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How do we interpret the effect of an experimental treatment on a principal component dimension, constructed from multiple outcomes in the experiment? Can this be interpreted substantively? What do we learn about the effect of treatment if it "causes" a principal component to decrease by 0.5?

The specific situation I am facing: I ran an experiment with one randomly assigned treatment (binary: control/treatment) and multiple outcomes of interest. I ran PCA on the outcomes. The treatment has a statistically significant effect (p<0.001) of about -0.5 on the first principal component.

The variables that load heavily in the first principal component vary together. As far as I understand, across all the PCs, the loadings times the treatment effect estimates for each PC sums to the overall treatment effect on each outcome. But I'm having trouble interpreting the effect of treatment on a single PC.

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Whether it can be interpreted depends entirely on whether the PCA is measuring something meaningful or not - if you have a substantive interpretation for the 1st component (and better if that's supported in previous work in the literature) then you may be ok - if it's just that there are a bunch of very different outcomes that co-vary then you probably shouldn't put them through a PCA in the first place.

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