Analyzing positively skewed continuous outcome variable

[chepchep]

Hello,

I am trying to model a continuous outcome variable which is highly skewed. I have several predictor variables in the model both continuous and categorical. The q-q plot of the residuals is shown below. As you can see, the normality assumption is clearly violated. I tried log transforming the outcome variable but it doesn't seem to fix the problem. Any body has an idea of how to remedy this issue ? Does the central limit theorem apply here?

Thanks.

Here is the code used:

proc glmselect data=b;
class a b  c d e / param=reference;
model y=a  b c d e f ;
output out=check r=residuals;
run;

proc univariate data=check;
var residuals;
histogram residuals / normal kernel;
qqplot residuals / normal(mu=est sigma=est);
run;

[StatDave]

Positively valued and skewed responses are often modeled using the gamma or inverse gaussian distribution as are available with the DIST= option in PROC GENMOD. 

[chepchep]

Thanks for your input. So how do I know which of the two to use, can either one of them work?

[StatDave]

You can use PROC SEVERITY in SAS/ETS to assess the fit of several distributions, including gamma and inverse gaussian and others. For example:

proc severity data=b crit=aicc;
   loss y;
   dist _predefined_;
run;

[chepchep]

Thank you so much for your input. So I did use the proc severity to select the the distribution that best fits my data and the Burr distribution was selected. This is a distribution that I am not very familiar with. How do I fit a Burr distribution in SAS?

Here is the partial output from proc severity:

Distribution Converged AICC SelectedBurrExpGammaIgaussLognParetoGpdWeibull

Yes	361869	Yes
Yes	382973	No
Yes	371532	No
Yes	368401	No
Yes	365528	No
Yes	382977	No
Yes	382975	No
Yes	377067	No

[StatDave]

I suggest you look at the plots (CDF/EDF and PDF) to visually assess how close the other distributions are to the EDF of the observed data. It's not so much a matter of picking the one with the lowest AICC as it is rejecting distributions that clearly don't fit well and picking one that does fit reasonably well.

[chepchep]

Thank you so much!

[PGStats]

What does the histogram of the residuals look like? Is there more than one mode? This would signal that you are missing some important effect, or some important interaction(s).

[chepchep]

This is how the histogram looks like:

[Rick_SAS · Accepted solution]

Personally, I would work on developing a better model. Use regression diagnostic plots to analyze whether you should include second-order interaction terms in the model. Since you are using PROC GLMSELECT, you can add in all second-order terms and use variable selection to see if any interactions improve the fit enough to make it into the final model.