I've often heard it said that linear regression assumes variables were measured on an interval or ratio scale. I understand why normality/homoskedasticity/independence are required (in order to keep alpha levels at 0.05), but why does regression assume interval or ratio data? Is it simply because the interpretation of the parameters change? Or is alpha also effected?

Put another way: suppose we have two measures of a response variable. Let's also assume both measure the same thing, but one is on an interval scale and the other is on an ordinal scale. If we run two separate regression models, what would be the cost of not meeting the interval assumption? Bias? Inflated (or deflated) standard errors?

And, most importantly for my curiosity, why?

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  • $\begingroup$ You will probably want to edit to say linear regression. There are, by the way, regression methods like logistic regression that do not assume interval or ratio scale! $\endgroup$ – kjetil b halvorsen Feb 8 '17 at 14:52
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    $\begingroup$ There is no such assumption about predictor variables. Of course if you transform the predictors things change, that is the point of transforming them but that applies whatever type of variable they are. $\endgroup$ – mdewey Feb 8 '17 at 14:56
  • $\begingroup$ Good point, @kjetilbhalvorsen. I've made that change. $\endgroup$ – dfife Feb 8 '17 at 15:03
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    $\begingroup$ Regression can be broadened further, but the plain (or perhaps vanilla) idea is arguably conditional averaging: how does the mean response vary with conditions? And taking an average is (usually) regarded as requiring interval scale. (I'll skate by the fact that many books tell you that you shouldn't average ordinal scales and then do precisely that.) $\endgroup$ – Nick Cox Feb 8 '17 at 15:08
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    $\begingroup$ I disagree with @kjetilbhalvorsen: logistic regression focuses on how the mean of a binary response (construed as a probability) changes. We are not in {nominal, ordinal} world any longer as soon as we do that. $\endgroup$ – Nick Cox Feb 8 '17 at 15:10

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