Beyond the hazard ratio
What question does this help answer?
How much longer, in time rather than in ratio, and whether the benefit is constant, delayed, or early and then lost.
Restricted mean survival time
The area under the survival curve up to a chosen horizon: the average event-free time over, say, five years. The difference in restricted means between arms is the extra event-free months the treatment buys. It requires no proportional hazards assumption, it is estimable from the curves themselves, and it is in units a patient can use. The horizon must be prespecified, since choosing it after seeing the curves is a choice about the answer.
The curves RMST measures the area under
Other approaches
- Milestone survival: the proportion event-free at a fixed time, simple and interpretable, though it ignores everything happening between times.
- Time-varying coefficients in a Cox model: let the hazard ratio change with time and report it by period.
- Flexible parametric models: smooth the baseline cumulative hazard with splines, which allows the hazard function itself to be plotted and non-proportional effects to be modelled directly.
- Accelerated failure time models: model the effect as stretching or shrinking the time scale, which some find more interpretable than a hazard ratio.
- Weighted log-rank tests: more sensitive to a delayed effect than the ordinary log-rank, and they must be chosen in advance for the pattern expected.
Check the assumption, and plan for its failure
Look at the log-minus-log plot, the scaled Schoenfeld residuals and the shape of the curves. Delayed effects are now common enough, particularly with immunotherapies, that the analysis plan should say what will be done if hazards are non-proportional rather than discovering the problem afterwards.
Why Mallard may recommend it
Mallard may name restricted mean survival time or a time-varying analysis when the description suggests a delayed or waning effect, and will not present a single hazard ratio as the whole answer when the assumption behind it is doubtful.
Weibull and accelerated failure time models
A Weibull model is a parametric survival model: it assumes the hazard changes steadily over time, rising, falling or staying constant, according to a single shape parameter. It is unusual in being both a proportional hazards model and an accelerated failure time model, so the same fit can be reported as a hazard ratio or as a time ratio. A time ratio of 1.5 means event times are stretched by half in the exposed group: the typical time to the event is 50 percent longer. Because it describes the whole survival curve it uses the data efficiently, but only if the assumed shape is right. Check it against the data, for example by plotting the log cumulative hazard against log time, which should be roughly a straight line; where the hazard rises and then falls, a Weibull model fits badly and a flexible parametric model is the better choice. A curve extended beyond the observed follow-up is an assumption, not an estimate.
What to watch out for
A single hazard ratio assumes the effect is constant over the whole follow-up, and when the curves cross or converge that one number can describe no period of the study. Worse, it remains estimable and looks perfectly reasonable when the assumption fails, so nothing in the output announces the problem. And a hazard ratio is a relative quantity on a scale most clinicians do not think in: it cannot tell a patient how much longer they might live.
Clinical example
Question: Does a therapy with a delayed effect improve survival? Analysis: Difference in restricted mean survival time at a horizon fixed in the protocol, alongside the Kaplan-Meier curves, with the proportional hazards assumption examined and the hazard ratio reported by period rather than as one number.
- Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects (opens in a new tab) · Stat Med · doi 10.1002/sim.1203
- Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome (opens in a new tab) · BMC Med Res Methodol · doi 10.1186/1471-2288-13-152
- The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis (opens in a new tab) · Stat Med · doi 10.1002/sim.4780111409