Signature
t_T = TR * t_C
| Inputs | Definition | Unit |
|---|---|---|
TR | Time ratio of treatment to reference from the accelerated failure time model | ratio, no unit |
t_C | Time by which the reference group reaches a stated survival proportion, for example the median | time, for example months |
t_T | Time by which the treated group reaches the same survival proportion as the reference group at t_C | time, same unit as t_C |
|---|
Function
Accelerated failure time function
Models the logarithm of the time to an event as a linear function of covariates plus a scaled error term whose distribution defines the model, such as Weibull, log-logistic or log-normal. A covariate multiplies time itself, so the survival curve of the treated group is the reference curve stretched or shrunk along the time axis: S(t|x = 1) equals S_0(t / TR), where TR is the time ratio.
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Implementations
Excel
Treated percentile time from named cells
Excel multiplies the reference time at the chosen percentile by the time ratio.
=TimeRatio*ReferenceTime
Assumptions
Time ratio holds at the chosen percentile
The model's constant time ratio implies the same multiplication at every percentile, so S_T(t_T) equals S_C(t_C). Mean survival is multiplied by the same factor; a restricted mean over a fixed horizon is not.
Worked examples
Median survival with a time ratio of 1.5
A reference median of 12 months and a time ratio of 1.5 give a treated median of 18 months, as in the article. Equivalently, S(18 | x = 1) equals S_0(12), which is 0.5.
TR = 1.5; t_C = 12; t_T = 18
Common errors
Scaling the restricted mean or life-years within a horizon
Life-years accrued within a fixed horizon are an area under the curve truncated at that horizon, so they do not scale by the time ratio. Multiplying modelled life-years by 1.5 overstates the gain.
Multiplying survival probabilities by the time ratio
The time ratio acts on the time axis, not on the probability axis. Multiplying a survival probability of 0.60 by 1.5 gives 0.90, which has no meaning under the model.
Sources
Stretching the survival curve along the time axis
Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436. Section on accelerated failure time models (the proportion event-free in one group at t_1 equals that in the other at a constant multiple of t_1).
Canonical Identity
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