# 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).

1. 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?

2. 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?

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.

• Thank you! This is incredibly helpful. In short, it sounds like my friend is way off in the analysis he was suggesting, and that mediation is the correct approach. – SASNewb Jul 5 '17 at 1:11
• just for my own understanding and advancement -- when he says that the change in B & C from baseline to follow-up will explain the change in D from baseline to follow-up (let's ignore the fact that there's an intervention going on) -- which of the three models I mentioned above would be preferable to demonstrate that? Or it would be this fourth one? D2 = intercept + Beta1(D1) + Beta2(B2) + Beta3(B1) + Beta4(C2) +Beta5(C1) – SASNewb Jul 5 '17 at 1:17
• I missed that B and C represent change scores as well. So the general essentially state that treatment influences change in B and C each of which in turn drive change in D? If that is the case I am going to edit my post above to move to a mediation model based on latent change scores. I still tend to see this as a path analysis problem, just a more complex one as there are multiple change processes being modeled simultaneously. – Matt Barstead Jul 5 '17 at 2:01
• Yes, exactly - treatment influences B and C, and they influence D. The updated model you posted makes sense! I need to clarify with my friend how he hopes to proceed -- but in the event he wants to ignore the mediation (and treatment) to start with and just examine how change in B and C relate to D, do you have an insight about which of the four models I proposed above would be most appropriate for this interpretation? – SASNewb Jul 5 '17 at 2:55
• If you want to use a regression approach you would likely need to test mediation hypotheses for B and C separately. Meaning that you would do something like D2 = D1 + B1 in a first model. Then D2 = D1 + B1 + B2 in a second model. Ideally, B2 will remain a significant predictor of D2, controlling for D1 and B1. If that is the case, you can then add Tx to the mix and get D2 = D1 + B1 + B2 + Tx and see if the estimate for B2 changes appreciably. All is predicated on the fact that Tx has to be related to D2 controlling for D1 in the first place. This approach is antiquated though. Good luck. – Matt Barstead Jul 5 '17 at 3:06