ARe simple linear regressions robust enough to not be affected by non normality in a small sample, n=89

Question

Are simple linear regressions robust enough to not be affected by non normality in a small sample, n=89?

I am being challenged by a journal reviewer on my use of simple regression. 

For my study I ran correlation, then simple regression on two variables that showed they may be possible predictors. The original data was found to be skewed (all Shapiro-Wilks’ lower than W=.96, p = .01); therefore, violating the normality assumption. While the data was found to be non-normal simple linear regressions are robust enough to not be affected by non-normality. Since multiple regressions were not used in this analysis homoscedasticity of residuals applies less to this situation. However, collinearity diagnostics

The original data was found to be skewed (all Shapiro-Wilks’ lower than W=.96, p = .01); therefore, violating the normality assumption. While the data was found to be non-normal simple linear regressions are robust enough to not be affected by non-normality. Since multiple regressions were not used in this analysis homoscedasticity of residuals applies less to this situation. However, collinearity diagnostics were ran on both predictors. The diagnostics suggest that collinearity is not an issue since the variance differences for both predictors were found to be less than .98.

Answer 1

Hi Jane,

 

My advice is to check the normality of your residuals, not the original data for the normality assumption. I've published papers by doing that. I read in the literature that the residuals' distribution is what really matters. If the residuals are not normally distributed, you may consider using the non-parametric options. 

 

Hope this helps. Good luck!