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I wanted to use a crossover design to compare two treatments (a supplement versus a placebo) on human participants. The advantage of this design is that I don't have to worry about the effects of the confounders, such as age, gender, ethnicity, etc, when I compare the supplement with the placebo. However, my collaborator just told me that he is not only interested in how treatment affects the outcome, but also how age and gender affect the outcome, and what is the interaction between age, gender and treatment. In that case, can I still use crossover design? If not, what would be a good alternative? Thanks!

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What you can do is build an experimental design that tells you whether age and gender are effect modifiers: that is, do they predict how well your treatment will work? You can still use a crossover design, and you'll have (for each participant) a difference $Y_i = Y_{trt, i} - Y_{pla, i}$. You then have essentially an observational dataset where you can regress $Y_i$ on factors like age and gender.

(Edit: As Phil notes, the resulting coefficients will measure interactions. To see how age affects outcomes in the absence of treatment, you could look at just $Y_{pla, i}$ or just $Y_{trt, i}$ versus age, rather than their difference.)

But, beware subgroup analysis: there's a major multiple comparison issue there, and false-positive horror stories abound. Try to pre-specify a few tests of particular interest to you, or reserve a subset of data for confirmatory analysis.

Also, in a crossover design, you need a suitable wash-out period or other scheme to limit the possibility of treatment effects persisting from one arm to the next.

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    $\begingroup$ It's worth noting that the coefficients for age and gender then would be the age:treatment and gender:treatment interactions respectively, and not the main effects of age and gender. The main effects would not be identified in a cross-over model (unless they are time-varying). $\endgroup$
    – Phil
    Oct 29 '18 at 8:20

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