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I am quite an amateur in statistics. I have data where I would like to test if I can perform parametric or non-parametric testing depending on if the data is normally distributed or not. The data has one dependent variable "time to fatigue" that I wish to study and two grouping factors: "length" and "velocity".

The grouping factor "length" has 3 levels: 15, 30, and 100.
The grouping factor "velocity" has 2 levels: 35 and 45.

In each level, there are 10 values of the dependent variable i.e., "time to fatigue".

Now if I wish to answer the question of whether "length" and "velocity" and "length: velocity" affect the "time to fatigue", I can perform a 2-way ANOVA.

However, I must make sure if I can perform parametric 2-way ANOVA or not.

My question now is on which data I must check the normality assumption.

  1. Should I check if the 10 values of "time to fatigue" for each level are normally distributed? OR
  2. Should I check if the "time to fatigue" within each grouping factor is normally distributed? OR
  3. Should I check if the "time to fatigue" (taking all time to fatigue values at once) are normally distributed?

What is the right approach? And what would it signify if one or two out of the above three options are normally distributed?

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  • $\begingroup$ "test for normality" should be straightforward, and web-searches (as well as R's ??normality) will often suggest the shapiro.test. As for testing it within each group or not, that sounds like it should be a question on Cross Validated (which is focused on statistics) and not here on SO (which is focused on programming). $\endgroup$
    – r2evans
    Commented Aug 21, 2022 at 12:26
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    $\begingroup$ Hi, it's actually the residuals that need to be normally distribution; this is somewhere between your 1) and your 2). It sounds a little funny, but you actually need to fit the model first before deciding if it works or not (fitting an ANOVA will give you these residuals you need). $\endgroup$
    – John Madden
    Commented Aug 21, 2022 at 21:31
  • $\begingroup$ My honest advice is to look at an analysis of experiments textbook, and see what they list as assumptions of a general linear model (including a two-way anova). One book you might find is D.C. Montgomery, Design and Analysis of Experiments. You'll see that model assumes that the errors are normally distributed. Whether you look at a plot to confirm that this is a reasonable assumption, or treat it as an assumption is up to you. Homoscedasticity among groups may be a more important concern. $\endgroup$
    – Sal Mangiafico
    Commented Aug 22, 2022 at 18:25
  • $\begingroup$ Perhaps see also: https://stats.stackexchange.com/questions/352866/two-way-non-parametric-test-for-non-normal-data $\endgroup$
    – Sal Mangiafico
    Commented Aug 22, 2022 at 18:33
  • $\begingroup$ At the time of writing, the following presentation, slide 8, and then slides 19 - 22, include information from Montgomery on the assumptions of a general linear model. $\endgroup$
    – Sal Mangiafico
    Commented Aug 22, 2022 at 18:39

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Why check for Normality? It is about the various statistics and the F test, and it used to provide p-values. So as stated above (John Madden) the Normality test is on the residuals. Typically the various statistics used in Anova are quite robust to Non-Normality so it should not be a major concern.

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  • $\begingroup$ I think it's not good advice to suggest ignoring the normality assumption of anova. Anova is somewhat robust when populations are not conditionally-normal, but this isn't going to hold in all cases. It's better to either understand or assess your data to know what analysis is appropriate. $\endgroup$
    – Sal Mangiafico
    Commented Aug 22, 2022 at 18:40
  • $\begingroup$ Please see stats.stackexchange.com/questions/5680/… $\endgroup$
    – dario
    Commented Oct 16, 2022 at 22:49

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