Regression to Assess if Change in Variables B & C is related to Change in Variable D...vs. Mediation? 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). 


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*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?
 A: I see this as a path analysis problem given the presence of more than one potential mediator, if I understand you correctly that is. Using the variables you describe, you would include D1 scores as a covariate and set the model up as follows: 

Note that I have omitted error variances in the model for those familiar with SEM path diagrams just for simplicity's sake. 
This model implies the following (abbreviated) equations: 
Tx = D1 + e
B = D1 + Tx + e
C = D1 + Tx + e
D2 = D1 + Tx + B + C + e
The e's are generic placeholders for the error terms associated with each variable. A number of stats programs can handle this setup, with your primary interest being potentially the indirect effect of Treatment (Tx) through B (and C) on time 2 scores, controlling for time 1 scores. Relevant programs include Mplus, AMOS, LISREL, R (lavaan package), and EQS. There may be others, but those are the ones I am familiar with. In each case you will have to do some extra work to setup your tests of indirect effects, but it will differ from program to program exactly how you have to go about doing so. 
UPDATED
This update is designed to reflect the comment below that it is changes in B and C as a function of treatment that are hypothesized to drive change in D. The approach I would advocate in this circumstance is to use a mediation model built on latent change scores. 
First a simple example of a latent change score. In this case several values are constrained to ensure that all of the change (and variance in change) is linked to the latent variable. 

You would repeat the same specification to create latent change scores for variables B and C. Then add treatment as a predictor in the model similar to the blue portion of the first image I created (above). 

Alternatively, you could specify the same model as a multi-group mediation model. In any case, this would allow you to look at change in B and/or C as potential mediators of the relation between treatment group and change in D. 
