# Best regression correcting for non-normality, outliers and heteroskedasticity

We are performing a regression on cross-sectional data for $Y$ = subjective well-being (scale 0-10) and $X$ = working hours (divided into 5 dummy categories; less than 27 hours, 27-32 hours etc).

After having performed statistical tests we have the following:

• Non-normality in the residuals
• Heteroskedasticity (when control variables are included)
• Outliers and leverage

Our question is now whether OLS still can be applied to our regression, despite the high kurtosis in the residuals (violation of the non-normality assumption)?
In that case, which is the best OLS regression to run that corrects for all the violations mentioned above (e.g. PROCREG)?

We have read that quantile regression can be appropriate as it does not require normality in the residuals. We are however only familiar with OLS regressions, and thus we do not really know what implications it will have for the other tests. Would be great to get some tips about how to best proceed now.

Furthermore, how do we perform a simple test for spatial regression in SAS (EG)?

• A 0-10 scale cannot be normal. You will need to use ordinal logistic regression. The UCLA help site has a tutorial here. – gung May 7 '16 at 15:13
• If you categorise an inherently continuous variable like working hours you will see strange patterns in your plots (as you only have 10 different predicted values. There is also a danger that your model is mis-specified – mdewey May 7 '16 at 15:14
• Gung- My understanding is that it's pretty common to use OLS with even ordinal scales having as few as five categories. Can you say more about how a 10-point scale, if its distribution was approximately normally shaped, "cannot" be normal? Thanks. – Rico May 7 '16 at 15:27
• @Rico its simple really, the normal distribution is continuous over the full set of real number. The outcome considered is bounded 0-10, it is very discrete and limited in range and clearly non normal – Repmat May 7 '16 at 15:40
• @Rico oh sorry I didn't mean to imply that OLS is useless without normality. Inference is fully valid without normality, given the gauss markov assumptions. In fact it's valid under even weaker assumptions, but without efficiency guaranteed. My point, and I suspects gung point, was there is no point in arguing for normality in this, because those residuals (y really) will never become normal. – Repmat May 7 '16 at 16:47

I am no expert of the wellbeing literature, but I guess that a viable route is to transform the outcome variable into a dummy variable. It could equal 0 if the original y is lower than the median and equal 1 is it is higher than the median.

I think this transformation is good starting point. Usually, wellbeing is around 7.5/10 whatever country whatever study; this is why the distribution is skewed and dummy based on the median of y is better.

Of course, the second step would be to use the ordered logit model...but beware of all the related problems. As someone has already suggested, better to start from the UCLA website to seek for information on this econometric model.

• Thank you for your reply! We would prefer to avoid transforming the Y-variable into a binary variable, as all similar studies we have seen use the scale to better catch individual differences. We consider to potentially show results from both OLS with a robust regression technique as well as ordered probit, as both methods have been used in previous studies. Are there also problems with ordered probit, as you indicate for ordered logit? Can we use a robust regression technique for ordered probit to correct for outliers and heteroskedasticity? – Josephine May 9 '16 at 12:38
• The assumptions of ordered logit/probit models are often violated: 1) Errors may not be homoskedastic (but with robust standard errors you might solve this problems) 2) The parallel lines/proportional odds assumption often does not hold.There are specific tests for this assumption. To solve this problem, you should use a generalized ordered logit/probit models, see stata.com/meeting/germany08/GSUG2008.pdf (from where I substantially copied/pasted the two points I have listed) – Fuca26 May 10 '16 at 10:45
• Thanks a lot! Taking all the tips here into account, we have come to the conclusion to direct our full focus on making the best out of the OLS regression. It seems like the results will be unbiased despite the non-normality in the residuals. We will look further into which type of ROBUSTREG is best suited for our leverage points and outliers. As we understand it, the PROC ROBUSTREG in SAS should give robust SEs both when it comes to outliers as well as for heteroskedasticity. Using this, it seems like we should obtain unbiased estimates. Does this sound right to you? – Josephine May 10 '16 at 14:27
• Never categorize a continuous dependent variable – Frank Harrell Jul 23 '17 at 13:38
• Think the other way around. You can treat a continuous variable as an ordinal variable, without any categorization. It is hazardous to treat an ordinal variable as continuous. – Frank Harrell Nov 3 '17 at 3:25

I'm not familiar with quantile regression. However, a typical approach to the problems you are facing is to transform the data.

Have you tried transforming the outcome to get it to approximate a normal distribution?

Have you examined the bivariate relationships with the (transformed) outcome with the predictors and explored transforming the predictors in order to get a more linear relationship? (It would certainly by my inclination to start with the IV ungrouped and used as a continuous variable if that makes sense theoretically and practically.)

These operations do make interpretability of the result more difficult, but they may allow you to use OLS, which is well known by most producers (and consumers) of research.

Next, do you mean a test for spatial autocorrelation? There is an explanation for doing this in SAS here: http://support.sas.com/kb/22/944.html

EDIT: I hope it's reasonable to say that there are almost always methods that better fit the theory, research question, dataset, and so on. However, better methods are sometimes beyond the technical capability of the analyst. If that is the case, and it is not remediable, a good practice is to be explicit about things like failures of assumptions and to use the best possible method.

• Thank you! We tried log transformation of the Y-variable in accordance to your suggestion, but the residuals remain non-normally distributed. An alternative would indeed be to have the X as a continuous variable, we are however interested in examining the differences between different levels of work (e.g. 6 h/day vs. 8 h/day), with full-time work as the reference group. According to our understanding, it seems alright with non-normal residuals when using a robust technique. What do you think about using PROC ROBUSTREG? Will this correct for both heteroskedastcity and outliers/leverage? – Josephine May 9 '16 at 12:49
• This cries out for ordinal regression. Comments about its assumptions above don't recognize that other methods make more assumptions or are less powerful. – Frank Harrell Jul 23 '17 at 13:39
• @FrankHarrell What if her wellbeing scale was obtained from a visual analogue scale where, although bound between 0 and 10cm, the values are continuous? – bobmcpop Feb 27 '18 at 14:41
• Ordinal regression would still work fine. – Frank Harrell Feb 27 '18 at 21:54