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Thread: Using residuals as predictors to adjust for effects of covariate in regression

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    Using residuals as predictors to adjust for effects of covariate in regression




    I was wondering if this is appropriate:

    I have two predictors that are highly correlated and collinear, x1 and x2, and are both associated with Y. I also have other covariates that I would like to include in the model. As a way of "adjusting" the association between Y and x1 for x2, I was wondering if it's reasonable to take the residuals from Y = x2 and the residuals from x1=x2, and run a regression as Y_residuals = x1_residuals + covariates.

    It seems to me this is kind of like calculating partial correlations, but adding additional covariates afterward. I'm a little concerned this may introduce bias since I'm not "adjusting" the other covariates for x2, as would be the case with Y = x1 + x2 + covariates; but I was trying to get around the collinearity between x1 and x2 and I'm really only concerned with adjusting the association between x1 and Y for x2. I don't really want to get into PCA, since I think that might muddle the interpretation and x1 and x2 likely have similar and independent biological activity (as well as influencing each other). Ideally I'd perform a validation study for regression calibration, but that isn't an option so I'm trying to do the best with the data that's available. I'd greatly appreciate any suggestions or references.

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    Re: Using residuals as predictors to adjust for effects of covariate in regression

    Not familiar with your approach. How correlated are the two IVs? Also, I believe centering the variables may help with the co-linearity issue. Have you explored this approach.
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    Re: Using residuals as predictors to adjust for effects of covariate in regression

    They are fairly highly correlated, Pearson's ~ 0.8. Centering had no effect on the correlation or collinearity between x1 and x2.

    I have a feeling that the approach isn't appropriate, but I'm concerned that just including x1 and x2 as covariates isn't appropriate either.

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    Re: Using residuals as predictors to adjust for effects of covariate in regression

    It is fine to keep them both, however you will just get huge SE. I would question the utility of including both if they are 0.8 correlated.


    Not something I have done, but I would also examine if they can be joined together.
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    Re: Using residuals as predictors to adjust for effects of covariate in regression


    Yeah, I guess I'll just deal with the inflated SE. The reason to include both is that both x1 and x2 have been previously shown to influence Y, perhaps both independently or through the same mechanisms (still a lot of unknowns).

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