Competing Risks Model with Short-Term and Long-Term Covariate Effects for Cancer Studies

Abstract

Patients are frequently exposed to failure from several mutually exclusive causes, leading to a competing risk setting. Standard methods concerning the effects of covariates on the cause-specific hazards assume constant hazard ratios across time. This assumption, however, is violated in several applications. To address this issue and test the effect of covariates on multiple risks, we develop a new regression model allowing for nonconstant hazard ratios over time. The proposed model allows explicit specification of the short-term and long-term covariate effects, which can be of clinical interest. We develop a statistically efficient nonparametric likelihood methodology for estimation and inference concerning the parameters of interest and compare it to the existing methods. We investigate the performances of the proposed methods using simulations and apply them to a European study on a registry cohort of patients with acute leukemia undergoing bone marrow transplantation. Our proposed model detects the differences in short-term and long-term risks of primary relapse between patients with and without acute lymphoblastic leukemia.

Appendix: Asymptotic properties of the proposed NPMLEs

In this Appendix, we outline the asymptotic properties of the proposed NPMLEs. Let $\varvec{\xi }_0=\{({\varvec{\beta }}_{0j},\varvec{\gamma }_{0j}), \quad j=1, \ldots, k\}$ and $\varvec{\varLambda }_0 = \{\varLambda _{0j}, \quad j=1,\ldots,k\}$ denote the true parameters of $\varvec{\xi }= \{({\varvec{\beta }}_j,\varvec{\gamma }_j),\quad j=1,\ldots,k\}$ and $\varvec{\varLambda }= \{\varLambda _j,\quad j=1,\ldots,k\}$, respectively. Let $\widehat{\varvec{\xi }} = \{(\widehat{{\varvec{\beta }}}_{j},\widehat{\varvec{\gamma }}_{j}),\quad j=1,\ldots,k\}$ and $\widehat{\varvec{\varLambda }} = \{\widehat{\varLambda }_{j}, \quad j=1,\ldots, k\}$ denote the proposed NPMLEs. To establish the consistency and asymptotic normality of the proposed NPMLEs, denoted by $\widehat{\varvec{\xi }} = \{(\widehat{{\varvec{\beta }}}_{j},\widehat{\varvec{\gamma }}_{j})\quad j=1,\ldots,k\}$ and $\widehat{\varvec{\varLambda }} = \{\widehat{\varLambda }_{j}\quad j=1,\ldots, k\}$, the following regularity conditions are needed:

• (C1) If there exist a constant c and a constant vector $\mathbf{b}$ such that

  $$\mathbf{b}^T\mathbf{X}=c,$$

  with probability one, then $c=0$ and $\mathbf{b}=\mathbf{0}$. In addition, the support of $\mathbf{X}$ contains 0.

• (C2) The failure time T and the censoring time C are conditionally independent given ${\mathbf{X}}$.

• (C3) There exists some positive constant number $\delta _0$ such that $P(C\ge \tau |{\mathbf{X}}) = P(C=\tau |{\mathbf{X}})\ge \delta _0$ almost surely, where $\tau$ is a constant denoting the end of the study.

• (C4) The true regression parameter values of $\varvec{\xi }_0$ belong to a known compact set $\mathcal {B}_0$ in $R^{2kp}$.

• (C5) The true baseline cumulative distribution functions $\varLambda _{0j}, j=1,...,k$ belong to the following class

  $$\begin{aligned} \mathcal {A}_0= & {} \{\varLambda : \varLambda \text { is a strictly increasing function and is continuously} \\&\text {differentiable in } [0,\tau ] \text { with }\varLambda (0)=0, \varLambda '(0)>0\text { and } \varLambda (\tau )<\infty \}. \end{aligned}$$

All the above conditions are standard in the semiparametric regression analysis of right-censored competing risks data. Since the nonparametric likelihood $L_n(\varvec{\theta })$ is bounded from above, the NPMLEs exist.

The following theorem establishes the consistency of the proposed NPMLEs.

Theorem 1

Under conditions (C1)-(C5), $||\widehat{\varvec{\xi }}-\varvec{\xi }_{0}||\rightarrow 0$ and for any $j=1,...,k$, $\underset{t\in [0,\tau ]}{\sup }|\widehat{\varLambda }_j(t)-\varLambda _{0j}(t)| \rightarrow 0$ almost surely, where $||\cdot ||$ is the Euclidean norm.

The proof of Theorem A.1 closely resembles the arguments in [13]. We outline the main steps below. We first show that the parameters $\varvec{\theta }$ is identifiable. We then show that $\widehat{\varLambda }_j(\tau )$ cannot diverge, for any $j=1,...,k$. Therefore, by Helly’s selection theorem, there exists a subsequence of $\widehat{\varLambda }_j$ converging pointwise to a bounded monotone function $\varLambda ^*_j$ in $[0,\tau ]$ and $\widehat{\varvec{\xi }}$ converges to some $\varvec{\xi }^*$. Let $\varvec{\varLambda }^* = \{ \varLambda _j^*, j=1,...,k\}$. For any $j=1,...,k$, as in [13], we construct a step function $\widetilde{\varLambda }_j$ with jumps at the observed failure times with the jth cause converging to $\varLambda _{0j}$. Write $\widetilde{\varvec{\varLambda }}= \{\widetilde{\varLambda }_j, j=1,...,k\}$. Because

$$\begin{aligned} L_n(\widehat{\varvec{\xi }},\widehat{\varvec{\varLambda }}) \ge L_n(\varvec{\xi }_0, \widetilde{\varvec{\varLambda }}), \end{aligned}$$

by taking the limits on both sides, we can show that the Kullback–Leibler information between the true density and the density indexed by $(\varvec{\xi }^*,\varvec{\varLambda }^*)$ is nonpositive. Hence, these two densities must be equal. The consistency of the NPMLEs will then follow from the identifiability result.

The second theorem establishes the asymptotic properties of the NPMLEs.

Theorem 2

Under conditions (C1)–(C5), the random element $\sqrt{n}(\widehat{\varvec{\xi }}-\varvec{\xi }_{0}, \widehat{\varvec{\varLambda }}-\varvec{\varLambda }_0)$ converges weakly to a zero mean Gaussian process in the metric space $l^\infty (\mathcal {H})$, the space of all bounded linear functionals on $\mathcal {H}$, where

$$\begin{aligned} \mathcal {H}= & {} \{(\mathbf{h}_1,\mathbf{h}_2):\mathbf{h}_1\in R^{2kp},\mathbf{h}_2=\{h_{2j}, j=1,...,k\}, h_{2j} \text { is a function on }[0,\tau ]; \\&||\mathbf{h}_1|| \le 1, |h_{2j}|_V\le 1\}, j=1,...,k.\}\\ \end{aligned}$$

and $|h|_V$ denotes the total variation of h in $[0,\tau ]$. Furthermore, $\widehat{\varvec{\xi }}$ is asymptotically efficient.

To prove Theorem A.2, we verify the four conditions in Theorem 3.3.1 of [29]. The technical details are similar to the proof of [13, Theorem 2].