Hello!
I have a non-normally distributed variable (income) and although I tried to transform it to a normally distributed variable skewness and kurtosis values are still so high and there is lots of outliers on it. But can't delete the outliers because it is about nature of income variable. So I didn't delete a single one (by the way N=9918, I am not sure it is acceptable to delete 200 or 300 of them). I read about after conducting the OLS if residuals are distibuting normally it is acceptable to use OLS results. But I couldn't find any academic source/strong reference about it.
I wonder that when I have normally-distributed residuals can I use OLS results even if the variable has outliers and have higher skewness and kurtosis values? If this is an acceptable way to conduct this analysis, can you suggest an academic resource that I can reference to support this usage?
Thank you in advance.
If you look up the assumptions for an OLS general linear model, you'll see that there is no assumption that the dependent variable is normally distributed.
Usually the assumption is written as the errors from the model being normally distributed. These are approximated by the residuals.
Pretty much any textbook on the design and analysis of experiments should be a reference for this.
Slides in the following presentation are taken from Montgomery, Design and Analysis of Experiments. See slide 8, and then slides 19 - 22.
https://staff.emu.edu.tr/adhammackieh/Documents/courses/ieng581/lecture-notes/ch03.pdf
Thank you for your response. But I should use income variable as an independent variable. Is that possible?
Having a skewed variable as an independent variable doesn't necessarily violate model assumptions, and may not be a bad thing. But sometimes transforming the independent variable is helpful.
If you want to force a (numeric) variable to be normally distributed, inverse normal scores transformation usually works.
Yes, this is possible (just an addition to Sal Mangiafico answer). The more observations you have the closer the hypothetical distribution of the statistic of interest approximates the normal distribution (https://en.wikipedia.org/wiki/Central_limit_theorem). Ofcourse this is not theoretically possible if income is bound between 0 and +Infinity. But, it might be (as I do not know the the data) approximated closely enough to be useful in the context of your study. I have some "heuristic-rules-of-tumb", such as don use "small" sample sizes and values of the test statistic "close to 0" might obtain wonky results.
If you are working with R the following code might help. We create "count" data and fit an OLS (m1) plus GLM with poisson distribution and identity link (m2). The coefficients for both m1 and m2 very similair and the qq-plot as well. Note that a qq-plot will suffice only fully in relation to the normal. Indeed, a GLM with Poisson error distribution and identity link would be the "better" choice. Yet, this does not mean the choice for an OLS would necessarily be "bad". At least, in this case.
#Create data
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Inversing the independent variable quite lowered the skewness value but kurtosis is still high (around 4). Dear Sal Mangiafico, how can I understand whether this problem is violating model assumptions?
Thank you for your response dear Wim Kaijser
Actually I am working with SPSS. This software also showing the qq plots but I think not in a way you describe. SPSS drawed the qq plot but it seemed there is quite departure from the center of the graph's line. How can I understand on SPSS wheter this condition of the variable is effecting the OLS model significantly? (By the way I standardize values with t scores, because I am triying to create an index. That's why values of the variable range on min. 41 and max 61,3).
There's no assumption about the distribution of the independent variable. See the link I included in my first response for the assumptions for a general linear model.
There is no Problem with, as long as the assumptions of OLS are not violated. The residuals beeing normal distributed is a good sign.
In addition: a binary variable is also not normally distributed, no problem at all!
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