# Regression on a non-normal dependent variable

I need to do a regression with a non-normal DV for which no proper non-linear transformation (that I know of) exists:

It is a score ranging from 10 to 50, with a high peak at 10, a drop at 11 and a regular decline from 11 to 50. The distribution of residuals is not normal.

There are more or less 500 subjects in the study, and the covariates are all dichotomous.

I thought of using n-tiles and performing an ordinal regression on the resulting variable, but then I end up with a high proportion of empty cells -- not because of a low sample size, but rather because of the several covariates that need to be in the model.

The best I could do for now is to remove covariates and use tertiles (instead of, say, quintiles) to minimize the number of empty cells. I am not sure, though, to what extent avoiding empty cells is THAT important. Compared to the original analysis on a raw score, regressing on tertiles with less covariates seems like a lot of sacrifices.

Based on the decent sample size, and given the fact that the distribution of residuals is not THAT far from normal, I am wondering if the results of the regression would be reliable as they are.

• What is the purpose of the regression? Prediction, exploration, model fitting, something else?
– whuber
May 26, 2011 at 4:50
• That makes sense. In reflecting on it, I was wondering how you could even formulate a definite hypothesis when you don't yet have a specific model for the data behavior? What kind of hypothesis could that be? I'm not trying to be pedantic or difficult; it just seems that this could be an important point. It might constrain your options and require adjustments to the p-values.
– whuber
May 26, 2011 at 17:54
• @whuber Not sure I'm following your thinking, but I'll try to be more explicit... We are interested in measuring an effect (change in Y associated with presence of, say, X1). As we were expecting, the effect of X1 is significant, as suggested by the first, somewhat bulky regression, and further asserted by the ordinal regression that followed (in both cases, we have p<.0005). However, before ruling out linear regression and reporting the ordinal regression results, I wanted to hear some points of views here. May 26, 2011 at 22:59
• @whuber Actually, I should have said "estimation" as the main purpose. Sorry for the confusion. I deleted my previous comment saying it was hypothesis testing (not being possible to edit it). May 27, 2011 at 16:57

The normality assumption is a convenient property of model's residuals, since it enables correct inferences about the estimated parameters and critical values of many other tests are also dependent on this assumption (therefore some corrections should be made, or you may roughly take more strict rule-of-thumb criteria, increasing the acceptable range of your tests), however it doesn't ruin the regression estimators.

Thus it may (you still need to check the other assumptions) produce well behaved predictions, but data-mining and hypothesis testing would be a bit more difficult. At this point I do agree with Huber that you need to clarify the purpose of the model.

Regarding some tips:

At first glance it seems that your distribution after $Y-10$ transformation, could be approximated by some truncated versions of continuous distributions: exponential (Gamma), log-normal, Pareto or some other. So in log-normal case you still may move to something close to normality.

Another option could be to try something like fitting the combo of generalized logistic function and logistic regression. Since you DO know the upper and lower limits it seems feasible.

• Please see my previous comment to whuber to get a better idea of the purpose of all of this. For the transformation suggestions, did you mean after Y - 9 transformation? Otherwise I lose a whole bunch of cases, ending up with zeroes. The last option you suggest is quite advanced... Not sure I can do this properly for now. May 26, 2011 at 23:19
• Are your scores discrete values ($10, 11, \dots, 50$) or some real numbers? For continuous distributions it is possible to start with many zeros (try exponential or Pareto approximations). If you want just a simple logarithmic transformation, you may go for $X:=Y-9$, since for log-normal approximation the values are defined for $X>0$ only. May 27, 2011 at 11:37
• @Dmitij - The scores represent discrete values.The transformations with Y-9 did not lead to anything close enough to normal, unfortunately. May 27, 2011 at 16:04
• @dominic, indeed it doesn't, for log-normal approximation you make $\log{Y-9}$ transformation actually, where $\log$ stands for natural logarithm. May 28, 2011 at 14:48
• :) Yes, I know. What I meant was that the log or ln of $Y-9$ did not approximate anything close to gaussian. Anyhow, it seems, after further reading, that for effect estimation I can trust my raw coefficients. May 30, 2011 at 5:20

Ordinal regression is not affected by empty cells of Y. Quantile grouping is not required unless you just want to reduce computational burden. Proportional odds or continuation ratio ordinal logistic models are likely to be able to handle the distribution of Y you plotted (with no grouping of Y).

• (+1) it would be great to have some general links to the models you mentioned. May 27, 2011 at 11:41
• May 27, 2011 at 13:31
• Thank you Frank I will look into this. I don't feel so confident in interpreting the coefficients of ordinal regression yet, but I guess I'll have to go that way. May 27, 2011 at 16:10