Propensity Score Matching:
A Primer for Educational
Researchers
Forrest Lane, Ph.D.
Department of Educational Studies & Research
Aims
• Recognize the implications for self-selection and
non-randomization in quasi-experimental
research,
• Understand key terms and theory behind the
propensity score matching,
• Identify strategies and resources for
implementing propensity score matching into
research.
Overview
• Theoretical Framework
• Propensity Score Matching Process
• Implications & Practical Guidance
Introduction
Experimental design has historically been
considered the “gold standard” for causal inference
(West, 2009).
Introduction
The problem is that experimental design may not be
possible in practice
There are many ethical, political, or financial arguments
against them (Cook, 2002). Some suggest experimental
designs:
– Can rarely be mounted in schools
– Sacrifice internal for external validity
– Creates a rational
– decision-making model that does not describe how
schools actually make decisions
Introduction
“Interventions conducted under laboratory
conditions with selective participant criteria do not
necessarily generalized well in real world of human
services” (Levant & Hasan, 2008, p. 658).
Quasi-Experiment Alternative
Allow for group comparisons but do not allow for
causal inferences
Groups may systematically differ from one another
based on number of covariates and therefore
cannot be directly compared.
– Non-randomized studies may lead to effect size bias
when interpreting treatment effects.
Problem
Increasing calls for evidence of a program’s or
intervention’s effectiveness.
– Psychology: Bauer (2007); Collins, Leffingwell, &
Belar (2007); Levant & Hasan (2008)
– Education: Rudd & Johnson (2008); Slavin (2002)
Quasi-experiments may not meet this aim
Experimental
Quasi-Experimental
• Better estimates of
treatment effects with
limited generalizability
• Biased estimates of
treatment effects with
greater generalizability
Counterfactuals
• Is a conceptual framework for investigating
causality.
• Two well-known frameworks include the
approaches taken by Campbell (1957) and Rubin
(1974; 2005)
DIMENSION
CAMPBELL
RUBIN
Domain
Psychology, Education
Medicine, Economics
Outcome Definition
Constructs
Operations
Key Feature
Threats to Validity
Precise Assumptions
Approach
Inductive, Scientific
Deductive, Mathematical
Primary Methods
Prevention of Threat
Assumption Checking,
Sensitivity Analysis
Causal Effect Estimate
Direction Only
Exact Magnitude
Role of Measurement
Strong Emphasis
Less Emphasis
*Table taken from West and Thoemmes (2010)
Propensity Score Matching
• Propensity score matching (PSM) is a statistical
technique that aims to controls for self-selection
bias and thus extend causal inference into nonrandomized or quasi-experimental studies
(Rosenbaum & Rubin, 1983).
• Grounded in the Rubin (1794; 2005)
counterfactual framework.
Propensity Score Matching
• The method uses statistical techniques to reduce
differences in the likelihood of group assignment
by matching participants on their likelihood of
group assignment.
• PSM assumes, once groups are well matched,
systematic differences between groups have been
removed and causal inference can be extended.
Propensity Score Matching
“For more than two decades, advanced statistical
methods known as propensity score (PS)
techniques, have been available to aid in the
evaluation of cause-effect hypotheses in
observational studies. None the less, PS techniques
have not yet been used widely in psychological
research” (Harder, Stuart, & Anthony, 2010).
Articles Using PSM
Figure taken from Thoemmes & Kim (2011)
PSM in the Literature
• Grunwald & Mayhew (2008) examined the development of
moral reasoning in young adults and demonstrated a
significant reduction is the overestimation of effects.
• Morgan (2001) used propensity score matching and
demonstrated the effect of private school education on
math and reading achievement is actually larger than
findings in non-matched samples.
• Other similar studies have been demonstrated in
economics (Dehejia & Wahba, 2002), medicine (Schafer &
Kang, 2008), and sociology (Morgan & Harding, 2006).
Defining a Propensity Score
• Defined as the conditional probability of assignment
to a particular treatment or control given a set of
covariates (Rosenbaum & Rubin, 1983).
𝑒𝑖 𝑋𝑖 = 𝑃(𝑇𝑖 = 1|𝑋𝑖 )
Propensity Scores
• Propensity scores incorporate covariates into a
singular scalar variable ranging from 0 to 1
which can then be used to match participants in
treatment groups.
• Once matched, treatments effects should be
more reflective of the true effect and analogous
to interpretation of randomized designs
Propensity Score Matching Process
Estimation/
Modeling
Strategy
Conditioning
Strategy
Balance
Evaluation
Estimation of
Treatment
Effects
Evaluation of
Hidden Bias
PSM Assumptions
• Strongly ignorable treatment assignment
– Assumes all systematic differences in group
assignment have been removed (Rosenbaum, 2010).
– matching techniques control only for systematic
differences due to observable covariates, not
unobservable covariates (Guo & Fraser, 2010)
Random Assignment
• To apply the Rubin counterfactual model, the
assumption of strongly ignorable treatment
assignment must be met.
(𝑌𝑜, 𝑌1 ) ⊥ 𝑇|𝑋
• In other words, conditional on a set of
covariates, the outcome for a participant must be
independent of treatment assignment (Guo &
Fraser, 2010)
Propensity Score Matching Process
Estimation/
Modeling
Strategy
Conditioning
Strategy
Balance
Evaluation
Evaluation of
Treatment
Effects
Post-hoc Test
for Hidden
Bias
Propensity Score Estimation
• The most commonly used method is logistic
regression (Thoemmes & Kim, 2011).
• Other methods include probit regression,
classification trees or ensemble methods such as
bagging, boosted regression trees, and random
forest (Shadish, Luellen, & Clark, 2006).
Modeling Strategy
• Non-Parsimonious
– All theoretically related variables included in PS
estimation
• Parsimonious
– Some variables can be ignored as a source of potential
bias
• Hierarchical Regression
• Stepwise Regression
Conditioning Strategy
• Matching
– One-to-one, One-to-many, Caliper
• Stratification
– stratification across quintiles may reduce
approximately 90% of bias due to covariates
(Shadish, Luellen, & Clark, 2005)
• Regression Adjustment
– The PS may be used as a covariate in ANCOVA
but must meet assumptions of the analysis.
Balance Evaluation
• The standardized difference in the mean
propensity score in the two groups should be
near zero (d < .20)
• The ratio of the variance of the propensity score
and continuous covariates in the two groups
should be near one, preferably between 0.80 and
1.25
Balance Evaluation
• Multivariate Measures
– Hansen and Bowers (2008) provide one test that assesses
simultaneously whether any variable or linear combination
of variables was significantly unbalanced after matching”
using a 𝜒 2 distribution (Thoemmes, 2012, p. 9).
• 𝑑2 𝑧; 𝑥1 , … , 𝑥𝑗 ≔ 𝑑 𝑧, 𝑥1 , … , 𝑑 𝑧, 𝑥𝑗
× 𝐶𝑜𝑣
𝑑(𝑍, 𝑥1 )
(𝑍, 𝑥𝑗 )
×
𝑑(𝑧, 𝑥1 )
𝑑(𝑧, 𝑥𝑗 )
– A measure ℒ, may also be used which assesses the balance
of all covariates including interaction effects (Iacus, King,
& Porro, 2011)
1
• ℒ1 = 2
𝓁1 … 𝓁𝑗 |𝑡𝓁1 … 𝓁𝑘 − 𝐶𝓁1 … 𝓁𝑘 |
Estimating Treatment Effects
• Treatment effects can be estimated on the
outcome variable(s) by testing in newly matched
sample through a t-test or appropriate multigroup equivalent analysis.
Common Support Region
• The shared overlap of between groups on the
distribution of propensity scores
• The common support region defines where the
estimation of causal effects may be inferred.
Hidden Bias
• Two participants measured on the same
covariates (x), should have the same probability
(P) of group assignment.
– When true, the ratio of the probability for group
assignment relative to non-group assignment should
be close to one.
– If false, probability of group assignment differs by a
multiplier or factor of Γ
Hidden Bias
• Rosenbaum (2010) suggested a Wilcoxon signed
rank test may be used to statistically test the
impact of various levels of Γ on the
interpretation of the treatment effect (i.e.,
sensitivity analysis).
Heuristic Scenario
• The content area reading strategies program (CARS)
was implement within Florida schools to improve
basic reading levels skills.
• Students were taught three animal science lessons
from the state approved curriculum and included
anatomy and physiology, nutrition, and reproduction.
– The lessons were taught over the course of 23 school days, or
nearly 1600 minutes of instruction” (Park & Osborne, 2007,
p. 57).
Heuristic Scenario
• The problem is that students could not be
randomly assigned to treatment and comparison
groups.
• Park and Osborne (2007) also suggested student
pre-test scores, grade level, grade point average,
gender, ethnicity, and standardized reading
levels were statistically significant predictors of
agricultural posttest scores (𝑅2 = .67).
Arguments Against ANCOVA
• ANCOVA is inappropriate when differences
between groups on covariates are large (Hinkle,
Wiersma, & Jurs, 2003).
• The outcome variable in ACOVA is an adjusted
score which makes interpretation difficult
• Potential mismatch between the research
question and analytic technique or Type IV error
(Fraas, Newman, & Pool, 2007).
Arguments Against ANCOVA
• The use of ANCOVA and propensity score
matching may result in a different interpretation
of the treatment effect (Fraas, Newman, & Pool,
2007).
Method
• Logistic regression was used to estimate
propensity scores
• One-to-one matching was the conducted using a
caliper width of 0.25 standard deviations of the
logit transformation of the propensity score
(Stuart & Rubin, 2007).
– Matched pairs exceeding the caliper width were
discarded from the analysis.
• Balanced was then examined on continuous
variables using NHST and effect sizes.
Pre-Matching Treatment Effect
N
M
SD
t
df
p
d
Non Participants
16
0.06
0.57
2.231
28
.034
.805
Participants
14
0.64
0.84
Biased
Treatment Effect
0
(0.06)
Comparison
(0.64)
Treatment
1
Likelihood of Receiving Treatment
N
M
SD
t
df
p
d
Non Participants
16
.33
.32
2.989
28
.006
1.12
Participants
14
.62
.24
Amount of Bias
0
Unlikely to be in
treatment group
(.36)
Comparison
(.59)
Treatment
1
Likely to be in the
treatment group
Matching Algorithms
• R
– MatchIt in R (Ho, Imai, King, and Stuart, 2007)
– Matching (Sekhon, 2011)
• Stata
– PSMATCH2 (Leuven & Sianesi, 2004)
– Pscore (Becker & Ichino, 2002)
• SAS
– SUGI 214-26 “GREEDY” (D’Agostino, 1998),
• SPSS
– PSM Matching_2.spd (Thoemmes, 2012)
Control
ID
Propensity
Score
Logit
Score
Treatment
ID
Propensity
Score
Logit
Score
d (Caliper)
2
.453
-0.190
26
.450
-0.200
-0.010
9
.201
-1.380
19
.195
-1.420
-0.030
12
.564
0.260
24
.575
0.300
0.040
11
.497
-0.010
29
.456
-0.180
-0.140
16
.081
-2.430
28
.111
-2.080
0.300
8
.533
0.130
23
.631
0.530
0.340
5
.817
1.500
18
.662
0.670
-0.700
10
.500
0.000
27
.730
0.990
0.850
6
.395
-0.430
21
.750
1.100
1.300
Assessing Balance
• The standardized difference in the mean
propensity score in the two groups should be
near zero (d < .20)
• The ratio of the variance of the propensity score
in the two groups should be near one, preferably
between 0.80 and 1.25 (Rubin, 2001).
Pre-Matching Group Differences
N
M
SD
t
df
p
d
Non Participants
16
.36
.22
2.989
28
.006
1.12
Participants
14
.59
.22
Amount of Bias
0
Unlikely to be in
treatment group
(.36)
Comparison
(.59)
Treatment
1
Likely to be in the
treatment group
Post-Matching Group Differences
N
M
SD
t
df
p
d
Non Participants
7
.44
.24
0.930
12
.930
.05
Participants
7
.46
.25
Amount of Bias
0
Unlikely to be in
treatment group
(.44) (.46)
1
Likely to be in the
treatment group
Pre-Matching Treatment Effect
N
M
SD
t
df
p
d
Non Participants
16
0.06
0.57
2.231
28
.034
.805
Participants
14
0.64
0.84
Biased
Treatment Effect
0
(0.06)
Comparison
(0.64)
Treatment
1
Post-Matching Treatment Effect
N
M
SD
t
df
p
d
Non Participants
7
0.14
0.69
0.630
12
.539
.338
Participants
7
0.43
0.98
Unbiased
Treatment Effect
(0.14)
0
(0.43)
1
Practical Guidance
• Some participants will be discarded as a result of
poor matching.
• As a result, larger samples are generally needed for
PSM (Luellen, Shadish, & Clark, 2005; Yanovitzky,
Zanutto, & Hornik, 2005).
– How many participants are needed is unclear (Luellen et
al., 2005, p. 548).
– N >100 may be too small (Akers, 2010), particularly as
prediction of group assignment improves (Lane, 2011).
Practical Guidance
• Examine improvement in prediction relative to the
null as there is some evidence to suggest this reduces
model sensitivity to hidden bias (Lane, 2011).
– Pearson 𝑥 2 goodness of fit, Hosmer-Lemeshow goodnessof-fit test and pseudo 𝑅2 have also been suggested for use
in evaluating propensity scores (Guo & Fraser, 2010)
– I index (Huberty & Holmes, 1983; Huberty & Lowman,
2000) may also provide a measure of effect size.
Practical Guidance
• Other methods beyond logistic regression are
available when estimating propensity scores
including classification trees, bagging, and
boosted regression trees(Austin, 2008; Shadish
et al., 2006).
• Each of these estimation methods were created
to help better inform covariate selection.
Practical Guidance
• Matching strategies seem to vary greatly in the
literature.
• However, other strategies exist (e.g., one-tomany matching) that may retain more
participants, improving statistical power and
perhaps generalizability of treatment results.
Useful Literature
• Caliendo and Kopeinig (2008) and Stuart (2010)
provide a thorough discussion on the
implementation of different matching methods.
• Thoemmes and Kim (2011) present a systematic
review of the various strategies employed by
social science researchers using PSM.
• Guo and Fraser (2010) provide an entire text
dedicated to propensity score matching.
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