# Which one is better for pre/post statistical analysis, “difference” or “ratio”?

I have confusion about a simple statistical analysis: one dependent variable measured before and after intervention in two groups. and the classical question is comparing effects between tow groups.

consider this case: (the value of independent vaiable presneted as 1 D vector)

group 1: Pre=[200 150 100 180] -> Post =[100 75 50 90]
group 2: Pre=[20 15 10 18] -> Post =[10 7.5 5 9]

the ratio(Post/pre) of change for both groups is 50%, which means no significant effect. while difference (Post-Pre) is significantly larger in the first group.

Suppose $$Y_{it}$$ is your outcome variable with $$t\in\{0,1\}$$ for pre and post, and $$T_{it}\in\{0,1\}$$ is the treatment indicator. (In your example we have that $$T_{it}=t$$.)
The question is, what is the conditional distribution of $$Y_{it}$$? If the conditional mean is $$E(Y_{it}\mid T_{it})=\alpha+\beta T_{it}$$, then you get $$\beta$$ by subtracting the pre mean from the post mean. If on the other hand $$E(Y_{it}\mid T_{it})=e^{\alpha+\beta T_{it}}$$, you get $$e^\beta$$ as $$E(Y_{it}\mid T_{it}=1)/E(Y_{it}\mid T_{it}=0)$$ which is the ratio, and $$\beta$$ by taking the log.