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A reviewer of a paper suggested that (Pearson product-moment) correlations cannot be used in input (predictors) in a regression model because they are "not interval levels of measurement" -- does anyone know how to interpret this statement?

Important: It is clear that correlations are not normally distributed, and it is also clear that they are bounded (by +/- 1, obviously) both of which require transforming them before using them in a regression model. Please note that we are not asking what to do with correlations to make them fit a regression model. Nor are we denying that they should not be used in regressions without some transformation. We fully understand the use of Fisher's Z transformation as shown here. Our question here is strictly conceptual.

Explicitly we would like to know what the level of measurement correlations are --- a la nominal, ordinal, interval, ratio; or the expanded range such as found in Chrisman (1998) --- or if the question is meaningless. The question seems somewhat nonsensical to me, but I have been unable to resolve the matter to my own satisfaction.

Citation:

Chrisman, N. R. (1998). Rethinking Levels of Measurement for Cartography. Cartography and Geographic Information Systems, 25(4), 231–242. (Sorry paywall blocks the link!)

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  • $\begingroup$ So what you're saying is that in your data set you have a column (variable) of correlations. And you're using that as a covariate in your analysis. Is that correct? $\endgroup$ – Jon Dec 7 '16 at 23:41
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    $\begingroup$ "It is clear that correlations are not normally distributed, and it is also clear that they are bounded (by +/- 1, obviously) both of which require transforming them before using them in a regression model." Not so. There is no objection to bounded or non-normal predictors in regression. If there were, then using indicator variables would be out of court, but it is utterly standard. There is no objection to non-normal responses as such. If you are trying to predict correlations, there could be a case for predicting them using e.g. Fisher's z as a scale. $\endgroup$ – Nick Cox Dec 7 '16 at 23:56
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    $\begingroup$ I would not worry about level of measurement here. The worry is just whether correlations will help in practice. I'd rather hear about why you decided to use them in the first place. $\endgroup$ – Nick Cox Dec 8 '16 at 0:00
  • $\begingroup$ @Jon Correct. They are being entered as predictors. $\endgroup$ – Doctorambient Dec 9 '16 at 19:00
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    $\begingroup$ @NickCox And as for using them, to be honest it is my colleague's analysis mostly. But we're using an analysis that is common in their literature. That's about all I know for this case. Thanks! $\endgroup$ – Doctorambient Dec 9 '16 at 19:35
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You need (maybe ...) interval level data to compute a (Pearson) correlation, but the correlation coefficient itself is a unitless number, and it is not clear that characterizing it by level of measurement is useful. If it can be used as predictor in a regression model is a more pragmatic choice, and you didn't tell us the context, so we have nothing to say. But I cannot understand that it should be prohibited to use. That would need quite strong arguments.

Otherwise, there is much wisdom in some of the comments, so I will just copy them here:

"It is clear that correlations are not normally distributed, and it is also clear that they are bounded (by +/- 1, obviously) both of which require transforming them before using them in a regression model." Not so. There is no objection to bounded or non-normal predictors in regression. If there were, then using indicator variables would be out of court, but it is utterly standard. There is no objection to non-normal responses as such. If you are trying to predict correlations, there could be a case for predicting them using e.g. Fisher's z as a scale. – Nick Cox

You say in another comment that As practice goes, people usually do prefer to enter correlations under the Fisher transform when building regression models due to the boundedness on Pearson's r. If this is intended as a comment about its use as predictor, it is difficult to see any good reason. How would you interpret the resulting coefficients?

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