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In my current experiment, I designed a 2 x 2 repeated measures factorial design.

Suppose that first condition has levels of (a / b), and the second condition has levels of (1 / 2). Thus, my factorial design has four groups (a1, a2, b1, b2). When I ran a statistical test, there were main effects of both conditions but no interaction effect.

However, I recently found that one confounding variable exists in my design. Suppose that the confounding variable has three levels (A / B / C). With the confounder added, my design was found to be four groups as (Aa2, Ba1, Bb2, Cb1).

Are there any methods for adjusting the confounding variables in my design? Or, are there any tests for finding the effect of the confounder?

It looks impossible but I cannot logically conclude. Any help would be appreciated!

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  • $\begingroup$ Have you tried reformulating your ANOVA into a regression can controlling for the confounder? $\endgroup$ – generic_user Dec 11 '17 at 18:39
  • $\begingroup$ Thank you for answering. But, could you please specify? $\endgroup$ – acrenaisy Dec 11 '17 at 18:45
  • $\begingroup$ Google has plenty of results when searching for "anova vs regression" $\endgroup$ – generic_user Dec 11 '17 at 18:51
  • $\begingroup$ Oh, sorry. I already know the issues of "anova vs regression". I just cannot figure out how reformulating ANOVA into a regression can "control the confounder". Could you let me know the concept or the term of that? $\endgroup$ – acrenaisy Dec 11 '17 at 19:01
  • $\begingroup$ Simply reformulate your anova into a regression and then add your confounder as a control variable. $\endgroup$ – generic_user Dec 11 '17 at 19:02
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Sadly, no. A is perfectly correlated with being in group a2, which means any difference between group a2 and the other groups could be attributable to either the effect of a2, the effect of A, or both, and within your experiment, it will be impossible to disentangle those.

Formally, your experiment lacks positivity. Positivity states that $0 < P(Z = z|X = x) < 1$ for $x \in X$ (i.e., for all levels of your confounder) and for all $z \in Z$ (i.e., for all levels of your treatment). Even ignoring the interaction (i.e., assuming there is no interaction in your treatments), all units in A are in a2, which means the "main effect" of 1 vs. 2 could be due at least partially to A vs. not A (even though are some Bs in 1 and 2), and the "main effect" of a vs. b could be due at least partially to A vs. not A.

As I see it, your only hope is to assume the proposed confounder is not actually a confounder and move on with your analysis as initially intended, possibly by assessing general sensitivity to an unobserved confounder. For example, you might consider how strong of an effect the confounder would have to have to change the substantive conclusion of your experiment. That would at least give readers an idea of how robust your experiment is to a potential confounder.

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