With the sample divided into strata of units that have similar propensity scores, we should be able to get clean estimates of treatment effects within each block. Aggregating these estimates appropriately gives the blocking estimator, which we illustrate in this post.
To begin, let's continue with the example data set, and use everything we have discussed previously on it:
>>> Y, D, X = vignette_data() >>> causal = CausalModel(Y, D, X) >>> causal.est_propensity_s() >>> causal.trim_s() >>> causal.stratify_s() >>> print(causal.strata) Stratification Summary Propensity Score Sample Size Ave. Propensity Outcome Stratum Min. Max. Controls Treated Controls Treated Raw-diff -------------------------------------------------------------------------------- 1 0.095 0.265 157 28 0.188 0.187 11.885 2 0.266 0.474 111 72 0.360 0.367 12.025 3 0.477 0.728 70 113 0.598 0.601 11.696 4 0.728 0.836 23 69 0.781 0.787 10.510 5 0.838 0.904 10 81 0.865 0.873 3.405
Now because each stratum contains only units that have similar propensity scores, the raw difference in average outcomes should be a decent estimate of the within-bin average treatment effect. There is no reason why we can't further control for covariates within each block, however.
One way we can do this — because each stratum in strata is a CausalModel instance in its own right — is simply to manually loop through strata and estimate the within-block average treatment effect via, say, least squares:
>>> for stratum in causal.strata: ... stratum.est_via_ols(adj=1)
We can collect these results together in a list:
>>> [stratum.estimates['ols']['ate'] for stratum in causal.strata] [10.379170390195274, 9.2918973715823867, 9.6787670925744589, 9.6722830043583361, 9.2239596078237778]
Note that these estimates are much more stable and closer to the true treatment effect of 10 than the within-bin raw differences in average outcomes that were reported in the stratification summary table, highlighting the virtue of further controlling for covariates even within blocks. Also note that, even though we are using least squares here, we are only doing so locally within each stratum, and therefore is not plagued by some of the issues we raised earlier when we discussed the standard least squares approach.
Taking the sample-weighted average of the above within-bin least squares estimates results in a propensity score matching estimator that is commonly known as the blocking estimator. However, instead of looping through the strata attribute and doing everything manually, we can also simply call est_via_blocking and have all of these operations performed automatically, with the results collected into the attribute estimates as usual:
>>> causal.est_via_blocking() >>> print(causal.estimates) Treatment Effect Estimates: Blocking Est. S.e. z P>|z| [95% Conf. int.] -------------------------------------------------------------------------------- ATE 9.702 0.381 25.444 0.000 8.954 10.449 ATC 9.847 0.527 18.701 0.000 8.815 10.879 ATT 9.553 0.332 28.771 0.000 8.903 10.204
As we can see in the results table, the estimated average treatment effect is now very close to the true value of 10.