Alternative Analysis Methods for Time to Event Endpoints Under Nonproportional Hazards: A Comparative Analysis

Abstract

The log-rank test is most powerful under proportional hazards (PH). In practice, non-PH patterns are often observed in clinical trials, such as in immuno-oncology; therefore, alternative methods are needed to restore the efficiency of statistical testing. Three categories of testing methods were evaluated, including weighted log-rank tests, Kaplan-Meier curve-based tests (including weighted Kaplan-Meier and Restricted Mean Survival Time, RMST), and combination tests (including Breslow test, Lee's combo test, and MaxCombo test). Nine scenarios representing the PH and various non-PH patterns were simulated. The power, type-I error, and effect estimates of each method were compared. In general, all tests control type I error well. There is not a single most powerful test across all scenarios. In the absence of prior knowledge regarding the PH or non-PH patterns, the MaxCombo test is relatively robust across patterns. Since the treatment effect changes overtime under non-PH, the overall profile of the treatment effect may not be represented comprehensively based on a single measure. Thus, multiple measures of the treatment effect should be pre-specified as sensitivity analyses to evaluate the totality of the data.

Sample size calculation for logrank test and prediction of number of events over time

“…Xu et al 18 propose the piecewise weighted logrank test optimal under the pre‐specified delayed effect scenario. Lin et al 19 compare the weighted log rank tests, Kaplan‐Meier curve‐based tests, and combination tests under non‐proportional hazards and recommend the use of the MaxCombo test for its relative robustness across patterns of non‐proportional hazards.…”

“…The critical value of the resulting MaxCombo test statistic can be calculated using the quantile function of the multivariate normal distribution qmvnorm from the R package mvtnorm, with entries of the correlation matrix obtained from (13). When non‐proportional hazards may take more complex patterns such as diminishing effect, crossing hazards, or delayed effect with converging tails, the MaxCombo test based on the FH's family of weighted logrank tests 19 can be considered. The critical value can be calculated using the multivariate normal quantile function with entries of the correlation matrix obtained from Equation (14).…”

Rejoinder to Letter to the Editor “The Hazards of Period Specific and Weighted Hazard Ratios”

“…We acknowledge that each effect measure has its values and limitations. As we highlighted in our article (Lin et al 2020), a single effect measure is unable to comprehensively describe the magnitude of the treatment effect over time because the effect changes over time under NPH.…”

“…Various NPH patterns with a mixture of delayed effect, longterm survival or crossing effect have been observed in recent studies (e.g., immuno-oncology trials). The alternative primary analysis strategy under NPH was proposed to prospectively address the challenges of unknown NPH patterns at the stage of study design (Roychoudhury et al 2019;Lin et al 2020). We recommended a stepwise approach which separated the confirmatory testing from the reporting of treatment effect.…”

The Hazards of Period Specific and Weighted Hazard Ratios

“…As Lin et al. ( 2019 ) note, quantification of treatment effects is of crucial importance in addition to hypothesis testing. Under NPH, Lin et al.…”