Treatment effect Analysis: What is Stratification and explanation/interpretation? In this paper by Angrist a stratification estimator is used (page 16 formula (4)) to calculate the Average Treatment Effect on the Treated (ATOT). The formula is given by:
\begin{align}
\widehat{ATOT}_{Stratification}=\sum_{k=1}^K \frac{\delta_{k} N_{1k}}{\sum_{k=1}^K\delta_{k} N_{1k}}(\overline{Y}_{1k}-\overline{Y}_{0k})
\end{align}
I do not understand this formula. My questions:


*

*What does it do? What is the idea?

*What is the difference to the general matching estimator, which uses a k-nearest neighbour matching or a kernel matching? 

*Is there any simple calculation example? I just found those papers, but with these huge datasets I cannot understand what this formula does, so I would be happy if I could calculate an example manually to understand what happens.
 A: (1) The right-most part of the formula (in parentheses) is calculating the difference between the treatment group mean ($\bar{Y}_{1k}$) and control group mean ($\bar{Y}_{0k}$) within each stratum $k$, of which there are $K$ strata. The part to the left of that is a weighting function that weights the mean-difference from each strata according to how many treatment group units there are in that strata (the stuff in the numerator) as a proportion of all treatment group units in the population (the stuff in the denominator). (See footnote 8 in the paper regarding population sizes versus sample sizes.)
(2) This is not really analogous to any of those things. Those are all procedures for creating matches (propensity scores or exact matching would be another approach). This is just the formula for calculating a mean-difference given a set of observations matched into strata on some discrete variable $X$. If a matching algorithm produces an index of discrete strata then this formula is basically directly applicable to those data.
(3) I don't really understand what you mean by your third question. This is the formula, which could be applied to any two-group (quasi-)experimental data with a covariate. Perhaps someone else can provide an example with real data.
