Weighted least-squares regression with competing risks data

Abstract

The semiparametric accelerated failure time (AFT) model linearly relates the logarithm of the failure time to a set of covariates, while leaving the error distribution unspecified. This model has been widely investigated in survival literature due to its simple interpretation and relationship with linear models. However, there has been much less focus on developing AFT-type linear regression methods for analyzing competing risks data, in which patients can potentially experience one of multiple failure causes. In this article, we propose a simple least-squares (LS) linear regression model for a cause-specific subdistribution function, where the conventional LS equation is modified to account for data incompleteness under competing risks. The proposed estimators are shown to be consistent and asymptotically normal with consistent estimation of the variance-covariance matrix. We further extend the proposed methodology to risk prediction and analysis under clustered competing risks scenario. Simulation studies suggest that the proposed method provides rapid and valid statistical inferences and predictions. Application of our method to two oncology datasets demonstrate its utility in routine clinical data analysis.

1 |. INTRODUCTION

Competing risks occur when subjects can experience one of multiple events or outcomes that compete with the event of interest. For example, cancer patients may die from the cancer under treatment or from other causes, such as cancer treatment complications, that may arise naturally or as an adverse effect of the treatment under study. If investigators are often interested in the first failure time and the corresponding cause of failure, a competing risks problem is created.¹ Since the presence of competing risks may either hinder the observation of the event of interest or modify the chance that this event occurs, all standard methods for univariate outcomes ignoring the competing event(s), such as the Kaplan–Meier method and Cox’s proportional hazards regression, are not directly applicable. In addition to the availability of a variety of nonparametric methods,^2–4 various regression modeling approaches exist to evaluate the relationship of covariates to cause-specific failures in competing risks; see Putter et al⁵ Lau et al,⁶ Bakoyannis and Touloumi,⁷ among others, for a comprehensive review on this topic.

The cause-specific hazard (CSH) function and cumulative incidence function (CIF) are two basic identifiable quantities for analyzing competing risks data. Let $T$ and $D\in \{1,\dots ,J\}$ be the failure time and the cause of failure, respectively, for which the $J$ causes are usually assumed to be observable. The CSH function for the event with $D=j$ at time $t$ represents the instantaneous risk for a particular cause, and defined as ${\lambda }_j(t)={\lim }_{dt\to 0}P(t\le T<t+dt,D=j)/(dt)$. On the other hand, the CIF measures the absolute cause-specific risk for the $j\text{th}$ event, given by $F_j(t)=P(T\le t,D=j)={\int }_0^t{S}_T(u-)d{\text{Λ}}_j(u)$, where ${\text{Λ}}_j(t)={\int }_0^t{\lambda }_j(u)$ is the cumulative hazards function for the $j\text{th}$ event and $S_T(t)=\exp \left\{-{\sum }_j{\text{Λ}}_j(t)\right\}$ is the overall survival function. Both do not require assumptions about the dependence among the events. There exists a nonlinear relationship between these two quantities in the presence of competing events, providing different perspectives for modeling failure times.⁵ Whenever ${\lambda }_1(t)<{\sum }_j{\lambda }_j(t)$, the $F_1$ is improper. Therefore, one can regard $F_1$ as the distribution function for the implied random variable $T_1^{*}=T\times I(D=1)+\infty \times I(D\ne 1)$, that is, $F_1(t)=P\left(T_1^{*}\le t\right)$, representing the cumulative probability of occurrence by time $t$ for the cause-1 failure in the presence of other risks.

With covariates, competing risks analysis conventionally involves modeling CSH⁸ or CIF⁹ under a proportional hazards (PH) assumption. The Fine–Gray approach,⁹ a cornerstone of CIF modeling, assumes that the complementary log-log of the CIF has a PH form. This approach is useful in predicting the probability of a given outcome at a given time for an individual patient. It can be seen, however, that the proportionality assumption generally does not hold simultaneously for these quantities under competing risks. Scheike et al¹⁰ developed a flexible semiparametric method for CIF with time-varying covariates. Another direct regression method for modeling the cumulative incidence probabilities include the pseudo-observation approach via jackknife resampling.¹¹ Fine¹² and Choi and Huang¹³ developed the linear transformation models, with the logit link as a special case to perform a mixture analysis of competing risks. The mixture approach is intended to provide cumulative incidence estimates for all failure causes from one common model, allowing for time-dependent patterns of covariate effects. Competing risks quantile regression^14,15 is an another useful tool that formulates the model based on conditional quantiles defined using the CIF, relaxing the constancy constraint on regression coefficients.

Although there has been extensive research on regression modeling in competing risk situations, little attention has been paid to the linear regression framework. Motivated from that, we develop estimation and inference procedures corresponding to the absolute-risk linear model for cause-specific failure times. Specifically, our approach is to consider the cause-1-specific linear regression model of the form

$$\log T_1^{*}=X^{\prime }\beta +\epsilon ,$$ (1)

where, $X$ is a $p\times 1$ vector of bounded and time-dependent covariates, $\beta$ is the corresponding $p\times 1$ vector of unknown regression parameters, and $\epsilon$ is an independent error term with unknown baseline subdistribution function $F_{10}(\cdot )$. Model (1) implies that the CIF takes the form $F_1(t;X)=P(T\le t,D=1\mid X)=F_{10}\left(t{e}^{-X^{\prime }\beta }\right)$, or alternatively, $h_1(t;X)=h_{10}\left(t{e}^{-X^{\prime }\beta }\right)e^{-X^{\prime }\beta }$, where, $h_{10}(t)$ is an unspecified baseline hazard and $h_1(t;X)$ is the subdistribution hazard function given $X$, that is, $h_1(t;X)={\lim }_{\delta t\to 0}P(t\le T\le t+\delta t,D=1\mid T\ge t\cup (T\le t\cap D\ne 1),X)=-d\log \left\{1-F_1(t;X)\right\}/dt$. In presence of a single cause of failure (ie, $J=1$), this model is often referred to as the accelerated failure time (AFT) model,^16–18 which has been extensively studied in statistical literature as an alternative to the Cox model. In the competing risks setting, this approach is intuitively appealing as it provides a meaningful direct characterization of the crude effects of covariates on the cause-1 failure times. Note that in this AFT model framework, unlike the Cox-type approach, the covariate effect fades away as time goes to infinity. To the best of our knowledge, there does not exist any valid regression method for assessing the crude-risk effects of covariates on failure times under model (1), other than the work by Choi and Cho,¹⁹ who compared this model to the usual cause-specific approach from a simulation perspective.

In this article, we address the methodological and implementation aspects of linear regression analysis on cumulative incidence probabilities in the presence of competing risks, properly accounting for censored observations. Specifically, in Section 2, we propose weighted least-squares (WLS) estimators for independent and clustered competing risks data, which can be easily implemented using standard softwares for linear regression. The resulting estimators are shown to be consistent and asymptotically normal with tractable estimation of the variance-covariance matrix. A separate procedure to predict the cumulative incidence probability for an individual (with certain covariates) is also discussed. Under clustered competing risks, we further extend our WLS technique to accommodate the informative cluster size (ICS) scenario,²⁰ where the cluster size may be correlated to the survival outcome of interest. In Section 3, we carry out extensive numerical studies using synthetic data to assess the finite-sample performance of the proposed estimators. We further illustrate our methodology in light of the cause-specific linear model for the CSH function via application to two real-life cancer datasets in Section 4. Finally, in Section 5, we present some concluding statements, and discuss relevant issues regarding the procedures presented in this article. We provide details of asymptotic results in the Appendix.

2 |. METHODS

2.1 |. Competing risks linear regression

In practice, time to events, or failure times observed in biomedical studies are typically subjected to additional right censoring. Let $C$ denote the censoring time, assumed independent of $T$ in our setup. For a random sample of size $n$, the competing risks data $\left({\tilde{T}}_i,{\delta }_i,{\delta }_i{D}_i,X_i\right)$, $i=1,\dots ,n$, are independent and identically distributed copies of $(\tilde{T},\delta ,\delta D,X)$, where ${\tilde{T}}_i=\min \left(T_i,C_i\right)$ and ${\delta }_i=I\left(T_i\le C_i\right)$ represent the observed failure time and the censoring indicator, respectively. We set the first entry in $X$ to 1, so that the first element of $\beta$ corresponds to the intercept. Let ${\text{Δ}}_i={\delta }_{i}I\left(D_i=1\right)$ denote the observed occurrence of failure from cause 1.

To motivate our estimating equation for $\beta$, we first consider a simple scenario without censoring, that is, ${\tilde{T}}_i=T_i$ and ${\text{Δ}}_i=I\left(D_i=1\right)$. Write $V_i=\log T_i$. Without censoring, the cause-1-specific LS estimator is easily obtained by minimizing the objective function ${\sum }_{i=1}^n{\text{Δ}}_i{\left(V_i-X_i^{\prime }\beta \right)}^2$ with respect to $\beta$, or equivalently by solving the cause-specific normal equation ${\sum }_{i=1}^n{\text{Δ}}_i{X}_i\left(V_i-X_i^{\prime }\beta \right)=0$. Now, to accommodate right censoring, we will consider a weighted estimating equation (WEE) with survivor function given $X$. Let $G(t)=P(C\ge t)$, the survival function for the censoring time. The proposed method is motivated by the inverse probability of censoring weighted (IPCW) argument, implying at failure time $T_i$, the $i\text{th}$ individual has a probability $G\left(T_i\right)$ of not being censored. The IPCW approach has been employed earlier in survival analysis and other settings with incomplete data, for example, in medical cost data analysis,²¹ and censored quantile regression.¹⁴ A weighted estimating function, using only uncensored individuals who have failed from cause-1 event, is thus given by

$$U_1(\beta )=\sum _{i=1}^n\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_i\right)}{X}_i\left({\tilde{V}}_i-X_i^{\prime }\beta \right),$$ (2)

where ${\tilde{V}}_i=\log {\tilde{T}}_i$ and $\widehat{G}(t)$ is the Kaplan–Meier estimator for the censoring survivor function $G(t)$, based on the data $\left({\tilde{T}}_i,1-{\delta }_i\right)$, $i=1,\dots ,n$, that is, $\widehat{G}(t)={\prod }_{u<t}\left\{1-d{N}^c(u)/Y(u)\right\}$, where $N^c(u)={\sum }_{i=1}^n{N}_i^c(u)={\sum }_{i=1}^{n}I\left({\tilde{T}}_i\le u,{\delta }_i=0\right)$ and $Y(u)={\sum }_{i=1}^n{Y}_i(u)=\sum _{i=1}^{n}I\left({\tilde{T}}_i\ge u\right)$. Unbiasedness of this estimating equation (2) follows easily using a conditioning argument:

$$E\left[\frac{\text{Δ}}{G(\tilde{T})}\right]=E\left[\left.E\left\{\frac{I(C\ge T,D=1)}{G(T)}\right\}\right|T,D\right]=E\left[\frac{I(D=1)G(T)}{G(T)}\right]=P(D=1),$$

where we assume $C⫫(T,D)$. This independence assumption can be readily relaxed to allow $C_i$ to depend on $X_i$ using a Cox model or other nonparametric survival ensembles.⁴ The solution to Equation (2) has a closed-form expression

$$\widehat{\beta }={\left\{\sum _{i=1}^n\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_i\right)}{X}_i^{\otimes 2}\right\}}^{-1}\sum _{i=1}^n\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_i\right)}{X}_i{\tilde{V}}_i,$$ (3)

where $a^{\otimes 2}=a^{\prime }a$. Under no censoring, the proposed estimator (3) reduces to the ordinary LS estimator for $\beta$. It can be easily shown that $n^{1/2}\left(\widehat{\beta }-{\beta }_0\right)$ weakly converges to a mean zero Gaussian process random vector with a variance-covariance matrix that can be consistently estimated by ${\widehat{A}}_1^{-1}{\widehat{B}}_1{\widehat{A}}_1^{-1}$, where

$$\begin{array}{l}{\widehat{A}}_1=n^{-1}\sum _{i=1}^n\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_i\right)}{X}_i^{\otimes 2},\\ {\widehat{B}}_1=n^{-1}\sum _{i=1}^n{\left[\frac{{\text{Δ}}_i{X}_i\left({\tilde{V}}_i-X_i^{\prime }\widehat{\beta }\right)}{\widehat{G}\left({\tilde{T}}_i\right)}-\left(1-{\delta }_i\right)Q_1\left({\tilde{T}}_i\right)+\sum _{l=1}^n\frac{\left(1-{\delta }_l\right)Y_i\left({\tilde{T}}_l\right)Q_1\left({\tilde{T}}_l\right)}{Y\left({\tilde{T}}_l\right)}\right]}^{\otimes 2},\end{array}$$

and

$$Q_1(t)=\frac{1}{Y(t)}\sum _{i=1}^n\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_i\right)}{Y}_i(t)X_i\left({\tilde{V}}_i-X_i^{\prime }\widehat{\beta }\right).$$

Remark 1.

The proposed LS estimator (3) measures the effect of covariate on the subdistribution hazard function $h_1(\cdot )$ by accelerating or decelerating the event time with covariates. Alternatively, one might be interested in quantifying covariate effects on the CSH ${\lambda }_1(\cdot )$ within the AFT framework, for which we can use

$$\overline{\beta }={\left\{\sum _{i=1}^n\frac{{\text{Δ}}_i}{\overline{G}\left({\tilde{T}}_i\right)}{X}_i^{\otimes 2}\right\}}^{-1}\sum _{i=1}^n\frac{{\text{Δ}}_i}{\overline{G}\left({\tilde{T}}_i\right)}{X}_i{\tilde{V}}_i.$$ (4)

Here, $\overline{G}(t)$ represents the nonparametric censoring survivor estimator, based on the data $\left\{{\tilde{T}}_i,1-\delta I\left(D_i=1\right)\right\}$, $i=1,\dots ,n$. Since $\widehat{G}(t)\le \overline{G}(t)$, $\widehat{\beta }$ imposes relatively lower weights on late event time than $\overline{\beta }$ and usually attenuates covariate effects as in Cox-based competing risks analyses.²² See Choi and Cho¹⁹ for simulation results for comparing $\widehat{\beta }$ and $\overline{\beta }$ in various scenarios.

Remark 2.

To further ease the strict independence assumption $C_i⫫\left(T_i,D_i\right)$ to the conditional independence assumption $C_i⫫\left(T_i,D_i\right)\mid X_i$, one might use Beran’s local Kaplan–Meier estimator²³ for $G\left(C_i\mid X_i\right)$, that is,

$$\widehat{G}(t\mid X)=\prod _{j=1}^n{\left\{1-\frac{B_{nj}(X)}{{\sum }_{k=1}^{n}I\left({\tilde{T}}_k\ge {\tilde{T}}_j\right)B_{nk}(X)}\right\}}^{I\left({\tilde{T}}_j\le t,{\delta }_j=0\right)},$$ (5)

where $B_{nj}(X)$ is a sequence of non-negative weights adding up to 1. For example, we can employ the commonly used Nadaraya–Watson’s type weight, that is, $B_{nj}(x)=K\left(\frac{X-X_j}{h_n}\right)/{\sum }_{k=1}^{n}K\left(\frac{X-X_k}{h_n}\right)$, where $K(\cdot )$ is a kernel density function and $h_n\in {ℝ}^{+}$ is the bandwidth converging to zero as $n\to \infty$. Alternatively, one can use a random forests method for nonparametric survival prediction.⁴ By plugging such a local Kaplan–Meier estimator into (3), we can obtain a nonparametric covariate-adjusted IPCW estimator.

2.2 |. Prediction of the cumulative incidence probabilities

We now consider the problem of predicting the CIF and its related quantities for future subjects with a given set of covariates. Following Gray² and Fine and Gray,⁹ we estimate the subdistribution hazard function, defined as $h_{10}(t)=-d\log \left\{1-F_{10}(t)\right\}/dt$, leading to the CIF defined as $F_{10}(t)=1-\exp \left\{-H_{10}(t)\right\}$, where $H_{10}(t)={\int }_0^t{h}_{10}(u)du$. Accordingly, we define counting processes for complete data

$$N_i^{\ast }(t,\beta )=I\left\{T_i(\beta )\le t,D_i=1\right\},Y_i^{\ast }(t,\beta )=1-N_i^{\ast }(t,\beta ),$$

and their weighted versions for incomplete data

$$N_i^w(t,\beta )={\delta }_{i}I\left\{{\tilde{T}}_i(\beta )\le t,D_i=1\right\}/\widehat{G}(t),Y_i^w(t,\beta )=1-N_i^w(t,\beta ),$$

where $T_i(\beta )=T_i{e}^{-X_i^{\prime }\beta }$ and ${\tilde{T}}_i(\beta )={\tilde{T}}_i{e}^{-X_i^{\prime }\beta }$. In presence of right censoring, $N_i^{\ast }(t,\beta )$ and $Y_i^{\ast }(t,\beta )$ are not fully observable, which motivates us to use $N_i^w(t,\beta )$ and $Y_i^w(t,\beta )$, the unbiased estimates of $N_i^{\ast }(t,\beta )$ and $Y_i^{\ast }(t,\beta )$, respectively.

Let $\tau$ be a preselected constant, such that $P\{{\tilde{T}}_1(\beta )>\tau \}>0$, for $\beta$ near ${\beta }_0$. Then, for $t<\tau$, the Nelson–Aalen²⁴ estimator where ${\widehat{H}}_{10}(t,\widehat{\beta })$, based on the data $\left\{\left({\tilde{T}}_i(\widehat{\beta }),{\text{Δ}}_i\right),i=1,\dots ,n\right\}$, is shown to be uniformly consistent for $H_{10}(t)$, where

$${\widehat{H}}_{10}(t,\beta )=n^{-1}\sum _{i=1}^n{\int }_0^t\frac{d{N}_i^w(u,\beta )}{{\overline{Y}}^w(u,\beta )},{\tilde{H}}_{10}^{*}(t,\beta )=n^{-1}\sum _{i=1}^n{\int }_0^t\frac{d{N}_i^{*}(u,\beta )}{{\overline{Y}}^{*}(u,\beta )},$$ (6)

with ${\overline{Y}}^w(t,\beta )=n^{-1}{\sum }_{i=1}^n{Y}_i^w(t,\beta )$ and ${\overline{Y}}^{*}(t,\beta )=n^{-1}{\sum }_{i=1}^n{Y}_i^{*}(t,\beta )$. Let $W(t)=n^{1/2}\left\{{\widehat{H}}_{10}(t,\widehat{\beta })-H_{10}(t)\right\}$. We prove in the Appendix that $W(t)$ converges weakly to a zero-mean Gaussian process. Since it is difficult to estimate the limiting distribution of $W(t)$ analytically, we appeal to a resampling approach.¹⁷ To this end, obtain ${\widehat{\beta }}^{\ast }$ by solving a perturbed estimating function

$$U_1^{\ast }(\beta )=\sum _{i=1}^n\frac{{\text{Δ}}_i}{{\widehat{G}}^{\ast }\left({\tilde{T}}_i\right)}{X}_i\left({\tilde{V}}_i-X_i^{\prime }\beta \right)Z_i,$$

where $Z_i$, $i=1,\dots ,n$, are independent positive random variables with unit mean and variance and ${\widehat{G}}^{\ast }(t)={\prod }_{u<t}\left\{1-\frac{Z_{i}d{N}_i^c(u)}{{\sum }_{j=1}^n{Z}_j{Y}_j(u)}\right\}$. We also show in the Appendix that $W(t)$ has the same limiting distribution as

$$\widehat{W}(t)=n^{1/2}\left\{{\widehat{H}}_{10}(t,\widehat{\beta })-{\widehat{H}}_{10}\left(t,{\widehat{\beta }}^{*}\right)+\sum _{i=1}^n{\widehat{H}}_i(t,\widehat{\beta })Z_i+\sum _{i=1}^n{\widehat{R}}_i(t,\widehat{\beta })Z_i\right\},$$

where ${\widehat{H}}_i(t,\beta )=n^{-1}{\int }_0^t\frac{d{M}_i^w(t,\beta )}{{\overline{Y}}^w(t,\beta )}$ with $M_i^w(t,\beta )=N_i^w(t,\beta )-{\int }_0^t{Y}_i^w(s,\beta )d{\widehat{H}}_{10}(t,\beta )$ and ${\widehat{R}}_i(t,\beta )$ is given in the Appendix. Thus, we may use the simulated distribution of $\widehat{W}(t)$ to make inferences about $H_{10}(t)$. The baseline CIF $F_{10}(t)$ can then be estimated by ${\widehat{F}}_{10}(t,\widehat{\beta })=1-\exp \left\{-{\widehat{H}}_{10}(t,\widehat{\beta })\right\}$. The 95% confidence interval for ${\widehat{F}}_{10}(t,\widehat{\beta })$ is $1-\exp \left\{-{\widehat{H}}_{10}(t,\widehat{\beta })\pm 1.96n^{-1/2}\widehat{\sigma }(t)\right\}$, where $\widehat{\sigma }(t)$ denoting the estimated standard error of $\widehat{W}(t)$.

To predict the cumulative incidence probability $F_1\left(t;x_0\right)=1-\exp \left\{-H_{10}\left(t;x_0\right)\right\}$ for subjects with a given covariate vector $X=x_0$, one may replace $X_i$ by $X_i-x_0$ in the dataset, and a consistent estimator for $F_1\left(t;x_0\right)$ is simply ${\widehat{F}}_{10}(t)$ for the underlying CIF with the modified data. Without transforming the dataset, we can estimate $F_1\left(t;x_0\right)$ by ${\widehat{F}}_{10}\left(t;x_0\right)=1-\exp \left\{-{\widehat{H}}_{10}\left(t;x_0\right)\right\}$, where ${\widehat{H}}_{10}\left(t;x_0\right)={\widehat{H}}_{10}\left(t{e}^{x_0^{\prime }\widehat{\beta }},\widehat{\beta }\right)$. The limiting distribution of $n^{1/2}\left\{{\widehat{H}}_{10}\left(t;x_0\right)-H_{10}\left(t;x_0\right)\right\}$ can be approximated by

$$n^{1/2}\left\{{\widehat{H}}_{10}\left(t{e}^{x_0^{\prime }\widehat{\beta }},\widehat{\beta }\right)-{\widehat{H}}_{10}\left(t{e}^{x_0^{\prime }{\widehat{\beta }}^{*}},{\widehat{\beta }}^{*}\right)+\sum _{i=1}^n{\widehat{H}}_i\left(t{e}^{x_0^{\prime }\widehat{\beta }},\widehat{\beta }\right)Z_i+\sum _{i=1}^n{\widehat{R}}_i\left(t{e}^{x_0^{\prime }\widehat{\beta }},\widehat{\beta }\right)Z_i\right\},$$

and a conventional delta method can be used to conduct inferences about $F_{10}\left(t;x_0\right)$.

2.3 |. Clustered competing risks: Accommodating informative cluster sizes

Clustered competing risks data²⁵ may arise when the subjects are sampled in clusters, such that the failure times within the same cluster tend to be correlated. Suppose there are $n$ clusters, with the $i\text{th}$ cluster having $m_i$ members. It is assumed that $m_i$ is relatively small compared to $n$. For the $k\text{th}$ member of the $i\text{th}$ cluster, let $T_{ik}$ and $C_{ik}$ denote the failure and censoring times, $D_{ik}$ the failure cause, and $X_{ik}$ the corresponding $p$-vector of covariates, in which $\left(T_{i1},\dots ,T_{i{m}_i}\right)$ and $\left(C_{i1},\dots ,C_{i{m}_i}\right)$ are assumed to be independent. The data consist of $\left\{\left({\tilde{T}}_{ik},{\delta }_{ik},D_{ik}{\delta }_{ik},X_{ik}\right),k=1,\dots ,m_i;i=1,\dots ,n\right\}$, where, ${\tilde{T}}_{ik}=\min \left(T_{ik},C_{ik}\right)$ and ${\delta }_{ik}=I\left(T_{ik}\le C_{ik}\right)$. We also write ${\tilde{V}}_{ik}=\log {\tilde{T}}_{ik}$ and ${\text{Δ}}_{ik}={\delta }_{ik}I\left(D_{ik}=1\right)$.

Suppose that the marginal distribution of the implied event time $T_{ij}^{\ast }=T_{ij}\times I\left(D_{ij}=1\right)+\infty \times I\left(D_{ik}\ne 1\right)$ satisfies the AFT model

$$\log T_{ik}^{\ast }=X_{ik}^{\prime }\beta +{\epsilon }_{ik};k=1,\dots ,m_i,i=1,\dots ,n,$$

where $\beta$ is a $p$-vector of unknown regression parameters and $\left({\epsilon }_{i1},\dots ,{\epsilon }_{i{m}_i}\right)$, $i=1,\dots ,n$, are independent random vectors. For each $i$, the error terms ${\epsilon }_{i1},\dots ,{\epsilon }_{i{m}_i}$ are assumed to be exchangeable with a common marginal subdistribution of $F_{10}$. Under the working independence assumption, we obtain the estimator $\tilde{\beta }$ for $\beta$ by solving the following weighted estimating function

$$U_2(\beta )=\sum _{i=1}^n{w}_i\sum _{k=1}^{m_i}\frac{{\text{Δ}}_{ik}}{\widehat{G}\left({\tilde{T}}_{ik}\right)}{X}_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\beta \right),$$ (7)

where, $w_i$ is a known weight to adjust the possible informativeness of cluster sizes.^26,27 For the marginal analysis of clustered survival data, we conventionally use $w_i=1$, which tends to overweight the large clusters, because each individual observation contributes equally in the estimating equation. When cluster sizes are informative to the outcome of interest, we can incorporate the inverse of cluster sizes as a weight in the estimating function, letting, for example, $w_i=1/m_i$ or $w_i=1/m_i^{\alpha }$ for some $0<\alpha \le 1$.²⁷ Solving equation (7) again yields an explicit solution

$$\tilde{\beta }={\left\{\sum _{i=1}^n{w}_i\sum _{k=1}^{m_i}\frac{{\text{Δ}}_i}{\widehat{G}\left({\tilde{T}}_{ik}\right)}{X}_{ik}^{\otimes 2}\right\}}^{-1}\sum _{i=1}^n{w}_i\sum _{k=1}^{m_i}\frac{{\text{Δ}}_{ik}}{\widehat{G}\left({\tilde{T}}_{ik}\right)}{X}_{ik}{\tilde{V}}_{ik}.$$ (8)

Assuming a common censoring distribution, we pool data across clusters to estimate $G(t)=P\left(C_{ik}\ge t\right)$ with the Kaplan–Meier estimator, since finite cluster sizes precludes consistent estimation of the censoring distribution. Thus, we estimate $G(t)$ with $\widehat{G}(t)={\prod }_{u<t}\left\{1-d{N}^c(u)/Y(u)\right\}$, where $N^c(u)={\sum }_{i=1}^n{w}_i{\sum }_{i=1}^{m_i}{N}_{ik}^c(u)={\sum }_{i=1}^n{w}_i{\sum }_{i=1}^{m_i}I\left({\tilde{T}}_{ik}\le u,{\delta }_{ik}=0\right)$ and $Y(u)={\sum }_{i=1}^n{w}_i{\sum }_{i=1}^{m_i}{Y}_{ik}(u)={\sum }_{i=1}^n{w}_i{\sum }_{i=1}^{m_i}I\left({\tilde{T}}_{ik}\ge u\right)$. Zhou and Fine²⁵ showed that $\widehat{G}(\cdot )$ converges in probability to $G(\cdot )$ uniformly on $[0,\tau ]$ and $n^{1/2}(\widehat{G}-G)$ converges weakly to a tight Gaussian process. Utilizing this fact, we derive the consistency and asymptotic normality of $\tilde{\beta }$ in the Appendix. Moreover, the variance-covariance matrix of the limiting normal distribution for $n^{1/2}\left(\tilde{\beta }-{\beta }_0\right)$ can be approximated by ${\widehat{A}}_2^{-1}{\widehat{B}}_2{\widehat{A}}_2^{-1}$, where

$$\begin{array}{l}{\widehat{A}}_2=n^{-1}\sum _{i=1}^n{w}_i\sum _{k=1}^{m_i}\frac{{\text{Δ}}_{ik}}{\widehat{G}\left({\tilde{T}}_{ik}\right)}{X}_{ik}^{\otimes 2},\\ {\widehat{B}}_2=n^{-1}\sum _{i=1}^n{\left[w_i\sum _{k=1}^{m_i}\left\{\frac{{\text{Δ}}_{ik}{X}_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\tilde{\beta }\right)}{\widehat{G}\left({\tilde{T}}_{ik}\right)}-\left(1-{\delta }_{ik}\right)Q_2\left({\tilde{T}}_{ik}\right)+\sum _{l=1}^n\frac{\left(1-{\delta }_{lk}\right)Y_{ik}\left({\tilde{T}}_{lk}\right)Q_2\left({\tilde{T}}_{lk}\right)}{Y\left({\tilde{T}}_{lk}\right)}\right\}\right]}^{\otimes 2},\end{array}$$

and

$$Q_2(t)=\frac{1}{Y(t)}\sum _{i=1}^n{w}_i\sum _{k=1}^{m_i}\frac{{\text{Δ}}_{ik}}{\widehat{G}\left({\tilde{T}}_{ik}\right)}{Y}_{ik}(t)X_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\tilde{\beta }\right).$$

3 |. SIMULATION STUDIES

In this section, we conducted extensive simulation studies using synthetic data to examine the finite-sample performance of the proposed methods, under independent and correlated competing risks scenarios.

3.1 |. Scenario 1: Independent competing risks

Our studies involve two causes of failure and two covariates $X_i=\left(x_{1i},x_{2i}\right)$, where $x_{1i}\sim N(0,1)$ and $x_{2i}\sim \text{Bernoulli}(0.5)$. In the first simulation, we specify the baseline subdistribution function for cause-1 failures as $F_{10}(t)=1-\exp \left\{-\theta {\text{Ψ}}_0(t)\right\}$, where ${\text{Ψ}}_0(t)$ denotes the distribution function of standard exponential (Exp), log-normal (LN) or Chi-squared (Chi) distribution, and generate cause-1 failure times from

$$P\left(T_i\le t,D_i=1\mid X_i\right)=F_{10}\left(t{e}^{{\beta }_1{x}_{1i}+{\beta }_2{x}_{2i}}\right),$$

where $\left(\theta ,{\beta }_1,{\beta }_2\right)=(-\log (0.4),0.5,-0.5)$. In this setup, $P\left(D_i=1\right)=1-e^{-\theta }=0.6=1-P\left(D_i=2\right)$. Cause-2 failure times are generated from the standard exponential distribution for $P\left(T_i\le t\mid D_i=2\right)$. Censoring times follow an $\text{Exp}\left({\lambda }_0\right)$ distribution, where ${\lambda }_0$ is chosen to yield the desired censoring rates (15% and 30%).

Table 1 summarizes the results of these studies from 2000 random replications with $n=300$ and $500$, including the average bias of the proposed IPCW estimators (Bias), the sampling standard errors (SSE), the averaged standard errors (ASE), and the coverage probability of 95% nominal Wald-type confidence interval (CP). It can be seen that the proposed estimators are virtually unbiased under moderate censoring rates and the variance estimators agree well with the true variations. As the censoring proportion increases, the proposed estimators seem to deviate slightly from the true value as expected, but this censoring effect reduces with increased sample sizes. Our experience is that the IPCW estimator is quite sensitive to the amount of censoring as well as the censoring distribution, even for usual survival data without competing risks. Thus care should be taken when the IPCW methods are applied in the presence of heavy censoring. In Figure 1, we plotted the estimated baseline cumulative hazards under 15% censoring, in addition to the competing risks. Although our method slightly overestimates the true curve, the estimated curves are generally within the 95% confidence intervals. Bias reduction is not much noticeable when the sample size increases from 300 to 500, but the associated standard errors decrease accordingly.

TABLE 1.

Simulation study (Scenario 1): Summary statistics corresponding to the fixed-effects parameters for various choices of sample sizes, the baseline subdistribution function and censoring levels

				$n=300$				$n=500$
Dist	Cens	Par	True	Bias	SSE	ASE	CP	Bias	SSE	ASE	CP
Exp	15%	${\beta }_1$	0.5	–0.010	0.112	0.109	0.949	−0.004	0.088	0.086	0.948
		${\beta }_2$	−0.5	0.003	0.221	0.218	0.945	0.009	0.172	0.170	0.944
	30%	${\beta }_1$	0.5	−0.037	0.134	0.126	0.916	−0.029	0.102	0.100	0.920
		${\beta }_2$	−0.5	0.026	0.254	0.252	0.939	0.019	0.202	0.201	0.944
LN	15%	${\beta }_1$	0.5	−0.014	0.085	0.083	0.934	−0.013	0.068	0.067	0.928
		${\beta }_2$	−0.5	0.020	0.175	0.168	0.938	0.013	0.131	0.133	0.954
	30%	${\beta }_1$	0.5	−0.039	0.100	0.097	0.908	−0.035	0.080	0.079	0.909
		${\beta }_2$	−0.5	0.034	0.206	0.204	0.946	0.033	0.167	0.165	0.943
Chi	15%	${\beta }_1$	0.5	−0.017	0.198	0.194	0.938	−0.019	0.149	0.153	0.953
		${\beta }_2$	−0.5	0.013	0.399	0.388	0.946	0.024	0.304	0.303	0.945
	30%	${\beta }_1$	0.5	−0.074	0.226	0.220	0.924	−0.070	0.181	0.179	0.919
		${\beta }_2$	−0.5	0.080	0.445	0.451	0.948	0.052	0.377	0.366	0.940

Abbreviations: ASE, average of estimated standard errors; Bias, average bias of estimators; CP, coverage probability of 95% confidence intervals; SSE, sampling standard errors.

FIGURE 1.

Estimation of the baseline cumulative hazard function under the exponential, log-normal, and chi-squared distributions. The curves represent true (solid), estimated and 95% confidence-intervals for the sample sizes $n=300$ (dotted) and $n=500$ (dashed)

3.2 |. Scenario 2: Clustered competing risks

We further consider the situation, where the competing risks event times are correlated within a cluster. In a similar simulation configuration, let the cause-1 cumulative incidence function for cluster $k$ satisfy

$$P\left(T_{ik}\le t,D_{ik}=1\mid X_{ik},{\omega }_i\right)=F_{10}\left({\omega }_{i}t{e}^{{\beta }_1{x}_{1ik}+{\beta }_2{x}_{2ik}}\right),$$

where $x_{1ik}\sim N(0,1)$, $x_{2ik}\sim \text{Bernoulli}(0.5)$, $\left({\beta }_1,{\beta }_2\right)=(0.5,-0.5)$, and the frailty variable ${\omega }_i$ follows a gamma distribution with mean 1 and variance $v\in \{0.5,1,2\}$. Note $T_{ik}$ follows an AFT model, conditioning or not conditioning on ${\omega }_i$. Censoring times are generated from an exponential distribution to yield approximately 45% of cause-1 events, 30% of cause-2 events, and 25% of censoring rates. We further consider two cases: (i) noninformative and (ii) informative cluster size. For case (i), we fix the cluster size as $K=2$ and $5$. For case (ii), we let $m_i$ depend on ${\omega }_i$ such that $m_i=(s/10)+2$, if $q_s\le {\omega }_i<q_{s+10}$, for $s=0,10,\dots ,90$, where $q_s$ is the $s\text{th}$ percentile of the frailty distribution. In this setup, cluster sizes vary from 2 to 11, and are positively correlated with the frailty variable. As in Scenario 1, we generated 2000 simulated datasets of clustered competing risks, with $n=300$ and $500$.

Tables 2 and 3 presents the simulation summary statistics corresponding to (i) noninformative and (ii) informative cluster sizes, respectively. In Table 2, where cluster sizes are not adjusted, the proposed IPCW estimator is compared to the complete-data estimator where failure times from all causes are assumed to be known. The results suggest that the proposed method performs very well across the combinations of cluster and sample sizes. The standard error estimates agree well with the empirical standard errors, which improves as the sample size increases. However, relative efficiency (RE) is in the range of 0.21–0.58, which shows that the IPCW method suffers severe information loss when competing risks are present. Other approaches, such as rank estimation, are worth investigating further to gain more efficiency. Finally, Table 3 presents summary estimates from the IPCW method adjusted for the informative cluster size (ie, $w_i=1/m_i$), or when unadjusted (ie, $w_i=1$). When cluster sizes are not adjusted, we observe that the model-based standard error estimation tends to underestimate the empirical standard errors. Clearly, by adjusting the cluster size, we can obtain smaller biases and more robust standard errors, resulting in improved coverage probabilities.

TABLE 2.

Simulation study (Scenario 2, Case (i)): Summary statistics corresponding to the fixed-effects parameters for various choices of sample size, the cluster size $\left(K\right)$, and the variance parameter $\left(v\right)$ of the Gamma frailty for clustered competing risks with noninformative cluster size

					IPCW				Complete-data
$n$	$K$	$v$	Par	True	Bias	SSE	ASE	CP	Bias	SSE	ASE	CP	RE
300	2	0.5	${\beta }_1$	0.5	−0.016	0.117	0.116	0.945	0.001	0.076	0.077	0.942	0.43
			${\beta }_2$	−0.5	0.007	0.126	0.124	0.940	−0.006	0.083	0.083	0.945	0.43
		1	${\beta }_1$	0.5	−0.017	0.143	0.142	0.938	−0.002	0.090	0.091	0.962	0.39
			${\beta }_2$	−0.5	0.011	0.152	0.155	0.946	−0.004	0.106	0.105	0.943	0.48
		2	${\beta }_1$	0.5	−0.035	0.204	0.196	0.940	−0.001	0.129	0.126	0.951	0.41
			${\beta }_2$	−0.5	0.029	0.221	0.213	0.938	−0.005	0.156	0.154	0.947	0.50
	5	0.5	${\beta }_1$	0.5	−0.006	0.092	0.088	0.953	−0.002	0.046	0.049	0.951	0.25
			${\beta }_2$	−0.5	−0.006	0.102	0.099	0.951	−0.006	0.062	0.064	0.948	0.37
		1	${\beta }_1$	0.5	−0.007	0.109	0.104	0.936	−0.001	0.059	0.057	0.948	0.29
			${\beta }_2$	−0.5	0.013	0.122	0.122	0.938	0.000	0.088	0.087	0.948	0.52
		2	${\beta }_1$	0.5	−0.015	0.152	0.143	0.937	0.002	0.081	0.079	0.935	0.29
			${\beta }_2$	−0.5	0.023	0.182	0.181	0.952	0.006	0.139	0.138	0.950	0.58
500	2	0.5	${\beta }_1$	0.5	−0.014	0.100	0.101	0.944	−0.003	0.058	0.060	0.948	0.33
			${\beta }_2$	−0.5	0.005	0.110	0.105	0.941	−0.004	0.069	0.065	0.933	0.39
		1	${\beta }_1$	0.5	−0.014	0.121	0.121	0.943	−0.005	0.071	0.071	0.947	0.34
			${\beta }_2$	−0.5	0.010	0.128	0.128	0.938	0.002	0.084	0.082	0.936	0.43
		2	${\beta }_1$	0.5	−0.017	0.170	0.165	0.942	0.003	0.102	0.098	0.945	0.34
			${\beta }_2$	−0.5	0.024	0.181	0.178	0.939	0.000	0.122	0.120	0.942	0.45
	5	0.5	${\beta }_1$	0.5	0.003	0.074	0.073	0.955	−0.001	0.037	0.038	0.959	0.25
			${\beta }_2$	−0.5	0.004	0.077	0.079	0.948	0.000	0.048	0.049	0.959	0.40
		1	${\beta }_1$	0.5	−0.005	0.096	0.089	0.935	−0.003	0.046	0.045	0.942	0.23
			${\beta }_2$	−0.5	0.001	0.103	0.102	0.950	0.003	0.069	0.068	0.956	0.45
		2	${\beta }_1$	0.5	−0.007	0.134	0.122	0.939	−0.001	0.061	0.062	0.951	0.21
			${\beta }_2$	−0.5	0.009	0.143	0.148	0.961	−0.006	0.104	0.107	0.953	0.53

Note: The proposed IPCW estimator is compared to the complete-data estimator with all event times known.

TABLE 3.

Simulation study (Scenario 2, Case (ii)): Summary statistics corresponding to the fixed-effects parameters for various choices of sample size, and the variance parameter $v$ of the Gamma frailty, for clustered competing risks with informative cluster size

				Adjusted				Unadjusted
$n$	$v$	Par	True	Bias	SSE	ASE	CP	Bias	SSE	ASE	CP
300	0.5	${\beta }_1$	0.5	−0.012	0.106	0.103	0.937	−0.033	0.090	0.088	0.918
		${\beta }_2$	−0.5	0.012	0.104	0.103	0.948	0.036	0.090	0.088	0.912
	1	${\beta }_1$	0.5	−0.011	0.105	0.103	0.936	−0.036	0.088	0.087	0.918
		${\beta }_2$	−0.5	0.011	0.103	0.103	0.949	0.033	0.089	0.088	0.922
	2	${\beta }_1$	0.5	−0.008	0.080	0.081	0.953	−0.027	0.073	0.071	0.916
		${\beta }_2$	−0.5	0.015	0.085	0.081	0.938	0.037	0.074	0.071	0.889
500	0.5	${\beta }_1$	0.5	−0.009	0.082	0.081	0.940	−0.029	0.072	0.071	0.918
		${\beta }_2$	−0.5	0.010	0.082	0.081	0.937	0.031	0.073	0.071	0.908
	1	${\beta }_1$	0.5	−0.011	0.080	0.081	0.956	−0.030	0.072	0.072	0.908
		${\beta }_2$	−0.5	0.011	0.080	0.080	0.948	0.032	0.071	0.071	0.907
	2	${\beta }_1$	0.5	−0.009	0.081	0.080	0.948	−0.029	0.073	0.071	0.910
		${\beta }_2$	−0.5	0.011	0.081	0.080	0.944	0.033	0.070	0.071	0.913

Note: Cluster sizes are either adjusted (ie, $w_i=1/m_i$), or unadjusted (ie, $w_i=1$).

4 |. APPLICATIONS

4.1 |. Follicular lymphoma data

We first applied the proposed method to data from a study of follicular lymphoma.²⁸ This study compared over 25 years of radiation plus chemotherapy to radiation alone in 541 individuals with stage I or II follicular lymphoma. For clinicians, the probabilities of failure from the disease (defined as either no response to treatment, or cancer relapse), or death from other causes are crucial to assess the outcomes associated with the treatment options and guide patient management. Of 423 patients on radiation only, 226 had either no favorable treatment response or experienced cancer relapse and 66 died without relapsing. Of 118 patients on radiation plus chemotherapy, 46 had either no favorable treatment response or experienced cancer relapse and 10 died without relapsing. Covariates include treatment (1, radiation plus chemotherapy; 0, radiation only), disease stage (1, stage I; 0, stage II), age group (1, age < 60 years; 0, age ≥ 60 years), and hemoglobin (Hgb) level (1, Hgb < 145 g/L; 0, Hgb ≥ 145 g/L). The median follow-up time for patients who were still alive and remained disease-free was 10.3 years and their median age was 58 years. R code for this data example is available at the GitHub link (https://github.com/taehwa015/cr-wls).

To determine whether or not censoring is related to the covariates, we first fit univariate Cox models to the censoring times with treatment, stage, age and hemoglobin level as a covariate. The analyses give estimated hazard ratios (95% confidence intervals) that are all significantly different from one; 2.84 (2.08, 3.86), 1.67 (1.20, 2.34), 0.70 (0.51, 0.97), and 0.59 (0.43, 0.80) for treatment, stage, age and hemoglobin level, respectively, when they are dichotomized as above. We fit the multivariate regression models with all the variables for disease relapse and death without relapse. Table 4 reports points estimates (95% confidence intervals) for the regression coefficients in the proposed model using the weights from Kaplan–Meier, Cox regression and random survival forest methods, respectively, and the subdistribution hazards ratios from the Fine–Gray model. Overall, three estimates from AFT models are quite similar with consistent implication, implying that patients who received radiation plus chemotherapy tend to experience both failure types sooner than those on radiation alone. Addition of chemotherapy significantly reduces expected survival times in the linear model for disease relapse (p < 0.001), while the Fine–Gray model indicates that it leads to lower subdistribution hazard, although marginally nonsignificant (p = 0.054).

TABLE 4.

Follicular lymphoma data: Point estimate (95% CI) of regression coefficients in models for the competing risks, disease relapse (event 1) and death without relapse (event 2)

Cause	AFT model			SPH model (Fine–Gray) HR (95% CI)
	KM weight	Cox weight	RSF weight
	Est (95% CI)	Est (95% CI)	Est (95% CI)
Event 1: Disease relapse
Chemotherapy	−1.63 (–2.57, −0.70)	−1.64 (–2.57, −0.71)	−1.62 (–2.55, −0.69)	0.72 (0.51, 1.01)
Disease stage	0.92 (0.39, 1.45)	0.93 (0.41, 1.46)	0.91 (0.38, 1.43)	0.59 (0.45, 0.77)
Age	0.89 (0.41, 1.37)	0.92 (0.44, 1.40)	0.88 (0.41, 1.35)	0.68 (0.53, 0.88)
Hemoglobin	−0.46 (−0.90, −0.02)	−0.45 (−0.89, 0.00)	−0.47 (−0.90, −0.03)	0.98 (0.77, 1.25)
Event 2: Death without relapse
Chemotherapy	−0.47 (−1.13, 0.19)	−0.52 (−1.07, 0.03)	−0.43 (−1.11, 0.26)	0.71 (0.36, 1.40)
Disease stage	−0.45 (−0.96, 0.07)	−0.47 (−0.98, 0.03)	−0.40 (−0.88, 0.09)	1.06 (0.66, 1.70)
Age	0.72 (0.27, 1.17)	0.79 (0.35, 1.23)	0.65 (0.21, 1.08)	0.38 (0.24, 0.60)
Hemoglobin	0.36 (−0.16, 0.88)	0.49 (−0.02, 0.99)	0.27 (−0.21, 0.75)	1.15 (0.71, 1.88)

Note: The results are based on the AFT models with weights from Kaplan–Meier (KM), Cox regression (Cox), and random survival forests (RSF), respectively, and the subdistribution proportional hazards (SPH) model.

Nonparametric methods (not shown here) represent that the two cumulative incidence curves for the chemotherapy and nonchemotherapy groups cross at around one year; more patients on chemotherapy experience disease relapse early in time but their cumulative incidences become lower once they survive treatment with chemotherapy. Thus, the proportionality assumption is unrealistic, and the proposed method provides alternative perspective to evaluate the treatment effect. Disease stage and age are important predictive factors on crude mean survival times for disease relapse, while hemoglobin level seems not significant for both failure types. In Figure 2, we plot the nonparametrically estimated CIF corresponding to disease relapse event, for patients with disease stages I and II (separately), with the estimates from the proposed AFT and Fine–Gray models overlayed. We observe that the two nonparametric curves tend to approach the same level as time increases, which corroborates to nonproportionality of the hazards assumption. This suggests that the AFT model is subjectively better than the Fine–Gray model in early years for the given variable, as evidenced in Figure 2. However, there is a deviation in the right tail between the nonparametric curve and our method, mainly from the fact that the treatment effects diminish in the limit under the AFT framework.

FIGURE 2.

Follicular lymphoma data: Estimated cumulative incidence probabilities for disease relapse. Lower and upper incidence curves correspond to patients stage I and II diseases, respectively. Grey solid lines are nonparametric estimates, black solid lines are from the crude-risk accelerated failure time model, while black dotted lines are from the Fine–Gray model

4.2 |. Acute myeloid leukemia data

Next, we consider the acute myeloid leukemia (AML) data, generated from the multicenter bone marrow transplant registry of European blood and bone marrow transplant (EBMT) groups. This study was mainly intended to identify important predictors for two competing events: (i) acute GvHD, grade II or chronic GvHD, and (ii) death and relapse without GvHD. Initially, there were a total of 2952 patients from 244 centers with 1385 GvHD and 629 competing events. For this dataset, Zhou et al²⁵ applied the subdistribution PH (SPH) model while treating the centers as clusters. In our analysis, after excluding rows containing missing values and centers with one patient only, we consider a subsample of 320 patients with two covariates, (i) female donor to male recipient (FM) and (ii) whether source of stem cells is peripheral blood (PBSC), or not. In our analysis, we have 153 centers with 2–9 patients per center (median = 3). Table 5 summarizes the analysis results from the AFT models for clustered data, where the cluster size is adjusted or unadjusted. As before, we consider three weighting methods from Kaplan–Meier (KM), Cox regression (Cox) and random survival forest (RSF), and also include the results from SPH model²⁵ for comparison. These three weighting methods yield quite consistent results, which implies that the censoring distribution may have low to no correlation with the covariates. The covariate effects change slightly with adjustment of the cluster size. Despite the insignificant importance of cluster size, comparing the two methods is recommended in a regular analysis to prevent a few large clusters from dominating the results.

TABLE 5.

AML data: Parameter estimates (95% confidence intervals) and associated p-values in the competing risks modeling, with acute or chronic GvHD (event 1) and death free of GVHD (event 2)

		Event 1: Acute, or chronic GVHD				Event 2: Death free of GvHD
		FM		PBSC		FM		PBSC
Method	Weight	Est (SE)	p-value	Est (SE)	p-value	Est (SE)	p-value	Est (SE)	p-value
Adjusted	KM	0.92 (0.41)	0.01	0.48 (0.31)	0.06	0.43 (0.31)	0.08	0.11 (0.23)	0.32
	Cox	0.96 (0.42)	0.01	0.47 (0.32)	0.07	0.44 (0.31)	0.08	0.10 (0.23)	0.33
	RSF	0.92 (0.41)	0.01	0.48 (0.31)	0.06	0.42 (0.31)	0.09	0.11 (0.23)	0.32
Unadjusted	KM	0.84 (0.38)	0.01	0.54 (0.28)	0.03	0.45 (0.33)	0.09	0.16 (0.22)	0.23
	Cox	0.87 (0.39)	0.01	0.53 (0.28)	0.03	0.48 (0.34)	0.08	0.15 (0.22)	0.25
	RSF	0.83 (0.38)	0.01	0.55 (0.27)	0.02	0.44 (0.33)	0.09	0.17 (0.22)	0.22
SPH	–	0.29 (0.15)	0.03	−0.22 (0.14)	0.06	0.14 (0.14)	0.16	−0.09 (0.14)	0.26

Note: Cluster sizes are adjusted or unadjusted in the AFT fit and adjusted in the SPH fit.²⁵

5 |. CONCLUSION

The competing risks problem is inherent to biomedical research when multiple mutually censoring endpoints are available. As a result of its direct physical interpretation, the AFT model is an attractive alternative to the popular subdistribution PH model⁹ for the regression analysis of competing risks data. These two models have very different characteristics in that in the long run the covariate effects fade away under the AFT model but remain unchanged under the PH model. In routine competing risks analysis, it is recommended to consider both approaches since they may present different perspectives on given data. Although the LS technique may appear as a natural approach here, it is applicability is hindered by the presence of censoring. The inference procedures developed in this paper represent a practical way of implementing the LS principles for competing risks data. Our methods can be further extended by permitting dependence between $C$ and $X$ in many ways. In our current setup, we model the censoring distribution using PH²¹ or (nonparametric) recursive partitioning methods via survival trees⁴ with inversely weighted estimating function using the censoring distribution from the fitted model.²⁹ However, it is important to point out that the proposed weighted LS estimator is sensitive to the amount of right censoring as well as the censoring distribution, rendering it inappropriate under heavy censoring. In such cases, we may alternatively seek a more robust inference procedure, based on log-rank estimating approach.¹⁷ One may use a rank estimator by using the proposed estimator as the initial value, and solving a rank estimating function for competing risks. Furthermore, our proposed methods may be challenging while modeling (clustered) competing risks endpoints observed in large electronic health records data, such as in kidney transplantation.³⁰ Investigation along these directions merit future research and is under consideration by the current authors.

APPENDIX A: ASYMPTOTIC RESULTS

A.1. Asymptotic properties of $\tilde{\beta }$

We first prove the asymptotic normality of $\tilde{\beta }$ for correlated competing risks data, stated in Section 2.2. The theory for $\widehat{\beta }$, given in Section 2.1, is a special case with $m_i=1$. For brevity, we assume the same cluster size in what follows, that is, $m_i=m$ for $i=1,\dots ,n$. To begin, define $N^c(u)={\sum }_{i=1}^n{\sum }_{i=1}^m{N}_{ik}^c(u)={\sum }_{i=1}^n{\sum }_{i=1}^{m}I\left({\tilde{T}}_{ik}\le u,{\delta }_{ik}=0\right)$ and $Y(u)={\sum }_{i=1}^n{\sum }_{i=1}^m{Y}_{ik}(u)={\sum }_{i=1}^n{\sum }_{i=1}^{m}I\left({\tilde{T}}_{ik}\ge u\right)$. We estimate $G(t)$ with $\widehat{G}(t)={\prod }_{u<t}\left\{1-d{N}^c(u)/Y(u)\right\}$, a naive Kaplan–Meier estimator of the survival function of the censoring variable $C_i$, where the dependence among individuals within clusters is ignored. Let the associated censoring martingale process $M_{ik}^c(t)=N_{ik}^c(t)-{\int }_0^t{\lambda }^c(u)Y_{ik}(u)du$, where ${\lambda }^c(u)$ is the hazard function of the censoring distribution.

The left-hand side of (7) can be written as $U_2(\beta )=U_{21}(\beta )+U_{22}(\beta )$, where

$$\begin{array}{l}{U}_{21}(\beta )=\sum _{i=1}^n\sum _{k=1}^m\frac{{\text{Δ}}_{ik}}{G\left({\tilde{T}}_{ik}\right)}{X}_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\beta \right),\\ U_{22}(\beta )=\sum _{i=1}^n\sum _{k=1}^m\frac{G\left({\tilde{T}}_{ik}\right)-\widehat{G}\left({\tilde{T}}_{ik}\right)}{G\left({\tilde{T}}_{ik}\right)\widehat{G}\left({\tilde{T}}_{ik}\right)}{\text{Δ}}_{ik}{X}_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\beta \right).\end{array}$$ (A1)

Let $y(t)=P\left({\tilde{T}}_{ik}\ge t\right)$. Using the martingale integral representation of the Kaplan–Meier estimator (Gill, 1980) and the uniform convergence of $\widehat{G}(\cdot )$ to $G(\cdot )$, we can express

$$n^{-1/2}{U}_{22}(\beta )=-n^{-1}\sum _{i=1}^n\sum _{k=1}^m{\int }_0^{\infty }{Q}_2(u)d{M}_{ik}^c(u)+o_p(1),$$

where $Q_2(t)=n^{-1}{\sum }_{i=1}^n{\sum }_{i=1}^m{\text{Δ}}_{ik}{Y}_{ik}(t)X_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\widehat{\beta }\right)/\left\{y(t)\widehat{G}\left({\tilde{T}}_{ik}\right)\right\}$. By the law of large numbers, $Q_2(t)$ converges to a well-defined limit, say $q_2(t)$, as $n\to \infty$, and therefore,

$$n^{-1/2}{U}_{22}(\beta )=-n^{-1}\sum _{i=1}^n\sum _{k=1}^m{\int }_0^{\infty }{q}_2(u)d{M}_{ik}^c(u)+o_p(1).$$ (A2)

Combining (A2) with (A1), we have $n^{-1/2}{U}_2(\beta )=n^{-1/2}{\sum }_{i=1}^n{\xi }_i+o_p(1)$, where

$${\xi }_i=\sum _{k=1}^m\left[\frac{{\text{Δ}}_{ik}}{G\left({\tilde{T}}_{ik}\right)}{X}_{ik}\left({\tilde{V}}_{ik}-X_{ik}^{\prime }\beta \right)-{\int }_0^{\infty }{q}_2(u)d{M}_{ik}^c(u)\right].$$

Since ${\xi }_i$, $i=1,\dots ,n$, are $n$ independent zero-mean random variables, $n^{-1/2}{U}_2(\beta )$ converges in distribution to a zero-mean normal random variable with the limiting variance, given by $B_2={\lim }_{n\to \infty }{n}^{-1}{\sum }_{i=1}^n{\xi }_i{\xi }_i^{\prime }$. By the Taylor series expansion, $n^{1/2}\left(\tilde{\beta }-{\beta }_0\right)=n^{-1/2}{\widehat{A}}_2^{-1}{U}_2(\beta )$, in which ${\widehat{A}}_2=n^{-1}{\sum }_{i=1}^n{\sum }_{i=1}^m\left\{{\text{Δ}}_{ik}/\widehat{G}\left({\tilde{T}}_{ik}\right)\right\}{X}_{ik}^{\otimes 2}$ converges in probability to $A_2\equiv E\left\{X_{ik}^{\otimes 2}I\left(D_{ik}=1\right)\right\}$. It then follows from the aforementioned asymptotic normality of $n^{-1/2}{U}_2(\beta )$ that $n^{1/2}\left(\tilde{\beta }-{\beta }_0\right)$ converges in distribution to a zero-mean normal random variable whose limiting variance is $A_2^{-1}{B}_2{A}_2^{-1}$ with the sample variance estimator ${\widehat{A}}_2^{-1}{\widehat{B}}_2{\widehat{A}}_2^{-1}$. The consistency of ${\widehat{B}}_2$ for $B_2$ follows from the law of large numbers, together with the consistency of $\tilde{\beta }$ and $\widehat{G}$.

A.2. Asymptotic properties of $W(t)$ and $\widehat{W}(t)$

We next deal with the weak convergence of $W(t)$ and $\widehat{W}(t)$. For simplicity, we let $m_i=1$, but the idea can be extended straightforwardly to clustered data. By the uniform strong law of large numbers and the uniform convergence of the Kaplan–Meier estimator,

$$n^{-1}\sum _{i=1}^n{N}_i^w(t,\beta ){\to }_{p}E\left\{N_1(t,\beta )\right\},n^{-1}\sum _{i=1}^n{Y}_i^w(t,\beta ){\to }_p{y}^{*}(t)\equiv E\left\{Y_1(t,\beta )\right\}$$

uniformly in $u\in [0,\tau ]$ and $\beta$ as $n\to \infty$. Therefore,

$${\widehat{H}}_{10}(t,\beta ){\to }_p{\int }_0^t\frac{E\left\{d{N}_1(u,\beta )\right\}}{E\left\{Y_1(u,\beta )\right\}}=H_{10}(t,\beta ),$$

establishing the consistency of ${\widehat{H}}_{10}(t,\beta )$. Consider $W(t)=n^{1/2}\left\{{\widehat{H}}_{10}(t,\widehat{\beta })-H_{10}(t)\right\}$ for $t\in [0,\tau ]$ and decompose it as

$$\begin{array}{l}\left[n^{1/2}\left\{{\widehat{H}}_{10}(t,\widehat{\beta })-{\tilde{H}}_{10}(t,\widehat{\beta })\right\}\right]+\left[n^{1/2}\left\{{\tilde{H}}_{10}(t,\widehat{\beta })-{\tilde{H}}_{10}^{*}(t,\widehat{\beta })\right\}\right]\\ +\left[n^{1/2}\left\{{\tilde{H}}_{10}^{*}(t,\widehat{\beta })-{\tilde{H}}_{10}^{*}\left(t,{\beta }_0\right)\right\}\right]+\left[n^{1/2}\left\{{\tilde{H}}_{10}^{*}\left(t,{\beta }_0\right)-H_{10}(t)\right\}\right],\end{array}$$ (A3)

where ${\widehat{H}}_{10}(t,\beta )$ and ${\tilde{H}}_{10}^{*}(t,\beta )$ are given in (6), and ${\tilde{H}}_{10}(t,\beta )=n^{-1}{\sum }_{i=1}^n{\int }_0^t\frac{d{N}_i^G(u,\beta )}{{\overline{Y}}^G(u,\beta )}$ with $N_i^G(t)={\delta }_{i}I\left(T_i\le t,D_i=1\right)/G(t)$ and ${\overline{Y}}^G=n^{-1}{\sum }_{i=1}^n{Y}_i^G(t)=n^{-1}{\sum }_{i=1}^n\left\{1-N_i^G(t)\right\}$. By using Gill’s martingale representation and the uniform convergence of $\widehat{G}(\cdot )$ to $G(\cdot )$, it can be seen that the first two terms in (A3) are asymptotically equivalent to $n^{-1/2}{\sum }_{i=1}^n{\int }_0^{\infty }{w}_0(u)d{M}_i^c(u)+o_p(1)$ where $w_0(t)={\lim }_{n\to \infty }{n}^{-1}{\sum }_{i=1}^n{\text{Δ}}_i{Y}_i(t)I\left({\tilde{T}}_i\le t\right)/\left\{y(t)y^{*}(t)\widehat{G}\left({\tilde{T}}_i\right)\right\}$. By applying the techniques of Ying¹⁶ and Lin, Wei, and Ying,¹⁷ we can show that, although ${\tilde{H}}_{10}^{*}(t,\beta )$ is not continuous, asymptotically there exists a deterministic function $b_0(t)$ such that, for $\widehat{\beta }$ near ${\beta }_0$,

$$n^{1/2}\left\{{\tilde{H}}_{10}^{*}(t,\widehat{\beta })-{\tilde{H}}_{10}^{*}\left(t,{\beta }_0\right)\right\}=b_0^{\prime }(t)n^{1/2}\left(\widehat{\beta }-{\beta }_0\right)+o_p(1)=b_0^{\prime }(t)A_1^{-1}{n}^{-1/2}U\left({\beta }_0\right)+o_p(1),$$ (A4)

where $A_1=E\left\{X_1^{\otimes 2}I\left(D_1=1\right)\right\}$ and $b_0(t)=-{\int }_0^t{\overline{X}}_0^{*}(u)d\left\{H_{10}^{\prime }(u)u\right\}$ with ${\overline{X}}_0^{*}(t)={\lim }_{n\to \infty }\frac{{\sum }_{i=1}^n{Y}_i^{*}\left(t,{\beta }_0\right)X_i}{{\sum }_{l=1}^n{Y}_i^{*}\left(t,{\beta }_0\right)}$. Finally, the fourth term of (A3) is asymptotically equivalent to $n^{-1/2}{\sum }_{i=1}^n{\int }_0^t\frac{d{M}_i^{*}\left(u,{\beta }_0\right)}{{\overline{Y}}^{*}\left(u,{\beta }_0\right)}$, where $M_i^{*}(t,\beta )=N_i^{*}(t,\beta )-{\int }_0^t{Y}_i^{*}(u,\beta )d{H}_{10}(u)$ is a martingale process based on a properly defined filtration. Combining these results, we have, uniformly in $t\in [0,\tau ]$, that

$$W(t)=n^{-1/2}\sum _{i=1}^n\left[{\int }_0^t\frac{d{M}_i^{*}\left(u,{\beta }_0\right)}{E\left\{Y_1^{*}\left(u,{\beta }_0\right)\right\}}+{\int }_0^{\infty }{w}_0(u)d{M}_i^c(u)+b_0^{\prime }(t)A_1^{-1}E\left\{n^{-1}U\left({\beta }_0\right)\right\}\right]+o_p(1).$$

By the multivariate central limit theorem with a straightforward covariance calculation, we observe that $W(\cdot )$ converges to a zero-mean Gaussian process with covariate function

$$\begin{array}{l}\sigma (s,t)=E\left(\left[{\int }_0^s\frac{d{M}_1^{*}\left(u,{\beta }_0\right)}{E\left\{Y_1^{*}\left(u,{\beta }_0\right)\right\}}+{\int }_0^{\infty }{w}_0(u)d{M}_1^c(u)+b_0^{\prime }(s)A_1^{-1}E\left\{n^{-1}U\left({\beta }_0\right)\right\}\right]\right.\\ \left.\times \left[{\int }_0^t\frac{d{M}_1^{*}\left(u,{\beta }_0\right)}{E\left\{Y_1^{*}\left(u,{\beta }_0\right)\right\}}+{\int }_0^{\infty }{w}_0(u)d{M}_1^c(u)+b_0^{\prime }(t)A_1^{-1}E\left\{n^{-1}U\left({\beta }_0\right)\right\}\right]\right).\end{array}$$

Now considering equation (A4) by replacing $\widehat{\beta }$ with ${\widehat{\beta }}^{*}$, we observe

$$n^{1/2}\left\{{\tilde{H}}_{10}^{*}(t,\widehat{\beta })-{\tilde{H}}_{10}^{*}\left(t,{\widehat{\beta }}^{*}\right)\right\}=b_0^{\prime }(t)n^{1/2}\left(\widehat{\beta }-{\widehat{\beta }}^{*}\right)+o_p(1)=b_0^{\prime }(t)A_1^{-1}{n}^{-1/2}{U}_1\left({\widehat{\beta }}^{*}\right)+o_p(1),$$

where the second equality follows from the asymptotic linearity of $U_1(\beta )$ for $\beta$ and $U_1(\widehat{\beta })=0$. Thus,

$$\widehat{W}(t)=n^{-1/2}\sum _{i=1}^n\left[n{\widehat{H}}_i(t,\widehat{\beta })+n{\widehat{R}}_i(t,\widehat{\beta })+b_0^{\prime }(t)A_1^{-1}{u}_{1i}(\widehat{\beta })\right]Z_i+o_p(1),$$

where $u_{1i}(\beta )=\left\{{\text{Δ}}_i/\widehat{G}\left({\tilde{T}}_i\right)\right\}{X}_i\left({\tilde{V}}_i-X_i^{\prime }\beta \right)$ and

$${\widehat{R}}_i(t,\beta )=\left(1-{\delta }_i\right)w_0^{*}\left({\tilde{T}}_i\right)-\sum _{l=1}^n\frac{\left(1-{\delta }_l\right)Y_i\left({\tilde{T}}_l\right)w_0^{*}\left({\tilde{T}}_l\right)}{Y\left({\tilde{T}}_l\right)}$$

with $w_0^{*}(t)=n^{-2}{\sum }_{i=1}^n{\text{Δ}}_i{Y}_i(t)I\left({\tilde{T}}_i\le t\right)/\left\{nY(t){\overline{Y}}^w(t)\widehat{G}\left({\tilde{T}}_i\right)\right\}$. Then, by the multivariate central limit theorem and a simple covariance calculation, $\widehat{W}$ converges in finite-dimensional distribution to the limiting Gaussian process of $W$.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are openly available in

1. R library: randomForestSRC

2. R library: crrSC.