Quantifying the average of the time-varying hazard ratio via a class of transformations

Abstract

The hazard ratio derived from the Cox model is a commonly used summary statistic to quantify a treatment effect with a time-to-event outcome. The proportional hazards assumption of the Cox model, however, is frequently violated in practice and many alternative models have been proposed in the statistical literature. Unfortunately, the regression coefficients obtained from different models are often not directly comparable. To overcome this problem, we propose a family of weighted hazard ratio measures that are based on the marginal survival curves or marginal hazard functions, and can be estimated using readily available output from various modeling approaches. The proposed transformation family includes the transformations considered by Schemper et al. (Statist Med 28:2473–2489, 2009) as special cases. In addition, we propose a novel estimate of the weighted hazard ratio based on the maximum departure from the null hypothesis within the transformation family, and develop a Kolmogorov$-$Smirnov type of test statistic based on this estimate. Simulation studies show that when the hazard functions of two groups either converge or diverge, this new estimate yields a more powerful test than tests based on the individual transformations recommended in Schemper et al. (Statist Med 28:2473–2489, 2009), with a similar magnitude of power loss when the hazards cross. The proposed estimates and test statistics are applied to a colorectal cancer clinical trial.

Appendix

We state here the conditions needed to establish consistency of $\hat{\theta }_G$ and to derive its asymptotic distribution.

• (C.1) $G$ is thrice-continuously differentiable and is strictly increasing. Additionally, $\varOmega (t;s,v)$ is twice-continuously differentiable in $(t,s,v)$.

• (C.2) $\sqrt{n}(\hat{S}_1-S_{10},\hat{S}_0-S_{00})$ converges in distribution to a bivariate mean-zero Gaussian process, denoted by $(\mathcal{G}_1, \mathcal{G}_0)$ in $BV[0,\tau ]\times BV[0,\tau ]$, where $BV[0,\tau ]$ denotes the spaces consisting of functions that have finite total variation in $[0,\tau ]$ and $S_{k0}$ is the true survival function in treatment arm $k$. Here, $\tau$ is the study duration.

• (C.3) $S_{k0}$ is strictly decreasing and thrice-continuously differentiable in $[0,\tau ]$. Moreover, $S_{k0}(\tau )>0$.

• (C.4) The kernel function $K(x)$ is differentiable, symmetric with respect to 0, and has compact support on $[-1,1]$.

• (C.5) The bandwidth $a_n$ satisfies $n a_n^2\rightarrow \infty$ and $na_n^4\rightarrow 0$.

• (C.6) Conditional on the data, $\sqrt{n} (\hat{S}_1^*-\hat{S}_1, \hat{S}_0^*-\hat{S}_0)$ converges in distribution to $(\mathcal{G}_1, \mathcal{G}_0)$, where $(\hat{S}_1^*, \hat{S}_0^*)$ are resampled statistics for $(\hat{S}_1, \hat{S}_0)$.

Condition (C.6) is an assumption regarding the consistency and asymptotic distribution of the estimates generated by the boostrap procedure. Chapter 20 of Kosorok (2007) validates this assumption for several survival modeling techniques including the Cox proportional hazards model.

Lemma 1

Under Conditions (C.1)–(C.5), $\sup _{t\in [0,\tau ]}\vert \hat{h}_k(t)-h_{k0}(t)\vert \rightarrow _p 0, \ \ k=0,1,$ where $h_{k0}$ denotes the true hazard function in treatment arm $k$.

Proof (of Lemma 1)

First, we note that $\hat{h}_k(t)=-a_n^{-1}\int K(x) d\log \hat{S}_k(t+a_n x).$ By carrying out an integration by parts, we can rewrite $\hat{h}_k(t)$ as $\hat{h}_k(t)=\int a_n^{-1}\log \hat{S}_k(t+a_n x) K'(x)dx.$ Moreover, we can continuously extend defining $\hat{S}_k$ and $S_{k0}$ to $[-a_n,\tau +a_n]$ so that (C.2) still holds. Thus, (C.2) implies $\sup _{t\in [-a_n, \tau +a_n]} \vert \hat{S}_k(t)- S_{k0}(t)\vert =O_p(n^{-1/2}).$ (C.3) further gives $\sup _{t\in [-a_n, \tau +a_n]} \vert \log \hat{S}_k(t)-\log S_{k0}(t)\vert =O_p(n^{-1/2}).$ Therefore,

$$\begin{aligned} \sup _{t\in [0,\tau ]}\vert \hat{h}_k(t)-h_{k0}(t)\vert&\le \int a_n^{-1}\sup _{t\in [-a_n, \tau +a_n]} \vert \log \hat{S}_k(t)- \log S_{k0}(t)\vert |K'(x)|dx\\&\le O_p(1/\surd {na_n^2}) \end{aligned}$$

goes to 0 by condition (C.5). This proves the lemma. $\square$

Proof (of Theorem 1)

Using Lemma 1 and noting $\inf _{t\in [0,\tau ]} h_0(t)>0$, we obtain that uniformly in $t\in [0,\tau ]$, as $n \rightarrow \infty , G\left\{ \frac{\hat{h}_1(t)}{\hat{h}_0(t)}\right\} \rightarrow _p G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} ,$ and $\varOmega \{t; \hat{S}_0(t), \hat{S}_1(t)\}\rightarrow _p \varOmega \{t;S_{00}(t), S_{10}(t)\}.$ Thus, it is clear that $\hat{\theta }_G \rightarrow _p \theta _G$ as $n \rightarrow \infty$. This establishes consistency.

To derive the asymptotic distribution of $\hat{\theta }_G$, by the mean-value theorem, we obtain

$$\begin{aligned}&n^{1/2}(\hat{\theta }_G-\theta _G)\\&\quad =n^{1/2} \left( (G^{-1})'\left[ \int G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} \varOmega \{t; S_{00}(t), S_{10}(t)\}dt\right] +o_p(1)\right) \\&\qquad \times \left[ \!\int \! G\left\{ \frac{\hat{h}_{1}(t)}{\hat{h}_{0}(t)}\right\} \varOmega \{t; \hat{S}_{0}(t), \hat{S}_{1}(t)\}dt\!-\! \int \! G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\!\right\} \varOmega \{t; S_{00}(t), S_{10}(t)\}dt\!\right] \!. \end{aligned}$$

Using the mean-value theorem again, we have

$$\begin{aligned}&G\left\{ \frac{\hat{h}_{1}(t)}{\hat{h}_{0}(t)}\right\} \varOmega \{t; \hat{S}_{0}(t), \hat{S}_{1}(t)\}- G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} \varOmega \{t; S_{00}(t), S_{10}(t)\}\\&\quad =\left[ G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} \varOmega \{t; S_{00}(t), S_{10}(t)\}+o_p(1)\right] \\&\qquad \times \left[ \frac{\hat{h}_1(t)-h_{10}(t)}{h_{00}(t)}-\frac{h_{10}(t)\{\hat{h}_0(t)-h_{00}(t)\}}{h_{00}(t)^2}\right] \\&\qquad +\left[ G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} \frac{\partial }{\partial S_0(t)} \varOmega \{t; S_{00}(t), S_{10}(t)\}+o_p(1)\right] \{\hat{S}_0(t)-S_{00}(t)\}\\&\qquad +\left[ G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} \frac{\partial }{\partial S_1(t)} \varOmega \{t; S_{00}(t), S_{10}(t)\}+o_p(1)\right] \{\hat{S}_1(t)-S_{10}(t)\}, \end{aligned}$$

where $o_p(1)$ is a random element converging in probability to zero uniformly in $t\in [0,\tau ]$.

For convenience, we denote $H=(G^{-1})'\{\int Q_0(t)Q_1(t)dt\}$, $Q_0(t)= G\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} , Q_1(t)= \varOmega \{t; S_{00}(t), S_{10}(t)\}$, $Q_2(t)=\frac{\partial }{\partial S_0(t)} \varOmega \{t; S_{00}(t), S_{10}(t)\}$, and $Q_3(t)=\frac{\partial }{\partial S_1(t)} \varOmega \{t; S_{00}(t), S_{10}(t)\}$. Then, after combining the above results, we obtain

$$\begin{aligned}&n^{1/2}(\hat{\theta }_G-\theta _G)\\&\quad =n^{1/2} \left\{ H+o_p(1)\right\} \left[ \int G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)\{\hat{h}_1(t)-h_{10}(t)\}/h_{00}(t) dt\right] \\&\qquad -\,n^{1/2} \left\{ H+o_p(1)\right\} \left[ \int G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)h_{10}(t)\{\hat{h}_0(t)-h_{00}(t)\}/h_{00}(t)^2 dt\right] \\&\qquad +\,n^{1/2} \left\{ H+o_p(1)\right\} \left[ \int Q_0(t)Q_3(t)\{\hat{S}_1(t)-S_{10}(t)\} dt\right] \\&\qquad +\,n^{1/2} \left\{ H+o_p(1)\right\} \left[ \int Q_0(t)Q_2(t)\{\hat{S}_0(t)-S_{00}(t)\} dt\right] . \end{aligned}$$

The last two terms on the right-hand side both take the form $n^{1/2} \int [A(t)+o_p(1)](\hat{S}_k(t)-S_{k0}(t))dt$ for some bounded function $A(t)$ so that they converge to a normal distribution by condition (C.2). Thus, we only focus on the first two terms, namely (I) and (II), on the right-hand side. Using the definition of $\hat{h}_1(t)$, we can rewrite the first term (I) as

$$\begin{aligned}&n^{1/2} \left\{ H+o_p(1)\right\} \int G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)\left\{ -\int K_{a_n}(s-t)d\log \hat{S}_1(s) \right. \\&\left. +\int K_{a_n}(s-t)d\log S_{10}(s)- \int K_{a_n}(s-t)d\log S_{10}(s)-h_{10}(t)\right\} /h_{00}(t) dt. \end{aligned}$$

Since

$$\begin{aligned}&- \int K_{a_n}(s-t)d\log S_{10}(s)-h_{10}(t)=\int K_{a_n}(s-t)h_{10}(s)ds-h_{10}(t) \\&\quad = \int K(x)\left\{ h_{10}(t+a_nx)-h_{10}(t)\right\} dx=O(a_n^2) \end{aligned}$$

and $na_n^4\rightarrow 0$, (I) becomes

$$\begin{aligned}&n^{1/2} \left\{ H+o_p(1)\right\} \int G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t) \\&\quad \times \left[ -\int K_{a_n}(s-t)d\{\log \hat{S}_1(s) -\log S_{10}(s)\}\right] /h_{00}(t) dt+o_p(1), \end{aligned}$$

which is also equal to

$$\begin{aligned}&-n^{1/2} \left\{ H+o_p(1)\right\} \int _s d\{\log \hat{S}_1(s) -\log S_{10}(s)\} \\&\quad \times \int _t G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)K_{a_n}(s-t)/h_{00}(t)dt+o_p(1). \end{aligned}$$

However, since

$$\begin{aligned} \int _t G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)K_{a_n}(s-t)/h_{00}(t)dt =G'\left\{ \frac{h_{10}(s)}{h_{00}(s)}\right\} Q_1(s)/h_{00}(s)+O(a_n^2), \end{aligned}$$

it follows that

$$\begin{aligned} (I)=-n^{1/2} H \int _s G'\left\{ \frac{h_{10}(s)}{h_{00}(s)}\right\} Q_1(s)/h_{00}(s)d\{\log \hat{S}_1(s) -\log S_{10}(s)\}+o_p(1). \end{aligned}$$

Similarly, we obtain

$$\begin{aligned} (II)\!=\!n^{1/2} H \int _s G'\left\{ \frac{h_{10}(s)}{h_{00}(s)}\right\} Q_1(s)h_{10}(s)/h_{00}(s)^2d\{\log \hat{S}_0(s)\! -\! \log S_{00}(s)\}\!+\!o_p(1). \end{aligned}$$

Thus, we have

$$\begin{aligned}&n^{1/2}(\hat{\theta }_G-\theta _G) \nonumber \\&\quad =n^{1/2} \int A_1(t)d \{\log \hat{S}_1(t)\!-\!\log S_{10}(t)\} +n^{1/2} \int A_0(t)d\{\log \hat{S}_0(t)\!-\!\log S_{00}(t)\} \nonumber \\&\qquad +\,n^{1/2} \int B_1(t) \{\hat{S}_1(t)-S_{10}(t)\}dt+n^{1/2} \int B_0(t)\{\hat{S}_0(t)-S_{00}(t)\}dt+o_p(1),\nonumber \\ \end{aligned}$$

(15)

where $A_0(t)=H G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)h_{10}(t)/h_{00}(t)^2$, $A_1(t)=-H G'\left\{ \frac{h_{10}(t)}{h_{00}(t)}\right\} Q_1(t)/h_{00}(t)$, $B_0(t) =H Q_0(t)Q_2(t)$, and $B_1(t)=H Q_0(t)Q_3(t)$. Hence, Theorem 1 follows from condition (C.2). $\square$

Proof of Theorem 2

Let the weight function $\varOmega (t)$ be independent of $S_0(t)$ and $S_1(t)$ and satisfy $\int \varOmega (t) dt=1$. We write the hazard function of the treatment arm as $h_1(t)/h_0(t)=1+\epsilon \lambda (t)$, where $\lambda (t)$ is a function of $t$. When $h_1(t)$ is in the neighborhood of the null hypothesis with $h_1(t)=h_0(t)$, i.e., $\epsilon$ is close to zero, the Taylor’s series expansion of $\theta _a$ is given by

$$\begin{aligned} \theta _a&= G_a^{-1}\left\{ \int _0^\tau G_a(1) \varOmega (t) dt\right\} + (G_a^{-1})' \left\{ \int _0^\tau G_a(1) \varOmega (t) dt \right\} \\&\int _0^\tau G_a'(1) \varOmega (t) \lambda (t) dt \times \epsilon + o(\epsilon ) \nonumber \\&= 1+ (G_a^{-1})'\{G_a(1)\} \times G_a'(1)\int _0^\tau \varOmega (t) \lambda (t) dt \times \epsilon + o(\epsilon ). \end{aligned}$$

Moreover, according to (15), the variance of $\hat{\theta }_a$ around the null hypothesis can be written as $(G_a'(t))^2 B$, where $B$ is a positive value independent of $a$. Therefore, the local power of $\theta _a$ can be written as

$$\begin{aligned}&P\left( \left| \frac{\hat{\theta }_{a}-1}{sd(\hat{\theta }_{a})} \right| > Z_{1-\alpha /2} \mid h_0(t)=h_1(t) \right) \\&\quad = P\left( \left| (G_a^{-1})'\{G_a(1)\} \epsilon \int _0^\tau \varOmega (t) \lambda (t) dt + o(\epsilon )\right| > B Z_{1-\alpha /2}\right) . \end{aligned}$$

For the transformation family in (2) with $a \in [-1,1]$, we can optimize the local power by maximizing $|(G_a^{-1})'\{G_a(1)\}|=|(1+a)^{1+a}|$ for $a \in [-1,1]$, resulting in an optimal value at $a=1$.

When $\int \varOmega (t) \lambda (t) dt=0$ for the crossing hazards case, we need a higher order Taylor’s series expansion, given by

$$\begin{aligned} \theta _a&= G_a^{-1}\left\{ \int _0^\tau G_a(1) \varOmega (t) dt\right\} + (G_a^{-1})' \left\{ \int _0^\tau G_a(1) \varOmega (t) dt \right\} \\&\times \int _0^\tau G_a'(1) \varOmega (t) \lambda (t) dt \times \epsilon \nonumber +\, (G_a^{-1})'' \left\{ \int _0^\tau G_a(1) \varOmega (t) dt \right\} \\&\times \left\{ \int _0^\tau G_a'(1) \varOmega (t) \lambda (t) dt\right\} ^2 \times \epsilon ^2 +\, (G_a^{-1})' \left\{ \int _0^\tau G_a(1) \varOmega (t) dt \right\} \nonumber \\&\times \left\{ \int _0^\tau G_a''(1) \varOmega (t) \lambda ^2(t) dt \right\} \times \epsilon ^2 + o(\epsilon ^2) \nonumber \\&= 1+ (G_a^{-1})'\{G_a(1)\} \times G_a''(1) \times \int _0^\tau \varOmega (t) \lambda ^2(t) dt \times \epsilon ^2 + o(\epsilon ^2) \nonumber \end{aligned}$$

In this case, the local power of $\theta _a$ is given by

$$\begin{aligned} P\left( \left| (G_a^{-1})'\{G_a(1)\} \times \frac{G_a''(1)}{G_a'(1)} \times \int _0^\tau \varOmega (t) \lambda ^2(t) dt \times \epsilon ^2 + o(\epsilon ^2)\right| > B Z_{1-\alpha /2}\right) \end{aligned}$$

where $G_a''(x)=-\left( \frac{a+1}{a+x} \right) G_a'(x)$. Again, this local power is maximized at $a=1$. $\square$

Proof of Theorem 3

The proof is based on the same linearization given in equation (15) but on the right hand side of equation (15), expressions $A_1(t), A_0(t), B_1(t)$, and $B_0(t)$ are indexed by $a \in [0,1]$. Additionally, $o_p(1)$ on the right hand side of (15) converges in probability to zero uniformly in $a$. It is easy to check from the explicit expressions of $A_1, A_0, B_1, B_0$ that they all belong to a bounded set in $BV[0,\tau ]$ for any $a \in [0,1]$. Thus, condition (C.2) and the results in Gill and Vaart (1993) yield that $n^{1/2} (\hat{\theta }_a-\theta _a)$, as a stochastic process indexed by $a \in [0,1]$, converges weakly to a Gaussian process. Theorem 3 thus follows from the continuity theorem. $\square$