12.7: Assumptions of One-way ANOVA
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Like any statistical test, analysis of variance relies on some assumptions about the data. There are three key assumptions that you need to be aware of: normality , homogeneity of variance and independence . If you remember back to Section 14.2.4 – which I hope you at least skimmed even if you didn’t read the whole thing – I described the statistical models underpinning ANOVA, which I wrote down like this:
H 0 :Y ik =μ+ϵ ik
H 1 :Y ik =μ k +ϵ ik
In these equations μ refers to a single, grand population mean which is the same for all groups, and μ k is the population mean for the k-th group. Up to this point we’ve been mostly interested in whether our data are best described in terms of a single grand mean (the null hypothesis) or in terms of different group-specific means (the alternative hypothesis). This makes sense, of course: that’s actually the important research question! However, all of our testing procedures have – implicitly – relied on a specific assumption about the residuals, ϵ ik , namely that
ϵ ik ∼Normal(0,σ 2 )
None of the maths works properly without this bit. Or, to be precise, you can still do all the calculations, and you’ll end up with an F-statistic, but you have no guarantee that this F-statistic actually measures what you think it’s measuring, and so any conclusions that you might draw on the basis of the F test might be wrong.
So, how do we check whether this assumption about the residuals is accurate? Well, as I indicated above, there are three distinct claims buried in this one statement, and we’ll consider them separately.
- Normality . The residuals are assumed to be normally distributed. As we saw in Section 13.9, we can assess this by looking at QQ plots or running a Shapiro-Wilk test. I’ll talk about this in an ANOVA context in Section 14.9.
- Homogeneity of variance . Notice that we’ve only got the one value for the population standard deviation (i.e., σ), rather than allowing each group to have it’s own value (i.e., σ k ). This is referred to as the homogeneity of variance (sometimes called homoscedasticity) assumption. ANOVA assumes that the population standard deviation is the same for all groups. We’ll talk about this extensively in Section 14.7.
- Independence . The independence assumption is a little trickier. What it basically means is that, knowing one residual tells you nothing about any other residual. All of the ϵ ik values are assumed to have been generated without any “regard for” or “relationship to” any of the other ones. There’s not an obvious or simple way to test for this, but there are some situations that are clear violations of this: for instance, if you have a repeated-measures design, where each participant in your study appears in more than one condition, then independence doesn’t hold; there’s a special relationship between some observations… namely those that correspond to the same person! When that happens, you need to use something like repeated measures ANOVA. I don’t currently talk about repeated measures ANOVA in this book, but it will be included in later versions.
robust is ANOVA?
One question that people often want to know the answer to is the extent to which you can trust the results of an ANOVA if the assumptions are violated. Or, to use the technical language, how robust is ANOVA to violations of the assumptions. Due to deadline constraints I don’t have the time to discuss this topic. This is a topic I’ll cover in some detail in a later version of the book.