In the statistical analysis of observational data, propensity score matching 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. PSM
attempts to reduce the bias due to confounding variables that could be found in an estimate
of the treatment effect obtained from simply comparing outcomes among units that received
the treatment versus to those that did not. The technique was first published by Paul
Rosenbaum and Donald Rubin in 1983, and implements the Rubin causal model for observational studies.
The possibility of bias arises because the apparent difference in outcome between these
two groups of units may depend on characteristics that affected whether or not a unit received
a given treatment instead of due to the effect of the treatment per se. In randomized experiments,
the randomization enables unbiased estimation of treatment effects; for each covariate,
randomization implies that treatment-groups will be balanced on average, by the law of
large numbers. Unfortunately, for observational studies, the assignment of treatments to research
subjects is, by definition, not randomized. Matching attempts to mimic randomization by
creating a sample of units that received the treatment that is comparable on all observed
covariates to a sample of units that did not receive the treatment.
For example, one may be interested to know the consequences of smoking or the consequences
of going to university. The people 'treated' are simply those—the smokers, or the university
graduates—who undergo in course of everyday life whatever it is that is being studied
by the researcher. In both of these cases it is unfeasible to randomly assign people
to smoking or a university education, so observational studies are required. The treatment effect
estimated by simply comparing a particular outcome—rate of cancer or life time earnings—between
those who smoked and did not smoke or attended university and did not attend university would
be biased by any factors that predict smoking or university attendance, respectively. PSM
attempts to control for these differences to make the groups receiving treatment and
not-treatment more comparable.
Overview PSM is for cases of causal inference and simple
selection bias in non-experimental settings in which: few units in the non-treatment comparison
group are comparable to the treatment units; and selecting a subset of comparison units
similar to the treatment unit is difficult because units must be compared across a high-dimensional
set of pretreatment characteristics. In normal Matching we match on single characteristics
that distinguish treatment and control groups. But If the two groups do not have substantial
overlap, then substantial error may be introduced: E.g., if only the worst cases from the untreated
“comparison” group are compared to only the best cases from the treatment group, the
result may be regression toward the mean which may make the comparison group look better
or worse than reality. PSM employs a predicted probability of group
membership e.g., treatment vs. control group—based on observed predictors, usually obtained from
logistic regression to create a counterfactual group. Also propensity scores may be used
for matching or as covariates—alone or with other matching variables or covariates.
General procedure 1.Run logistic regression:
Dependent variable: Y = 1, if participate; Y = 0, otherwise.
Choose appropriate confounders Obtain propensity score: predicted probability
or log[p/(1 − p)]. 2. Check that propensity score is balanced
across treatment and comparison groups, and check that covariates are balanced across
treatment and comparison groups within strata of the propensity score.
Use standardized differences or graphs to examine distributions
3.Match each participant to one or more nonparticipants on propensity score:
Nearest neighbor matching Caliper matching
Mahalanobis metric matching in conjunction with PSM
Stratification matching Difference-in-differences matching
Exact matching 4. Verify that covariates are balanced across
treatment and comparison groups in the matched or weighted sample
5. Multivariate analysis based on new sample Use analyses appropriate for non-independent
matched samples if more than one nonparticipant is matched to each participant
Note: When you have multiple matches for a single treated observation, it is essential
to use Weighted Least Squares rather than OLS.
Formal definition A propensity score is the probability of a
unit being assigned to a particular treatment given a set of observed covariates. Propensity
scores are used to reduce selection bias by equating groups based on these covariates.
Suppose that we have a binary treatment T, an outcome Y, and background variables X.
The propensity score is defined as the conditional probability of treatment given background
variables:
Let Y(0) and Y(1) denote the potential outcomes under control and treatment, respectively.
Then treatment assignment is unconfounded if potential outcomes are independent of treatment
conditional on background variables X. This can be written compactly as
where denotes statistical independence. If unconfoundedness holds, then
Judea Pearl has shown that existence of a simple graphical criterion called backdoor
path implies the presence of confounding variables. To estimate the effect of treatment, backdoor
paths need to be blocked. This blocking can be done either by adding the confounding variable
as a control in regression, or by matching on the confounding variable.
Advantages and disadvantages 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. If units in the treatment and control were balanced on
a large number of covariates one at a time, large numbers of observations would be needed
to overcome the "dimensionality problem" whereby the introduction of a new balancing covariate
increases the minimum necessary number of observations in the sample geometrically.
One disadvantage of PSM is that it only accounts for observed covariates. Factors that affect
assignment to treatment and outcome but that cannot be observed cannot be accounted for
in the matching procedure. Another issue is that PSM requires large samples, with substantial
overlap between treatment and control groups. Any hidden bias due to latent variables may
remain after matching because the procedure only controls for observed variables.
General concerns with matching have also been raised by Pearl, who has argued that hidden
bias may actually increase because matching on observed variables may unleash bias due
to dormant unobserved confounders. Similarly, Pearl has argued that bias reduction can only
be assured by modelling the qualitative causal relationships between treatment, outcome,
observed and unobserved covariates. Confounding occurs when the experimental controls do not
allow the experimenter to reasonably eliminate plausible alternative explanations for an
observed relationship between independent and dependent variables.
References
External links Introduction to Propensity Score Matching:
A New Device for Program Evaluation. Lecture notes 2004
Implementing Propensity Score Matching Estimators with STATA. Lecture notes 2001
[1]. Methods for Constructing and Assessing Propensity Scores.
See also Heckman correction