Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis

Abstract

In a longitudinal clinical study to compare two groups, the primary end point is often the time to a specific event (eg, disease progression, death). The hazard ratio estimate is routinely used to empirically quantify the between-group difference under the assumption that the ratio of the two hazard functions is approximately constant over time. When this assumption is plausible, such a ratio estimate may capture the relative difference between two survival curves. However, the clinical meaning of such a ratio estimate is difficult, if not impossible, to interpret when the underlying proportional hazards assumption is violated (ie, the hazard ratio is not constant over time). Although this issue has been studied extensively and various alternatives to the hazard ratio estimator have been discussed in the statistical literature, such crucial information does not seem to have reached the broader community of health science researchers. In this article, we summarize several critical concerns regarding this conventional practice and discuss various well-known alternatives for quantifying the underlying differences between groups with respect to a time-to-event end point. The data from three recent cancer clinical trials, which reflect a variety of scenarios, are used throughout to illustrate our discussions. When there is not sufficient information about the profile of the between-group difference at the design stage of the study, we encourage practitioners to consider a prespecified, clinically meaningful, model-free measure for quantifying the difference and to use robust estimation procedures to draw primary inferences.

Fig 1.

Estimated survival curves, hazard ratio, and restricted mean survival times with data from the Eastern Cooperative Oncology Group E4A03 study. (A) Kaplan-Meier curves for low-dose (blue) and high-dose (gold) groups. (B) Estimate of the ratio of hazard functions (low dose over high dose) over time and corresponding 0.95 point-wise confidence band. (C) Estimate of restricted mean survival time (blue area) and the restricted mean time lost (gray area) up to 40 months for the low-dose group.

Fig 2.

Estimated survival curves and the hazard ratio with reconstructed data for comparing single-agent pemetrexed (P) with carboplatin plus pemetrexed (CP) in patients with advanced non–small-cell lung cancer. (A) Kaplan-Meier curves for CP (blue) and P (gold). (B) Estimate of the ratio of hazard functions (CP over P) over time and the corresponding 0.95 point-wise confidence band.

Fig 3.

Estimated survival curves and the hazard ratio with reconstructed data for comparing modified FOLFOX6 (mFF6) plus bevacizumab (Bev) with mFF6 in patients with stage II to III colon cancer. (A) Estimated survival curves for mFF6 + Bev (blue) and mFF6 (gold). (B) Estimate of the ratio of hazard functions (mFF6 + Bev over mFF6) and the corresponding 0.95 point-wise confidence band.