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?