Consider linear mixed models.

Consider the following:

https://www.researchgate.net/post/Is-there-a-non-parametric-equivalent-of-Repeated-Measures-ANOVA

Repeated measures ANOVA is regularly used with nonnormal data, especially with large sample sizes. Examine this issuel

Thanks for recommending that thread David Coker, that looks very useful

Rachel Cann, It seems to me that you have a 2x2 mixed design. Repeated measures on one factor and independent groups on another. It would easily be factorial ANOVA but since parametric analysis is not preferred you can try bootstrap procedure or some kind of robust variant of ANOVA. It is better than several non-parametric tests as it will allow for testing interaction effects that you should check out for in any factorial design. Andy Field covers robust tests in his book on using R for data analysis so I guess it might help. Look for WRS2 package tutorials. You can always fall back to two Mann-Whitney tests for an independent factor and two Wilcoxon tests for a dependent factor but it is suboptimal, problematic with family-wise error rate and difficult to identify possible interaction effect, so putting up an effort to do a robust ANOVA would be a good choice.

Hi Anuraj Nayarisseri thanks so much for your answer. I had not thought about bootstrapped procedures. I have the Andy Field book for SPSS, and now I'm keen to get his book on R as it sounds like that will cover more of what I need. Thanks