# Multiple regression approach strategies for non-normal dependent variable

I'm hoping to analyze the influence of a set of variables on a continous dependent variable (between 0-1). The independent variables are a mix of both categorical, continuous and discrete features.

The dependent variable is not normally distributed: In my eyes, this doesn't even qualify for a log-normal distribution?

However, zooming in on the interval from 0-0.01 the picture improves a bit: The remaining interval (0.01-1.0) is distributed accordingly. Log transforming the dependent variable in the entire interval yields: What would be a sensible regression strategy in this case? Not all relationships between individual independent variables and the dependent variable are linear.

Would I better off defining a categorical outcome variable with e.g. 3 groups defined based on the log-transformation? Or should I look into non-linear regression analysis? Recently discovered GAMs...but with my lack of experience in regression in general, this seems a bit daunting.

Any guidance is immensely appreciated!

• Most of these solutions are a not good idea for your situation. You can use a generalized linear model with logit link and appropriate standard errors, even though the variable is continuous, or beta regression. Some people use keywords such as quasi-maximum likelihood here On the whole, it is more important to use a regression flavour that respects the bounds of the variable than to worry about the exact form of the distribution. In any case, no regression method makes assumptions about the marginal distribution of the response. – Nick Cox May 13 '15 at 10:10
• Complementary log-log link is another possibility. – Nick Cox May 13 '15 at 10:50
• And see here. As @NickCox has said:-"no regression method makes assumptions about the marginal distribution of the response". – Scortchi - Reinstate Monica May 13 '15 at 11:33

## 1 Answer

I think your best best is probably quantile regression. Given the highly skewed nature of your dependent variable, the mean is unlikely to be the best measure of central tendency, so why try to model it?

And, rather than a transformation, quantile regression lets you model any quantile that you like.

I wrote a paper about it. The paper uses SAS, but it also covers the ideas behind quantile regression.

Beta regression, mentioned in the comment by Nick Cox, is also good, but I am not sure whether it will deal with the oddities as well as quantile regression.

You might want to try both.

• Thank you all for your feedback. @Peter, I've been looking at quantile regression since yesterday and quite like its capabilities. However, in much of the resources I've looked at, QR is mentioned as a good tool for inspecting marginal zones of the dependent variable. But given the nature of the quantile function, if I wanted to inspect the effects in a quantile-interval (e.g. what happens between the 80-90 percentile), I should look at the difference in QR results for tau 0.8 and tau 0.9 right? – tschmitty May 14 '15 at 8:28
• I am not sure what you mean by "between the 80-90 %tile". You could, of course, look at the 85th percentile (or 81st or whatever). I am not sure what the differences between the parameters at two different percentiles would show. – Peter Flom May 14 '15 at 11:49