# Directly take the difference of the metric between 2 groups after the treatment or doing a difference-in-difference?

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?

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 ($$Y_i-Y'_i$$).