See previous: http://econometricsense.blogspot.com/2013/05/selection-bias-and-rubin-causal-model.html

The problem of selection bias can be well characterized within
the Rubin causal model or potential outcomes framework (Angrist and
Pischke,2008; Rubin, 1974; Imbens and Wooldridge, 2009, Klaiber &
Smith,2009). In a previous post I explained how selection bias can overpower the
actual treatment effect and leave the naïve researcher to conclude that the
intervention or treatment was ineffectual or lead them to under or overestimate
the true treatment effects depending on the direction of the bias.

However, according to the conditional
independence assumption (CIA) ( Rubin, 1973; Angrist & Pischke, 2008; Rosenbaum and Rubin, 1983;Angrist and
Hahn,2004) conditional on covariate comparisons may remove selection bias, giving us
the estimate of the treatment effect we need:

E[Y

_{i}|x_{i},d_{i}=1]- E[Y_{i}|x_{i},d_{i}=0]= E[Y_{1i}-Y_{0i}|x_{i}] or Y_{1i},Y_{0i}⊥ d_{i}|*x*_{i}
The last term
implies that treatment assignment ( d

_{i}) and response (Y_{1i},Y_{0i}) are conditionally independent given covariates*x*. This conclusion provides the justification and motivation for utilizing matched comparisons to estimate treatment effects. Matched comparisons imply balance on observed covariates, which ‘recreates’ a situation similar to a randomized experiment where all subjects are essentially the same except for the treatment(Thoemmes and Kim,2011). However, matching on covariates can be complicated and cumbersome. An alternative is to implement matching based on an estimate of the probability of receiving treatment or selection. This probability is referred to as a propensity score. Given estimates of the propensity or probability of receiving treatment, comparisons can then be made between observations matched on propensity scores. This is in effect a two stage process requiring first a specification and estimation of a model used to derive the propensity scores, and then some implementation of matched comparisons made on the basis of the propensity scores. Rosenbaum and Rubin’s propensity score theorem (1983) states that if the CIA holds, then matching or conditioning on propensity scores (denoted p(x_{i}_{i}) ) will also eliminate selection bias, i.e. treatment assignment ( d_{i}) and response (Y_{1i},Y_{0i}) are conditionally independent given propensity scores p(x_{i}):
Y

_{1i},Y_{0i}⊥d_{i}| x_{i}= Y_{1i},Y_{0i}⊥ d_{i}|p(x_{i})
In fact,
propensity score matching can provide a more asymptotically efficient estimator
of treatment effects than covariate matching (Angrist andHahn,2004).

So the idea is to first generate propensity scores by specifying a
model that predicts the probability of receiving treatment given covariates x

_{i}
p(x

_{i}) = p(d_{i}=1|x_{i})
There are many possible functional forms for estimating propensity
scores. Logit and probit models with the binary treatment indicator as the
dependent variable are commonly used. Hirano et. al find that an efficient
estimator can be achieved by weighting by a non-parametrically estimated
propensity score (Hirano, et al, 2003). Millimet and Tchernis find evidence that
more flexible and over specified estimators perform better in propensity score
applications (Millimet and Tchernis , 2009). A comparative study of propensity
score estimators using logistic regression, support vector machines, decision trees,
and boosting algorithms can be found in Westreich et al (Westreich et al ,
2009).

Once these probabilities, or ‘propensity scores’ are generated for
each individual, matching is accomplished by identifying individuals in the
control group with propensity scores similar to those in the treated group.
Types of matching algorithms include 1:1 and nearest neighbor methods. Differences between matched cases are
calculated and then combined to estimate an average treatment effect. Another method that implements matching based
on propensity scores includes stratified comparisons. In this case treatment
and control groups are stratified or divided into groups or categories or bins of propensity scores. Then comparisons
are made across strata and combined to estimate an average treatment effect.
Matched comparisons based on
propensity score strata are discussed in
Rosenbaum and Rubin (1984). This method can remove up to 90% of bias due to
factors related to selection using as few as five strata (Rosenbaum and Rubin,
1984).

**References:**

Angrist, J. D., & Hahn, J. (2004). When to control for
covariates? Panel-Asymptotic Results for Estimates
of Treatment Effects.

*Review of Economics and Statistics*. 86, 58-72.
Angrist, J. D. & Pischke J. (2008).

*Mostly harmless econometrics: An empiricist's companion*. Princeton University Press.
Hirano, K. &
Imbens, G.W. & Ridder, G. (2003).
Efficient estimation of average treatment effects using
the estimated propensity score.

*Econometrica,*Vol. 71, No. 4, 1161–1189.
Klaiber, H.A.
& Smith,V.K. (2009

*). Evaluating Rubin's causal model for measuring the capitalization of environmental amenities.*NBER Working Paper No 14957. National Bureau of Economic Research.
Imbens, G. W. & Wooldridge,
J.M.(2009). Recent developments in the econometrics of program
evaluation.

*Journal of Economic Literature*, 47:1, 5–86
Millimet , D. L. & Tchernis,
R.(2009). On the specification of propensity scores, with applications to the analysis of trade
policies.

*Journal of Business & Economic Statistics,*Vol. 27, No. 3
Rosenbaum , R. &. Rubin,
D.B.(1983). The central role of the propensity score in observational studies for causal effects.

*Biometrika,*Vol. 70, No. 1, pp. 41-55
Rosenbaum , R. &. Rubin,
D.B.(1984). Reducing Bias in Observational Studies Using Sub classification on the Propensity Score.

*Journal of the American Statistical Association,*Vol. 79, Issue. 387, pp.516-524
Rubin, D. B.(1974). Estimating
causal effects of treatments in randomized and nonrandomized studies.

*Journal of Educational Psychology,*Vol 66(5), Oct 1974, 688-701
Rubin, Donald B. (1973).
Matching to remove bias in observational studies.

*Biometrics,*29, 159-83.
Thoemmes, F. J. & Kim, E.
S. (2011). A systematic review of propensity score methods in the social sciences.

*Multivariate Behavioral Research, 46(1)*, 90-118.
Westreich, D. , Justin L., & Funk, M.J. (2010).
Propensity score estimation: machine learning and classification methods as
alternatives to logistic regression.

*Journal of Clinical Epidemiology,*63(8): 826–833.
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