Semiparametric Analysis of Treatment Effect via Failure Probability Ratio and the Ratio of Cumulative Hazards

Abstract

For clinical trials with time-to-event data, statistical inference often employs the constant hazard ratio assumption. When the hazards are possibly non-proportional, the hazard ratio function is often the focus of analysis and it gives a visual inspection of proportionality assumption or how severe of a deviation there is from it. However, the hazard ratio does not directly reflect the treatment effect on survival or event occurrence. The failure probability ratio and the ratio of cumulative hazards are two measures that relate to the survival experience and supplement the hazard ratio in helping assess the treatment effect. For these ratios, although simple nonparametric estimators are available through the Nelson-Aalen estimator of the cumulative hazard and the Kaplan–Meier estimator of the survival function, often they are not very smooth and can be quite unstable near the beginning of the data range. In this article, point estimates, point-wise confidence intervals and simultaneous confidence intervals of the two ratios are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are illustrated for data from two clinical trials.

Appendices

Appendix A: Consistency

The following regularity conditions will be assumed throughout the Appendices:

• Condition 1. $\lim \frac{n_1}{n}=\rho \in (0,1).$

• Condition 2. The survivor function G _i of C _i given Z _i is continuous and satisfies

  $$\begin{aligned} \frac{1}{n}\sum _{i\leq n_1} G_i(t) \rightarrow \Gamma_1,\ \frac{1}{n}\sum _{i> n_1} G_i(t) \rightarrow \Gamma_2,\end{aligned}$$

  uniformly for $t\leq \tau$, for some Γ₁, Γ₂, and $\tau<\tau_0$ such that $\Gamma_j(\tau)>0,\ j=1,2.$

• Condition 3. The survivor functions S _C and S _T are absolutely continuous and $S_C(\tau)>0$.

Under these conditions, the strong law of large numbers implies that (21.3) is satisfied.

For $t\leq \tau,$ define

$$\begin{aligned} L(t)&= \Gamma_1 S_C+ \Gamma_2 S_T,\\ U_j(t;{\bf b})&= {\int^t_0} \Gamma_1 dF_C+ {\rm exp}\Big(-b_j\Big) {\int^t_0} \Gamma_2dF_T, j= 1,2,\\ \Lambda_j(t;{\bf b})&={\int^t_0} \frac{dU_j(s;{\bf b})}{L(s)},\ j= 1,2,\\ P(t;{\bf b})&={\rm exp}\{-\Lambda_2(t;{\bf b})\}, R(t;{\bf b})=\frac{1}{P(t;{\bf b})}{\int^t_0} P(s;{\bf b})d\Lambda_1(s;{\bf b}),\\ f_j^0(t;{\bf b})&=\frac{{\rm exp}(-b_j)R^{j-1}(t;{\bf b})}{{\rm exp}(-b_1)+{\rm exp}(-b_2)R(t;{\bf b})},\ j=1,2,\\ m_j({\bf b})&= \left\{{\int^{\tau}_0} f_j^0 \Gamma_2(t)dF_T(t) -\!{\int^{\tau}_0} \frac{f_j^0\Gamma_2(t) S_T(t)dR(t;{\bf b})} {{\rm exp}\Big(-b_1\Big) +{\rm exp}\Big(-b_2\Big)R(t;{\bf b})}\right\},\ j\!=\!1,2,\end{aligned}$$

and $m({\bf b})=\Big(m_1({\bf b}),m_2({\bf b})\Big)^\prime$. We will also assume

• Condition 4. The function $m({\bf b})$ is non-zero for $b\in \cal{B}-\{{\bf \beta}\}$, where $\cal{B}$ is a compact neighborhood of ${\bf \beta}$.

Theorem 1.

Suppose that Conditions 1 ∼ 4 hold. Then, (i) the zero ${\hat{\bf \beta}}$ of $Q({\bf b})$ in $\cal{B}$ is strongly consistent for ${\bf \beta}$ ; (ii) $\widehat{RR}(t)$ is strongly consistent for $RR(t)$ , uniformly for $t\in [0, \tau]$ , and $\widehat{\textit{CHR}}(t)$ is strongly consistent for $\textit{CHR}(t)$ , uniformly on $t\in [0, \tau]$ ; (iii) $\hat{\Omega}$ converges almost surely to a limiting matrix $\Omega^*$.

Proof.

Under Conditions $1 \sim 3$, the limit of ${\sum^n_{i=1}} I\Big(X_i\geq t\Big)/n$ is bounded away from zero on $t\in [0,\tau].$ Thus, it can be shown that, with probability 1,

$$\frac{{\sum^n_{i=1}} \delta_i e^{-b_jZ_i}I\Big(X_i=t\Big)} {{\sum^n_{i=1}} \delta_i I\Big(X_i\geq t\Big)}\rightarrow 0,\ j=1,2, | \Delta{\hat{P}}(t;{\bf b})|\rightarrow 0,\ | \Delta{\hat{R}}(t;{\bf b})|\rightarrow 0,$$

(21.16)

uniformly for $t\in [0,\tau]$ and $b\in \cal{B},$ where Δ indicates the jump of the function in t.

Define the martingale residuals

$$\begin{aligned} {\hat{M}}_{i}(t;{\bf b})=\delta_iI\Big(X_i\leq t\Big)-{\int^t_0} I\Big(X_i\geq s\Big) \frac{{\hat{R}}(ds;{\bf b})}{e^{-b_1Z_i}+e^{-b_2Z_i} {\hat{R}}(s;{\bf b})}, 1\leq i \leq n.\end{aligned}$$

From (21.16) and the fundamental theorem of calculus, it can be shown that, with probability 1,

$$Q({\bf b})={\sum^n_{i=1}} {\int^{\tau}_0} \Big\{f_{i}(t;{\bf b})+o(1)\Big\}{\hat{M}}_i(dt;{\bf b}),$$

(21.17)

uniformly in $t\leq \tau$, $b\in \cal{B}$ and $i\leq n$, where $f_i=(f_{1i},f_{2i})^T$, with

$$\begin{aligned} f_{1i}(t;{\bf b})=\frac{Z_ie^{-b_1Z_i}}{e^{-b_1Z_i}+ e^{-b_2Z_i}{\hat{R}}(t;{\bf b})},\ f_{2i}(t;{\bf b})=\frac{Z_ie^{-b_2Z_i}{\hat{R}}(t;{\bf b})}{e^{-b_1Z_i}+ e^{-b_2Z_i}{\hat{R}}(t;{\bf b})}.\end{aligned}$$

From the strong law of large numbers (Pollard 1990, p. 41) and repeated use of Lemma A1 of Yang and Prentice (2005), one obtain, with probability 1,

$$ {\hat{P}}(t;{\bf b})\rightarrow{\hat{P}}(t;{\bf b}),\ {\hat{R}}(t;{\bf b})\rightarrow R(t;{\bf b}),\ Q({\bf b})/n \rightarrow m({\bf b}),$$

(21.18)

uniformly in $t\leq \tau$ and ${\bf b} \in \cal{B}$. From these results and Condition 4, one obtains the strong consistency of $\widehat{RR}(t)$ and $\widehat{\textit{CHR}}(t)$, and almost sure convergence of $\hat{\Omega}$.

Appendix B: Weak Convergence

Let $\xi_0(t)=1+ R(t), \xi(t)=e^{-{\beta}_1}+e^{-{\beta}_2} R(t),\hat{\xi}_0(t)=1+ {\hat{R}}(t;{\bf \beta}), \hat{\xi}(t)=e^{-{\beta}_1}+e^{-{\beta}_2} {\hat{R}}(t;{\bf \beta}),$ and define

$$\begin{aligned} \nonumber K_1(t)&=\sum_{i\leq n_1}I(X_i\geq t),\ K_2(t)=\sum_{i>n_1}I(X_i\geq t),\\ \nonumber H(t)&=\frac{1}{\hat{\xi} (t)} (e^{-{\beta}_1}, e^{-{\beta}_2} {\hat{R}}(t;{\bf \beta}))^T,\\ J(t)&= \int^\tau_t \frac{H(s)K_1(s)K_2(s)}{\hat{\xi} (s){\hat{P}(s;{\bf \beta})}} \left(\frac{e^{-{\beta}_2}}{\xi(s)}-\frac{1}{\xi_0(s)}\right)dR(s).\end{aligned}$$

Similarly, to the proof of Theorem 1, it can be shown that, with probability 1,

$$ Q({\bf \beta})= \sum_{i\leq n_1}\int_0^\tau \{\mu_1(t) +o(1)\} dM_i(t) +\sum_{i>n_1}\int_0^\tau \Big\{\mu_2(t) +o(1)\Big\} dM_i(t),$$

(21.19)

uniformly in $t\leq \tau$ and $i\leq n$, where

$$\begin{aligned} \mu_1 (t)& =-\frac{\hat{\xi}_0(t)H(t)K_2(t)}{\hat{\xi}(t)K(t)} +\frac{\hat{\xi}_0(t)\hat{P}_{-}(t;{\bf \beta})}{K} J(t), \nonumber\\ \mu_2(t)& = H(t) \frac{K_1(t)}{K(t)} + \frac{\hat{\xi}(t) \hat{P}_{-}(t;{\bf \beta})} {K(t)} J(t),\end{aligned}$$

(21.20)

$$\begin{aligned} M_{i}(t)&=\delta_iI(X_i\leq t)-{\int^t_0} I(X_i\geq s) \frac{dR(s)}{e^{-{\beta}_1Z_i}+ e^{-{\beta}_2Z_i}R(s)},\ i=1,\dots,n.\nonumber\end{aligned}$$

By Lemma A3 of Yang and Prentice (2005),

$$ \sqrt{n}({\hat{R}}(t;{\bf \beta})-R(t))=\frac{1}{\sqrt{n}{\hat{P}}(t;{\bf \beta})} \left(\sum_{i\leq n_1}\int_0^t \nu_1dM_i +\sum_{i>n_1}\int_0^t \nu_2dM_i\right)$$

(21.21)

where

$$\begin{aligned}\nu_1(t)&=\frac{n\xi_0(t) {\hat{P}}_-(t;{\bf \beta})}{K(t)},\ \nu_2(t)=\frac{n\xi(t){\hat{P}}_-(t;{\bf \beta})}{K(t)}.\end{aligned}$$

Define

$$\begin{aligned} A_{RR}(t)&=\!\left(\frac{\hat{S}_T(t)}{\hat{F}_C(t)\hat{\xi}(t)}- \frac{\hat{F}_T(t)\hat{S}_C^2(t)}{\hat{F}^2_C(t)} \right)\! \frac{\partial{\hat{R}}(t;{\bf \beta})} {\partial{\bf \beta}} +\frac{\hat{S}_T(t)}{\hat{F}_C(t)} \left(\frac{R(t)}{\xi(t)}, {\Lambda}_T(t)-\frac{R(t)}{\xi(t)}\right)^T\!,\\ B_{RR}(t)&= \frac{1}{{\hat{P}}(t;{\bf \beta})} \left(\frac{\hat{S}_T(t)}{\hat{F}_C(t)\hat{\xi}(t)}- \frac{\hat{F}_T(t)\hat{S}_C^2(t)}{\hat{F}^2_C(t)}\right),\\ A_{\textit{CHR}}(t)&=\!\left(\frac{1}{\Lambda_C(t)\hat{\xi}(t)}- \frac{\Lambda_T(t)\hat{S}_C(t)}{\Lambda_C^2(t)} \right)\! \frac{\partial{\hat{R}}(t;{\bf \beta})} {\partial{\bf \beta}} +\frac{1}{\Lambda_C(t)} \!\left(\frac{R(t)}{\xi(t)}, {\Lambda}_T(t)-\frac{R(t)}{\xi(t)}\right)^T\!\!\!,\\ B_{\textit{CHR}}(t)&=\frac{1}{{\hat{P}}(t;{\bf \beta})} \left(\frac{1}{\Lambda_C(t)\hat{\xi}(t)}- \frac{\Lambda_T(t)\hat{S}_C(t)}{\Lambda_C^2(t)}\right).\end{aligned}$$

For $A_{RR}(t),\ B_{RR}(t),\ A_{\textit{CHR}}(t),\ B_{\textit{CHR}}(t),\ mu_1(t), \mu_2(t), \nu_1(t),\ \nu_2(t)$, let $A_{RR}^*(t),\ B_{RR}^*(t),{\ldots}$ etc. be their almost sure limit. In addition, let L _j be the almost sure limit of $K_j/n,\ j=1, 2.$ For $0\leq s, t<\tau,$ let

$$\begin{aligned} \nonumber&\sigma_{RR}(s,t)\nonumber\\&= A_{RR}^{*T}(s) \Omega^* \left(\int^\tau_0 \frac{\mu_1^*\mu_1^{*T}} {1+R} L_1dR + \int^\tau_0 \frac{\mu_2^*\mu_2^{*T}} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right) \Omega^{*T} D^*(t)\nonumber\\&\quad+ B_{RR}^*(s)B_{RR}^*(t) \left(\int^s_0 \frac{\nu_1^{*2}} {1+R} L_1dR + \int^s_0 \frac{\nu_2^{*2}} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right)\nonumber\\&\quad+ B_{RR}^*(t) A_{RR}^{*T}(s)\Omega^* \left(\int^t_0 \frac{\mu_1^*\nu_1^*} {1+R} L_1dR + \int^t_0 \frac{\mu_2^*\nu_2^*} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right)\\&\quad +B_{RR}^*(s) A_{RR}^{*T}(t) \Omega^* \left(\int^s_0 \frac{\mu_1^*\nu_1^*} {1+R} L_1dR + \int^s_0 \frac{\mu_2^*\nu_2^*} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right),\end{aligned}$$

(21.22)

and

$$\begin{aligned} \nonumber&\sigma_{\textit{CHR}}(s,t)\nonumber\\=& A_{\textit{CHR}}^{*T}(s) \Omega^* \left(\int^\tau_0 \frac{\mu_1^*\mu_1^{*T}} {1+R} L_1dR + \int^\tau_0 \frac{\mu_2^*\mu_2^{*T}} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right) \Omega^{*T} D^*(t)\nonumber\\&+ B_{\textit{CHR}}^*(s)B_{\textit{CHR}}^*(t) \left(\int^s_0 \frac{\nu_1^{*2}} {1+R} L_1dR + \int^s_0 \frac{\nu_2^{*2}} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right)\nonumber\\&+ B_{\textit{CHR}}^*(t) A_{\textit{CHR}}^{*T}(s)\Omega^* \left(\int^t_0 \frac{\mu_1^*\nu_1^*} {1+R} L_1dR + \int^t_0 \frac{\mu_2^*\nu_2^*} {e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right)\\&+B_{\textit{CHR}}^*(s) A_{\textit{CHR}}^{*T}(t) \Omega^* \left(\int^s_0 \frac{\mu_1^*\nu_1^*} {1+R} L_1dR + \int^s_0 \frac{\mu_2^*\nu_2^*}{e^{-{\beta}_1}+e^{-{\beta}_2}R}L_2dR\right).\end{aligned}$$

(21.23)

For $A_{RR}(t),\ B_{RR}(t), {\ldots}$, etc. define corresponding estimator $\hat{B}_{RR}(t), \hat{A}_{RR}(t),$ … by replacing ${\bf \beta}$ with ${\hat{\bf \beta}}$, R(t) with ${\hat{R}}(t;{\hat{\bf \beta}})$. Defince $\hat{\sigma}_{RR}(s,t)$ and $\hat{\sigma}_{\textit{CHR}}(s,t)$ by repalcing $B_{RR}(t),\ A_{RR}(t)$, $\mu_1(t)$, $\mu_2(t)$, $\nu_1(t),\ \nu_2(t)$,… in $\sigma_{RR}(s,t)$ and $\sigma_{\textit{CHR}}(s,t)$ by $\hat{B}_{RR}(t), \hat{A}_{RR}(t), {\ldots}$ etc.

Theorem 2.

Suppose that Conditions 1 ∼ 4 hold and that the matrix $\Omega^*$ is non-singular. Then, (i) U _n is asymptotically equivalent to the process $\tilde{U}_n$ in (21.6) which converges weakly to a zero-mean Gaussian process $U^*$ on $[0,\tau]$ , with covariance function $\sigma_{RR}(s,t)$ in (21.22). $\sigma_{RR}(s,t)$ can be consistently estimated by $\hat{\sigma}_{RR}(s,t).$ In addition, $\hat{U}_n(s)$ given the data converges weakly to the same limiting process $U^*$ . (ii) $V_n(t)$ is asymptotically equivalent to the process $\tilde{V}_n$ in (21.7) which converges weakly to a zero-mean Gaussian process $V^*$ on $[0,\tau]$ , with covariance function $\sigma_{\textit{CHR}}(s,t)$ in (21.23). $\sigma_{\textit{CHR}}(s,t)$ can be consistently estimated by $\hat{\sigma}_{\textit{CHR}}(s,t).$ In addition, $\hat{V}_n(s)$ given the data converges weakly to the same limiting process $V^*$.

Proof.

(i) As in the proof for Theorem A2 (ii) in Yang and Prentice (2005), by the strong embedding theorem and (21.19), $Q({\bf \beta})/\sqrt{n}$ can be shown to be asymptotically normal. Now Taylor series expansion of $Q({\bf b})$ around ${\bf \beta}$ and the non-singularity of $\Omega^*$ imply that $\sqrt{n}\Big({\hat{\bf \beta}}-{\bf \beta}\Big)$ is asymptotically normal. From the $\sqrt{n}$- boundedness of ${\hat{\bf \beta}}$,

$$\begin{aligned} \sqrt{n}\Big(\hat{R}\Big(t;{\hat{\beta}}\Big) -{\hat{R}}\Big(t;{\beta}\Big)\Big) =\frac{\partial R(t;\beta)}{\partial \beta} \sqrt{n} \Big({\hat{\beta}} -{\beta}\Big)+o_p(1),\end{aligned}$$

uniformly in $t\leq \tau.$ These results, some algebra and Taylor series expansion together show that U _n is asymptotically equivalent to $\tilde{U}_n$. Similarly, to the proof of the asymptotic normality of $Q({\bf \beta})/\sqrt{n}$, one can show that $\tilde{U}_n$ converges weakly to a zero-mean Gaussian process. Denote the limiting process by $U^*$. From the martingale integral representation of $\tilde{U}_n$, it can be shown that the covariation process of $U^*$ is given by $\sigma(s,t)$ in (21.22). The consistency of $\hat{\sigma}_{RR}(s,t)$ can be shown similarly to the proof of Theorem 1.

By checking the tightness condition and the convergence of the finite-dimensional distributions, it can be shown that $\hat{U}_n(s)$ given the data also converges weakly to $U^*$.

(ii) The assertions on V _n , $\tilde{V}_n$, etc. can be proved similarly to the case for U _n , $\tilde{U}_n$, etc. in (i).