I am a little bit confused on what the assumptions of linear regression are.
So far I checked whether:
- all of the explanatory variables correlated linearly with the response variable. (This was the case)
- there was any collinearity among the explanatory variables. (there was little collinearity).
- the Cook's distances of the datapoints of my model are below 1 (this is the case, all distances are below 0.4, so no influence points).
- the residuals are normally distributed. (this may not be the case)
But I then read the following:
violations of normality often arise either because (a) the distributions of the dependent and/or independent variables are themselves significantly non-normal, and/or (b) the linearity assumption is violated.
Question 1 This makes it sound as if the independent and depend variables need to be normally distributed, but as far as I know this is not the case. My dependent variable as well as one of my independent variables are not normally distributed. Should they be?
Question 2 My QQnormal plot of the residuals look like this:
That slightly differs from a normal distribution and the
shapiro.test also rejects the null hypothesis that the residuals are from a normal distribution:
> shapiro.test(residuals(lmresult)) W = 0.9171, p-value = 3.618e-06
The residuals vs fitted values look like:
What can I do if my residuals are not normally distributed? Does it mean the linear model is entirely useless?