Propensity Score

Propensity score adjustment is a method of adjusting for all covariates in an observational case-control study, using scalar matching. In case-control studies, the goal is typically to determine if one group (cases) of subjects has a different outcome than another group (controls). These groups might be defined by which treatment they received or which factor they were exposed to, and the purpose of the study is to determine if the treatments or factors result in different outcomes in the cases than in the controls. For example, we can observe people who smoke (cases) and people who don't smoke (controls) and compare the rates of cancer between the two groups. Because people cannot ethically be assigned to one or the other condition (smoking or nonsmoking) we have to accept the groupings that exist. However, because random assignment to condition was not used, the two groups very likely differ on other factors, which can introduce bias into the study. Historically epidemiologists have dealt with this issue by matching subjects in the case group with subjects in control group based on observed covariates, for example, age category and gender, to attempt to remove the influence of these factors on the outcome by equalizing their distribution in the two groups. However, matching can be performed on only a limited number of covariates before the sample size within each matching group becomes too small for statistical analysis. For instance, if we divided age into four categories (e.g., < 30, 31–50, 51–70, ≥ 71), then matching on age and gender would divide the sample into eight separate subgroups. In addition, by dividing a continuous variable (age) into categories, we are losing some of the information contained in the variable.

Propensity score adjustment overcomes this limitation using scalar matching as follows. Let the data measured on subjects be classified into three sets of variables: X is the set of all covariates to be adjusted for, Y is the group membership (case or control, smoker or nonsmoker in this example), and Z is the outcome (cancer or no cancer in this example). Each subject [Page 847]observed has the set of variables (X, Y, Z) measured on them. Note that the covariates X maybecontinuous, categorical, or both, and the outcome variable Z may also be either categorical (e.g., develop cancer or not) or continuous (e.g., number of pounds lost).

In the simplest propensity-score-matching approach, a logistic regression model is fit using all the covariates in X to predict the group membership Y. Note that this analysis excludes the use of the outcome Z in the model fitting. The logistic regression model assigns each subject a predicted log-odds value for belonging to the smoking group (case), whether they smoked or not. Matching cases with controls can then be done based on the log-odds of smoking number (the scalar) calculated for each subject. Matching cases with controls who have a similar log-odds of smoking value results in an adjustment for all the covariates in X.

After matching cases with controls, a comparison of the outcome Z in the matched sets is done using standard statistical methods. If matching is done on a one-to-one basis, with a categorical outcome, as in the smoking example, a paired test of proportion (McNemar's test) can be used to determine if death is more likely to occur in the smokers than in the nonsmokers). Stratified matching can also be done by grouping subjects according to the distribution of the log-odds scalar. For instance, to form five strata in the smoking study, the first group would be the cases and controls whose log-odds of smoking are in the lowest 20th percentile, the next group in the 21st to 40th percentile, and so on. The analysis of the outcome can then be performed on each subgroup separately, using Fisher's exact test, or over all the subgroups, using the Mantel-Haenzel test.

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