Signature
H_t = -log(S_t); F_t = 1 - S_t
| Inputs | Definition | Unit |
|---|---|---|
S_t | Probability of being event free beyond time t, read from a fitted or estimated survival curve, above 0 and up to 1 | probability |
H_t | Accumulated hazard from the time origin to t, zero or above with no upper limit | none |
|---|---|---|
F_t | Probability of having had the event by time t; the cumulative incidence when no other event competes | probability |
Function
Cumulative hazard function linking the hazard, survival and cycle probabilities
Maps a hazard function over follow-up to its running total from the time origin, the cumulative hazard H(t). Survival is the exponential of minus H(t), so H(t) can be read off a survival curve and turned back into survival, and the rise in H(t) across a model cycle gives the probability of the event in that cycle for people event free at its start. On the log scale a Weibull cumulative hazard is a straight line in log time, which is the basis of the log-cumulative hazard plot. The notation follows the Cumulative Hazard article.
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Implementations
Excel
Cumulative hazard and event probability from a named survival cell
With the survival probability in a cell named SurvProb, the first formula returns the cumulative hazard, held in a cell named CumHaz, and the second the probability of having had the event. LN is the natural logarithm.
=-LN(SurvProb); =1-SurvProb
Assumptions
Single event type for the cumulative hazard and survival link
S_t is the survival function for one event type, so H_t equals minus log S_t exactly. With competing events, minus log of an all-cause survival gives the all-cause cumulative hazard, and the cause-specific cumulative hazard does not convert to a cause-specific probability through 1 minus exp(minus H).
Survival estimated with non-informative censoring
When S_t comes from censored trial data, censored patients are assumed to have the same risk of the event as those still followed, the condition TSD 14 states for the standard survival methods.
Worked examples
Cumulative hazard of 0.8 from four-year survival of 0.4493
In the article's illustrative Weibull example, survival to year 4 is 0.4493, so the cumulative hazard is about 0.8001 (exactly 0.8 before survival was rounded) while the probability of having died by year 4 is 0.5507.
S_t = 0.4493; H_t = 0.8001; F_t = 0.5507
Cumulative hazard above 1 at five-year survival of 0.3269
At year 5 survival is 0.3269 and the cumulative hazard about 1.118: a cumulative hazard, not a probability of 112%. The probability of having died is 0.6731.
S_t = 0.3269; H_t = 1.1181; F_t = 0.6731
Cumulative hazard of 0.5 against an event probability of 0.39
A survival probability of 0.6065 gives a cumulative hazard of about 0.5, while the probability of having had the event is about 0.39, the gap the article warns misleads quietly.
S_t = 0.6065; H_t = 0.5001; F_t = 0.3935
Common errors
Reading the cumulative hazard as the proportion who have had the event
A cumulative hazard of 1.118 is not a 112% probability, and one of 0.5 corresponds to an event probability of about 0.39, not 0.5. Treating H_t as a probability overstates the share who have had the event, more so as H_t grows.
Base-10 logarithm in place of the natural logarithm
Excel's LOG function is base 10. Minus LOG(0.4493) gives about 0.3475 instead of the cumulative hazard of 0.8000, and every later conversion through EXP is then wrong (computed here for illustration).
Sources
Log cumulative hazard as log of minus log survival in TSD 21
Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020. Section 3.2.2, equation 1: log[H(t)] = log[minus log[S(t)]], so the cumulative hazard is minus the log of the survival function.
Log cumulative hazard scale in the flexsurv documentation
Jackson C. flexsurv: Flexible parametric survival and multi-state models. R package version 2.3. Help page flexsurvspline. Details: the log(minus log) link g(S(t,z)) = log(minus log(S(t,z))) = log(H(t,z)), the log cumulative hazard.
Non-informative censoring for standard survival methods
Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2011, last updated March 2013. Section 2: the standard survival analysis methods are suitable only if censoring is uninformative.
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