Semiparametric estimation of quality adjusted lifetime distribution in semi-Markov illness–death model

Abstract

In this work, we consider semiparametric estimation of quality adjusted lifetime (QAL) distribution using Cox proportional hazards model for the sojourn time in each health state. The regression coefficients are estimated by maximizing the corresponding partial likelihood and the baseline cumulative hazards are estimated by using the method of Breslow (Biometrics 30:89–99, 1974). The estimate of QAL distribution is obtained by using these estimates in the theoretical expression of QAL distribution. The asymptotic normality of the proposed estimator is established. The performance of the proposed estimator is studied using Monte Carlo simulation. A real data example of the Stanford Heart Transplant Program is used to illustrate the proposed method. Extension to a general model is also discussed and illustrated with an analysis of International Breast Cancer Study Group (IBCSG) Trial V data.

Appendix

Let us assume the regularity conditions 1–3 of Shu et al. (2007). Theorem 1 is Lemma 1(ii) of their work.

Proof of Theorem 2

Using Lemmas 1 and 2 of Shu et al. (2007),

$$ \begin{array}{rll} &&\sqrt{n}\left[\hat{S}_{0}\left(\frac{q}{w_0}, {\bf Z}_0\right)-S_{0}\left(\frac{q}{w_0}, {\bf Z}_0\right)\right]\\ &&{\kern12pt}\quad = n^{-1/2} \displaystyle \sum\limits_{i=1}^n \left\{W_{1i}^{(0)}(q)+W_{2i}^{(0)}(q)+ W_{3i}^{(0)}(q)\right\}+o_p(1), \label{P1} \end{array} $$

(15)

where

$$ \begin{array}{rll} W_{1i}^{(0)}(q)&=& Q^{(0)}\left(\frac{q}{w_0}, \beta \right)^{T} \Omega^{-1} \displaystyle \sum\limits_{hj} \int_0^{\infty} \{{\bf Z}_{hji}-E_{hji}(\beta, u)\} dM_{hji}(u),\\ Q^{(0)}\left(\frac{q}{w_0}, \beta \right) &=& -S_0\left(\frac{q}{w_0}; {\bf Z}_0 \right) \left[\displaystyle \int_0^{\frac{q}{w_0}}\{{\bf Z}_{010}-e_{01}(\beta, u)\}d\Lambda_{01}(u;{\bf Z}_0) \right. \\ & &\,\,\,\,\,\qquad\qquad\qquad + \left. \int_0^{\frac{q}{w_0}}\{{\bf Z}_{020}-e_{02}(\beta, u)\}d\Lambda_{02}(u;{\bf Z}_0)\right], \\ W_{2i}^{(0)}(q) &=& -nS_0\left(\frac{q}{w_0}; {\bf Z}_0 \right) \exp\left(\beta^{T} {\bf Z}_{010}\right) \displaystyle \int_0^{\frac{q}{w_0}} \frac{J_0(u) dM_{01i}(u)}{S_{01}^{(0)}(\beta, u)},\\ W_{3i}^{(0)}(q) &=& -nS_0\left(\frac{q}{w_0}; {\bf Z}_0 \right)\exp\left(\beta^{T} {\bf Z}_{020}\right) \displaystyle \int_0^{\frac{q}{w_0}} \frac{J_0(u) dM_{02i}(u)}{S_{02}^{(0)}(\beta, u)}. \end{array} $$

Note that, for each q, the right hand side of Eq. 15 is essentially a sum of n independent and identically distributed zero-mean random variables. Using the arguments of Shu et al. (2007), we conclude that $\sqrt{n}\big[\hat{S}_0(\cdot ; {\bf Z}_0)-S_0(\cdot; {\bf Z}_0)\big]$ converges weakly to a zero-mean Gaussian process with covariance function at $\left(\frac{q}{w_0}, \frac{q'}{w_0}\right)$ given by

$$ \begin{array}{rll} &&\psi^{(0)}\left(\frac{q}{w_0}, \frac{q'}{w_0}\right)\\ &&{\kern12pt} = \frac{1}{n} \displaystyle \sum\limits_{i=1}^n \mbox{cov}\left\{W_{1i}^{(0)}(q)+W_{2i}^{(0)}(q)+W_{3i}^{(0)}(q), W_{1i}^{(0)}(q')\!+\! W_{2i}^{(0)}(q') \!+\! W_{3i}^{(0)}(q')\right\}\\ &&{\kern12pt} = Q^{(0)}\left(\frac{q}{w_0}, \beta \right)^{T} \Omega^{-1} \frac{1}{n}E\left[\displaystyle \sum\limits_{hj} \int_0^{\infty} \left\{ \frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)} -E_{hj}(\beta, u)^{\otimes 2} \right\} \right. \\ & & \qquad\qquad\qquad\qquad\,\,\,\,\left.\phantom{\frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)}}\times S_{hj}^{(0)}(\beta, u) d\Lambda_{hj0}(u)\right] \Omega^{-1} Q^{(0)}\left(\frac{q'}{w_0}, \beta \right) \\ & &\qquad+\, n S_0\left(\frac{q}{w_0}; {\bf Z}_0\right)S_0\left(\frac{q'}{w_0}; {\bf Z}_0\right)\{\exp\left(\beta^{T} {\bf Z}_{010}\right)\}^2 E \!\left\{\!\displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}} \frac{J_0(u)d \Lambda_{01}(u)}{S_{01}^{(0)}(\beta, u)}\right\}\\ & & \qquad+\, n S_0\left(\frac{q}{w_0}; {\bf Z}_0\right) S_0\left(\frac{q'}{w_0}; {\bf Z}_0\right)\{\exp\left(\beta^{T} {\bf Z}_{020}\right)\}^2 E \!\left\{\!\displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}} \frac{J_0(u)d \Lambda_{02}(u)}{S_{02}^{(0)}(\beta, u)}\right\} \end{array} $$

which, for q = q′, can be estimated uniformly consistently by Eq. 7 in Section 3.□

Proof of Theorem 3

Following the similar decomposition technique as that used in Voelkel and Crowley (1984) and Shu et al. (2007), we have

$$ \begin{array}{rll} &&{\kern-1pc}\sqrt{n}\left[\hat{P}_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)-P_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)\right]\\ && = n^{-1/2} \displaystyle \sum\limits_{i=1}^n \left\{W_{1i}^{(12)}(q)+W_{2i}^{(12)}(q)+W_{3i}^{(12)}(q) +W_{4i}^{(12)}(q)\right\}+o_p(1), \label{P12} \end{array} $$

(16)

where

$$ \begin{array}{rll} W_{1i}^{(12)} &=& Q^{(12)}\left(\frac{q}{w_0}, \beta \right)^{T} \Omega^{-1} \displaystyle \sum\limits_{hj} \int_0^{\infty} \{{\bf Z}_{hji}-E_{hji}(\beta, u)\} dM_{hji}(u), \\ W_{2i}^{(12)}(q)&=& n\displaystyle \int_0^{\frac{q}{w_0}} \left\{S_0(u; {\bf Z}_0) S_{12}\left(\frac{q-w_0u}{w_1}; {\bf Z}_0 \right) \right. \\ & & \,\,\quad\qquad\left. - \int_u^{\frac{q}{w_0}} S_0(x; {\bf Z}_0) S_{12}\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x)\right \},\\ & & \times \,\exp(\beta^T {\bf Z}_{010}) \frac{J_0(u) dM_{01i}(u)}{S_{01}^{(0)}(\beta, u)}, \end{array} $$

$$ \begin{array}{rll} W_{3i}^{(12)}(q)&=& -n\displaystyle \int_0^{\frac{q}{w_0}} \left\{ \int_u^{\frac{q}{w_0}} S_0(x; {\bf Z}_0) S_{12}\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x)\right \}\\ & & \times \,\exp\left(\beta^T {\bf Z}_{020}\right)\frac{J_0(u) dM_{02i}(u)}{S_{02}^{(0)}(\beta, u)}, \\ W_{4i}^{(12)}(q)&=& -n\displaystyle \int_0^{\frac{q}{w_1}} \left\{ \int_0^{\frac{q-w_1 u}{w_0}} S_0(x; {\bf Z}_0) S_{12}\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{12}(x)\right \} \\ & & \times \,\exp\left(\beta^T {\bf Z}_{120}\right)\frac{J_1(u) dM_{12i}(u)}{S_{12}^{(0)}(\beta, u)}, \\ Q^{(12)}\left(\frac{q}{w_0}, \beta \right) & =& \displaystyle \int_0^{\frac{q}{w_0}} S_0\left(u; {\bf Z}_0 \right) S_{12}\left(\frac{q-w_0 u}{w_1}; {\bf Z}_0\right)\\ & &\times\,\left[\big\{{\bf Z}_{010}-e_{01}(\beta, u)\big\} - \int_0^{u}\big\{{\bf Z}_{010}-e_{01}(\beta, x)\big\}d\Lambda_{01}(x;{\bf Z}_0) \right. \\ & & -\int_0^{u}\big\{{\bf Z}_{020}-e_{02}(\beta, x)\big\}d\Lambda_{02}(x;{\bf Z}_0) \\ & & \left. -\int_0^{u}\big\{{\bf Z}_{120}-e_{12}(\beta, x)\big\}d\Lambda_{12}(x;{\bf Z}_0) \right] d\Lambda_{01}(u;{\bf Z}_0). \end{array} $$

Note that, for each q, the right hand side of Eq. 16 is essentially a sum of n independent and identically distributed zero-mean random variables. Using the arguments of Shu et al. (2007), we conclude that $\sqrt{n}\left[\hat{P}_{12}(\cdot)-P_{12}(\cdot)\right]$ converges weakly to a zero-mean Gaussian process with covariance function at $\left(\frac{q}{w_0}, \frac{q'}{w_0} \right)$ given by

$$ \begin{array}{rll} \psi^{(12)}\left(\frac{q}{w_0}, \frac{q'}{w_0}\right)&=& \frac{1}{n} \displaystyle \sum\limits_{i=1}^n \mbox{cov}\left\{W_{1i}^{(12)}(q)+W_{2i}^{(12)}(q)+W_{3i}^{(12)}(q)+W_{4i}^{(12)}(q), \right. \\ & &\left. W_{1i}^{(12)}(q')+W_{2i}^{(12)}(q')+W_{3}^{(12)}(q')+ W_{4i}^{(12)}(q')\right\} \\ &=& Q^{(12)}\left(\frac{q}{w_0}, \beta \right)^{T} \Omega^{-1} \\ &&\times\,\frac{1}{n}E\left[\displaystyle \sum\limits_{hj} \int_0^{\infty}\left\{ \frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)} -E_{hj}(\beta, u)^{\otimes 2} \right\} \right. \\ & &\quad \times \left. S_{hj}^{(0)}(\beta, u) d\Lambda_{hj0}(u)\vphantom{\sum_{hj} \int_0^{\infty}\left\{ \frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)} -E_{hj}(\beta, u)^{\otimes 2} \right\}}\right] \Omega^{-1} Q^{(12)}\left(\frac{q'}{w_0}, \beta \right) \end{array} $$

$$ \begin{array}{rll} & &+\, nE\left[ \!\displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}} \!\left\{S_0(u; {\bf Z}_0) S_{12}\left(\frac{q-w_0u}{w_1}; {\bf Z}_0\right) \!-\!\! \int_u^{\frac{q}{w_0}}\! S_0(x; {\bf Z}_0)\right. \right. \\ & &\qquad\qquad\qquad \left. \times\, S_{12}\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x; {\bf Z}_0)\vphantom{\int_u^{\frac{q}{w_0}}}\right \} \\ & &\qquad\qquad \times \left\{S_0(u; {\bf Z}_0) S_{12}\left(\frac{q'-w_0u}{w_1}; {\bf Z}_0\right) \vphantom{\int_u^{\frac{q'}{w_0}}}\right. \\ & &\qquad\qquad\qquad \left. \left. - \int_u^{\frac{q'}{w_0}} S_0(x; {\bf Z}_0) S_{12}\left(\frac{q'-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x; {\bf Z}_0)\right \} \right. \\ & & \left. \times \left\{\exp(\beta^{T} {\bf Z}_{010})\right\}^2 J_0(u) \frac{d \Lambda_{01}(u)}{S_{01}^{(0)}(\beta, u)}\vphantom{\displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}}}\right] \\ & & +\, nE\left[ \displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}} \left\{\int_u^{\frac{q}{w_0}}S_0(x; {\bf Z}_0) S_{12}\right.\right.\\ &&\qquad\qquad\qquad\qquad\qquad \left.\times\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x; {\bf Z}_0)\vphantom{\displaystyle \int_0^{\frac{q}{w_0} \wedge \frac{q'}{w_0}}}\right\} \\ & &\qquad\qquad\qquad \left. \times \left\{\int_u^{\frac{q'}{w_0}}S_0(x; {\bf Z}_0) S_{12}\left(\frac{q'-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x; {\bf Z}_0)\right\} \right. \\ & &\qquad\qquad\qquad \left. \times \left\{\exp\left(\beta^T {\bf Z}_{020}\right)\right\}^2 J_0(u) \frac{d \Lambda_{02}(u)}{S_{02}^{(0)}(\beta, u)}\right]\\ & & +\, n E\left[\displaystyle \int_0^{\frac{q}{w_1} \wedge \frac{q'}{w_1}} \left\{\int_0^{\frac{q-w_1 u}{w_0}}S_0(x; {\bf Z}_0) S_{12}\right.\right.\\ &&\qquad\qquad\qquad\qquad \left.\times\left(\frac{q-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x; {\bf Z}_0)\right\} \\ & & \left. \times \left\{\int_0^{\frac{q'-w_1 u}{w_0}}S_0(x; {\bf Z}_0) S_{12}\left(\frac{q'-w_0x}{w_1}; {\bf Z}_0\right) d \Lambda_{01}(x)\right\} \right. \\ & & \left. \times \, \left\{\exp\left(\beta^T {\bf Z}_{120}\right)\right\}^2 J_1(u) \frac{d \Lambda_{12}(u)}{S_{12}^{(0)}(\beta, u)}\right] \end{array} $$

which, for q = q′, can be estimated uniformly consistently by Eq. 8 in Section 3.□

Proof of Theorem 4

Note that

$$ \begin{array}{rll} \sqrt{n}\left[\hat{S}_{Q}\left(q; {\bf Z}_0\right)-S_{Q}\left(q; {\bf Z}_0\right)\right]&=&\sqrt{n}\left[\hat{S}_{0}\left(\frac{q}{w_0}; {\bf Z}_0\right)-S_{0}\left(\frac{q}{w_0}; {\bf Z}_0\right)\right] \\ & & +\sqrt{n}\left[\hat{P}_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)-P_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)\right]. \end{array} $$

Hence, by some rearrangement of terms and following the techniques used in the proofs of Theorems 2 and 3, $\sqrt{n}\left[\hat{S}_{Q}\left(q; {\bf Z}_0\right)-S_{Q}\left(q; {\bf Z}_0\right)\right]$ can be written as a sum of n independent and identically distributed zero mean random variables. The weak convergence result follows by using similar arguments. The covariance term in Theorem 4 is given by

$$ \begin{array}{rll} &&{\kern-6pt} \mbox{cov}\left[\!\sqrt{n}\left\{\hat{S}_{0}\left(\frac{q}{w_0}; {\bf Z}_0 \right)\!-\!S_{0}\left(\frac{q}{w_0}; {\bf Z}_0\right)\right\}, \sqrt{n}\left\{\hat{P}_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)\!-\!P_{12}\left(\frac{q}{w_0}; {\bf Z}_0\right)\!\right\}\!\right]\\ &&=\frac{1}{n} \displaystyle \sum\limits_{i=1}^n \mbox{cov}\left\{W_{1i}^{(0)}(q)+W_{2i}^{(0)}(q)+W_{3i}^{(0)}(q), \right.\\ & & \qquad\qquad\qquad\left. W_{1i}^{(12)}(q)+W_{2i}^{(12)}(q)+W_{3i}^{(12)}(q)+W_{4i}^{(12)}(q)\right\} +o_p(1)\\ &&= Q^{(0)}\left(\frac{q}{w_0}, \beta \right)^{T} \Omega^{-1}\frac{1}{n}E\left[\displaystyle \sum\limits_{hj} \int_0^{\infty} \left\{ \frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)} -E_{hj}(\beta, u)^{\otimes 2} \right\} \right. \\ & &\qquad\qquad\qquad\qquad\qquad \times \left. S_{hj}^{(0)}(\beta, u) d\Lambda_{hj0}(u)\vphantom{\displaystyle \sum_{hj} \int_0^{\infty} \left\{ \frac{S_{hj}^{(2)}(\beta, u)}{S_{hj}^{(0)}(\beta, u)} -E_{hj}(\beta, u)^{\otimes 2} \right\}}\right] \Omega^{-1} Q^{(12)}\left(\frac{q'}{w_0}, \beta \right) \\ &&\quad -\, n S_{0}\left(\frac{q}{w_0}; {\bf Z}_0\right) \left\{\exp(\beta^T {\bf Z}_{010})\right\}^2\\ &&\quad\times E\,\left[ \displaystyle \int_0^{\frac{q}{w_0}} \left\{S_0(u; {\bf Z}_0) S_{12}\left(\frac{q-w_0u}{w_1}; {\bf Z}_0\right) \right. \right.\\ & &\qquad\qquad\qquad \left. \left. - \int_u^{\frac{q}{w_0}} S_0(x; {\bf Z}_0) S_{12}\left(\frac{q-w_0x}{w_1}; {\bf Z}_0 \right) d \Lambda_{01}(x)\right \} J_0(u) \frac{d \Lambda_{01}(u)}{S_{01}^{(0)}(\beta, u)} \right]\\ &&\quad +\, nS_{0}\left(\frac{q}{w_0}\right) \{\exp(\beta^T {\bf Z}_{020})\}^2 \\ &&\quad\times E\,\left[\displaystyle \int_0^{\frac{q}{w_0}} \left\{\int_u^{\frac{q}{w_0}} S_0(x) S_{12}\left(\frac{q-w_0 x}{w_1}\right) d \Lambda_{01}(x)\right \} J_0(u) \frac{d \Lambda_{02}(u)}{S_{02}^{(0)}(\beta, u)} \right] + o_p(1), \end{array} $$

which can be estimated uniformly consistently by Eq. 10 in Section 3.□