This question is mainly because I see many tutorials, whether on YouTube videos or on websites, forums, and the like, where when using software like SPSS, they always apply normality tests directly to the data. But that seems to contradict the idea that normality tests are performed on errors, not on the data itself.
I don't know if the software already calculates errors in the background even if it doesn't display them, if it's a widespread error, if it's some kind of consensus, or if I'm just missing the point. I would appreciate in-depth or concise explanations on this matter, because I'm trying to write a guide on ANOVAs and I've realized I've been explaining normality poorly.
Yes (to not be confusing with exceptions) a simple ANOVA with three groups is formulated as y = b0•x0+b1•x2+b3•x2+error. The error=N(0, sigma^2). The assumprion being that the behaviour of b0, b1, b2 is approx. asymptotically normal. Meaning, hat{b0}~N(b0, SE^2). This assumption is reasonably satisfied when error=N(0, sigma^2). Yet, this is not entirely the case. Because the residuals could also be t-distributed and the hypothetical behaviour of b0, b1, b2 can still be normal. Actually, the distribution should be approx. symetrical. This means that lower/upper bounds in the domain of the respose variable are not severely restricted.
First and foremost, without beating around the bush, it is a necessary condition for the application of the ANOVA test, as this ensures that any inferences made will be valid.
Now, It is only used in the construction of models. In this case, ANOVA is a linear model, so it must be applied to the residuals. Since the model estimates a mean for each Ti (treatment), the data behavior is expected to be random, either above or below this value, so that the mean adequately represents the dataset corresponding to each Ti.
Here, the experimental design is relevant, and there are many differences between pure experimental designs and quasi-experimental designs. However, you don’t need to worry too much about meeting the normality assumption, since there are alternative tests (often mistakenly called non-parametric) such as Kruskal-Wallis or Jonckheere-Terpstra.
the models
1. Completely Randomized Design (CRD) (oneway)
2. Randomized Complete Block Design (RCBD) (twoways)
3. Factorial Design (FD) - Two Factors with Interaction (
where
mu = overall mean
Tau_i = effect of treatment
Alpha_i =effect of factor A
Beta_i = effect of Block or factor B (according to the model)
e_ij =residual
https://rpubs.com/CJG/1327603
These days, you can simply use PERMANOVA and general additive models, which make no distributional assumptions :)
Yes, you calculate the ANOVA, and then analyze the normality based on those residuals. Although, as always, Jochen is right: the procedures and results are mainly for those who don’t know as much, who are usually the ones that question things the most.
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