Non-parametric alternative to multiple linear regression?

Question

I'm working a study to assess the relationship between psychotic symptoms and suicide ideation on adolescents. A battery of instruments was applied to 1635 adolescents from 2 contexts (Group A=School sample, n=1520; Group B=Clinical sample, n=115) to assess psychotic symptoms and suicide ideation.

I want to know if higher scores on psychotic symptoms are able to predict higher scores on suicide ideation. The thing is, none of the variables have a normal distribution.

My question is, do simple and/or multiple linear regression models need the variables to have a normal distribution? I've read both that they do and that they don't. If they don't, why? If they do, what kind of non-parametric analysis should I use?

Thanks in advance.

-César

Answer 1

I have spent some time answering questions regarding the assumptions of the classical regression model (and I am sure other users as well) so please take a look at, for instance, How incorrect is a regression model when assumptions are not met?

The gist is that normality in the errors is not essential, except if your sample is small and you would like to use classical inference procedures, e.g. t-tests or ANOVA. The optimality properties of the OLS estimator within the class of linear unbiased estimators are retained regardless. For the predictors it hardly ever matters, as the regression model is conditional on their values.

Now, if you still have doubts about your model and you would like to turn to a non-parametric procedure, the easiest (and also the oldest) would be the Nadaraya-Watson estimator and its variants, which basically only differ with respect to the manner in which the weights are generated. This is an entirely non-parametric estimator and that adapts the idea of kernel density estimation to the regression setting.

A more elaborate procedure is local polynomial smoothing, which includes the Nadaraya-Watson estimator as a special case. The theory for this procedure is significantly more complicated, but its performance is also superior. Rarely would I go there, however, as the NW estimator usually does the trick.