Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

As an assumption of linear regression, the normality of the distribution of the error is sometimes wrongly "extended" or interpreted as the need for normality of the y or x.

Is it possible to construct a scenario/dataset that where the X and Y are non-normal but the error term is and therefore the obtained linear regression estimates are valid?

share|improve this question
4  
Trivial example: X has a Bernoulli distribution (ie, taking the values 0 or 1); Y = X + N(0, 0.1). Neither X nor Y is normally distributed on its own, but regressing Y on X still works. –  Hong Ooi Feb 17 at 8:45
    
I guess you are thinking about the distribution of the residuals, not the distribution of the variables. –  tashuhka Feb 17 at 10:03
2  
I have an example worked out here: What if residuals are normally distributed but Y is not? –  gung Feb 17 at 14:52
add comment

1 Answer 1

Expanding on Hong Oois comment with an image. Here is an image of a dataset where none of the marginals are normally distributed but the residuals still are, thus the assumptions of linear regression are still valid:

enter image description here

The image was generated by the following R code:

library(psych)
x <- rbinom(100, 1, 0.3)
y <- rnorm(length(x), 5 + x * 5, 1)

scatter.hist(x, y, correl=F, density=F, ellipse=F, xlab="x", ylab="y")
share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.