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Is simply including a covariate in a model (ANCOVA) enough for variance in the DV due to it to be factored away? Or do the higher order interactions of the covariate with the other factors of the model have to be modelled as well?

In some software (e.g. Statistica), you have to specifically ask what factorial effects you want to have tested, and it seems to make a huge difference in the results whether you just ask for the main effect of the covariate or also ask for its crossings with all other categorical IVs.

I suppose it's up to the researcher/situation if you would then decompose any of those computed interactions (it would presumably depend on whether you consider the covariate to be 'of interest' or not), but the question is, which specific interactions do you "ask for" in the first place? Is there a trade-off between how informative an effect would be if included in the model VS the cost it would then have on inference power?

Also, does the covariate (continuous) factor have the same 'status' among the other (categorical) factors, i.e. is it somehow less important to include covariate-related interactions than it is to include categorical-factor-related interactions? Or does the selection of the requested interactions simply depend on hypotheses rather than being a simple answer such as "all possible interactions should be requested"?

Thanks a lot for any help!

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It's an assumption made in regression that there are no interactions between covariates and predictors, so you should test for them, and if they are statistically significant (or if they actually make a difference) you should interpret those interactions.

It's a bit of a tricky issue because, as you say, there's a tradeoff. If you have two covariates, you have an interaction of covariates as well. And you should include things like quadratic effects of covariates, and quadratic interactions, and so on. Heteroscedasticity can also be a sign that you have omitted covariate interactions (perhaps covariates that you didn't even measure). Once you have a lot of covariates, you're then at risk of overfitting by including things like interaction effects. If I'm worried about this, I like to use something like boosted regression.

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  • $\begingroup$ Thanks! But assuming I have a simple ANCOVA with 2 factors A and B (let's say A is between-subjects and B is within) and one co-variate C, then, after adding the full factorial for A and B (i.e., main effect of A and of B, and AB), and the main effect of C, would I then also add AC, BC and AB*C? Or would that depend on whether that interaction is of interest? $\endgroup$ – z8080 Mar 4 '14 at 19:34
  • $\begingroup$ You assume that those interactions do not have an effect when you interpret the effects. (I wouldn't call it a 'simple' ancova when you have a repeated measures factor.) $\endgroup$ – Jeremy Miles Mar 5 '14 at 18:34
  • $\begingroup$ Sorry, still not clear. So I should assume that all interactions involving the covariate (AC, BC and ABC) won't be significant - but does that mean I add them to (request them of) the model and then ignore whether they're significant or not, or do I simply not define them in my ANCOVA? $\endgroup$ – z8080 Mar 6 '14 at 17:17
  • $\begingroup$ Add them, see if they're significant, if they're not, take them out. $\endgroup$ – Jeremy Miles Mar 6 '14 at 17:34

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