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I have a paper in front of me (peer reviewed, etc.), which states "comparison of the P values for [the covariates] gives a good indication of the relative importance of each [covariate] for determining the [dependent variable]". The supporting point made is that for a single regression, the confounding sample size issue drops out.

Having done a bit of reading around what the p-value is and is not (or might be), in the specific context of regression coefficient estimators, does this have any merit?

I can see a large p-value being used as a basis for covariate de-selection, but what about 0.001 being 'more important' than 0.01? Surely the conditioning on the rest of the model confounds the latter case as well as the former?

This is related, but seems not to resolve itself ... p-value as a distance?

This question seems to suggest the paper is riding the p-value rather too hard, so it would be useful to confirm that interpretation... Can p values be used to show impact of treatment

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