The primary goal for conducting clinical studies is to obtain robust, clinically interpretable results with respect to risk-benefit perspectives for individual patients in a well-defined target population via efficient and reliable inference procedures. Unfortunately, most conventional inference procedures somehow are not readily translatable.
Hajime Uno, a faculty member at DFCI and HSPH, has been working on this issue for the past few years. A notable recent contribution from Dr. Uno relates to the concept of survival analysis, especially regarding the quantification of treatment efficacy from clinical trials. Dr. Uno, whose efforts have been recognized by regulatory agencies and the drug industry, has published several important articles on this topic in the Annals of Internal Medicine and Journal of Clinical Oncology.
On August 26, 2016, Dr. Uno was invited by the FDA to give a one-day short course on survival analysis in conjunction with Professor Lee-Jen Wei. See the talk abstract below.
Alternatives to the hazard ratio in survival analysis – moving beyond the comfort zone
H. Uno and L.J. Wei, Harvard University
For analyzing time to event observations from a single population, the Kaplan-Meier estimate is routinely utilized to approximate the survival function. Commonly used summary measures of this function are the median survival time estimate and the event-rate estimate at specific time point(s). If the censoring is heavy, the median survival time may not be estimable. The event rate at a specific time point may not catch the temporal profile effectively. For the two-sample problem with survival time data, the difference of median survival times is generally used to summarize the size of the group difference, but again, it may not be estimable. There are very few model-free measures available for summarizing the group contrast in the presence of censored observations. This is probably one of the main reasons that the hazard ratio estimate elegantly proposed by Sir David [1] has been so popular for the past 40 years as a summary of the group difference in survival analysis.
The validity of the hazard ratio estimate is based on a strong model assumption, that is, the ratio of two hazard functions is constant over time. Even when the proportional hazards (PH) assumption is plausible, the physical/clinical interpretation of a hazard ratio estimate, say, 0.7, between the control and treatment, may be difficult. The hazard is the “force of mortality” and is not a probability measure. Moreover, since a hazard function can vary quite bit over time and its scale is not easy to interpret clinically, a 30% reduction from a very low hazard of the control arm would not represent a clinically relevant difference between two groups. On the other hand, if the control arm’s hazard at some time point is high, then a 30% reduction would be quite impressive clinically. To present a single hazard ratio without having a benchmark from the control is not informative. Lastly, it is important to note that the precision of the hazard ratio estimator depends almost exclusively on the number of observed events, not on the study patients’ exposure times or the number of patients involved. This property is not ideal when we deal with a comparative, non-inferiority, safety study, especially when the event rates are low. For a safety study, the patients’ exposure times are more relevant than the observed number of events from the study.
When the PH assumption is not valid, the Cox estimate is difficult to interpret. It is not a simple, weighted average of the true hazard ratios over time. In fact, the weights depend on the study patient’s follow-up time distribution, an undesirable feature as a summary measure of two survival functions as the censoring distribution is considered a nuisance parameter and unrelated to the actual between-group difference.
In this short course, we will present a well-known alternative, that is, the restricted mean survival time (RMST), up to a specific time point, say, t-years. For instance, if t = 48 months and RMST of the treated patient is 45 months, this means that when patients are followed for 48 months, on average they would enjoy 45 months of event-free time. The RMST is the area under the survival curve up to 48 months and can be easily estimated via the area under the KM curve. This measure can be used as a summary of a single survival function. The choice of the time “t-years” can be extended as long as the KM curve is still stable. For two-sample problems, one can use the difference or ratio of two RMSTs to quantify the group difference. This measure is purely model-free. Furthermore, this contrast measure, coupled with the RMST estimates for both groups, is more informative for decision making on the treatment selection.
The RMST metric has been studied in the statistical literature extensively, but did not get much attention among practitioners. Recently, this measure has been discussed in the clinical literature [2 – 4] for superiority and safety studies in survival analysis. We will use several recent trial data (both for efficacy and safety) to illustrate this approach. We will also apply this approach to deal with the classical competing risks problems. Lastly, we will discuss regression models for RMST [5] as well as extensions where RMST methods have been used to estimate lifetime benefits of therapy based on short-term clinical trial follow-up data [6].
Since the recent official statement from the American Statistical Association denouncing the conventional practice of decision-making based on the presence of a p-value < 0.05, in this short course, we will concentrate on the estimation issues. However, all the above estimation procedures can be converted as test statistics to obtain our old friend: p-values. Empirically, the statistical significance of the standard HR-based test and an RMST-based test are largely concordant [4]. On the other hand, a RMST based estimate appears to be easier to interpret to quantify the size of the group difference than the HR estimate.
References:
- Cox, DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), 187–220, 1972.
- Uno H, Claggett B, Tian L, et al: Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. J Clin Oncol 32:2380-2385, 2014
- Uno H, Wittes J, Fu H, et al: Alternatives to hazard ratios for comparing the efficacy or safety of therapies in noninferiority studies. Ann Intern Med 163:127-34, 2015
- Trinquart L, Jacot J, Conner SC, et al: Comparison of Treatment Effects Measured by the Hazard Ratio and by the Ratio of Restricted Mean Survival Times in Oncology Randomized Controlled Trials. J Clin Oncol: online ahead of print, 2016. http://doi.org/10.1200/JCO.2015.64.2488
- Tian L, Zhao L, & Wei LJ. Predicting the restricted mean event time with the subject’s baseline covariates in survival analysis. Biostatistics, 15(2), 222–233, 2014.
- Claggett, B., Packer, M., McMurray, J.J., Swedberg, K., Rouleau, J., Zile, M.R., Jhund, P., Lefkowitz, M., Shi, V. and Solomon, S.D., 2015. Estimating the Long-Term Treatment Benefits of Sacubitril–Valsartan. New England Journal of Medicine, 373(23), pp.2289-2290.
