left(e_{\beta ,11}\left(t,U_i\left(t\right)\right)\right)}^{-1}{\tilde{X}}_i\left(t\right)\right\}d{N}_i\left(t\right) \\ =E{\int }_0^{\tau }{W}_i\left(t\right)\left[\phi \left\{{\vartheta }_0^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\right\}-\phi \left\{{\vartheta }_{\beta }^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }^T{X}_{2i}\left(t\right)\right\}\right]\times \left\{X_{2i}\left(t\right)-{\left(e_{\beta ,12}\left(t,U_i\left(t\right)\right)\right)}^T{\left(e_{\beta ,11}\left(t,U_i\left(t\right)\right)\right)}^{-1}{\tilde{X}}_i\left(t\right)\right\}{\xi }_i\left(t\right){\lambda }_i\left(t\right)dt \\ \equiv u\left(\beta \right), \end{gathered}$$ (A.2)

where β₀ is the unique root of u(β). Then by Theorem 5.9 of van der Vaart (1998), $\widehat{\beta }\stackrel{\mathcal{P}}{\to }{\beta }_0$ as n → ∞.

By the Glivenko-Cantelli theorem and Lemma 3, we obtain that as n → ∞,

$$\begin{gathered} -n^{-1}\frac{\partial U_{\beta }\left(\beta \right)}{\partial \beta }|{}_{\beta ={\beta }_0} \\ =n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right)\dot{\phi }\{{\tilde{\vartheta }}^T\left(t,U_i\left(t\right),{\beta }_0\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\}\{{\left(\frac{\partial \tilde{\vartheta }\left(t,U_i\left(t\right),{\beta }_0\right)}{\partial \beta }\right)}^T{\tilde{X}}_i\left(t\right)+X_{2i}\left(t\right)\}d{N}_i\left(t\right)+o_p\left(1\right) \\ \stackrel{\mathcal{P}}{\to }E\left[{\int }_0^{\tau }{w}_i\left(t\right){\dot{\mu }}_i\left(t\right){\left\{X_{2i}\left(t\right)-{\left(e_{12}\left(t,U_i\left(t\right)\right)\right)}^T{\left(e_{11}\left(t,U_i\left(t\right)\right)\right)}^{-1}{\tilde{X}}_i\left(t\right)\right\}}^{\otimes 2}d{N}_i\left(t\right)\right]\equiv A_{\beta }. \end{gathered}$$

Now we show that $n^{-1/2}{U}_{\beta }\left({\beta }_0\right)$ converges in distribution to a normal distribution. By the Taylor expansion,

$$\begin{gathered} \phi \left\{{\tilde{\vartheta }}^T\left(t,U_i\left(t\right),{\beta }_0\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\right\}-\phi \left\{{\vartheta }_0^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\right\} \\ ={\dot{\mu }}_i\left(t\right)\left\{{\tilde{\vartheta }}^T\left(t,U_i\left(t\right),{\beta }_0\right)-{\vartheta }_0^T\left(t,U_i\left(t\right)\right)\right\}{\tilde{X}}_i\left(t\right)+O_p({\Vert \tilde{\vartheta }\left(t,U_i\left(t\right),{\beta }_0\right)-{\vartheta }_0\left(t,U_i\left(t\right)\right)\Vert }^2. \end{gathered}$$ (A.3)

By Lemma 3 and Lemma 4,

$$\begin{gathered} n^{-1/2}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right)\left[\phi \left\{{\tilde{\vartheta }}^T\left(t,U_i\left(t\right),{\beta }_0\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\right\}-\phi \left\{{\vartheta }_0^T\left(t,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)+{\beta }_0^T{X}_{2i}\left(t\right)\right\}\right]\left\{{\left({\tilde{X}}_i\left(t\right)\right)}^T\frac{\partial \tilde{\vartheta }\left(t,U_i\left(t\right),{\beta }_0\right)}{\partial \beta }+{\left(X_{2i}\left(t\right)\right)}^T\right\}d{N}_i\left(t\right) \\ =n^{-1/2}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){\dot{\mu }}_i\left(t\right)\left\{{\tilde{\vartheta }}^T\left(t,U_i\left(t\right),{\beta }_0\right)-{\vartheta }_0^T\left(t,U_i\left(t\right)\right)\right\}\left\{{\tilde{X}}_i\left(t\right){\left({\tilde{X}}_i\left(t\right)\right)}^T\frac{\partial \tilde{\vartheta }\left(t,U_i\left(t\right),{\beta }_0\right)}{\partial \beta }+{\tilde{X}}_i\left(t\right){\left(X_{2i}\left(t\right)\right)}^T\right\}d{N}_i\left(t\right)+o_p\left(1\right) \\ =o_p\left(1\right). \end{gathered}$$ (A.4)

Hence,

$$\begin{gathered} n^{-1/2}{U}_{\beta }\left({\beta }_0\right) \\ =n^{-1/2}\sum _{i=1}^n{W}_i\left(t\right){ϵ}_i\left(t\right)\left\{{\left(\frac{\partial \tilde{\vartheta }\left(t,U_i\left(t\right),{\beta }_0\right)}{\partial \beta }\right)}^T{\tilde{X}}_i\left(t\right)+X_{2i}\left(t\right)\right\}d{N}_i\left(t\right)+o_p\left(1\right) \\ =n^{-1/2}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)d{N}_i\left(t\right)+o_p\left(1\right), \end{gathered}$$

which converges in distribution to a normal distribution with variance ${\Sigma }_{\beta }=E{\left({\int }_{t_1}^{t_2}{w}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)\hspace{1em}d{N}_i\left(t\right)\right)}^{\otimes 2}$. Hence, $n^{1/2}\left(\widehat{\beta }-{\beta }_0\right)\stackrel{\mathcal{D}}{\to }N\left(0,A_{\beta }^{-1}{\Sigma }_{\beta }{A}_{\beta }^{-1}\right)$ as n → ∞.

Proof of Theorem 2.

• aSince $\widehat{\vartheta }\left(t_0,u_0\right)=\tilde{\vartheta }\left(t_0,u_0,\widehat{\beta }\right)$, we have that as n → ∞, $\widehat{\vartheta }\left(t_0,u_0\right)\stackrel{\mathcal{P}}{\to }{\vartheta }_0\left(t_0,u_0\right)$ uniform in t ∈ [0, τ ] and u ∈ [u₁, u₂] by Lemma 1 and Theorem 1. Then

$$\begin{gathered} \underset{t_0\in \left[t_1,t_2\right]}{\sup }\left|\widehat{\vartheta }\left(t_0\right)-{\vartheta }_0\left(t_0\right)\right|=\underset{t_0\in \left[t_1,t_2\right]}{\sup }\left|n^{-1}\sum _{j=1}^n\left\{\widehat{\vartheta }\left(t_0,U_j\left(t_0\right)\right)-{\vartheta }_0\left(t_0,U_j\left(t_0\right)\right)\right\}\right| \\ \le \underset{t_0\in \left[t_1,t_2\right],u_0\in \left[u_1,u_2\right]}{\sup }\left|\widehat{\vartheta }\left(t_0,u_0\right)-{\vartheta }_0\left(t_0,u_0\right)\right|=o_p\left(1\right). \end{gathered}$$

• bFollowing the proof of Lemma 4, we obtain

$$\begin{gathered} \sqrt{nhb}\left\{\tilde{\alpha }\left(t_0,u_0,{\beta }_0\right)-{\alpha }_0\left(t_0,u_0\right)\right\} \\ =-{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,u_0\right)\sqrt{\frac{hb}{n}}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){ϵ}_i\left(t\right){\tilde{X}}_i\left(t\right)K_h\left(t-t_0\right)K_b\left(U_i\left(t\right)-u_0\right)d{N}_i\left(t\right) \\ +\frac{1}{2}\sqrt{nhb}{h}^2{v}_2{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,u_0\right)b_{\alpha }\left(t_0,u_0\right)+\frac{1}{2}\sqrt{nhb}{b}^2{v}_2{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,u_0\right)b_{\gamma }\left(t_0,u_0\right)+o_p\left(\sqrt{nhb}\left(h^2+b^2\right)\right). \end{gathered}$$

Note that $e_{11}^{-1}\left(t_0,u_0\right)b_{\gamma }\left(t_0,u_0\right)$ is zero for the first p₁ components and $e_{11}^{-1}\left(t_0,u_0\right)b_{\alpha }\left(t_0,u_0\right)$ is $\ddot{\alpha }\left(t_0\right)$ for the first p₁ components. Then

$$\begin{gathered} \sqrt{nh}\left\{\widehat{\alpha }\left(t_0\right)-{\alpha }_0\left(t_0\right)-\frac{1}{2}{h}^2{v}_2\ddot{\alpha }\left(t_0\right)\right\} \\ =\sqrt{nh}\left\{\widehat{\alpha }\left(t_0,{\beta }_0\right)-{\alpha }_0\left(t_0\right)-\frac{1}{2}{h}^2{v}_2\ddot{\alpha }\left("t_0\right)\right\}+\sqrt{nh}\left\{\widehat{\alpha }\left(t_0,\widehat{\beta }\right)-\widehat{\alpha }\left(t_0,{\beta }_0\right)\right\}. \end{gathered}$$ (A.5)

By Lemma A.1 of Yin et al. (2008),

$$\frac{1}{n}\sum _{j=1}^n{e}_{11}^{-1}\left(t_0,U_j\left(t_0\right)\right)K_b\left(u-U_j\left(t_0\right)\right)=e_{11}^{-1}\left(t_0,u\right)+O_p\left(\log b/\left(\sqrt{nb}\right)\right)+O\left(b^2\right),$$

uniformly in t ∈ [t₁, t₂] and u ∈ [u₁, u₂]. The first term of (A.5) becomes

$$\begin{gathered} -\sqrt{\frac{h}{n}}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){ϵ}_i\left(t\right)\left\{n^{-1}\sum _{j=1}^n{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,U_j\left(t_0\right)\right){\tilde{X}}_i\left(t\right)K_b\left(U_i\left(t\right)-U_j\left(t_0\right)\right)\right\}{K}_h\left(t-t_0\right)d{N}_i\left(t\right) \\ =\sqrt{\frac{h}{n}}\sum _{i=1}^n{\int }_0^{\tau }{W}_i\left(t\right){ϵ}_i\left(t\right){\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)K_h\left(t-t_0\right)d{N}_i\left(t\right)+o_p\left(1\right). \end{gathered}$$

In addition, by applying Lemma 3 to the second term of (A.5), we have

$$\sqrt{nh}\left\{\widehat{\alpha }\left(t_0\right)-{\alpha }_0\left(t_0\right)-\frac{1}{2}{h}^2{v}_2\ddot{\alpha }\left(t_0\right)\right\}=n^{-1/2}{h}^{1/2}\sum _{i=1}^n{g}_i\left(t_0\right)+o_p\left(1\right),$$

where

$$\begin{gathered} g_i\left(t_0\right)={\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(t\right){\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)K_h\left(t-t_0\right)d{N}_i\left(t\right) \\ +E\left\{-{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,U_i\left(t_0\right)\right)e_{12}\left(t_0,U_i\left(t_0\right)\right)\right\}{A}_{\beta }^{-1}{\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)d{N}_i\left(t\right). \end{gathered}$$

Following the arguments of Lemma 2 of Sun (2010), we obtain that as n → ∞,

$$\sqrt{nh}\left(\widehat{\alpha }\left(t_0\right)-{\alpha }_0\left(t_0\right)-\frac{1}{2}{h}^2{v}_2\ddot{\alpha }\left(t_0\right)\right)\stackrel{\mathcal{D}}{\to }N\left(0,{\Sigma }_{\alpha }\left(t\right)\right),$$ (A.6)

where ${\Sigma }_{\alpha }\left(t_0\right)={\lim }_{n\to \infty }hE{\left[{\int }_0^{\tau }\left\{Y_i\left(t\right)-{\mu }_i\left(t\right)\right\}{\mathcal{I}}_1{e}_{11}^{-1}\left(t_0,U_i\left(t\right)\right){\tilde{X}}_i\left(t\right)K_h\left(t-t_0\right)d{N}_i\left(t\right)\right]}^{\otimes 2}$.

Proof of Theorem 3.

Following the same argument as the proof of Theorem 2, we have that as n → ∞, $\widehat{\gamma }\left(u\right)\stackrel{\mathcal{P}}{\to }{\gamma }_0\left(u\right)$ uniformly in u ∈ [u₁, u₂], and $\sqrt{nb}\left(\widehat{\gamma }\left(u\right)-{\gamma }_0\left(u\right)-\frac{1}{2}{b}^2{v}_2\ddot{\gamma }\left(u\right)\right)\stackrel{\mathcal{D}}{\to }N\left(0,{\Sigma }_{\gamma }\left(u\right)\right)$. □

Proof of Theorem 4.

By Lemma 4,

$$\sqrt{nh}\left\{\widehat{\gamma }\left(u\right)-{\gamma }_0\left(u\right)-\frac{1}{2}{b}^2{v}_2\ddot{\gamma }\left(u\right)\right\}=\sqrt{nh}\left\{\tilde{\gamma }\left(u,{\beta }_0\right)-{\gamma }_0\left(u\right)-\frac{1}{2}{b}^2{v}_2\ddot{\gamma }\left(u\right)\right\}-\sqrt{nh}{\mathcal{I}}_{3}E\left\{{\left(e_{11}\left(U_j^{-1}\left(u\right),u\right)\right)}^{-1}{e}_{12}\left(U_j^{-1}\left(u\right),u\right)\right\}\left\{\widehat{\beta }-{\beta }_0\right\}.$$

Thus,

$$G_n\left(u\right)=n^{-1/2}\sum _{i=1}^n[{\int }_{u_1}^u{\int }_0^{\tau }{w}_i\left(t\right){ϵ}_i\left(t\right){\mathcal{I}}_3{e}_{11}{\left(t,s\right)}^{-1}{\tilde{X}}_i\left(t\right)K_b\left(U_i\left(t\right)-s\right)d{N}_i\left(t\right)ds-{\int }_{u_1}^u{\mathcal{I}}_{3}E\left\{e_{11}^{-1}\left(U_j^{-1}\left(s\right),s\right)e_{12}\left(U_j^{-1}\left(s\right),s\right)\right\}ds{A}_{\beta }^{-1}{\int }_{t_1}^{t_2}{w}_i\left(t\right){ϵ}_i\left(t\right)R_i\left(t\right)d{N}_i\left(t\right)]+o_p\left(1\right),$$

which converges weakly to a mean-zero Gaussian process by central limit theorem. □