ANCOVA with randomized and nonrandomized groups

I was recently discussing a study and I was presented with an interesting scenario.

In this hypothetical, there is a crossover RCT where all participants have pre-test values obtained on a physiological outcome variable (continuous) then a drug/placebo is administered for period of time and the participants return for a post-test (washout period and then the participants complete the other treatment condition drug/placebo). Now imagine there are two groups within the study (let’s just call them A & B). The groups A & B naturally differ on the outcome variable (pre-test differences are expected) and there is reason to believe that group A may have a different response to the drug. The research question here would be this: does group (A & B) moderate the response to treatment (drug). So overall, the theoretical study can be thought of as a 2 x 2 design (treatment x group) with pre & post values in every subject being obtained twice (placebo and drug conditions). In all my reading on the topic (which is limited), this would make the “ANCOVA” approach biased because these two groups already differ on the outcome variable, but the change score approach doesn’t seem appropriate either (maybe I’m overthinking this).

So what do all think: how would you approach analyzing this data and why?

(I also realize there are some related threads here, but thought this particular scenario was unique.)


This is an excellent question. Vern Chincilli of Penn State has experience with these kinds of designs. I’ve contacted Vern to see if has time to respond here.

In a two-period crossover it is customary to compute B-A differences and to test against a mean of zero. Your design could just turn that into a two-sample problem instead of a one-sample problem, to see if the differences differ by group. At least I think so.

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It is not clear to me whether the groups A and B are determined prior to randomization and identified in the protocol. If that is the case, then a stratified analysis of the 2 x 2 crossover design (stratified by group) is appropriate in which you could investigate the modifying effect of group on the drug versus placebo difference. If groups A and B are determined after study completion, then you are in the realm of post-hoc analyses. You still could apply the modification analysis described above, but you would need to be cautious when interpreting and describing the results because of the post-hoc nature of the analysis.

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Thank you all for the quick responses!

In this case the groups would be determined a priori and are balanced (e.g., 25 participants from A and B).

If I am understanding both you correctly an appropriate statistical model here would be something like what I have written out below.
(Y_{drug}-Y_{placebo}) = \beta_0\cdot X_{0i} + \beta_1\cdot X_{1i} + \epsilon_i

With \beta_0 and \beta_1 representing the regression coefficients for the intercept and treatment effect respectively. The pretest values (e.g., measure of outcome Y prior to receiving drug or placebo) could then be added as covariate?

Yes, this is how I would approach the problem.