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I have read through the other responses on this form regarding the "best" non-parametric test to run when you were planning to run a 2-way anova and you determine that your dataset (residuals) are not normal even after attempts at transformation (shapiro test = 0.001032, prior to attempts at transformation). The p value is not far off from being significant assuming significance at p 0.05. I thought transforming the data would solve the problem, but it made it "less" normal (i.e. much smaller p value).

My goal is to determine if there are significant differences in mean monthly chloride data over a period of 31 years. I have single monthly measurements of chloride and was going to use "years" as my replicates/blocks.

The data (not residuals) have a gamma distribution. The residuals overall follow a linear pattern (see figure).Residuals plotted.

This is the dataset not transformed. I also have added the histogram of residuals. Histogram of data

To solve the issue I thought I could run a GLM or run a a regression analysis across months to generate a predictive curvilinear model. I have also been told that since my p value is relatively close to "normal" its okay to run a two way anova. I don't know how to choose the best test and am looking for some general feedback.

I can post my data, but I don't know that it is necessary. Thank you.

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Correct me if I'm wrong, but it sounds like you are trying to justify the assumptions of anova by appealing to the shaprio wilk test.

From your residual QQ plot, I think things look fine. Statistical tests for diagnostics are problematic because with large data you can easily reject the null even when the departure from normality is small. I mean, the null for the shapiro wilk test is "Do my data come from a mathematical abstraction (the normal distribution)?" so it is strictly always "No" since Normal distributions to not exist in nature in the same ways that perfect circles do not exist in nature.

In any case, rejecting the null of the shaprio wilk is fine. Your residuals look good to me, but it is hard to say without the data. In the future, I would avoid assessing normality assumptions based on statistical tests.

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  • $\begingroup$ Thank you that is helpful. Do you think its reasonable to just look at the QQ plot? I also plotted the residuals with a histogram and it looks "ok." I added the histogram into my original question. $\endgroup$ – Ellen Dec 21 '20 at 6:51
  • $\begingroup$ @Ellen Looking at the QQ plot and the residuals is the preferred way. I would say so long as the qq plot does not strongly deviate from the dashed line and the residuals are symmetric about 0, you're fine. Again, I'd need the data to be certain. $\endgroup$ – Demetri Pananos Dec 21 '20 at 15:05

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