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I’m running a linear regression where my response variable is mean satisfaction score of GP practices, while one of the predictor variables is funding received. I used a cube root transformation for this predictor but now I’m a bit lost as to the interpretation (a one unit change in funding is associated with ??? change in satisfaction).

I trawled through the site, and I can see the question has been asked before, but I haven’t been able to find a direct answer.

Luka Crnjakovic
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  • what was your reason for using a cube root transformation for the predictor? Was a log transformation (which has a more straightforward interpretation) not a good fit in your model? – Marquis de Carabas Aug 15 '15 at 19:46
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    Transformations are like drugs .. some are good for you and some are not ....See when and why http://stats.stackexchange.com/questions/18844/when-and-why-to-take-the-log-of-a-distribution-of-numbers/18852#18852 – IrishStat Aug 15 '15 at 21:13
  • @marquisdecarabas - the predictor has negative and zero values so I cannot use a log transformation as I understand. I also understand that quadric will treat negative and positive values alike, and cubic ill make skewness much worse – Luka Crnjakovic Aug 16 '15 at 08:54

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Since you took a cube root transformation of funding you now have to say that a one unit increase in the cube root of funding is related to the dependent variable. The effect of a one unit increase in the original variable on satisfaction will be different at different levels of funding.

An increase in funding from 2 to 8 would be associated with the same change in satisfaction as an increase from 8 to 27 or 27 to 64.

Transformations such as this are often unnecessary now that there are statistical methods that allow for non-normal residuals (e.g. robust regression, quantile regression, spline fits) one of those may work better for you.

Peter Flom
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    You might be thinking of transforming the response variable, which is the usual application of a cube root. However, the OP states that the *predictor* variable was transformed. That will only indirectly affect the residuals; it is done to achieve a linear relationship, not to stabilize or symmetrize residuals. – whuber Aug 15 '15 at 23:12
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    Thank you Peter for that very helpful description. I'm more worried about non-linearity than residuals. I ran a test for normality of residuals and it seems fine. When it comes to other transformations, as the funding variable has zero and negative values I understand I cannot use log, quadric or cubic transformations. – Luka Crnjakovic Aug 16 '15 at 09:08
  • There are other possibilities, like spline regression, but they will have the same issue that a one unit change in the IV does not have the same meaning at different levels of the IV. That's more or less what non-linearity means. – Peter Flom Aug 16 '15 at 12:33