I am currently working on a 2 x 2 repeated-measures ANOVA (rmANOVA), and I am unsure as to which assumptions I need to investigate and how to investigate them. It's a behavioral experiment. I have 30 subjects and 4 (2 x 2) measurements from each of them. There are equal sample sizes (30) in each of the four conditions. The dependent variable is measured at the continuous level (reaction time).

I have consulted a lot of sources (Field's "Discovering Statistics" textbook being a major one) to get to the bottom of this, but I get confused because different sources talk about different assumptions, and they talk about different ways of handling them. Another issue is that people often talk about "ANOVA assumptions", but they do not specify whether the assumptions in question is relevant for between-subjects ANOVA or rmANOVA or both, nor do they say whether they are relevant for one-way ANOVAs or two-way ANOVAs or both.

To mention the assumptions I think I have figured out first:

Now, for the assumptions I am less sure about.

If anybody knows anything about one or more of these assumptions (or other ones I might have missed), your help would be greatly appreciated.

I have consulted a lot of sources (Field's "Discovering Statistics" textbook being a major one) to get to the bottom of this, but I get confused because different sources talk about different assumptions, and they talk about different ways of handling them. Another issue is that people often talk about "ANOVA assumptions", but they do not specify whether the assumptions in question is relevant for between-subjects ANOVA or rmANOVA or both, nor do they say whether they are relevant for one-way ANOVAs or two-way ANOVAs or both.

To mention the assumptions I think I have figured out first:

**- Sphericity.**This should not be an issue as you need at least 3 levels of an independent variable for sphericity to be relevant.**- Homogeneity of variance/homoscedasticity.**I am not entirely sure about this one as some people mention it as relevant for rmANOVA and others do not, but Field (and others) says that it's only relevant if you have unequal sample sizes, which I do not have. Therefore I assume I don't have to worry about this assumption.**- No significant outliers.**I haven't looked a lot into this specifically yet as I have already done a pretty thorough screening for outliers, but I get the gist of it.Now, for the assumptions I am less sure about.

**- Normality.**This assumption should, according to Field, be checked visually. I think this is a bit iffy as it's not objective, but since he is quite detailed I think I will be able to do it. I was considering a normality test such as Shapiro-Wilk, but Field argues quite heavily against doing this. The think that bothers me about this one is that Field says you don't have to worry about this assumption if you have a large enough sample size, and that 30 often is "large enough" – but*not always.*It's this "not always" thing that bothers me, as I don't know whether I fall into this "not always" category or not. I have sort of decided to just to the visual inspection "just in case", but it would be nice to understand this a bit better.**- Additivity and linearity.**Field mentions the assumption of linearity briefly, but does not say much about it. He gets back to it later when talking about how to check for homoscedasticity (visually, as he advices against using Levene's test), as he says you can check for linearity and homoscedasticity at the same time. I think I understand how to do this (although this too feels a bit iffy as it's checked visually). My question is whether I need to check linearity at all, since I did not have to check for homoscedasticity. As for additivity, I can't see him mentioning this ever after first mentioning how important it is – is there any specific way to check for this? Or is additivity ok if linearity is ok?**- Independence.**I think I understand this assumption when you're talking about a between-subjects design, but is it relevant for a repeated-measures design as well? Some people seem to talk about as relevant for rmANOVA as well, but I don't quite understand how you investigate or check for it.**- Multicolinearity.**Mentioned by some (for instance: https://www.statisticssolutions.com/assumptions-of-the-factorial-anova/), but not a lot of sources. The source in the paranthesis there makes me think that this and independence is two sides of the same coin, but I honestly don't know. In any case, I don't find a lot of sources that explain in detail how you check this assumption. And since a lot of the "big" sources (Field, Howell) don't mention it for ANOVA (only regression), I am unsure as to whether I should really check it at all.If anybody knows anything about one or more of these assumptions (or other ones I might have missed), your help would be greatly appreciated.

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