Is there a way to determine difference in slopes with non-normal data?

[#1]

For my model, I have one categorical effect variable, one continuous effect variable, and two continuous role (Y) variables. The residual distributions of my variables appear relatively normal with the Q-Q plot, but the Anderson-Darling test showed that the distributions are not normal. I read a previous post that said you should do an ANCOVA with interaction and use indicator parameterization estimates to compare slopes. Is there a non-parametric way to do this? Thanks! 

[#2]

Personally, if the Q-Q plot of the residuals looks good, I would not worry about the Anderson-Darling test. How many data points do you have? With enough data, any statistical test can show as statistically significant difference. Your other option, rather than non-parametric is to use a general linear model that will allow you to specify the error distribution rather than assuming a normal error distribution.

[#3]

Hi Dan,

Thanks for your help. For my model, each group is a different size. Some of the group sizes are small, but there is a large number of total data points/observations because each individual sample has many data points. I plotted simple linear regression models of the data (including all data points from all samples in each group), which totals to at least 500 data points per group included in the linear regression. I was also wondering if this means I need to do a mixed effects model to account for more than one data point from each sample in the group? Or could I still perform a simple linear regression? Thank you!