I'm trying to help a colleague with a proposal to assess the impact of an intervention (let's call that "Variable A") and understand why the intervention is working. The intervention has three conditions (one of the conditions is a control group). My friend thinks that the intervention will impact Variables B, C, and D. He has baseline and follow-up data on B, C, and D, before and after the intervention.
I'm very confused about his analytic plan. My understanding is that he wants to understand why the intervention is working -- in other words, he wants to show that the type of intervention condition has an impact on Variables B and C, and he thinks that the change in Variables B and C from baseline to follow-up will explain the change in Variable D from baseline to follow-up (in other words - that change in Variable B and C will be mediators between the intervention and Variable D?).
-But - in his proposal, my friend first hypothesized that Variables B and C will predict CHANGE in Variable D from baseline to follow-up. Which would be (if 2 = follow-up and 1 = baseline):
D2 - D1 = intercept + Beta1(B2) + Beta2(C2)
-Then he reiterated elsewhere that he would conduct a regression to assess if CHANGES in "B and C" are related to CHANGES in D. Which would be
D2 - D1 = intercept + Beta1(B2-B1) + Beta2(C2-C1)
-And then later, he proposed that he would regress follow-up Variable D scores on baseline Variable D scores, controlling for residual change in B and C. Which would be (if I understand it):
D2 = intercept + Beta1(D1) + Beta2(B2-B1) + Beta3(C2-C1)
Or perhaps it would be?
D2 = intercept + Beta1(D1) + Beta2(B2) + Beta3(B1) + Beta4(C2) +Beta5(C1)
These seem to me like four totally different models, and they all seem different from what he wants to assess (showing that change in B and C explain why the intervention improves D).
So - my question is, can we confirm that the proposal to regress follow-up Variable D scores on baseline Variable D scores, controlling for "residual change" in B and C, is not the correct analysis to see if "changes in B and C" are related to "changes in D"? And then is it correct that we actually want a mediation analysis (I'm thinking Model 4, in SPSS Process?) is what we want to understand how the intervention is having an impact?
EDIT - My third question is, if we were ignoring the fact that there was an intervention and someone wanted to see if changes in B and C are related to the changes in D, which of the four models I have up here would be correct, and how are they different?