Non-parametric test of recurrent cumulative incidence functions for competing risks models

Abstract

Recurrent competing risks data are common in survival studies. In such contexts the effects of competing risks on lifetime outcomes are important problem of study. In this work we introduce recurrent cumulative incidence function and then propose a non-parametric test for comparing recurrent cumulative incidence functions. Asymptotic distribution of the test statistic is derived. A simulation study is carried out to assess the performance of the proposed test statistic. The proposed method is applied to an auto-mobile warranty data.

Appendix

Proof of Theorem 2.1

To find the asymptotic distribution of the test statistic $Z(\tau )$, first consider the quantity

$$\begin{aligned} v_l(t)=\int _{0}^{t} w(u)d\left( {\hat{F}}_l(u)-\frac{{\hat{F}}(u)}{k}\right) ,l=1,2,...,k. \end{aligned}$$

(A-1)

After adding and subtracting $\int _{0}^{t} w(u)dF_{l}(u)+ \int _{0}^{t} w(u)\frac{dF(u)}{k}$, (A-1) can be modified as,

$$\begin{aligned} v_l(t)&=\int _{0}^{t}w(u)d({\hat{F}}_{l}(u)-\frac{{\hat{F}}(u)}{k}) +\int _{0}^{t}w(u)dF_l(u)-\int _{0}^{t}w(u)dF_l(u)\nonumber \\&\quad +\int _{0}^{t}w(u) \frac{dF(u)}{k}-\int _{0}^{t}w(u)\frac{dF(u)}{k}.\nonumber \\&= \int _{0}^{t} w(u)d\left( {\hat{F}}_l(u)-F_l(u)\right) +\int _{0}^{t} w(u)d\left( F_l(u)-\frac{F(u)}{k}\right) \nonumber \\&\quad +\int _{0}^{t}\frac{w(u)}{k}d\left( F(u)-{\hat{F}}(u)\right) . \end{aligned}$$

(A-2)

under $H_0$, $F_l(t)=\frac{F(t)}{k}$. Then second term in the (A-2) becomes zero. Then

$$\begin{aligned} v_l(t )=\int _{0}^{t}w(u)d\left( {\hat{F}}_l(u)-F_l(u)\right) +\int _{0}^{t}\frac{w(u)}{k}d\left( F(u)-{\hat{F}}(u)\right) . \end{aligned}$$

(A-3)

It can be shown that the difference between ${\hat{F}}_l(t)$ and $F_{l}(t)$ can be approximated by the sum of independent and identically distributed random variables. For this we need to prove that

$$\begin{aligned} \hat{\Lambda _{l}}(t)-\Lambda _{l}(t)=-\frac{1}{n}\sum _{i=1}^{n}\psi _{il}(t)+o_{p}(n^{-\frac{1}{2}}). \end{aligned}$$

(A-4)

This can be shown by the similar method used in the Wang and Chang [18]. Note that

$$\begin{aligned} {\hat{F}}_{l}(t)-F_{l}(t)=\int _{0}^{t}S(u)d\left\{ \hat{\Lambda _{l}}(u)-\Lambda _{l}(u) \right\} +\int _{0}^{t} \left\{ {\hat{S}}(u)-S(u)\right\} d{{\hat{\Lambda }}}_{l}(u). \end{aligned}$$

(A-5)

We can shown that ${\hat{F}}_{l}(t)-F_{l}(t)$ is sum of inedependent and identically distributed random varibles by applying (A-4) in (A-5) and following the techniques used in Hyun et al. [7]. Then,

$$\begin{aligned} {\hat{F}}_l(t)-F_l(t)=-n^{-1}\sum _{i=1}^{n}\epsilon _{li}(t)+o_p(n^{-\frac{1}{2}}), \end{aligned}$$

(A-6)

where $\epsilon _{li}(t)=\int _{0}^{t}[\sum _{l=1}^{k}F^{c}_{l}(t,u)+S(u)]\psi _{il}(u)$ and $F^{c}_{l}(t,u)=F_{l}(t)-F_{l}(u)$. For fixed t, $\epsilon _{li}(t)$ are independent and identically distributed mean zero random variables and $\sqrt{n}({\hat{F}}_l(t)-F_l(t))$ follows a mean zero Gaussian process with variance $\sigma ^{2}_{l}(t)=E(\epsilon _{li}(t))^{2}$ and variance-covariance function $\sigma _l(t_1,t_2)=E[\epsilon _{li}(t_1)\epsilon _{li}(t_2)],l=1,2,...,k.$

From Wang and Chang [18], we can shown that the ${\hat{F}}(t)-F(t)$ is a sum of independent and identically distributed random variables.

$$\begin{aligned} {\hat{F}}(t)-F(t)=-n^{-1}\sum _{i=1}^{n}S(t)\phi _{i}(t)+o_p(n^{-\frac{1}{2}}), \end{aligned}$$

(A-7)

where $\phi _i(t)=\int _{0}^{t}\{H_a(u)\}^{-2}\{\frac{a_i}{m^{*}_{i}}\sum _{j=1}^{m^{*}_{i}} I(y_{ij}\ge u )\}dG_a(u)-\frac{a_iI(m_i\ge 2)}{m^{*}_{i}}\sum _{j=1}^{m^{*}_{i}}\frac{1}{H_a(y_{ij})}I(y_{ij} <t)$.

The random variables $S(t)\phi _{i}(t)$ are uniformly bounded in $0 \le t \le t^{*}$ when $a_{i}$ is bounded by a constant, where $t^{*}$ is a non negative constant satisfying $t^{*} < sup \{ t: S(t)G(t) >0\}$. Then by the multivariate central limit theorem, the finite dimensional distributions of the random process $\sqrt{n}\{{\hat{F}}(t)-F(t)\}$ follows mean zero Gaussian process with variance covariance function $\sigma (t_1,t_2)$.

Then

$$\begin{aligned} v_l(t)&=-\int _{0}^{t}w(u)d\left( n^{-1}\sum _{i=1}^{n}\epsilon _{li}(u)\right) +\int _{0}^{t}\frac{w(u)}{k}d\left( n^{-1}\sum _{i=1}^{n}S(u)\phi _{i}(u)\right) +o_p(n^{-\frac{1}{2}}).\nonumber \\&= -\frac{1}{n}\left[ \int _{0}^{t}w(u)d(\sum _{i=1}^{n}\epsilon _{li}(u))-\int _{0}^{t}\frac{w(u)}{k}d(\sum _{i=1}^{n}S(u)\phi _{i}(u))\right] +o_p(n^{-\frac{1}{2}}).\nonumber \\&= -\frac{1}{n} \sum _{i=1}^{n}\int _{0}^{t}\left[ w(u)d\left( \epsilon _{li}(u)-\frac{S(u)\phi _{i}(u)}{k}\right) \right] +o_p(n^{-\frac{1}{2}}). \end{aligned}$$

(A-8)

$\square$