Signature
T_bar = (1 - S_1) / (-log(S_1)) + S_1 / (-log(q_2))
| Inputs | Definition | Unit |
|---|---|---|
S_1 | Probability of surviving the first year; above 0 and below 1 | probability |
q_2 | Probability of surviving year 2 given alive at one year, also applied to every later year; above 0 and below 1 | probability |
T_bar | Mean survival in years under the two-piece constant hazard | years |
|---|
Function
Inverse probability of censoring weighting for dependent censoring
Maps patients who remain under observation to weights equal to the inverse of their probability of having remained uncensored, given their covariate history, so that they also represent similar patients who were censored. Weighted versions of the Kaplan-Meier estimator, the Cox model and mean cost estimators then remove the bias that dependent censoring causes, provided every factor predicting both censoring and outcome is measured. The notation follows the Dependent Censoring article and its switchers-censored control arm.
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Implementations
Excel
Two-piece mean survival from named survival probabilities
With SurvYearOne and CondSurvTwo named, the formula returns mean survival, held in MeanSurv.
=(1-SurvYearOne)/(-LN(SurvYearOne))+SurvYearOne/(-LN(CondSurvTwo))
Assumptions
Constant hazard within year 1 and a constant hazard after year 1
Survival is exponential within the first year and from year 1 onwards with the year-2 hazard, an illustrative extrapolation; mean survival is the area under the extrapolated curve.
Year-2 conditional survival free of censoring bias
q_2 estimates what would have happened without switching only if it comes from an analysis that removes the dependence, such as IPCW.
Worked examples
Mean survival without switching or with IPCW
With S_1 of 0.75 and q_2 of 0.78 the hazards are 0.2877 and 0.2485, so mean survival is 0.869 plus 0.75 / 0.2485, about 3.8876 years (3.89 in the article).
S_1 = 0.75; q_2 = 0.78; T_bar = 3.8876
Mean survival with switchers censored
With q_2 of 0.825 the year-2 hazard is 0.1924 and mean survival about 4.7677 years (4.77 in the article), 0.88 life-years more than the target, as in the article.
S_1 = 0.75; q_2 = 0.825; T_bar = 4.7677
Common errors
Extrapolating a hazard biased by dependent censoring
A 3.4-point gap in two-year survival becomes 0.88 life-years once the censored-analysis hazard is extrapolated, inflating control survival and shrinking the incremental QALY gain.
Using one minus a survival probability as the hazard
Taking each hazard as 1 minus the survival probability, 0.25 and 0.22, gives 1.0 plus 3.41, about 4.41 years, instead of 3.89; hazards are minus the log of survival.
Sources
Mean life expectancy as the area under the extrapolated survival curve
Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: Partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2017. Section 1: in partitioned survival analysis state membership comes from non-mutually exclusive survival curves, and the area under the (extrapolated) overall survival curve provides an estimate of mean life expectancy; section 2: analysing quality-adjusted survival with standard survival methods induced informative censoring, as people with worse quality of life were more likely to be censored earlier.
Canonical Identity
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