# Do dichotomous or continuous moderators lead to most statistical power when testing interactions?

My understanding is that, in general, continuous variables offer more power over dichotomous variables. For example, doing a between groups (e.g., men vs. women) t-test testing height would be better than doing a chi-square test (e.g., men vs. women) testing height with dichotomous groups (e.g., tall vs. short).

I am interested in testing moderation, namely, whether depression (moderator) affects outcomes differentially across two different anxiety treatments. I think depressed people will do better in one treatment over the other.

I was going to test for moderation using a continuous depression variable, but landed upon the following statement (http://davidakenny.net/cm/moderation.htm): "Power for tests of moderation is very low when one or both of the variables are continuous (McClelland & Judd, 1993)." I took a look at McClelland & Judd, 1993 and could not find the rational for this. I am now wondering whether I should split depression into high/low groups and examine slopes in each group rather than test for interaction using a continuum

Do dichotomous or continuous moderators lead to most statistical power when testing interactions?

The question is quite tricky as it depends on different properties of the data set.

The paper of McClelland and Judd is a very bad choice of a reference (even though it is cited a lot) as there is numerous problems in the paper. It is also not about dichotomous versus continuous variables even though there is an example on the matter, i.e., the "field study" vs "experiment study", which, btw, is very poorly designed if you look closely, and most of the paper is based on it.

The first problem (or lack of power thereof) that can occur is when the "natural" scale of the variable is continuous but is dichotomized. This leads to reduce variance in the variable, which reduces the effect. This problem is similar to median-spliting, for instance, which is not a practice recommended. This is however hard to capture as it more theoretically-orientated than statistical.

The second consideration is how the variables are correlated to each other, which is independent on their scale, but may or may not be artificially enhanced by the scaling.

The third consideration is that, dichotomous variable that are fixed (let say 50-50 like in an experimental design) will have no sampling error in the long run, which will reduce the sampling error, which increases power. This may be an increase in power associated to dichotomous variables.

I have not found yet a good account as to what cause the increase in sampling error in moderation analysis. It is somewhat related to this still open post with no definitive (or convincing) answers.

Artificially dichotomizing a continuous variable is generally agreed to lead to less power. However, if the only difference between the analyses is the continuous vs. binary variables (e.g., all the effect sizes are the same and the variances are the same) then there is hardly any difference. The example in McClelland & Judd, 1993 is weird because they make such sweeping statements, but in their simulated example the binary variables have an SD of 1, but the continuous variables have and SD of 0.5 (pg. 379, 3rd paragraph). So of course the continuous interaction is worse. They seem to have made this choice so that the range of the continuous and and binary variables is the same, without considering the implications of different standard deviations.