# Can I use a difference score as my outcome variable (pre-post change) in a moderated multiple regression equation?

I am wanting to examine whether religiosity moderates intervention effects on stigmatized attitudes. Here are my variables: X = group (1 = experimental; 0 = control) Z = religiosity (14-item scale - using total score) Y = either post-test scores or pre-post change scores - Social Distance; Genderism/Transphobia

I realize that I need to center my moderator variable and will also need to center my outcome variable (if I use post-test scores).

My question is whether to use post-test scores or to use pre-post test change scores (and if I use the latter would I still center)

• hm, my hunch would be to include interaction between treatment (X) and religiosity (Z) in the model rather than between religiosity and the pre-treatment measure of genderism. I wonder what it is I am missing here... Commented Feb 1, 2017 at 21:05

I see at least three reasonable options, although there is one I tend to do.

• Compute the difference score, $D = Pre - Post$ and then predict that. The regression equation being something like: $$\hat{D}_{i} = b_0 + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}*Genderism_{i}$$ One thing that is attractive about this is that it is straightforward to do.
• Predict the post scores using your model, but also controlling for pre scores. This regression equation would look something like: $$\hat{Post}_i = b_0 + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}*Genderism_{i} + b_5Pre_{i}$$ This tends to be my preferred approach. It does not require you to compute any new variables (which is not a big issue but can be slightly annoying). It also includes in the results an estimate of the strength of association between pre and post scores. It works because your treatment, $X$, predicts that portion of the post scores that are not explained by pre scores. The only drawback I see is that if you have a very small sample size, you lose an additional degree of freedom controlling for pre scores. That seems reasonable to me; however, because there is measurement error at both time points anyway, so it is not like computing the difference scores has zero error.
• The last approach I see is to reshape the data from wide to long, and fit a mixed effects model. This would look something like: $$\hat{Outcome}_{ij} = b_0 + u_{0i} + b_1X_{i} + b_2Z_{i} + b_3Genderism_{i} + b_4Z_{i}*Genderism_{i} + b_5time_{ij} + b_6time_{ij}*X_{i}$$ The outcome is the score at the jth time point for the ith individual. The model includes a random intercept (captured by $u_{0i}$) for each individual. The time captures the change over time, and the interaction between time and the group variable is the "treatment" effect. Although kind of cool, I think this model is far too much work in the simple case where you only have pre and post scores (if you had 3+ time points, it would make sense).

By the way, I do not particularly think that you need to center your variables prior to using them as moderators. I know many people teach that, but the models should work out the same. It can be slightly convenient as it makes the simple effects potentially more interpretable (0 = mean, therefore they are the effect of the variable at the mean of the other), but aside from that I see little gain. In more complex models, the reduction in collinearity between the variables and their product can also be helpful, but I have only ever see that matter in complicated random effects models or in some parallel latent growth models I fit once.

• in the first method you spell out, where you treat the differences as the outcome variable, do you have a reference for this approach? That seems perfect because then b1b1 estimates the treatment effect and interactions could be used to explore moderators. For some reason, I've rarely/ever seen people use that approach and I'm wondering why, and I'm wondering what the possible downsides could be. Commented Jan 30, 2017 at 23:32
• Nevermind. I read about this. Apparently treating the pre-post diffs as the outcome variable ignores possible regression to the mean, so it's suggested to model the raw follow-up measurement and adjust for baseline as a covariate (bmj.com/content/bmj/323/7321/1123.full.pdf)... When there are multiple follow-ups, that approach overestimates the treatment effect, so it is suggested to only adjust the first follow-up for the baseline value, but not the others (Twisk, Jos WR, and Wieke De Vente. European journal of epidemiology 23.10 (2008): 655-660) Commented Jan 31, 2017 at 16:56
• @not_bonferroni that is right, difference scores do not capture regression to the mean. A commonly (but not always) observed phenomenon is that initial scores are correlated with the degree of change. Controlling for baseline allows one to test a predictor over and above the effects of baseline. If the predictor is randomized, this is not as big of an issue as baseline scores will be orthogonal to the predictor, anyway, but especially in other contexts it is important to show that a predictor predicts change beyond just its correlation with baseline scores. Commented Feb 1, 2017 at 1:05
• Thanks for following up @Joshua. I didn't realize until after I asked that this post was over 4 years old Commented Feb 1, 2017 at 21:07
• In the original answer by Joshua, he listed equations which include interaction term between the moderator (Z=religiosity) and the pre-test value of the dependent variable (genderizm). But I think the interaction term should be between the moderator Z and the treatment X. Is that clearer? Commented Feb 3, 2017 at 1:46