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Assume we are doing a randomized experiment. The dependent variable is $Y$. Usually, we randomly put some of the subjects in the treatment group ($n$ subjects) and the rest to the control group ($m$ subject). Then, before the treatment, we measure $Y_i$ for $i\in T$ (treatment) and $Y_j$ for $j\in C$ (control). Then we give the treatment group the real treatment and give the control group placebo. Then after some time, we measure the $Y'_i$ for $i\in T$ (treatment) and $Y'_j$ for $j\in C$ (control).

Now we consider two ways of measuring the treatment effect.

  1. directly take the difference between the 2 groups after the treatment $\sum_{i=1}^nY'_i/n-\sum_{j=1}^mY'_j/m$

  2. take the difference in difference $\sum_{i=1}^n(Y'_i-Y_i)/n-\sum_{j=1}^m(Y'_j-Y_j)/m=\sum_{i=1}^nY'_i/n-\sum_{j=1}^mY'_j/m-(\sum_{i=1}^nY_i/n-\sum_{j=1}^mY_j/m)$

If the randomization is perfect and if $n$ and $m$ are very large, then we should have $\sum_{i=1}^nY_i/n-\sum_{j=1}^mY_j/m=0$, which makes the 2 methods the same. However,if $n$ and $m$ are not very large, then $\sum_{i=1}^nY_i/n-\sum_{j=1}^mY_j/m\neq 0$, then which method should I use? Is there a superiority of one over the other?

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Assume your dependent variable $Y$ is continue. Then

1: $Pr(\sum_{i=1}^nY_i/n-\sum_{j=1}^mY_j/m=0) = 0$

2: $Var(\sum_{i=1}^nY'_i/n-\sum_{j=1}^mY'_j/m) \ge Var(\sum_{i=1}^n(Y'_i-Y_i)/n-\sum_{j=1}^m(Y'_j-Y_j)/m))$ given $Cov(Y_i,Y'_i) > 0.5Var(Y_i)Var(Y'_i)$, and it is very common.

  1. So should use DID in estimate and test. The test can be carried out by performing the grouped (trt vs control) t-test on the individual difference ($Y_i-Y'_i$).
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