Inferences on cumulative incidence function for middle censored survival data with Weibull regression

Abstract

This article considers the problem of competing risks analysis in the presence of middle censoring scheme. In this censoring, the exact lifetime of the subject under study becomes unobservable when it falls in a random censoring interval and the associated competing causes of failure can be identified in later inspection. We consider the cumulative incidence function for evaluation of competing risks by assuming a Weibull cause specific hazard function with the common shape parameter for all competing causes. Maximum likelihood and midpoint approximation methods are employed for point estimation of unknown parameters and cumulative incidence functions. Asymptotic confidence intervals of maximum likelihood estimators are computed. In addition, the Bayes estimators and the associated credible intervals are also considered. The Monte Carlo simulation and an illustrative example on bone marrow transplant data are given for comprehensive comparison of proposed methods.

Appendix

Here we provide the details of system of score equations used in Sect. 3.1. First, we define some notations for reducing the length of expressions

$$\begin{aligned} \varPsi= & {} \sum _{j=1}^{p}(\theta _{j}^{\alpha }e^{{\varvec{\beta }}_{j}^{\top }{\varvec{x}}_{i}}),\\ S(\varPsi ;u,v)= & {} \text {exp}\left( -\left( \sum _{j=1}^{p}\theta _{j}^{\alpha }e^{{\varvec{\beta }}_{j}^{\top }{\varvec{x}}_{i}}\right) u_{i}^{\alpha }\right) -\text {exp}\left( -\left( \sum _{j=1}^{p}\theta _{j}^{\alpha }e^{{\varvec{\beta }}_{j}^{\top }{\varvec{x}}_{i}}\right) v_{i}^{\alpha }\right) . \end{aligned}$$

From Eq. (9) we obtain the partial derivatives of log-likelihood with respect to each parameter by the following equations.

$$\begin{aligned} \frac{\partial \ell ({\varvec{\varTheta }})}{\partial \alpha }= & {} \frac{n_{1}}{\alpha }+\sum _{j=1}^{p}n_{1j} \log \theta _{j}+\sum _{i=1}^{n_{1}}\log t_{i}-\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial \alpha }t_{i}^{\alpha }-\sum _{i=1}^{n_{1}}\varPsi t_{i}^{\alpha }\log t_{i}\nonumber \\&+\sum _{j=1}^{p}n_{2j}\log \theta _{j} -\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial \varPsi /\partial \alpha }{\varPsi } +\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial S(\varPsi ;u,v)/\partial \alpha }{S(\varPsi ;u,v)}=0\qquad \end{aligned}$$

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$$\begin{aligned} \frac{\partial \ell ({\varvec{\varTheta }})}{\partial \theta _{j}}= & {} \frac{n_{1j}\alpha }{\theta _{j}} -\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial \theta _{j}}t_{i}^{\alpha } +\frac{n_{2j}\alpha }{\theta _{j}}-\sum _{i=n_{1}+1}^{n_{1}+n_{2}} \frac{\partial \varPsi /\partial \theta _{j}}{\varPsi }\nonumber \\&+\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial S(\varPsi ;u,v)/\partial \theta _{j}}{S(\varPsi ;u,v)}=0 \end{aligned}$$

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$$\begin{aligned} \frac{\partial \ell ({\varvec{\varTheta }})}{\partial {\varvec{\beta }}_{j}}= & {} \sum _{i=1}^{n_{1j}} {\varvec{x}}_{i}-\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial {\varvec{\beta }}_{j}}t_{i}^{\alpha }+\sum _{i=n_{1}+1}^{n_{1}+n_{2j}} {\varvec{x}}_{i}-\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial \varPsi /\partial {\varvec{\beta }}_{j}}{\varPsi }\nonumber \\&+\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial S(\varPsi ;u,v)/\partial {\varvec{\beta }}_{j}}{S(\varPsi ;u,v)}=0 \end{aligned}$$

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The observed Fisher information matrix is given by

$$\begin{aligned} {\varvec{I}}_{O}({\varvec{\varTheta }})= \begin{pmatrix} -\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial \alpha ^{2}}&{} -\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial \alpha \partial \theta _{j}} &{}-\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial \alpha \partial {\varvec{\beta }}_{j}} \\ - &{}-\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial \theta _{j}^{2}} &{}-\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial \theta _{j}\partial {\varvec{\beta }}_{j}}\\ -&{}-&{}-\frac{\partial ^{2}\ell ({\varvec{\varTheta }})}{\partial {\varvec{\beta }}_{j}\partial {\varvec{\beta }}_{j}^{\top }}\end{pmatrix}_{{\varvec{\varTheta }}=\hat{{\varvec{\varTheta }}}} . \end{aligned}$$

The elements of ${\varvec{I}}_{O}({\varvec{\varTheta }})$ are given below.

$$\begin{aligned}&\frac{\partial \ell ({\varvec{\varTheta }})}{\partial \alpha ^{2}}= -\frac{n_{1}}{\alpha ^{2}} -\sum _{i=1}^{n_{1}}\frac{\partial ^{2}\varPsi }{\partial \alpha ^{2}}t_{i}^{\alpha } -2\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial \alpha }t_{i}^{\alpha }\log t_{i} -\sum _{i=1}^{n_{1}}\varPsi t_{i}^{\alpha }(\log t_{i})^{2}\nonumber \\&\quad -\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\varPsi \partial ^{2}\varPsi / \partial \alpha ^{2}}{\varPsi ^{2}} +\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{(\partial \varPsi /\partial \alpha )^{2}}{\varPsi ^{2}}+\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\nonumber \\&\quad \frac{S(\varPsi ;u,v)\partial ^{2} S(\varPsi ;u,v)/\partial \alpha ^{2} -(\partial S(\varPsi ;u,v)/\partial \alpha )^{2}}{[S(\varPsi ;u,v)]^{2}}\end{aligned}$$

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$$\begin{aligned}&\frac{\partial \ell ({\varvec{\varTheta }})}{\partial \theta _{j}\partial \alpha } =\frac{n_{1j}}{\theta _{j}}-\sum _{i=1}^{n_{1}}\frac{\partial ^{2}\varPsi }{\partial \theta _{j}\partial \alpha }t_{i}^{\alpha } -\sum _{i=1}^{n_{1}} \frac{\partial \varPsi }{\partial \theta _{j}}t_{i}^{\alpha }\log t_{i}+\frac{n_{2j}}{\theta _{j}}\nonumber \\&\quad -\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\varPsi \partial ^{2}\varPsi /\partial \theta _{j}\partial \alpha -\partial \varPsi /\partial \theta _{j}\partial \varPsi /\partial \alpha }{\varPsi ^{2}} +\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\nonumber \\&\quad \frac{S(\varPsi ;u,v)\partial ^{2} S(\varPsi ;u,v)/ \partial \theta _{j}\partial \alpha -(\partial S(\varPsi ;u,v)/\partial \theta _{j}) (\partial S(\varPsi ;u,v)/\partial \alpha )}{[S(\varPsi ;u,v)]^{2}}\nonumber \\ \end{aligned}$$

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$$\begin{aligned}&\frac{\partial \ell ({\varvec{\varTheta }})}{\partial {\varvec{\beta }}_{j}\partial \alpha } = -\sum _{i=1}^{n_{1}}\frac{\partial ^{2}\varPsi }{\partial {\varvec{\beta }}_{j} \partial \alpha }t_{i}^{\alpha }-\sum _{i=1}^{n_{1}} \frac{\partial \varPsi }{\partial {\varvec{\beta }}_{j}}t_{i}^{\alpha }\log t_{i}\nonumber \\&\quad -\sum _{i=n_{1}+1}^{n_{1}+n_{2}} \frac{\varPsi \partial ^{2}\varPsi /\partial {\varvec{\beta }}_{j}\partial \alpha -\partial \varPsi /\partial {\varvec{\beta }}_{j}\partial \varPsi /\partial \alpha }{\varPsi ^{2}} +\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\nonumber \\&\quad \frac{S(\varPsi ;u,v)\partial ^{2} S(\varPsi ;u,v)/ \partial {\varvec{\beta }}_{j}\partial \alpha -(\partial S(\varPsi ;u,v)/\partial {\varvec{\beta }}_{j}) (\partial S(\varPsi ;u,v)/\partial \alpha )}{[S(\varPsi ;u,v)]^{2}}\nonumber \\ \end{aligned}$$

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The other mixed partial derivatives can also be obtained.

Score equations under the MPA method are given below

$$\begin{aligned}&\frac{\partial \ell _\text {MPA}({\varvec{\varTheta }})}{\partial \alpha } =\frac{n_{1}}{\alpha }+\sum _{j=1}^{p}n_{1j}\log \theta _{j} +\sum _{i=1}^{n_{1}} \log t_{i}-\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial \alpha }t_{i}^{\alpha } -\sum _{i=1}^{n_{1}}\varPsi t_{i}^{\alpha }\log t_{i}\nonumber \\&+\frac{n_{2}}{\alpha }+\sum _{j=1}^{p}n_{2j}\log \theta _{j} +\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\log t_{i}^{*}-\sum _{i=n_{1}+1}^{n_{1} +n_{2}}\frac{\partial \varPsi }{\partial \alpha }t_{i}^{*\alpha }-\sum _{i=n_{1}+1}^{n_{1} +n_{2}}\varPsi t_{i}^{*\alpha }\log t_{i}^{*}=0 \nonumber \\ \end{aligned}$$

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$$\begin{aligned}&\frac{\partial \ell _\text {MPA}({\varvec{\varTheta }})}{\partial \theta _{j}} =\frac{n_{1j}\alpha }{\theta _{j}}-\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial \theta _{j}}t_{i}^{\alpha }+\frac{n_{2j}\alpha }{\theta _{j}} -\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial \varPsi }{\partial \theta _{j}}t_{i}^{*\alpha }=0 \end{aligned}$$

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$$\begin{aligned}&\frac{\partial \ell _\text {MPA}({\varvec{\varTheta }})}{\partial {\varvec{\beta }}_{j}} =\sum _{i=1}^{n_{1j}}{\varvec{x}}_{i}-\sum _{i=1}^{n_{1}}\frac{\partial \varPsi }{\partial {\varvec{\beta }}_{j}}t_{i}^{\alpha }+\sum _{i=n_{1}+1}^{n_{1}+n_{2j}} {\varvec{x}}_{i}-\sum _{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\partial \varPsi }{\partial {\varvec{\beta }}_{j}}t_{i}^{*\alpha }=0 \end{aligned}$$

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The variance formula of the CIF is given by

$$\begin{aligned}&\text {var}(\hat{F_{j}}(t;\hat{{\varvec{\varTheta }}},{\varvec{x}}))= \hat{F_{j}}(t;\hat{{\varvec{\varTheta }}},{\varvec{x}})^{2}\nonumber \\&\quad \left. \left( \frac{\partial \log {F_{j}}(t;{\varvec{\varTheta }},{\varvec{x}})}{\partial {\varvec{\varTheta }}} \right) \right| _{{\varvec{\varTheta }}=\hat{{\varvec{\varTheta }}}} {\varvec{I}}_O^{-1}(\hat{{\varvec{\varTheta }}}) \left. \left( \frac{\partial \log {F_{j}}(t;{\varvec{\varTheta }},{\varvec{x}})}{\partial {\varvec{\varTheta }}} \right) ^{\top } \right| _{{\varvec{\varTheta }}=\hat{{\varvec{\varTheta }}}}. \end{aligned}$$

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