Propensity Score Matching
The propensity score can be defined as the probability of treatment assignment given baseline covariates. Based on treatment assignment, the average treatment effect is defined as the difference between the potential outcome in the case of treatment versus the absence of treatment.
PSM operates on certain presumptions. The first is the premise of unconfoundedness which states that treatment assignment is independent of outcomes conditional on covariate variables. The second premise is overlap, also known as the common support condition. If these assumptions are met, we might use the average result of comparable people not exposed to the treatment versus people who are not exposed to treatment. The assignment method may be understood as if a perfectly random experiment was conducted among subpopulations of units with the same covariate value.
Propensity scores are used in both randomized and observational studies. In practice, the propensity score is derived by regressing treatment status on baseline variables. While most researchers use logistic regression to estimate the propensity score, once the score is computed, a researcher can use the score to match units across treatment and comparison groups. Once matching is done, one can assess the overlap to check the balance properties across treatment and comparison groups. Once the balancing property is satisfied, one can estimate the average treatment effect on treated (ATT), which is the difference between the mean observed outcome for treated and untreated individuals. The specific steps to execute PSM include:
Step 1: Finalising covariate for PSM
In the first stage, one must select variables/covariates used in the propensity score match model. It is important to note that the model should only include variables that affect both the outcome and participation decision. The researcher should have a solid knowledge of the previous research and information about the institutional setting to guide them in creating the model (Smith and Todd 2005 or Sianesi 2004).
Step 2: Estimating the Propensity Score
Propensity scores may be calculated using various techniques like logistic regression, Mahalanobis distance, etc. (Rosenbaum & Rubin, 1983a; Stuart, 2010; Stuart & Rubin, 2008a). Logistic regression is the most popular method among these for creating propensity scores (Austin, 2011a; Stuart, 2010).
| Box: How to use the propensity score as this will define PSA analysis The propensity score allows simultaneous balance on a wide range of variables between treated and untreated populations. After the propensity scores have been calculated, they can be used in four ways: match on the propensity scored, inverse probability weighting (IPTW), stratification on propensity scoring, and covariate adjustment with the propensity score. Matching and conventional stratification of the propensity score (also known as subclassification) accomplish balance by guaranteeing that the propensity scores of the treated and reference populations are, on average, equal. But weighing approaches use a propensity score function to reweigh the populations and create balance by making a fake population in which the treatment assignment is independent of the observed covariables. Lastly, weighing the propensity score is likely the most adaptable method for using propensity scores in the study, with various accessible modifications allowing for the inference-based targeting of certain groups. In addition to traditional approaches of propensity score weighing that employ inverse probability treatment weights (IPTW) or standardized mortality ratio weights (SMRW), several newer approaches (including propensity score fine stratification weights, matching weights, and overlap weights) have been proposed to address significant limitations of traditional weighing methods. |
Step 3: Select a method and algorithm for matching
Once propensity scores are computed, research can select a matching method for the comparison group. The first decision point is to match either using one-to-one or one-to-many matching.
Another option is to choose between an exact match and an approximate match.
The proposed propensity score matching approaches may be implemented using either a greedy or an optimal matching algorithm (Rosenbaum & Rubin, 1983a). Once a match is created via greedy matching, the matched units cannot be modified, and each pair of matched units is the best available pair. In optimal matching, past matches might be altered before the current match to attain the least or ideal distance. As a result, if the goal is to find well-matched groups, greedy matching may suffice; however, if the goal is to find well-matched couples, optimum matching may be preferable (Stuart, 2010).
Research can opt from several existing matching methods, like Nearest-Neighbor Matching (with or without caliper), Radius Matching, Kernel Matching, and Stratification Matching.
Step 4: Checking overlap/common support
The next step includes checking for overlap/common support. Importantly, considerable overlap in variables between the exposed and unexposed groups is required to draw causal conclusions from our data. We can examine the balance of our covariables with the assistance of a few tools:
- A histogram of the PS for exposed and unexposed groups may be created.
- We may evaluate the standard deviation.
- We may evaluate the decrease in bias.
Step 5: Estimating treatment effect and matching quality
After matching samples and checking for overlap/common support, researchers can compare treated and untreated subjects in the matched sample to determine treatment impact. If the result is continuous, researchers can assess the impact of intervention as the difference between treated and untreated subjects’ mean outcomes (Rosenbaum & Rubin, 1983a) (Rosenbaum & Rubin, 1983a). If the result is dichotomous, the impact may be evaluated as the difference in the matched sample’s proportion of treated vs. untreated participants. Given binary outcomes, intervention effects can alternatively be stated using relative risk (Austin, 2008a, 2010; Rosenbaum & Rubin, 1983a) (Austin, 2008a, 2010; Rosenbaum & Rubin, 1983a). Furthermore, it is important to note that standard errors may be calculated using bootstrap resampling methods.
Step 6: Sensitivity analysis
Propensity score methods analysis is prone to covariate measurement error. Researchers can use existing sensitivity analyses for unobserved confounding like propensity score calibration, VanderWeele, and Arah’s bias formulas, and Rosenbaum’s sensitivity analysis to address this problem.
References :
Austin P.C. The performance of different propensity score methods for estimating relative risks. Journal of Clinical Epidemiology. 2008a;61:537–545. doi:10.1016/j.jclinepi.2007.07.011.
Austin P.C. The performance of different propensity score methods for estimating difference in proportions (risk differences or absolute risk reductions) in observational studies. Statistics in Medicine. 2010;29:2137–2148. doi:10.1002/sim.3854.
Austin P.C. Optimal caliper widths for propensity-score matching when estimating differences in means and differences in proportions in observational studies. Pharmaceutical Statistics. 2011b;10:150–161. doi:10.1002/pst.433.
Rosenbaum P.R., Rubin D.B. The central role of the propensity score in observational studies for causal effects. Biometrika. 1983a;70:41–55.
Rosenbaum P.R., Rubin D.B. Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. Journal of the Royal Statistical Society, Series B. 1983b;45:212–218.
Sianesi, B. (2004): “An Evaluation of the Active Labour Market Programmes in Sweden,” The Review of Economics and Statistics, 86(1), 133–155.
Smith, J., and P. Todd (2005): “Does Matching Overcome LaLonde’s Critique of Nonexperimental Estimators?,” Journal of Econometrics, 125(1-2), 305–353.
Stuart, E. A., & Rubin, D. B. (2008a). Best practices in quasiexperimental designs: Matching methods for causal inferences. in J.W. Osborne (Ed.), Best practices in quantitative methods, pp. 155-176. Los Angeles, CA: SAGE Publications.
Stuart, E. A. (2010). Matching methods for causal inference: A review and a look forward. Statistical Science, 25, 1-21.
Kultar Singh – Chief Executive Officer, Sambodhi
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