In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment.
Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983, defining the propensity score as the conditional probability of a unit (e.g., person, classroom, school) being assigned to the treatment, given a set of observed covariates.
Unfortunately, for observational studies, the assignment of treatments to research subjects is typically not random.
By having units with similar propensity scores in both treatment and control, such confounding is reduced.
An observational study is required since it is unethical to randomly assign people to the treatment 'smoking.'
Check that covariates are balanced across treatment and comparison groups within strata of the propensity score.
Estimate effects based on new sample The basic case[1] is of two treatments (numbered 1 and 0), with N independent and identically distributed random variables subjects.
Let some subject have a vector of covariates X (i.e.: conditionally unconfounded), and some potential outcomes r0 and r1 under control and treatment, respectively.
A propensity score is the conditional probability of a unit (e.g., person, classroom, school) being assigned to a particular treatment, given a set of observed covariates.
Propensity scores are used to reduce confounding by equating groups based on these covariates.
Suppose that we have a binary treatment indicator Z, a response variable r, and background observed covariates X.
The propensity score is defined as the conditional probability of treatment given background variables: In the context of causal inference and survey methodology, propensity scores are estimated (via methods such as logistic regression, random forests, or others), using some set of covariates.
The following were first presented, and proven, by Rosenbaum and Rubin in 1983:[1] If we think of the value of Z as a parameter of the population that impacts the distribution of X then the balancing score serves as a sufficient statistic for Z.
Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if thinking of Z as a parameter of X. Lastly, if treatment assignment Z is strongly ignorable given X then the propensity score is a minimal sufficient statistic for the joint distribution of
Judea Pearl has shown that there exists a simple graphical test, called the back-door criterion, which detects the presence of confounding variables.
To estimate the effect of treatment, the background variables X must block all back-door paths in the graph.
Like other matching procedures, PSM estimates an average treatment effect from observational data.
The key advantages of PSM were, at the time of its introduction, that by using a linear combination of covariates for a single score, it balances treatment and control groups on a large number of covariates without losing a large number of observations.
Factors that affect assignment to treatment and outcome but that cannot be observed cannot be accounted for in the matching procedure.
[5] Another issue is that PSM requires large samples, with substantial overlap between treatment and control groups.
Similarly, Pearl has argued that bias reduction can only be assured (asymptotically) by modelling the qualitative causal relationships between treatment, outcome, observed and unobserved covariates.
[6] Confounding occurs when the experimenter is unable to control for alternative, non-causal explanations for an observed relationship between independent and dependent variables.