Signature
sw_t = sw_prev * p_num / p_den
| Inputs | Definition | Unit |
|---|---|---|
sw_prev | Stabilised weight carried from earlier intervals, 1 before the first | none |
p_num | Numerator probability of not switching in interval t, conditional on baseline covariates | probability |
p_den | Denominator probability of not switching in interval t, conditional on baseline and time-dependent covariates; above 0 | probability |
sw_t | Stabilised weight for a patient still uncensored and alive through interval t | none |
|---|
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
Stabilised IPCW weight from named cells
With PrevStabWeight, ProbUnswitchedBaseline and ProbUnswitchedFull named, the formula returns the stabilised weight, held in StabWeight.
=PrevStabWeight*ProbUnswitchedBaseline/ProbUnswitchedFull
Assumptions
Numerator model uses baseline covariates only
The numerator conditions on baseline covariates, which then also enter the outcome model; time-dependent covariates appear only in the denominator.
Same switching data for numerator and denominator models
Both probabilities are estimated on the same patients and intervals, so their ratio is 1 when time-dependent covariates add nothing.
Worked examples
Progressed non-switcher with stabilised weights
With no baseline covariates, the numerator is the overall share unswitched, 120 of 150 or 0.8, so a progressed non-switcher gets 0.8 / 0.5, which is 1.6, as in the article.
sw_prev = 1; p_num = 0.8; p_den = 0.5; sw_t = 1.6
Progression-free patient with stabilised weights
A progression-free patient, who cannot switch, gets 0.8 / 1, which is 0.8, as in the article.
sw_prev = 1; p_num = 0.8; p_den = 1; sw_t = 0.8
Common errors
Putting time-dependent covariates in the numerator
If the numerator also conditions on progression it equals the denominator, every weight becomes 1 and the analysis is simple censoring of switchers, 0.619 instead of 0.585 at two years in the example.
Leaving baseline covariates in the numerator out of the outcome model
Stabilised weights leave the baseline part of the censoring mechanism to the outcome model, so any prognostic baseline variables in the numerator must also be included in the weighted outcome model.
Sources
Stabilised weights recommended as more efficient, numerator on baseline covariates
Latimer NR, Abrams KR. NICE DSU Technical Support Document 16: Adjusting survival time estimates in the presence of treatment switching. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2014. Appendix A, IPCW: Robins and Finkelstein (2000) recommend stabilised inverse probability of censoring weights as more efficient; stabilised weights are the probability of having remained uncensored until t given baseline covariates divided by the probability given baseline and time-dependent covariates, and equal 1 for all t if the included prognostic factors do not affect the hazard of censoring (equation A1).
Numerator without time-dependent confounders and baseline variables repeated in the outcome model
Bell Gorrod H, Latimer NR, Abrams KR. NICE DSU Technical Support Document 24: Adjusting survival time estimates in the presence of treatment switching: an update to TSD 16. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2024. Section 4.2.2: the numerator model for stabilised weights must not include any time-dependent confounding variables and typically includes only prognostic baseline variables; any prognostic baseline variables included in the numerator should also be included in the weighted outcomes model; stabilised weights are likely to be less extreme, while unstabilised weights are more intuitive to apply to survival models for extrapolation.
IPCW applied to dependent censoring at stopping or switching therapy
Robins JM, Finkelstein DM. Correcting for noncompliance and dependent censoring in an AIDS clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests. Biometrics. 2000;56(3):779-788. doi:10.1111/j.0006-341x.2000.00779.x (abstract read). Abstract: subjects are regarded as dependently censored at the first time they stop or switch therapy, and the data are analysed with inverse probability of censoring weighted Kaplan-Meier and Cox partial likelihood estimators to adjust for the dependent censoring.
Canonical Identity
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