Which statistical test should I use if the assumptions of a 2-way ANOVA are not met?

Question

My study design consists of two factors (one with 2 levels, the other with 6) and a continuous response variable. In order to analyze the influence of both factors on the explanatory variable I built a linear model in the following format:

modela<-lm(response~factor1*factor2, data=dataset)

I was going to run a 2-way ANOVA in order to test the significance of each of the explanatory variables however, upon evaluating the assumptions of this test, I found that the assumption of normality of residuals was violated (shown via a significant p-value from a Shapiro-Wilk test). All other assumptions (independent observations, no significant outliers, homogeneity of variances) were met.

Given this assumption violation is there a nonparametric alternative test that would be more appropriate to analyze my data. I have also read that transforming the data might help but I'm not sure a) if this would be appropriate and b) which transformations I should use.

Any help anyone can provide would be greatly appreciated.

Edit 1 - Here is the Q-Q plot for my model:

Edit 2:

This is the output I got for the aligned ranks transformation ANOVA.

Call:
art(formula = Duration.egg ~ Temperature + Species + Temperature:Species, 
    data = egg.na.1)

Column sums of aligned responses (should all be ~0):
        Temperature             Species Temperature:Species 
                  0                   0                   0 

F values of ANOVAs on aligned responses not of interest (should all be ~0):
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.0000  0.0000  0.4130  0.1609  2.2636 
Warning message:
In summary.art(x) :
  F values of ANOVAs on aligned responses not of interest are not all ~0. ART may not be appropriate.
```

Answer 1

I think that the discussion in the comments largely addresses the question. Here are a few points.

• Residuals for an ordinary least squares regression (OLS) don't need to be perfectly normal. On the one hand, nothing in the real world has a perfectly normal distribution. But also, OLS regression is somewhat robust to deviations from normality.
• The data in this case appear to be discrete and bound on the upper and lower end. I don't know if I would recommend OLS in this case. My interpretation of the q-q plot is that the residuals are symmetric with light tails. OLS regression may be okay in this instance. I don't know.
• A traditional nonparametric test for a two-factorial anova is the Scheirer–Ray–Hare test. This test is sometimes criticized. Also, there may not be a clear method for conducting post-hoc tests in all cases. There also may situations (unbalanced designs, small sample size) where the test may not be ideal.
• A more contemporary nonparametric method is the aligned ranks transformation anova (ART anova). In the current implementation in R, there are methods for post-hoc analysis and effect size. However, it is advised to research situations in which the method may not perform well.
• There are other methods that may be applicable, like permutation anova or quantile regression.