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I am analyzing capacity values from an aging test of 24 battery cells, which have three categories:

  1. Battery Type: they can be 29V or 33V
  2. Storage Temperature: 10°C, 20°C or 30°C
  3. Storage SOC: 30% or 100%

Every test case has 2 batteries being tested (example: teo batteries in 29V-10°C-30% ; two batteries in 33V-10°C-30% etc).

To investigate the impact of test conditions on capacity (main effects and interactions), I performed three-way ANOVA, which met the requirements for the first batch (after 4 weeks of storage). I used Sharpiro-Wilk test and Anderson-Darling test to check normality and Levene's test to check equal variances.

However, for the second batch (after 9 weeks of storage), Sharpiro-Wilk test showed that Battery Type = 33V does not have a normal distribution. Anderson-Darling showed that battery type = 33V and SOC=100% don't have normal distribution. SOC=100% in Sharpiro test had a borderline result (p-value 0.07).

So, from all 7 groups, I considered 2 of them did not show normal distribution and, therefore, Battery Type and Storage SOC dont meet the normal distribution requirement.

My "problems":

  1. How to analyze the interaction between the test conditions? I performed a 3 way ANOVA (even knowing that Battery Type and SOC don't have normal distribution) and it didn't show any interaction. However I would need another alternative to confirm if that is really the case. A question recommended odds ordinal logistic regression model as a non-parametric alternative to 3 way-ANOVA. I tried to implement but as the variables have too much multi-collinearity the model did not work.

  2. How to analyze the main effects? I conducted a One way standard ANOVA on Temperature, as it has normal distribution, and it showed significant main effect. For SOC and Battery Type, I conducted the Kruskal Wallis Test which showed that both of them are not statistically significant. But I am a bit skeptical here: the boxplots and heatmap below might indicate that at least SOC is a significant factor. Therefore I am not confident about my approach on the main effects. enter image description here enter image description here Worth noting that we have a small sample size, although they are relatively similar (8 samples for each group in storage temperature and 12 samples for battery type and storage SOC).

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  • $\begingroup$ The new plots are helpful. I would also suggest (i) plotting the raw data points in the first plot (jittered if necessary) instead of boxplots, since you have just 24 points, and you can add colour and/or shape to indicate the levels of the other variables, and (ii) splitting the second plot into two panels (by voltage level or storage SOC), and (iii) using a sequential rather than a diverging colour scale for the second plot: blog.datawrapper.de/diverging-vs-sequential-color-scales $\endgroup$
    – mkt
    Commented Sep 27 at 12:23

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  1. It does not matter that the response variable is not normally distributed. And normality testing is not useful.

  2. Don't try to analyse or interpret interactions and main effects as separate things. Just plot the model output to visualise the combined effects of the different terms.

  3. You don't have enough data to meaningfully estimate 3-way interactions, and even 2-way interactions will be a bit of a stretch. Consider whether you need parameter estimates at all; perhaps simply visualising the data would achieve your goals.

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  • $\begingroup$ Thanks @mkt 1. As well said in that post, my goal here is to know if the data is normal enough for ANOVA to have trust-worthy results. So, I didnt quite get it: can I trust ANOVA results without performing any normality check, as it doesnt matter that the response variable isnt normally distributed? $\endgroup$
    – acdm
    Commented Sep 27 at 12:37
  • $\begingroup$ @AnnaClaraMorelli Normality checking is basically useless, let's set that aside. Whether you can trust the ANOVA results depends on whether you've got the right model structure and what you are trusting it to achieve. If you want reliable p-values, there are further assumptions involved. The simplest useful step would be to check the residuals and QQplot after fitting the model. But the dataset is small; you'll have to rely on some assumptions that we know are justified with large sample sizes but may not be for small sample sizes. $\endgroup$
    – mkt
    Commented Sep 27 at 12:46
  • $\begingroup$ @AnnaClaraMorelli Basically, I would treat the ANOVA as a useful way to summarise your data but wouldn't rely on inferences drawn from it. $\endgroup$
    – mkt
    Commented Sep 27 at 12:47

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