Regression analysis of multiplicative hazards model with time-dependent coefficient for sparse longitudinal covariates

Abstract

We study the multiplicative hazards model with intermittently observed longitudinal covariates and time-varying coefficients. For such models, the existing ad hoc approach, such as the last value carried forward, is biased. We propose a kernel weighting approach to get an unbiased estimation of the non-parametric coefficient function and establish asymptotic normality for any fixed time point. Furthermore, we construct the simultaneous confidence band to examine the overall magnitude of the variation. Simulation studies support our theoretical predictions and show favorable performance of the proposed method. A data set from Alzheimer’s Disease Neuroimaging Initiative study is used to illustrate our methodology.

1. Introduction

In clinical trials and epidemiological studies, it is of interest to explore the relationship between longitudinally collected covariates and time-to-event outcomes. The celebrated proportional hazards model postulates a multiplicative relationship between covariates and the hazard function:

$$\lambda \left\{t∣Z\right\}={\lambda }_0\left(t\right)\text{exp}\left\{{\beta }_0^{\text{T}}Z\right\},$$ (1.1)

where ${\lambda }_0\left(t\right)$ is an unspecified baseline hazard function, $Z\in {\mathbb{R}}^p$ is the covariate and ${\beta }_0\in {\mathbb{R}}^p$ is the unknown regression coefficient (Cox, 1972). In (1.1), it is assumed that the hazard ratio ${\beta }_0$ is constant. In practice, this assumption can be violated, as demonstrated in Stensrud and Hernán (2020). To accommodate hazard ratio that varies with respect to time, Tian et al. (2005) proposed a model that replaces ${\beta }_0$ by ${\beta }_0\left(t\right)$ in (1.1) and developed estimating equations for statistical inference. In Tian et al. (2005), the covariate $Z$ can be time-dependent, denoted as $Z\left(t\right)$, and it is assumed that the entire trajectory of $Z\left(t\right)$ is available. For longitudinally collected $Z\left(t\right)$, only intermittent values are available.

An ad hoc approach to deal with longitudinally collected $Z\left(t\right)$ is the last value carried forward, where the most recently observed longitudinal covariate is imputed as the current value for each subject. First, this assumes that $Z\left(t\right)$ does not change from the time of the last measurement. Second, the uncertainty inherent in the imputation is not considered, as the imputed value is treated indiscriminately with observed data. As a result, substantial bias can arise, which leads to erroneous inferences, as shown in Andersen and Liestøl (2003); Molnar et al. (2009); Cao et al. (2015a); Cao and Fine (2021).

As an alternative, a joint modeling strategy is commonly adopted (Wulfsohn and Tsiatis, 1997). In the joint modeling approach, the longitudinal measurement is assumed to follow a linear mixed model with normal measurement error (Laird and Ware, 1982), and the failure time is modeled through the proportional hazards model. The time-dependent covariate is taken as the unobserved longitudinal process (Tsiatis and Davidian, 2001). Statistical inference is carried out by likelihood or conditional likelihood. Furthermore, Song and Wang (2008) and Andrinopoulou et al. (2018) extended the constant hazard ratio in classic Cox model to a time-varying one. A recent review of the joint modeling approach can be found in Rizopoulos (2012). As a likelihood-based method, joint modeling imposes rather strong modeling assumptions, and inferences are quite complicated (Song et al., 2002; Rizopoulos et al., 2009). Furthermore, if the model is misspecified, bias occurs, and standard deviations may not be computable (Cao et al., 2015a; Arisido et al., 2019). Despite the fact that joint modeling imposes strong modeling assumptions and has complicated computation and inference, it allows summary trends, such as slope or spread, to enter the survival model as covariates. Such features make the model more coherent and interpretable.

In this paper, we propose a varying coefficient model for censored outcomes with intermittently observed time-dependent covariates. The hazard function is specified as follows:

$$\lambda \left\{t∣Z\left(r\right),r\le t\right\}={\lambda }_0\left(t\right)e^{{\beta }_0\left(t{)}^{\text{T}}Z\right(t)},$$ (1.2)

where ${\lambda }_0\left(t\right)$ is the baseline hazard function, $Z\left(t\right)$ is the longitudinally collected time-dependent covariates and ${\beta }_0\left(t\right)$ is a non-parametric function describing the multiplicative relationship between $Z\left(t\right)$ and the hazard function. This flexible modeling framework allows the hazard ratio to change over time, which is more informative and realistic for many practical situations. A naïve approach that imputes $Z\left(t\right)$ at failure time by the most recent longitudinal observation and implements the method in Tian et al. (2005) results in biased coefficient estimation. The bias does not attenuate with increased sample size, as demonstrated in our simulation studies. We propose an estimating equation-based weighting strategy without imposing stringent distributional assumptions on the longitudinal process. To estimate ${\beta }_0\left(t\right)$ in (1.2), we use a two-dimensional kernel function to do smoothing. At any fixed time point $t$, we estimate ${\beta }_0\left(t\right)$ by solving an estimating equation, where higher weights are given to the longitudinal observations that have measurement times close to the failure time, and lower weights are given to those that have measurement times far from the failure time. Unlike the last value carried forward method, which imputes the most recent observation regardless of the distance between its measurement time and the failure time, by adaptive weighting, we get an asymptotically unbiased coefficient estimation. With time-invariant coefficient ${\beta }_0$, Cao et al. (2015a); Cao and Fine (2021) studied the proportional hazards model (1.1) for sparse longitudinal covariates.

Moreover, we construct simultaneous confidence bands (SCBs) for ${\beta }_0\left(t\right)$. Specifically, for a pre-specified confidence level $1-\alpha$, we aim to find random functions $L\left(t\right)$ and $U\left(t\right)$ such that

$$P\left\{L\left(t\right)\le {\beta }_0\left(t\right)\le U\left(t\right),t\in \left[b_1,b_2\right]\right\}\to 1-\alpha$$

as the number of subjects $n\to \infty$ in some closed interval $\left[b_1,b_2\right]$. Unlike point-wise confidence intervals, a simultaneous confidence band covers the underlying non-parametric function with a pre-specified probability. For a non-parametric function, it is more informative to evaluate the overall pattern and magnitude of the variation. In addition, a confidence band is often graphically intuitive and versatile, especially when investigators do not know a priori the precise hypothesis of interest.

To construct SCBs, one has to derive the asymptotic distribution of the maximum deviations between the estimated and the true coefficient functions. The convergence of such asymptotics is slow, of log $n$ rate (Cao et al., 2018). In practice, a multiplier bootstrap, where standard normal variates are introduced into an appropriate statistic, keeping the data fixed, is usually adopted. The distribution of the limiting process is approximated via a large number of realizations, repeatedly generating standard normal variates (Lin et al., 1993; Lin, 1997). Later developments include using Poisson multipliers and other zero mean and unit variance multipliers (Tian et al., 2005; Beyersmann et al., 2013; Dobler and Pauly, 2014; Dobler et al., 2019).

The rest of the paper is organized as follows. In Section 2, we use a kernel-weighted estimating equation to estimate the non-parametric function ${\beta }_0\left(t\right)$ in (1.2). We establish its asymptotic normality for any fixed time point. Furthermore, we construct a simultaneous confidence band to evaluate the overall magnitude of variation. A series of simulation studies in Section 3 illustrate that the proposed method works well in finite samples and has improved performance over the last value carried forward approach and joint modeling approach. Section 4 applies the proposed method to analyze a dataset from the Alzheimer’s Disease Neuroimaging Initiative study. Concluding remarks are given in Section 5. All proofs are relegated to the Supplementary Material.

2. Estimation and Inference

2.1. Problem Set Up and Notations

Suppose that we have a random sample of $n$ independent subjects. For the $i$-th subject, let $T_i$ denote the failure time, and $C_i$ denote the censoring time. It is assumed that censoring is coarsened at random such that $T_i$ and $C_i$ are independent given the covariate process $Z_i(\cdot )$ Heitjan and Rubin, 1991). Denote $X_i=\text{min}\left(T_i,C_i\right)$ and ${\delta }_i=I\left(T_i\le C_i\right)$. The $p$-dimensional covariate process $Z_i(\cdot )$ may include both time-independent and time-varying covariates. It is assumed that the time-varying covariates are observed at the same time points within individuals. The longitudinal covariates are observed at $M_i$ observation times $R_{ik},k=1,\dots ,M_i$, where $M_i$ is assumed finite with probability one. The observed data consist of $n$ independent realizations of $\left(X_i,{\delta }_i,R_{ik},Z_i\left(R_{ik}\right),k=1,\dots ,M_i\right),i=1,\dots ,n$. We use the counting process to denote $N_i\left(t\right)=I\left(X_i\le t,{\delta }_i=1\right),Y_i\left(t\right)=I\left(X_i\ge t\right)$, and $N_i^{*}\left(t\right)=\sum _{k=1}^{M_i}I\left\{R_{ik}\le t\right\},i=1,\dots ,n$.

2.2. Estimation

We first write the partial likelihood with a time-varying coefficient function as follows.

$$L_n\left\{\beta \left(t\right),t\right\}=\prod _{i=1}^n{\left\{\frac{e^{\beta \left(t{)}^T{Z}_i\right(t)}}{\sum _{j=1}^n{Y}_j\left(t\right)e^{\beta \left(t{)}^T{Z}_j\right(t)}}\right\}}^{\Delta N_i\left(t\right)},$$

where

$$\Delta N_i\left(t\right)=\left\{\begin{array}{ll}1& \text{if}{N}_i\left(t\right)-N_i(t-)=1\\ 0& \text{otherwise}.\end{array}\right.$$

The log partial likelihood function is

$$l_n\left\{\beta \left(t\right),t\right\}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\left[\beta (t{)}^T{Z}_i(t)-\text{log}\left\{\sum _{j=1}^n{Y}_j\left(t\right)e^{\beta \left(t{)}^T{Z}_j\right(t)}\right\}\right]d{N}_i\left(t\right),$$ (2.3)

where $\tau$ is the pre-specified maximum observation time. With a time-independent coefficient $\beta$, Cao et al. (2015a) proposed a kernel smoothing approach to handle the sparse longitudinal covariates. When the entire trajectory of $Z\left(t\right)$ is assumed to be known, Tian et al. (2005) proposed to estimate the time-varying effect ${\beta }_0\left(t\right)$ in (1.2) through kernel smoothing. To simultaneously incorporate both the time-varying effect and sparsely observed longitudinal covariates, we use a bivariate kernel function to perform the smoothing. Specifically, we propose the following estimating equation

$$U_n\left\{\beta \right(s\left)\right\}=n^{-1}\sum _{i=1}^n\sum _{k=1}^{M_i}{\int }_0^{\tau }{K}_{h_1,h_2}\left(t-s,R_{ik}-s\right)\left[Z_i\left(R_{ik}\right)-\bar{Z}\left\{\beta \right(s),t\}\right]d{N}_i\left(t\right),$$ (2.4)

where $K_{h_1,h_2}(t,s)=K\left(t/h_1,s/h_2\right)/\left(h_1{h}_2\right),K(\cdot ,\cdot )$ is a bivariate kernel function, and

$$\bar{Z}\left\{\beta \right(s),t\}=\frac{S^{\left(1\right)}\left\{\beta \right(s),t\}}{S^{\left(0\right)}\left\{\beta \right(s),t\}},$$

$$S^{\left(l\right)}\left\{\beta \right(s),t\}=\frac{1}{n}\sum _{j=1}^n\sum _{k=1}^{M_j}{K}_{h_1,h_2}\left(t-s,R_{jk}-s\right)Y_j\left(t\right)Z_j{\left(R_{jk}\right)}^{\otimes l}\text{exp}\left\{\beta (s{)}^T{Z}_j\left(R_{jk}\right)\right\},l=\mathrm{0,1},2,$$

where $a^{\otimes 0}=1,a^{\otimes 1}=a$, and $a^{\otimes 2}=a{a}^T$.

Denote

$$\text{E}\left[d{N}^{*}\left(t\right)\right]={\lambda }^{*}\left(t\right)dt,$$ (2.5)

where ${\lambda }^{*}\left(t\right)$ is positive and twice continuously differentiable for all $t\in [0,\tau ]$. Define

$$s^{\left(l\right)}\left\{\beta \right(t),t\}=\text{E}\left[Y\left(t\right)Z(t{)}^{\otimes l}\text{exp}\left\{\beta \left(t{)}^{T}Z\right(t)\right\}\right]{\lambda }^{*}\left(t\right),$$

as the limit of $S^{\left(l\right)}\left\{\beta \right(t),t\},l=\mathrm{0,1},2$. For any fixed time point $s\in [0,\tau ]$, we weigh the contribution from the longitudinally observed covariate process and the failure time by their distances to $s$ using two bandwidths. Solving the estimating equation (2.4), we obtain $\hat{\beta }\left(s\right)$ as the estimation of the non-parametric regression function ${\beta }_0\left(s\right)$ in (1.2). In practice, we use a quasi-Newton method (Broyden, 1965) to solve the estimating equation (2.4). To state its asymptotic properties, we need the following conditions.

• (A1)(2.5) holds. The longitudinal observation process $N_i^{*}(\cdot )$ is independent of the observed longitudinal covariate $Z_i\left(R_{ik}\right),i=1,\dots ,n;k=1,\dots ,M_i$. In addition, censoring time is non-informative in the sense that $C_i$ is independent of longitudinal observation time $R_{ik}$ and the observed longitudinal covariate $Z_i\left(R_{ik}\right)$ in addition to that $T_i$ and $C_i$ are independent given the covariate process $Z_i(\cdot ),i=1,\dots ,n;k=1,\dots ,M_i$. Moreover, with the pre-specified constant $\tau ,N^{*}\left(\tau \right)$ is bounded by a finite constant. Additionally, we require $\tau$ to satisfy $P(X\ge \tau )>0$.
• (A2)
  For any fixed time point $t\in [0,\tau ],B\left\{{\beta }_0\left(t\right),t\right\}$ is non-singular, where

  $$B\left\{{\beta }_0\left(t\right),t\right\}=\left[s^{\left(2\right)}\left\{{\beta }_0\left(t\right),t\right\}-\frac{s^{\left(1\right)}{\left\{{\beta }_0\left(t\right),t\right\}}^{\otimes 2}}{s^{\left(0\right)}\left\{{\beta }_0\left(t\right),t\right\}}\right]{\lambda }_0\left(t\right).$$
• (A3)For any fixed time point $s\in [0,\tau ],\text{E}\left[Z\left(s+t_2\right)Y\left(s+t_1\right)e^{{\beta }_0{\left(s+t_1\right)}^{T}Z\left(s+t_1\right)}\right]$ and $\text{E}\left[\bar{Z}\left\{{\beta }_0\left(s\right),s+t_1\right\}Y\left(s+t_1\right)e^{{\beta }_0{\left(s+t_1\right)}^{T}Z\left(s+t_1\right)}\right]$ are twice continuously differentiable for $\left(t_1,t_2\right)\in [0,\tau {]}^{\otimes 2}$ and $\left(s+t_1,s+t_2\right)\in [0,\tau {]}^{\otimes 2}$.
• (A4)$K(x,y)$ is a symmetric bivariate density function. In addition, $\iint x^{2}K(x,y)\mathit{\text{dxdy}}<\infty ,\iint y^{2}K(x,y)\mathit{\text{dxdy}}<\infty$, and $\int K(x,y{)}^2\mathit{\text{dxdy}}<\infty$. Moreover, as $n\to \infty ,n{h}_1{h}_2\to \infty$ and ${\left(n{h}_1{h}_2\right)}^{1/2}\left(h_1^2+h_2^2\right)\to 0$.
• (A5)The covariate process $Z\left(t\right)$ has bounded total variation on $[0,\tau ]$ almost surely.

Condition (A1) assumes that the observational time and the censoring time is noninformative. Analogous assumptions have been assumed in (Cao and Fine, 2021; Sun et al., 2022). The assumption $P(X\ge \tau )>0$ guarantees that $E\left[Y\left(t\right)\text{exp}\left\{{\beta }_0\left(t{)}^{T}Z\right(t)\right\}\right]{\lambda }^{*}\left(t\right)>0,\forall t\in [0,\tau ]$, which implies that the limit of the denominator of $\bar{Z}\left\{{\beta }_0\left(t\right),t\right\}$ in equation (2.4) is bounded away from 0 on $[0,\tau ]$ (Andersen and Gill, 1982). Condition (A2) ensures the identifiability of ${\beta }_0\left(t\right)$ at any fixed time point $t\in [0,\tau ]$. Condition (A3) posits smoothness assumptions on the expectation of certain functions of the covariate process. Condition (A4) specifies valid kernels and bandwidths. Condition (A5) is common for time-dependent covariates.

The asymptotic property of the non-parametric function $\hat{\beta }(\cdot )$ is detailed in the following theorem:

Theorem 1. Under conditions (A1)-(A5), for any fixed time point $s\in [h,\tau -h]$, where $h=h_1\vee h_2$ is a small positive number, the asymptotic distribution of $\hat{\beta }\left(s\right)$ satisfies

$${\left(n{h}_1{h}_2\right)}^{1/2}B\left\{{\beta }_0\left(s\right),s\right\}\left\{\hat{\beta }\left(s\right)-{\beta }_0\left(s\right)\right\}\stackrel{d}{\to }N\left(0,\Sigma \left({\beta }_0\left(s\right),s\right)\right)$$

where

$$\Sigma \left({\beta }_0\left(s\right),s\right)=\iint K{\left(z_1,z_2\right)}^2\left\{s^{\left(2\right)}\left({\beta }_0\left(s\right),s\right)-\frac{s^{\left(1\right)}{\left({\beta }_0\left(s\right),s\right)}^{\otimes 2}}{s^{\left(0\right)}\left({\beta }_0\left(s\right),s\right)}\right\}{\lambda }_0\left(s\right)d{z}_{1}d{z}_2.$$

In practice, we use the estimating equation (2.4) and estimate $\Sigma \left({\beta }_0\left(s\right),s\right)$ by

$$\hat{\Sigma }\left(\hat{\beta }\left(s\right),s\right)=n^{-2}\sum _{i=1}^n{\left(\sum _{k=1}^{M_i}{\int }_0^{\tau }{K}_{h_1,h_2}\left(u-s,R_{ik}-s\right)\left[Z_i\left(R_{ik}\right)-\bar{Z}\left\{\hat{\beta }\left(s\right),u\right\}\right]d{N}_i\left(u\right)\right)}^{\otimes 2}.$$

The variance of $\hat{\beta }\left(s\right)$ can be estimated by the sandwich formula

$${\left({\left.\frac{\partial U_n\left\{\beta \right(s\left)\right\}}{\partial \beta \left(s\right)}\right|}_{\beta \left(s\right)=\hat{\beta }\left(s\right)}\right)}^{-1}\hat{\Sigma }\left(\hat{\beta }\right(s),s){\left({\left.\frac{\partial U_n\left\{\beta \right(s\left)\right\}}{\partial \beta \left(s\right)}\right|}_{\beta \left(s\right)=\hat{\beta }\left(s\right)}\right)}^{-1}.$$ (2.6)

In the proof presented in the Supplementary Material, the asymptotic bias is of order ${\left(n{h}_1{h}_2\right)}^{1/2}{O}_p\left(h_1^2+h_1{h}_2+h_2^2\right)$, which vanishes under (A4). Under (A4), the proposed estimator $\hat{\beta }\left(s\right)$ is consistent for any $s\in [h,\tau -h]$.

Corollary 1. Under conditions (A1)-(A5), the sandwich formula (2.6) consistently estimates the variance of $\hat{\beta }\left(s\right)$, for any fixed time point $s\in [h,\tau -h]$, where $h=h_1\vee h_2$ is a small positive number.

2.3. The Construction of Simultaneous Confidence Band

For the unknown function ${\beta }_0(\cdot )$, it is more informative to construct a simultaneous confidence band in addition to pointwise inference in Theorem 1. Specifically, for a pre-specified $\alpha$, and a smooth function $l\left(t\right)\in R^p$, we aim to find smooth random functions $L\left(t\right)$ and $U\left(t\right)$ that satisfy

$$P\left\{L\left(t\right)\le l\left(t{)}^T{\beta }_0\right(t)\le U(t),\forall t\in [h,\tau -h]\right\}\to 1-\alpha$$

as number of subjects $n\to \infty$. To obtain such SCBs for $l\left(t{)}^T{\beta }_0\right(t),t\in [h,\tau -h]$, we need to derive a large-sample approximation to the distribution of

$${𝒮}_{SCB}=\underset{t\in [h,\tau -h]}{\text{sup}}\hat{w}\left(t\right)\left|l(t{)}^T\left\{\hat{\beta }\left(t\right)-{\beta }_0\left(t\right)\right\}\right|,$$ (2.7)

where $\hat{w}\left(t\right)$ is a possibly data-dependent, positive weight function. It converges uniformly to a deterministic function. For instance, when ${\beta }_0\left(t\right)$ is a scalar, we can define $\left\{\hat{w}\right(t){\}}^{-1}$ as the estimated standard error of $\hat{\beta }\left(t\right)$. Such weighting prevents the domination of time points with large variances, which may lead to unnecessarily wider confidence band.

Inspired by Tian et al. (2005) and Dobler et al. (2019), we consider a stochastic perturbation of the estimating equation.

$${\tilde{U}}_n\left\{\beta \right(s\left)\right\}=n^{-1}\sum _{i=1}^n\sum _{k=1}^{M_i}{\int }_0^{\tau }{K}_{h_1,h_2}\left(t-s,R_{ik}-s\right)\left[Z_i\left(R_{ik}\right)-\bar{Z}\left\{\beta \right(s),t\}\right]{\xi }_{i}d{N}_i\left(t\right),$$ (2.8)

where ${\xi }_i(i=1,\dots ,n)$ are i.i.d. random variables with zero mean and unit variance and are independent of the data $\left\{\left(X_i,{\delta }_i,Z_i\left(R_{ik}\right),R_{ik},k=1,\dots ,M_i\right),i=1,\dots ,n\right\}$. For example, ${\xi }_i$ can be $N\left(\mathrm{0,1}\right)$ or the Rademacher variable, which takes values +1 and −1 with equal probability. Then, conditional on the data $\left\{\left(X_i,{\delta }_i,Z_i\left(R_{ik}\right),R_{ik},k=1,\dots ,M_i\right),i=1,\dots ,n\right\}$, the distribution of

$${\widetilde{𝒮}}_{SCB}=\underset{s\in [h,\tau -h]}{\text{sup}}\hat{w}\left(t\right)\left|l\left(s{)}^{T}I\right\{\hat{\beta }\left(s\right){\}}^{-1}{\tilde{U}}_n\left\{\hat{\beta }\right(s\left)\right\}\right|$$ (2.9)

can be used to approximate the unconditional distribution of ${𝒮}_{SCB}$ in (2.7), where

$$I\left\{\hat{\beta }\right(s\left)\right\}=\frac{-1}{n}\sum _{i=1}^n\sum _{k=1}^{M_i}{\int }_0^{\tau }{K}_{h_1,h_2}\left(t-s,R_{ik}-s\right)Y_i\left(t\right)\left[\frac{S^{\left(2\right)}\left\{\hat{\beta }\right(s),t\}}{S^{\left(0\right)}\left\{\hat{\beta }\right(s),t\}}-{\left(\frac{S^{\left(1\right)}\left\{\hat{\beta }\right(s),t\}}{S^{\left(0\right)}\left\{\hat{\beta }\right(s),t\}}\right)}^{\otimes 2}\right]dt.$$

We repeat this process $B$ times to obtain $\left\{{\widetilde{𝒮}}_{SCB}^{\left(1\right)},\dots ,{\widetilde{𝒮}}_{SCB}^{\left(B\right)}\right\}$. For instance, we can take $B=\mathrm{5,000}$. Denote its $1-\alpha$ empirical percentile as $c_{\alpha }$. The SCB for $l\left(t{)}^T{\beta }_0\right(t)$ can be written as

$$l\left(t{)}^T\hat{\beta }\right(t)\pm c_{\alpha }\{\hat{w}\left(t\right){\}}^{-1}.$$ (2.10)

This is computationally efficient as we do not need to find the non-linear function $\hat{\beta }\left(t\right)B$ times. We only need to evaluate ${\tilde{U}}_n\{\cdot \}$ at $\hat{\beta }\left(t\right)B$ times.

In practice, for instance, if we are interested in the regression coefficient function of the first covariate, we can take $l\left(t\right)=(1,\dots ,0)$. We provide a theoretical justification of this procedure in the Supplementary Material. Numerical studies show that this procedure works well with a moderate sample size, and the nominal coverage can be obtained.

2.4. Bandwidth Selection

Choosing a suitable bandwidth is practically important. We propose to estimate the bias and variability separately and choose the bandwidth that minimizes the mean squared error (CaO et al., 2015b). For a fixed time point $t\in [h,\tau -h]$, we regress $\hat{\beta }\left(h_1,h_2,t\right)$ on $b={\left(h_1^2,h_1{h}_2,h_2^2\right)}^T$ in a reasonable range of the bandwidths to obtain the slope estimate $\hat{C}={\left(c_1,c_2,c_3\right)}^T$ for bias term calculation. For the variance term, we split the data randomly into two parts and obtain regression coefficient estimates ${\hat{\beta }}_1\left(h_1,h_2,t\right)$ and ${\hat{\beta }}_2\left(h_1,h_2,t\right)$ based on each part. The variance of $\hat{\beta }\left(h_1,h_2,t\right)$ is then estimated by $\hat{V}\left(h_1,h_2,t\right)={\left\{{\hat{\beta }}_1\left(h_1,h_2,t\right)-{\hat{\beta }}_2\left(h_1,h_2,t\right)\right\}}^2/4$. Using both $\hat{C}$ and $\hat{V}\left(h_1,h_2,t\right)$, we thus calculate the mean-squared error as ${\left({\hat{C}}^{T}b\right)}^2+\hat{V}\left(h_1,h_2,t\right)$ for $t$. We repeat the above procedure at equally spaced time points and sum them up to obtain integrated mean-square errors. The optimal bandwidth is the one that minimizes this summation.

3. Numerical Studies

In this section, we evaluate finite sample performance of the proposed method through simulations. We repeatedly generate 1, 000 datasets with sample sizes 400 and 900. The end of follow-up time $\tau$ is specified as 1.

3.1. Data Generating Process

We first generate the time dependent covariate process $Z\left(t\right)$ based on a Gaussian process with mean $-1-2(t-1{)}^2$ and variance covariance matrix $\text{Cov}\left\{Z\right(t),Z(s\left)\right\}=e^{-|t-s|}$, where $t,s\in \left(\mathrm{0,1}\right)$. Specifically, we generate the covariate process through the piecewise constant function

$$Z\left(t\right)=\sum _{i=1}^{20}I\left\{i-1/20\le t<i/20\right\}{z}_i,$$

where ${\left(z_i\right)}_{i=1}^{20}$ follows a multivariate normal distribution with mean $-1-2\left(\right(i-1)/20-1{)}^2$ and variance 1. The covariance between $z_i$ and $z_j$ is $e^{-|i-j|/20}$. The number of observations is generated from $\text{Pois}\left(5\right)+1$, and the observational times are generated from $𝒰\left(\mathrm{0,1}\right)$, where Pois(5) means a Poisson distributed random variable with mean 5 and $𝒰\left(\mathrm{0,1}\right)$ means standard uniform distribution.

The failure time $T$ is generated from the multiplicative hazards model with varying-coefficient as follows.

$$\lambda \left\{t∣Z\left(s\right),s\le t\right\}={\lambda }_0\left(t\right)\text{exp}\left\{{\beta }_0\left(t{)}^{T}Z\right(t)\right\},$$

where ${\lambda }_0\left(t\right)=2+0.1t$ and ${\beta }_0\left(t\right)=0.5\text{sin}\left(2\pi t\right)$. To generate the failure time, we first generate a random variable $u$ from $𝒰\left(\mathrm{0,1}\right)$. Then, we solve $S\left(t\right)=u$, where the survival function takes the form

$$S\left(t\right)=\text{exp}\left\{-{\int }_0^t\lambda \left\{v∣Z\left(u\right),u\le v\right\}dv\right\}=\text{exp}\left\{-{\int }_0^t{\lambda }_0\left(v\right)\text{exp}\left\{{\beta }_0\left(v{)}^{T}Z\right(v)\right\}dv\right\}.$$

To approximate the integral in $S\left(t\right)$, we use Gauss Legendre quadrature (Abramowitz and Stegun, 1983), which is implemented using package gaussquad in R. The censoring time is generated from $\text{min}\left(1,C^{*}\right)$, where $C^{*}~𝒰(\gamma ,1.5)$ giving censoring percentages of 15% and 35%, respectively, by changing $\gamma$. For each fixed time point $t$, we construct confidence intervals by solving for (2.4) to obtain an estimate of ${\beta }_0\left(t\right)$ and an estimate of variance with sandwich form (2.6)

3.2. Comparison with Last Value Carried Forward and Joint Modeling Approach

With time-dependent covariate, if the longitudinal covariate is unavailable at the failure time, the most recently observed longitudinal covariate is used instead. Such imputation is intuitively appealing yet ignores the dynamics of the longitudinal process yielding biased results. Joint modeling is the most widely used method to simultaneously handle longitudinal covariates and censored outcome. Despite its popularity, the modeling assumptions are quite strong with complicated inferences and computations. In this subsection, we compare the ad-hoc last value carried forward method and the commonly used joint modeling approach with the proposed kernel weighting approach.

First, we compare our method with the last value carried forward method. We report the results with $h_1=n^{-0.35}$ and $h_2=n^{-0.35}$ or $n^{-0.45}$, since those bandwidths give stable results. Results based on the automatic bandwidth selection rule are also provided. We summarize the results based on the proposed method for different sample sizes and censoring rates in Table 1, where “auto” refers to bandwidths determined using the adaptive selection technique described in Section 2.4. The biases are small at different time points. The $\hat{\beta }\left(t\right)$ variance estimator is accurate. The coverage probabilities are close to the nominal one. Estimation based on data adaptive bandwidth selection has satisfactory performance. The performance of estimators under varied censoring rates does not differ much. The performance improves with increased sample size.

Table 1:

Simulation results of $\hat{\beta }\left(s\right)$.

$\mathit{s}$	$\mathit{n}$	${\mathit{h}}_1$	${\mathit{h}}_2$	Censoring rate is 15%				Censoring rate is 35%
				Bias	SE	SD	CP	Bias	SE	SD	CP
Proposed method
0.2	400	$n^{-0.35}$	$n^{-0.35}$	−0.072	0.171	0.169	91.6	−0.073	0.177	0.174	91.9
		$n^{-0.35}$	$n^{-0.45}$	−0.053	0.207	0.203	93.4	−0.053	0.217	0.209	92.9
		auto		−0.058	0.198	0.193	93.3	−0.060	0.207	0.197	92.2
	900	$n^{-0.35}$	$n^{-0.35}$	−0.042	0.150	0.146	93.3	−0.044	0.153	0.150	92.7
		$n^{-0.35}$	$n^{-0.45}$	−0.030	0.187	0.181	93.2	−0.031	0.193	0.186	92.2
		auto		−0.031	0.178	0.171	93.3	−0.033	0.182	0.174	92.5
0.4	400	$n^{-0.35}$	$n^{-0.35}$	−0.056	0.156	0.153	92.0	−0.053	0.189	0.176	90.6
		$n^{-0.35}$	$n^{-0.45}$	−0.052	0.194	0.183	91.3	−0.046	0.232	0.210	90.8
		auto		−0.055	0.190	0.178	90.2	−0.049	0.216	0.199	90.3
	900	$n^{-0.35}$	$n^{-0.35}$	−0.042	0.139	0.130	91.2	−0.040	0.156	0.147	91.5
		$n^{-0.35}$	$n^{-0.45}$	−0.040	0.173	0.162	92.0	−0.038	0.195	0.183	92.2
		auto		−0.040	0.164	0.154	91.6	−0.038	0.184	0.171	92.2
0.6	400	$n^{-0.35}$	$n^{-0.35}$	0.039	0.133	0.129	93.0	0.044	0.166	0.163	92.9
		$n^{-0.35}$	$n^{-0.45}$	0.035	0.160	0.153	92.4	0.042	0.198	0.194	93.0
		auto		0.036	0.152	0.147	93.1	0.041	0.187	0.184	93.0
	900	$n^{-0.35}$	$n^{-0.35}$	0.027	0.106	0.105	93.8	0.029	0.134	0.133	92.7
		$n^{-0.35}$	$n^{-0.45}$	0.021	0.134	0.131	93.3	0.023	0.167	0.165	92.6
		auto		0.021	0.126	0.124	93.8	0.025	0.158	0.154	92.1
0.8	400	$n^{-0.35}$	$n^{-0.35}$	0.038	0.201	0.189	91.3	0.022	0.285	0.250	90.0
		$n^{-0.35}$	$n^{-0.45}$	0.024	0.241	0.225	92.4	−0.006	0.347	0.307	90.8
		auto		0.027	0.231	0.214	91.5	0.006	0.322	0.282	90.1
	900	$n^{-0.35}$	$n^{-0.35}$	0.031	0.166	0.158	92.1	0.013	0.229	0.208	92.7
		$n^{-0.35}$	$n^{-0.45}$	0.018	0.209	0.197	92.6	−0.010	0.287	0.260	91.5
		auto		0.021	0.197	0.186	92.1	−0.003	0.269	0.243	91.5
LVCF
0.2	400	$n^{-0.35}$		−0.094	0.167	0.160	89.0	−0.096	0.174	0.164	88.4
	900	$n^{-0.35}$		−0.077	0.128	0.125	89.7	−0.078	0.133	0.129	88.9
0.4	400	$n^{-0.35}$		−0.082	0.127	0.123	88.2	−0.076	0.143	0.139	90.1
	900	$n^{-0.35}$		−0.071	0.093	0.095	88.5	−0.069	0.107	0.107	89.8
0.6	400	$n^{-0.35}$		0.088	0.095	0.095	83.7	0.096	0.119	0.119	85.4
	900	$n^{-0.35}$		0.080	0.069	0.071	78.8	0.084	0.088	0.089	83.0
0.8	400	$n^{-0.35}$		0.112	0.139	0.134	83.5	0.100	0.192	0.180	87.3
	900	$n^{-0.35}$		0.115	0.106	0.103	76.1	0.109	0.144	0.138	83.7

Note: “BD” represents different bandwidths, “Bias” is the difference between ${\beta }_0\left(s\right)$ and $\hat{\beta }\left(s\right)$, “SD” is the sample standard deviation, “SE” is the average of the standard error estimates, “CP”/100 represents the coverage probability of the 95% confidence interval for estimators of ${\beta }_0\left(s\right)$ at fixed $s$, and LVCF represents the last value carried forward method.

Table 1 also shows results based on the last value carried forward. We observe that bias occurs, which does not attenuate with increased sample size. As a result, the coverage probabilities are lower than the nominal ones. Paradoxically, the coverage probabilities get worse with a decreased censoring rate. The reason is that with lower censoring rate, more data are used for parameter estimation, producing a larger bias. We use covariates observed before the minimum of censoring and death time. When the observation process follows a homogeneous Poisson process, as time $s$ increases, the amount of observation decreases, and the information used to estimate $\beta \left(s\right)$ also decreases. Consequently, the standard deviation becomes larger. It is a coincidence that the bias changes with $s$. In other settings, such as the ones summarized in the Supplementary Material, the bias does not decrease with time.

Next, we compare the proposed method with joint modeling approach. Specifically, we employed the joint modeling approach proposed by Andrinopoulou et al. (2018), which utilizes P-splines to estimate the time-varying coefficient function. P-splines extend B-splines by incorporating a penalty term to regulate smoothness and mitigate overfitting (Eilers and Marx, 1996). The simulation setup follows the same configuration as above. In this scenario, the censoring rate is 15%. We use automatic bandwidth selection for our method.

Due to the extensive computational time required by the joint modeling approach, we present results based on 100 replications. Additionally, since Andrinopoulou et al. (2018) does not explicitly state the estimation of the time-varying coefficients ${\beta }_0(\cdot )$, its standard deviation or the construction of confidence intervals, we follow the approach described in Rizopoulos (2012). Specifically, we use the median of the post-burn-in Monte Carlo samples to estimate $\beta \left(s\right)$, the standard deviation of the post-burn-in Monte Carlo samples to estimate the standard deviation of $\hat{\beta }\left(s\right)$ (“SE” in Table 2), and the 0.025 and 0.975 quantiles of the samples to construct the 95% confidence intervals. The results based on the mean of the post-burn-in Monte Carlo samples are similar and thus omitted. The simulation results are summarized in Table 2. As shown in Table 2, the estimates obtained using the joint modeling approach exhibit substantial biases resulting in poor coverage probabilities. In comparison, the proposed method has decent performances. The bias is small, and decreases with increased sample size. Additionally, SE agrees with SD, and the coverage probabilities align well with the nominal level of 95%. Furthermore, the joint modeling approach requires a large number of iterations when employing Markov chain Monte Carlo method, making it computationally expensive. For instance, with a sample size of 100, using a computer equipped with an Intel Core i7-12700H CPU (4.7 GHz) and 64 GB of RAM, a single run of the joint modeling approach takes approximately 45 minutes. In contrast, the proposed method finishes the computation within a few seconds.

Table 2:

Simulation results of $\hat{\beta }\left(s\right)$.

$\mathit{n}$	$\mathit{s}$	Joint modeling approach				The proposed method
		Bias	SD	SE	CP	Bias	SD	SE	CP
100	0.2	−0.439	0.319	0.276	64.0	−0.123	0.233	0.213	86.0
	0.4	−0.225	0.278	0.255	83.0	−0.113	0.214	0.221	89.0
	0.6	0.304	0.362	0.305	79.0	0.033	0.192	0.189	93.0
	0.8	0.417	0.525	0.424	72.0	0.055	0.239	0.260	96.0
200	0.2	−0.456	0.265	0.270	62.0	−0.062	0.231	0.210	92.0
	0.4	−0.234	0.276	0.257	85.0	−0.030	0.191	0.194	94.0
	0.6	0.366	0.430	0.315	71.0	0.035	0.152	0.166	96.0
	0.8	0.517	0.639	0.432	68.0	0.027	0.254	0.225	91.0

Note: ‘Bias” is the difference between ${\beta }_0\left(s\right)$ and $\hat{\beta }\left(s\right)$, “SD” is the sample standard deviation, “SE” is the average of the standard deviation estimates, “CP”/100 represents the coverage probability of the 95% confidence interval for $\hat{\beta }\left(s\right)$.

3.3. Constructing Simultaneous Confidence Band

We are interested in constructing SCBs for ${\beta }_0\left(s\right)$ in this subsection. We use 50 equally spaced grid points in $[h,1-h]$ to calculate coverage probability. Here, all intervals are constructed based on $M=5000$ realizations of $\left\{{\xi }_i\stackrel{i.i.d.}{~}\text{Exp}\left(1\right)-1,i=1,\dots ,n\right\}$, where Exp(1) represents the exponential distribution with mean 1. We tried other ${\xi }_i,i=1,\dots ,n$, and the results are similar, and thus omitted. In the simulation study and the real data analysis, we use the inverse of the estimated standard error of $\hat{\beta }\left(s\right)$ as the weight $\hat{w}\left(s\right)$. In this case, the weighted estimation bias $\hat{w}\left(s\right)\left\{\hat{\beta }\left(s\right)-{\beta }_0\left(s\right)\right\}$ has the same distribution at all time points. Then, the contribution to the simultaneous confidence band is the same at different time points, even if the variance of the estimator is different. Such a choice can avoid the influence of points with larger variance dominating the influence of other points and can lead to a narrower band.

Figure 1 shows the estimate $\hat{\beta }\left(s\right)$ using “auto” bandwidth selection rule with the sample sizes 400 and 900. The left panel corresponds to 15% censoring, and the right panel corresponds to 35% censoring. The green dashed curve is the 95% SCB, and the blue dashed curve is the 95% point-wise confidence interval. Figure 1 shows that the SCBs become narrower and more accurate as the sample size increases. SCBs are wider than pointwise intervals, as they have uniform coverage throughout the time domain. Table 3 summarizes the probability of uniform coverage of the SCBs and point-wise confidence intervals based on the simulations of 1000 datasets. We observe that SCBs achieve coverage probabilities near the nominal level under different censoring rates. The performance improves with a larger sample size. On the other hand, the pointwise confidence interval is not valid for simultaneous inference, due to much lower uniform coverage probabilities.

Figure 1:

Estimates with “auto” bandwidth selection approach.

Table 3:

Uniform coverage probability based on different methods

$\mathit{n}$	Bandwidth	15% Censoring rate		35% Censoring rate
		SCB	CI	SCB	CI
400	$h_1=n^{-0.35},h_2=n^{-0.35}$	91.3	34.8	90.4	34.3
	$h_1=n^{-0.35},h_2=n^{-0.45}$	92.0	32.5	90.5	29.3
	auto	92.4	31.9	90.3	27.2
900	$h_1=n^{-0.35},h_2=n^{-0.35}$	93.1	28.0	93.4	26.0
	$h_1=n^{-0.35},h_2=n^{-0.45}$	92.9	23.2	93.0	21.0
	auto	93.8	17.6	91.9	16.6

Note: “SCB” refers SCB and “CI” means point-wise confidence interval.

4. Data Analysis

We apply the proposed method to a dataset from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study to demonstrate its practical application. ADNI is a large, ongoing study initiated in 2004. It supports the investigation and development of treatments that slow or stop the progresison of Alzheimer’s Disease (AD), a progressive neurodegenerative disorder that affects memory and cognitive function. We are interested in identifying possible risk factors and their dynamic effects for AD. Specifically, we look at APOE4 gene (coded as 0 for non-carriers, 1 for carriers of one allele, and 2 for carriers of two alleles), gender (coded as 1 for males and 0 for females), and the volumes of the hippocampus and entorhinal cortex (measured in cubic millimeters) on AD. While APOE4 and gender are time-independent, the volumes of the hippocampus and entorhinal cortex are measured longitudinally.

The dataset consists of 2430 subjects. After eliminating missing data, 2088 participants are used for analysis under missing at random assumption (Little and Rubin, 2019), among whom 736 (35.2%) experienced AD. The admission time of participants is set as time origin and the maximum follow-up time $\tau =6280$ days. For simplicity, we set $h_1=h_2=h$. Then, we adopt the proposed automatic bandwidth selection method to select the bandwidth between $9\left(Q_3-Q_1\right)n^{-1/2}\approx 454$ days and $9\left(Q_3-Q_1\right)n^{-1/6}\approx 5798$ days, where $Q_3$ is the 3rd quartile and $Q_1$ is the 1st quartile of the longitudinal measurement times, and $n$ is the total number of participants. Our theory suggests valid bandwidth should range from $O\left(n^{-1/2}\right)$ to $o\left(n^{-1/6}\right)$. In practice, we need to decide the constants, which we use $9\left(Q_3-Q_1\right)$ here, producing a wide range of time for bandwidth selection. We fit the data with the following varying-coefficient multiplicative hazards model:

$$\lambda \{t\mid \text{APOE}4,\text{Gender},\text{Hippocampus}(r),\text{Entorhinal}(r),r\le t\}={\lambda }_0\left(t\right)\text{exp}\left\{{\beta }_1\left(t\right)\text{APOE}4\right.+{\beta }_2\left(t\right)\text{Gender}+{\beta }_3\left(t\right)\text{Hippocampus}\left(t\right)+{\beta }_4\left(t\right)\text{Entorhinal}\left.\left(t\right)\right\}.$$

For a fixed time point $s\in [h,\tau -h]$, solving (2.4), we obtain $\hat{\beta }\left(s\right)$. Its variance is estimated by the sandwich formula (2.6). A 95% point-wise confidence interval can be constructed based on normal approximation. For the construction of a simultaneous confidence band, we use ${\beta }_1(\cdot )$ to illustrate. We generate $M=5000$ realizations of $\left\{{\xi }_i\stackrel{i.i.d.}{~}\text{Exp}\left(1\right)-1,i=1,\dots ,2088\right\}$, where Exp(1) represents the exponential distribution with mean 1, and plug them into (2.8). After obtaining ${\tilde{U}}_n\left\{\hat{\beta }\right(s\left)\right\}$ and $I\left\{\hat{\beta }\right(s\left)\right\}$, we use the inverse of the estimated standard error of ${\hat{\beta }}_1\left(s\right)$ as the weight $\hat{w}\left(s\right)$ in the calculation of $\left\{{\tilde{S}}_{SCB}^{\left(1\right)},\dots ,{\tilde{S}}_{SCB}^{\left(5000\right)}\right\}$ in (2.9). The SCB of ${\beta }_1(\cdot )$ is constructed using (2.10).

We summarize the estimated coefficient function, 95% point-wise confidence interval, and 95% simultaneous confidence band in Figure 2. We also plot a horizontal line to represent the 0 effect. Figure 2 shows that carrying the APOE4 gene significantly increases the hazard of developing AD, with the hazard progressively rising over time. This is consistent with the literature that APOE4 allele is a genetic risk factor for AD, with carriers of this allele having a higher likelihood of developing the disease (Yamazaki et al., 2019; Ding et al., 2024). Our analysis reveals that the detrimental effect of carrying APOE4 increases over time. The APOE4 gene contributes to AD pathogenesis by impairing amyloid-beta clearance, increasing its aggregation, and promoting neuroinflammation. Over time, the cumulative effects lead to a higher hazard of developing AD. Furthermore, Figure 2 suggests that women are more susceptible to AD than men, the effect is significant and stable over time. As highlighted by Pike (2017) and Cui et al. (2023), women exhibit a higher susceptibility to AD, attributed to both biological and hormonal factors. For instance, the loss of estrogen after menopause may exacerbate neurodegeneration and amyloid-beta pathology. Additionally, from Figure 2, we observe that a reduction in hippocampal volume is associated with a higher risk of AD, with the effect increasing over time. The hippocampus, a critical brain region involved in memory formation, is often one of the first areas to be affected by AD, leading to memory loss. A decrease in hippocampal volume reflects neuronal loss and synaptic degeneration, hallmark features of AD (Huijbers et al., 2020). This structural decline progressively disrupts cognitive function, explaining the increased risk of AD over time as hippocampal volume decreases. Finally, Figure 2 indicates that a reduction in entorhinal cortex volume is linked to an elevated risk of AD, consistent with the findings of Tran et al. (2022). The entorhinal cortex is involved in memory and spatial navigation and shows early signs of degeneration in AD, contributing to cognitive decline. We observe that the effect is more pronounced between 0–4000 days but tends to diminish between 4000–6000 days. The pronounced risk associated with its volume reduction in the earlier time frame (0–4000 days) may reflect its early involvement in the disease process, where pathological changes in this region trigger broader neurodegenerative cascades. The subsequent diminishment of risk (4000–6000 days) could suggest that by this stage, the disease has progressed to affect other brain regions, thereby distributing the risk factors more diffusely.

Figure 2:

Estimates with auto bandwidth selection approach.

5. Concluding Remarks

In this paper, we propose to estimate the time-varying effect of longitudinally collected covariates for the multiplicative hazards model. This allows us to examine the dynamic relationship between time-dependent covariates and time-to-event outcomes. For any fixed time point, we establish the asymptotic normality of the proposed estimator. To quantify the uncertainty of the non-parametric coefficient function, we further develop a simultaneous confidence band through multiplier bootstrap. Simulation studies demonstrate the favorable performance of the proposed method, and an analysis of a dataset from Alzheimer’s Disease Neuroimaging Initiative study reveals the dynamic relationship between APOE4 gene, gender, hippocampus volume and entorhinal cortex volume and the onset of AD.

We assume that the observational time of the covariate process is external. Our approach can be extended to the case of informative observational times, which may depend on the past covariate as in Cao et al. (2016). We assume that the observation times of each individual occur at random and we aggregate information across different individuals. Asymptotically, there would be enough data at any time point $s$ as we increase the sample size. In finite samples, it is possible that for a particular point of interest $s$, we do not have nearby observations to perform kernel smoothing. Our approach can be used to analyze the censored outcome with other models, such as the additive hazards model or the transformed hazards model (Sun et al., 2022, 2023). We leave these for future work. In practice, some covariates may have a time-independent coefficient, and some covariates may have a time-dependent coefficient, which warrants more research.