Download Propensity Score Weighting

Introduction to
Propensity Score Weighting
Weimiao Fan
10/10/2009
1
Background
Propensity Score Analysis (PSA) is used to adjust the confounding
effect when studying the treatment effect in observational studies.
Propensity score is the conditional probability of being in the
treatment group given a set of covariates.
There are four primary usages/methods of propensity scores
(Posner& Ash ):
1. Random selection within strata.
2. Matching on the propensity score.
3. Regression adjustment.
4. Propensity score weighting.
We are focusing on the weighting methods for this study.
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Propensity Score Weighting
• Weighted estimators (Lunceford, 2004; Robins, 1994) using
propensity score can be used to adjust the sample selection bias.
• Weighting is based in the inverse of the propensity score. Such
technique is referred to as ‘IPW’ , which denotes ‘inverse probablity
weighting’.
• Specificially, the weights are defined as:
Wt=1/ps for treatment group
Wc=1/(1-ps) for control group
• This means to give more weight to observations with lower
propensity scores for treatment group; and give more weight to
observations with higher propensity scores for control group.
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IPW Estimator
• Theoretically, the IPW estimator produces an unbiased estimate of
the true treatment effect.
• Suppose Y is the outcome variable, T is the treatement indicator, p(x,T)
is the propensity score, is the outcome for treated group.
YT
YT
YT
E(
)=E E
|X]] = E E
|X, T = 1]P(T = 1|X)]
p(x, T)
p(x, T)
p(x, T)
Yt
=EE
|X]p(x, t)] = E E Yt |X]] = E[Yt ]
p(x, t)
• Similar conclusion is obtained for the control group.
• This suggests that IPW approach can be used to adjust the sample
selection bias and obtain the true treatment effect.
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The IPW Estimators in Literature
There are different methods to estimate the average causal effects
(E(Yt)-E(Yc)) in the literature.
1. Rosenbaum and others (1998) proposed the following weighted
estimators of the average causal effects. IPW2 is sometimes known
as a ratio estimator in sampling literature, which normalizes the
weights so thatnthey add upnto 1 in each treatment group.
IPW1 = n−1
i=1
n
IPW2 =
i=1
Ti Yi
− n−1
psi
Ti
psi
−1
n
i=1
i=1
(1 − Ti )Yi
1 − psi
Ti Yi
–
psi
n
i=1
1 − Ti
1 − psi
−1
n
i=1
(1 − Ti )Yi
1 − psi
2. Another estimator with ‘double robustness’ is created by Robins,
Rotnitzky, and Zhao (1994).
n
−1
IPWDR = n
i=1
Ti Yi − (Ti − psi )m1 X i , β1
psi
n
−1
−n
i=1
(1 − Ti )Yi − (Ti − psi )m0 Xi , β0
1 − psi
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Weighting Within Strata and
Proportional Weighting Within Strata
•
•
One drawback of Inverse probability weighting approach is that it’s
very sensitive to extreme values. For an observation with extremely
small propensity score, the weight will be extremely large so that it
will be very influential to the estimate.
Posner and Ash proposed weighting within strata and proportional
weighting within strata as alternative weighting methods.
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Application – in progress
• The IPW weighting method is applied to the
birthweight data to determine the effect of
smoking.
• Some preliminary results of the smoke effect
are shown below
ipw1= -0.6542768
ipw2= -0.5678549
While loess-based estimate using loess.psa gives
an estimate of -0.5434
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References
•
Model selection, confounder control, and marginal structural models: Review
and new applications Joffe, MM
AMERICAN STATISTICIAN Volume: 58 Issue: 4 Pages: 272-279 Published: NOV 2004
•
A comparison of propensity score methods: A case-study estimating the
effectiveness of post-AMI statin use Austin PC, Mamdani MM
STATISTICS IN MEDICINE Volume: 25 Issue: 12 Pages: 2084-2106 Published:
JUN 30 2006
•
A Generalization of Sampling Without Replacement From a Finite Universe D. G.
Horvitz and D. J. Thompson Journal of the American Statistical Association, Vol. 47,
No. 260 (Dec., 1952), pp. 663-685
•
Estimation of Regression Coefficients When Some Regressors Are Not Always
Observed James M. Robins, Andrea Rotnitzky and Lue Ping Zhao Journal of the
American Statistical Association, Vol. 89, No. 427 (Sep., 1994), pp. 846-866
•
Rosenbaum PR. Propensity Score. In Encyclopida of Biostatistics, Armitage P,
Colton T (eds), vol. 5. Wiley: New York, 1998; 3551-3555
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