Adjusted survival curves with inverse probability weights
Introduction
In observational studies without random assignment of treatment, the unadjusted Kaplan–Meier survival (1-probability of death) curves may be misleading due to confounding. For example, if patients with better prognostic values were more likely to be assigned to the new treatment at baseline, then a higher survival curve could be found in the treated group than in the untreated group, even if treatment were not efficacious. A common approach to deal with this non-comparability problem is to display a separate pair of survival curves for each level of disease severity. Thus, only treated and untreated subjects with the same level of prognostic values at baseline would be compared. However, adjustment based on stratification of survival curves becomes unfeasible when the number of baseline covariates is large or when any of them are continuous. Note that the analytic issue motivating this problem is confounding; therefore an adequate graphic that correctly summarizes over subgroups is needed. In contrast, if the issue at hand were effect measure modification (i.e. statistical interaction), then summarizing over subgroups may not be appropriate.
Several proposed alternative methods to produce adjusted survival curves suffer from numerous shortcomings, including constraining the adjusted survival curves to be proportional, computation difficulty with numerous or time-varying covariates, and not allowing adjustment for continuous covariates [1]. Below, we describe a method based on inverse probability weights (IPW) that overcomes the above-mentioned problems and is easily implemented with standard software (e.g. SAS).
Section snippets
Method
It is well known that standardization is a general alternative to stratification-based methods of adjustment for covariates. In the simplest case with one dichotomous baseline covariate, it can be shown that direct standardization (to the combined study population) is equivalent to Robin’s non-parametric g-computation algorithm [2], which in turn is equivalent to non-parametrically estimated IPW [3]. Specifically, given n subjects with m exposed, of whom a are diseased and c are non-diseased,
Example
Table 1 provides the data from a study comparing disease-free survival in 76 Ewing’s sarcoma patients, 47 of whom received a novel treatment (S4) while 29 received one of three (S1–S3) standard treatments [8]. Panel A of Fig. 1 displays the unadjusted survival curves for treatment S4 versus S1–S3 combined. The estimated hazard (i.e. instantaneous risk) ratio comparing treatment S4 against S1–S3 from a Cox model was 0.5 (95% confidence interval: 0.3, 1.0; P=0.03). The unadjusted analysis
Acknowledgements
We thank Drs. Paul Allison, Sander Greenland, Alvaro Muñoz, James Robins and Patrick Tarwater for constructive suggestions. Portions of this paper were presented at the 35th annual meeting of the Society for Epidemiologic Research. Dr. Cole was partly supported by the National Institute of Allergy and Infectious Diseases by means of the data coordinating centers for the Multicenter AIDS Cohort (U01-AI-35043) and Women’s Interagency HIV studies (U01-AI-42590). Dr. Hernán was partly supported by
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