Sequential tests for non-proportional hazards data

Abstract

In clinical trials survival endpoints are usually compared using the log-rank test. Sequential methods for the log-rank test and the Cox proportional hazards model are largely reported in the statistical literature. When the proportional hazards assumption is violated the hazard ratio is ill-defined and the power of the log-rank test depends on the distribution of the censoring times. The average hazard ratio was proposed as an alternative effect measure, which has a meaningful interpretation in the case of non-proportional hazards, and is equal to the hazard ratio, if the hazards are indeed proportional. In the present work we prove that the average hazard ratio based sequential test statistics are asymptotically multivariate normal with the independent increments property. This allows for the calculation of group-sequential boundaries using standard methods and existing software. The finite sample characteristics of the new method are examined in a simulation study in a proportional and a non-proportional hazards setting.

Appendix: Proofs

Proof (Proof of Theorem 1)

Following the arguments of Kalbfleisch and Prentice (1981) for the Kaplan–Meier estimator, we have in the general case

$$\begin{aligned} n^{1/2}\left\{ \hat{x}_1(t) - x_1\right\}= & {} -\int _0^L n^{1/2}\left\{ \hat{S}_0(t,\,s) - S_0(s)\right\} S_1(ds)\nonumber \\&+ \int _0^L n^{1/2}\left\{ \hat{S}_1(t,\,s) - S_1(s)\right\} S_0(ds)\nonumber \\&- n^{1/2}\left\{ \hat{S}_1(t,\,L) - S_1(L)\right\} S_0(L) + o_p(1), \end{aligned}$$

(8)

and

$$\begin{aligned} n^{1/2}\{\hat{G}(t,\,L) - G(L)\}= & {} S_1(L)n^{1/2}\left\{ \hat{S}_0(t,\,L) - S_0(L)\right\} \nonumber \\&+ S_0(L)n^{1/2}\left\{ \hat{S}_1(t,\,L) - S_1(L)\right\} + o_p(1). \end{aligned}$$

(9)

Define the linear map $\phi {\text {:}}\,(D[0,\,L])^{2K} \rightarrow \mathbb {R}^{2K}$ by

$$\begin{aligned} \begin{pmatrix} X_{01} \\ \vdots \\ X_{0K} \\ X_{11} \\ \vdots \\ X_{1K} \end{pmatrix} \mapsto \begin{pmatrix} -\int _0^L X_{01}(s) S_1(ds) + \int _0^L X_{11}(s) S_0(ds) - X_{11}(L)S_0(L) \\ S_1(L)X_{01}(L)) + S_0(L)X_{11}(L)\\ \vdots \\ -\int _0^L X_{0K}(s) S_1(ds) + \int _0^L X_{0K}(s) S_0(ds) - X_{0K}(L)S_0(L) \\ S_1(L)X_{0K}(L) + S_0(L)X_{1K}(L) \end{pmatrix}. \end{aligned}$$

From Assumptions 1–4 it follows that $n^{1/2}\{\hat{S}_i(t_l,\,\cdot ) - S_i(\cdot )\}$ converges weakly to a mean zero Gaussian process $U_i(t_l,\, \cdot )$ with covariance function $\nu _i^{-1}\rho _i(t_l,\,\cdot ,\,\cdot )$ as $n\rightarrow \infty$ for each $l=1,\ldots ,K.$ The covariance of any two of these processes is

$$\begin{aligned} \text {cov}\left\{ U_i(t,\, s),\, U_j(t^\prime ,\, s^\prime )\right\} = I(i = j) \nu _i^{-1} \rho _i(t \wedge t^\prime ,\, s,\, s^\prime ), \end{aligned}$$

(10)

because of the independence of the samples and Assumption  3. Define the vectors

$$\begin{aligned} \mathbf {U}= & {} \left\{ U_0\left( t_1,\,\cdot \right) , \ldots , U_0\left( t_K,\,\cdot \right) ,\, U_1\left( t_1,\,\cdot \right) , \ldots , U_1\left( t_K,\,\cdot \right) \right\} ,\\ \mathbf {S^{(n)}}= & {} \left\{ \hat{S}_0\left( t_1,\,\cdot \right) , \ldots , \hat{S}_0\left( t_K,\,\cdot \right) ,\, \hat{S}_1\left( t_1,\,\cdot \right) , \ldots , \hat{S}_1\left( t_K,\,\cdot \right) \right\} ,\\ \mathbf {S}= & {} \left\{ S_0(\cdot ), \ldots , S_0(\cdot ),\, S_1(\cdot ), \ldots , S_1(\cdot )\right\} . \end{aligned}$$

It then follows from Eqs. (8) and (9) and the continuous mapping theorem, that as $n \rightarrow \infty$

$$\begin{aligned} \begin{pmatrix} n^{1/2}\{\hat{x}_1(t_1) - x_1\} \\ n^{1/2}\{\hat{G}(t_1,\, L) - G(L)\}\\ \vdots \\ n^{1/2}\{\hat{x}_1(t_K) - x_1\} \\ n^{1/2}\{\hat{G}(t_K,\, L) - G(L)\} \end{pmatrix} = n^{1/2}\left\{ \mathbf {\phi \left( \mathbf {S}^{(n)}\right) } - \mathbf {\phi (\mathbf {S})}\right\} \mathop {\longrightarrow }\limits ^{\mathcal {L}} \mathbf {\phi (\mathbf {U})}. \end{aligned}$$

Since $\phi$ is linear, the vector $\mathbf {\phi (\mathbf {U})}$ has a multivariate normal distribution with mean zero and covariance matrix

$$\begin{aligned} \mathbf {\Sigma _\theta } = \begin{pmatrix} \mathbf {\Sigma (t_1)} &{} \mathbf {\Sigma (t_2)} &{} \cdots &{} \mathbf {\Sigma (t_K)} \\ &{} \mathbf {\Sigma (t_2)} &{} &{} \vdots \\ &{} &{} \ddots &{} \\ &{} &{} &{} \mathbf {\Sigma (t_K)} \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \mathbf {\Sigma (t)} = \begin{pmatrix} V_x(t) &{} V_{xG}(t) \\ V_{xG}(t) &{} V_G(t) \end{pmatrix}. \end{aligned}$$

The covariance matrix $\mathbf {\Sigma _\theta }$ can be derived using Eq. (10) and Fubini’s theorem. The claimed asymptotic normality now follows by noting that

$$\begin{aligned} n(t)^{1/2}\left\{ \hat{\theta }_1(t) - \theta _1\right\} = \pi (t)^{1/2}n^{1/2}\left\{ \hat{\theta }_1(t) - \theta _1\right\} + o_p(1). \end{aligned}$$

Consistency of the variance estimator follows from the uniform consistency of $\hat{S}_i,$ by Assumption 2, and $\hat{\rho }_i,$ by Assumption 4 ($i=0,\,1$).