Propensity Score Methods for Causal Inference
John Pura
BIOS790
October 2, 2015
John PuraBIOS790
Propensity Score Methods for Causal Inference
Causal inference
Philosophical problem, statistical solution
Important in various disciplines (e.g. Koch’s postulates,
Bradford Hill criteria, Granger causality)
Good reference on history of causal inference: Paul Holland
“Statistics and Causal Inference” JASA, 1986
John PuraBIOS790
Propensity Score Methods for Causal Inference
What can we estimate?
Potential Outcomes Framework (Rubin’s Causal Model)
Notation:
Z (1=treated, 0=control), baseline covariates X = (X1 , ..., Xp ),
outcome Y
potential outcomes Y0 , Y1
We observe (Z , Y , X) for an individual
Y = ZY (1) + (1 ≠ Z )Y (0)
Causal effect of treatment: Y (1) ≠ Y (0)
Average causal effect:
= E [Y (1) ≠ Y (0)]
John PuraBIOS790
Propensity Score Methods for Causal Inference
What can we estimate?
Average causal effect
ACEAll or ATE = E [Y (1) ≠ Y (0)])
ACEExp or ATT = E [Y (1) ≠ Y (0)|Z = 1])
ACEUn or ATU = E [Y (1) ≠ Y (0)|Z = 0])
Estimand and statistical methods depends on the study
goal/question
John PuraBIOS790
Propensity Score Methods for Causal Inference
Assumptions
1. Z precedes Y
2. Stable Unit Treatment Value Assumption (SUTVA)
non-interference
no variation in treatment
3. Strongly Ignorable Treatment Assigment (SITA)
0 < P(Z = 1|X) < 1 (this is the propensity score)
(Y (0), Y (1)) ‹ Z |X (very strong assumption)
no unobserved confounders
John PuraBIOS790
Propensity Score Methods for Causal Inference
Randomized Controlled Trials vs. Observational Studies
RCTs
Treatment effects on outcome considered as causal
Z is determined for each participant at random,
(Y (0), Y (1)) ‹ Z
E [(Y |Z = 1) ≠ (Y |Z = 0)] is unbiased estimate of
ATT = ATE
Observational Study
Z is not controlled, (Y (0), Y (1)) ”‹ Z
E (Y |Z = 1) = E (Y (1)|Z = 1) ”= E (Y (1)). Cannot obtain
unbiased estimate by direct comparison. But...
John PuraBIOS790
Propensity Score Methods for Causal Inference
Potential Solution
In observational studies, assuming SITA assumption is met then
treatment assignment, Z, among individuals with particular X
is essentially random and independent of potential outcomes
Rosenbaum and Rubin (1983) - conditioning on the propensity
score (PS) we can identify E (Y (0)) and E (Y (1)) from the
observed data (Z , Y , X) and ultimately estimate .
John PuraBIOS790
Propensity Score Methods for Causal Inference
Propensity Score
Austin, 2011: “The propensity score is a balancing score:
conditional on the propensity score, the distribution of observed
baseline covariates will be similar between treated and
untreated subjects”
This is a large sample property
Unknown in practice, but can be estimated from the data,
given some assumptions on e(X) (e.g. parametric regression
model, CRTs) .
Mathematically: e(X) = P(Z = 1|X). R&R showed that
X ‹ Z |e(X) and in addition to the SITA assumption,
(Y (0), Y (1)) ‹ Z |e(X).
For theoretical properties see R&R (1983) and Lunceford and
Davidian (2004)
John PuraBIOS790
Propensity Score Methods for Causal Inference
The Propensity Score Model
Goal: Covariate balance
Popular method for estimating PS is logistic regression, though
others exist (e.g. tree-based methods, random forests, neural
networks, etc.)
Regress logit[P(Z = 1|X)] on X and obtain predicted
probabilities (ê(X))
R&R (1984) and Austin 2011 describe an iterative approach:
1. Specify an initial model to estimate ê(X)
2. Perform diagnostics to assess covariate balance for each
treatment
3. Modify PS by adding covariates, interactions, or using
non-linear terms
4. Important: Each step should not be motivated by statistical
significance but by objective
John PuraBIOS790
Propensity Score Methods for Causal Inference
The Propensity Score Model
Goal: Covariate balance
What covariates do we include?
Selection driven by subject-matter knowledge
Only baseline variables
Include all confounders and possible non-linear transformations
(e.g. interactions). Overfitting generally not an issue (unless
treatment is uncommon)
Always include variables that affect the outcome even if they
don’t affect treatment assignment (Brookhart et al. (2006))
John PuraBIOS790
Propensity Score Methods for Causal Inference
Diagnostics
How do we know the PS model has been adequately specified?
Assess standardized differences of each covariate between
treatment groups (very useful)
Assess PS distributions by treatment (need common support
condition)
Compare distributions of the covariates between treatments
Varies with PS method
Difficult in practice with high dimensional data
Assess the sensitivity of study conclusions to the SITA
assumption.
John PuraBIOS790
Propensity Score Methods for Causal Inference
Methods utilizing PS
Matching
Stratification
Inverse PS weighting
Covariate adjustment by PS
PS methods allow for estimation of the marginal treatment
effect.
The first three separate the design of the study from the
analysis of the study.
Can do subsequent regression adjustment to eliminate residual
imbalance in prognostically important covariates after first
three PS methods
John PuraBIOS790
Propensity Score Methods for Causal Inference
Matching
Simple formulation for ATT
For each treated subject, select single untreated subject
(without replacement) with same value of ê(X) or its logit
(R&R, 1985)
Take difference of outcomes for the matched pair and average
over all matched pairs
Calculating ATE and ATU require slightly different sampling,
possibly with replacement
Advantage: Eliminates large proportion of systematic
differences in baseline characteristics between treated and
untreated subjects
Disadvantage: Inexact matching may lead to bias. Unmatched
individuals are discarded, leading to loss in statistical power.
Discarding individuals may also alter our estimand (Hill, 2008)
John PuraBIOS790
Propensity Score Methods for Causal Inference
Stratification
Easily estimate ATT:
Create quantiles (e.g. quintiles) of the PS values, thereby
dividing the subjects into equal-sized strata
Within each stratum estimate treatment effect
Calculate weighted average of within-strata estimates of
treatment effect. Weight of each stratum is simply the percent
of the quantile
Estimating ATE and ATU require weighting by fraction of
treated or untreated individuals, respectively, per stratum
Advantage: Easy to construct and estimate causal effects.
Disadvantage: Small number of strata may result in residual
confounding within the strata, resulting in bias. ATT estimates
largely biased (compared to weighting)
John PuraBIOS790
Propensity Score Methods for Causal Inference
Inverse weighting
Weighted linear regression of outcome on treatment where
w=
Z
1≠Z
+
w1
w2
For ATE, w1 = e(X), w2 = 1 ≠ e(X); for ATT, w1 = 1,
e(X)
w2 = 1≠e(X)
; For ATU, w1 = 1≠e(X)
e(X) , w2 = 1. (Morgan &
Todd, 2008)
Advantage: Uses all available data; Can deal with more
complex non-linear link functions (e.g. odds ratio); generally
less biased than stratification (Lunceford & Davidian, 2004)
Disadvantage: An individual with PS close to 0 or 1 will have
unstable weights, leading to potentially spurious treatment
effects with high variance and wide CIs.
John PuraBIOS790
Propensity Score Methods for Causal Inference
Covariate adjustment using PS
Fit model: E (Y |Z , X) = – + —Z + “f (e(X)) (may include
interaction of Z and e(X))
Can obtain ATE, ATT, and ATU by evaluating — at different
values of ê(X)
Advantage: Allows for flexible relationship between PS and
outcome (e.g. use of splines for PS)
Disadvantage: Sensitive to whether PS has been accurately
estimated. Analyst may be tempted to work toward desired or
anticipated result, given that outcome is in sight.
John PuraBIOS790
Propensity Score Methods for Causal Inference
Final Thoughts
PS methods can be done without reference to outcome i.e. separate study design from analysis
Balance of covariates can be easily checked
PS methods more robust to model misspecification compared
to traditional outcome regression (all we care about is balance)
Measures a different quantity, namely, the marginal/population
treatment effect (vs. conditional/individual treatment effect in
traditional regression)
Important to distinguish the two in relation to study goals
Omitted variable bias affects internal validity of both
approaches similarly
Strategy so far is to balance covariates. Another idea is to find
an “instrument”" S that is randomly assigned and affects Y
only through Z
John PuraBIOS790
Propensity Score Methods for Causal Inference