Covariate-adjusted log-rank test: guaranteed efficiency gain and universal applicability

Summary

1 Introduction

In clinical trials, adjusting for baseline covariates has been widely advocated as a way to improve efficiency for demonstrating treatment effects ‘under approximately the same minimal statistical assumptions that would be needed for unadjusted estimation’ (ICH E9, 1998; EMA, 2015; FDA, 2023). In testing for an effect between two treatments with right-censored time-to-event outcomes, adjusting for covariates using the Cox proportional hazards model has been demonstrated to yield valid tests even if the Cox model is misspecified (Lin & Wei, 1989; Kong & Slud, 1997; DiRienzo & Lagakos, 2002). However, these tests may be less powerful than the log-rank test that does not adjust for any covariates when the Cox model is misspecified (Kong & Slud, 1997). Although efforts have been made to improve the efficiency of the log-rank test through covariate adjustment from semiparametric theory (Lu & Tsiatis, 2008; Moore & van der Laan, 2009), the solutions are complicated and their validity is established only under simple randomization, i.e., treatments are assigned to patients completely at random.

To balance the number of patients in each treatment arm across baseline prognostic factors in clinical trials with sequentially arrived patients, covariate-adaptive randomization has become the new norm. From 1989 to 2008, covariate-adaptive randomization was used in more than 500 clinical trials (Taves, 2010); among nearly 300 trials published in two years, 2009 and 2014, 237 of them applied covariate-adaptive randomization (Ciolino et al., 2019). The two most popular covariate-adaptive randomization schemes are the stratified permuted block (Zelen, 1974) and the Pocock–Simon minimization (Taves, 1974; Pocock & Simon, 1975). Other schemes can be found in the reviews of Schulz & Grimes (2002) and Shao (2021). Unlike simple randomization, covariate-adaptive randomization generates a dependent sequence of treatment assignments, which may render conventional methods developed under simple randomization not necessarily valid under covariate-adaptive randomization (EMA, 2015; FDA, 2023). For time-to-event data under covariate-adaptive randomization, Ye & Shao (2020) showed that some conventional tests including the log-rank test are conservative and Wang et al. (2023) showed that the Kaplan–Meier estimator of the survival function has reduced variance compared to that under simple randomization.

The discussion so far has brought up two issues in adjusting for covariates. First is the need for guaranteed efficiency gains over unadjusted methods, without requiring additional assumptions. Second is the need for methods with wide applicability to all commonly used covariate-adaptive randomizations. These issues have been well addressed when adjustments are made under linear working models for non-time-to-event data (Tsiatis et al., 2008; Zhang et al., 2008; Lin, 2013; Ye et al., 2022). Ye et al. (2022) also showed that adjustment via linear working models can achieve universal applicability in the sense that the same inference procedure can be universally applied to all commonly used covariate-adaptive randomization schemes, a desirable property for application. For right-censored time-to-event outcomes, to the best of our knowledge, no result has been established for covariate adjustment with guaranteed efficiency gain and universal applicability.

In this paper we propose a nonparametric covariate adjustment method for the log-rank test, which has a simple explicit form and can achieve the goal of guaranteed efficiency gain over the unadjusted log-rank test as well as universal applicability to simple randomization and all commonly used covariate-adaptive randomization schemes. The unadjusted log-rank test is not valid under covariate-adaptive randomization; although it can be modified to be applicable to some randomization schemes (Ye & Shao, 2020), the modification needs to be tailored to each randomization scheme, i.e., no universal applicability. Our main idea is to obtain a particular derived outcome for each patient from linearizing the log-rank test statistic and then apply the generalized regression adjustment or augmentation (Cassel et al., 1976; Lu & Tsiatis, 2008; Tsiatis et al., 2008; Zhang et al., 2008) to the derived outcomes. We also develop parallel results for the stratified log-rank test with adjustment for additional covariates. Our proposed tests are supported by novel asymptotic theory of the existing and proposed statistics under the null hypothesis and alternative without requiring any specific model assumption, and under all commonly used covariate-adaptive randomization schemes. Estimation and confidence intervals for treatment effects after testing are also discussed. Our theoretical results are corroborated by a simulation study that examines finite sample Type-I error and power of tests. A real data example is included for illustration.

2 Preliminaries

For a patient from the population under investigation, let T_j and C_j be the potential failure time and right-censoring time, respectively, under treatment j = 0 or 1, and W be a vector containing all observed baseline covariates. Suppose that a random sample of n patients is obtained from the population with independent $(Ti0,Ci0,Ti1,Ci1,Wi) (i=1,…,n)$⁠, identically distributed as $(T0,C0,T1,C1,W)$⁠. For each patient, only one of the two treatments is received. Thus, if patient i receives treatment j, then the observed outcome with possible right censoring is ${min(Tij,Cij),δij}$⁠, where δ_ij is the indicator of the event $Tij ⩽ Cij$⁠.

Let I_i be a binary treatment indicator for patient i and $0<π<1$ be the prespecified treatment assignment proportion for treatment 1. Consider the design, i.e., the generation of the I_i for n sequentially arrived patients. Simple randomization assigns patients to treatments completely at random with $pr(Ii=1)=π$ for all i, which does not make use of baseline covariates and may yield treatment proportions that substantially deviate from the target π across levels of some prognostic factors. Because of this, covariate-adaptive randomization using a subvector Z of W is widely applied, which does not use any model and is nonparametric. All commonly used covariate-adaptive randomization schemes satisfy the following mild condition (Baldi Antognini & Zagoraiou, 2015).

Although simple randomization is not counted as covariate-adaptive randomization, it also satisfies Condition 1.

We focus on testing the following null hypothesis of the no-treatment effect, which is the null hypothesis when the conventional log-rank test is applied: $H0:λ1(t)=λ0(t)$ for all times t, versus the alternative that H  ₀ does not hold, where $λj(t)$ is the unspecified hazard function of T_j, unconditional on covariates.

After data are collected from all patients, a test statistic $T$ is a function of observed data, constructed such that H  ₀ is rejected if and only if $|T|>zα/2$⁠, where α is a given significance level and $zα/2$ is the $(1−α/2)$th quantile of the standard normal distribution. A test $T$ is asymptotically valid if, under H  ₀, $limn→∞pr(|T|>zα/2) ⩽ α$⁠, with equality holding for at least one parameter value under the null hypothesis H  ₀. A test $T$ is asymptotically conservative if, under H  ₀, there exists an α  ₀ such that $limn→∞pr(|T|>zα/2) ⩽ α0<α$⁠.

The log-rank test $TL$ in (1) is valid under simple randomization and the following assumption.

Assumption 1 is needed for a valid nonparametric log-rank test without requiring any model on T_j or C_j (Kong & Slud, 1997; DiRienzo & Lagakos, 2002; Lu & Tsiatis, 2008; Parast et al., 2014; Zhang, 2015).

As $TL$ does not utilize any baseline covariate information, it is used as the benchmark in considering baseline covariate adjustment for efficiency gain, under the same Assumption 1 ‘that would be needed for unadjusted’ $TL$ (FDA, 2023).

There is a line of research weakening Assumption 1 to censoring at random (Robins & Finkelstein, 2000; Lu & Tsiatis, 2011; Díaz et al., 2019), under which, however, the log-rank test is not valid and needs to be replaced by a weighted log-rank test that requires a correctly specified censoring distribution as the weights are inverse probabilities of censoring. Thus, the conditions and properties of weighted log-rank tests are not comparable with those of the log-rank test. Furthermore, the validity of weighted log-rank tests has only been established under simple randomization. The study of weighted log-rank tests under covariate-adaptive randomization is left for future work.

3 Covariate-adjusted log-rank test

Let $X⊂W$ be the vector containing observed baseline covariates to be adjusted in the construction of tests, with a nonsingular covariance matrix $ΣX=var(X)$⁠. In this section, we develop a nonparametric covariate-adjusted log-rank test that has a simple and explicit formula, enjoys guaranteed efficiency gain over the log-rank test and is universally valid under all covariate-adaptive randomization schemes satisfying Condition 1.

Asymptotic properties of covariate-adjusted log-rank test $TCL$ in (6) are established in the following theorem. All technical proofs are given in the Supplementary Material. In what follows, $→d$ and $→p$⁠, respectively, denote convergence in distribution and in probability, as $n→∞$⁠.

The results under an alternative hypothesis in Theorem 1 are obtained without any specific model on the distribution of T_j or C_j, different from many published research articles that assume a specific model under an alternative hypothesis, such as the Cox proportional hazards model for T_j.

Theorem 1 shows that $TCL$ in (6) is applicable to all randomization schemes satisfying Condition 1 with a universal formula, if all levels of Z_i are included in X_i. Tests with universal applicability are desirable for application, as the complication of using tailored formulas for different randomization schemes is avoided.

To show that $TCL$ in (6) has a guaranteed efficiency gain over the benchmark $TL$ in (1), we establish an asymptotic result for $TL$ under covariate-adaptive randomization satisfying an additional condition.

Under simple randomization, Condition 1$†$ holds with $ν=π(1−π)$ and, hence, Theorem 2 also applies with $σL2(ν)=σL2$⁠. Under the local alternative specified in Theorem 1(c) with $πc1−(1−π)c0≠0$⁠, by Theorems 1(c) and 2(c), Pitman’s asymptotic relative efficiency of $TCL$ in (6) with respect to the benchmark $TL$ in (1) is $σL2/σCL2=1+π(1−π)(β1+β0)TΣX(β1+β0)/σCL2 ⩾ 1$ with the strict inequality holding unless $β1+β0=0$⁠. Thus, $TCL$ has a guaranteed efficiency gain over $TL$ under simple randomization.

The Pocock–Simon minimization satisfies Condition 1, but not necessarily Condition 1$†$ as the I_i are correlated across strata. Hence, under the Pocock–Simon minimization, Theorem 2 is not applicable and $TL$ may not be valid, whereas $TCL$ is valid according to Theorem 1, another advantage of covariate adjustment.

In the numerator of (6)  $U^CL$ is the same as the augmented score in Lu & Tsiatis (2008), which shares the same idea as those in Tsiatis et al. (2008) and Zhang et al. (2008) for noncensored data. However, the denominator $σ^CL$ in (6) is different from that used by Lu & Tsiatis (2008). The key difference between our result on guaranteed efficiency gain and the result in Lu & Tsiatis (2008) is that our result is obtained under covariate-adaptive randomization and an alternative hypothesis without any specific model on the distribution of T_j or C_j, whereas the result in Lu & Tsiatis (2008) is for simple randomization and an alternative hypothesis under a correctly specified Cox proportional hazards model for T_j.

After testing H  ₀, it is often of interest to estimate and construct a confidence interval for an effect size (Lu & Tsiatis, 2008; Parast et al., 2014; Zhang, 2015; Díaz et al., 2019). A commonly considered effect size is the hazard ratio $eθ$ under the Cox proportional hazards model $λ1(t)=λ0(t)eθ$⁠. The hazard ratio $eθ$ is interpretable only when the Cox proportional hazards model is correctly specified. Thus, in the rest of this section we consider covariate-adjusted estimation and a confidence interval for θ, assuming that $λ1(t)=λ0(t)eθ$⁠.

4 Covariate-adjusted stratified log-rank test

The stratified log-rank test (Peto et al., 1976) is a weighted average of the stratum-specific log-rank test statistics with finitely many strata constructed using a discrete baseline covariate. We consider stratification with all levels of Z_i. Results can be obtained similarly for stratifying on more levels than those of Z_i or fewer levels than those of Z_i with levels of Z_i not used in stratification included in X_i. Here, we remove the part of X_i that can be linearly represented by Z_i and still denote the remaining as X_i. As such, it is reasonable to assume that $E{var(Xi|Zi)}$ is positive definite.

$Y¯z1(t)=∑i:Zi=zIiYi(t)/n, Y¯z0(t)=∑i:Zi=z(1−Ii)Yi(t)/n$ and $Y¯z(t)=Y¯z1(t)+Y¯z0(t)$⁠.

With stratification, $TSL$ in (7) actually tests the null hypothesis $H˜0:λ1(t|z)=λ0(t|z)$ for all (t, z), where $λj(t|z)$ is the hazard function of T_j conditional on Z = z. Hypothesis $H˜0$ may be stronger than $H0:λ1(t)=λ0(t)$ for all t, the null hypothesis for unstratified log-rank test $TL$ and its adjustment $TCL$ considered in § 2–§ 3. In some scenarios, $H˜0=H0$⁠. For example, the two hypotheses are the same when there exists a transformation model $h{pr(T0 ⩾ t|W)}=θ+h{pr(T1 ⩾ t|W)}$ for all (t, W) and an unknown constant θ, where h is an increasing function that is possibly unknown (Cheng et al., 1995). This transformation model includes many commonly used semiparametric models as special cases, for example the Cox proportional hazards model with $h(s)=−log {−log(s)}$⁠.

The following theorem establishes the asymptotic properties of the stratified log-rank test $TSL$ and covariate-adjusted stratified log-rank test $TCSL$⁠.

Like $TCL$ in (6), both $TSL$ in (7) and $TCSL$ in (8) are applicable to all covariate-adaptive randomization schemes with universal formulas, i.e., they achieve the universal applicability. In terms of Pitman’s asymptotic efficiency under the local alternative specified in Theorem 3(c), $TCSL$ is always more efficient than $TSL$⁠, since $σCSL2 ⩽ σSL2$with the strict inequality holding unless $γ1+γ0=0$⁠.

The condition $CI⊥TI|(I,Z)$ in Theorem 3 for the stratified log-rank test and its adjustment is in general not comparable with Theorem 1(c) for the unstratified log-rank test.

Is $TSL$ or $TCSL$ more efficient than the unstratified log-rank test $TL$? The answer is not clear because, firstly, the null hypotheses $H˜0$ and H  ₀ may be different, as we discussed earlier, and secondly, even if $H˜0=H0$⁠, under the alternative, the asymptotic mean $(n1θ1−n0θ0)/n$ of $U^L$ may not be comparable with the asymptotic mean $∑z(nz1θz1−nz0θz0)/n$ of $U^SL$ or $U^CSL$⁠. In fact, the indefiniteness of relative efficiency between the stratified and unstratified log-rank tests is a standing problem in the literature.

There is also no definite answer when comparing the efficiencies of $TCL$ and the stratified $TCSL$⁠.

Similar to the discussion at the end of § 3, after testing hypothesis $H˜0$⁠, we can obtain a covariate-adjusted confidence interval for the effect size θ under a stratified Cox proportional hazards model $λ1z(t)=λ0z(t)eθ$ for every z; see the Supplementary Material for further details.

5 Simulations

To supplement the theory and examine finite sample Type-I error and power of tests $TL, TCL, TSL$ and $TCSL$⁠, we carry out a simulation study under the following four cases/models.