# 8Analysis of Variance

Analysis of variance or ANOVA is the technique we use when all the explanatory variables are categorical. The explanatory variables are called *factors*, and each factor has two or more *levels*. When there is a single factor with three or more levels we use one-way ANOVA. If we had a single factor with just two levels, we would use Student's *t* test (see p. 97), and this would give us exactly the same answer that we would have obtained by ANOVA (remember the rule that with just two levels then *F* = *t*^{2}). Where there are two or more factors, then we use two-way or three-way ANOVA, depending on the number of explanatory variables. When there is replication at each level in a multi-way ANOVA, the experiment is called a *factorial* design, and this allows us to study *interactions* between variables, in which we test whether *the response to one factor depends on the level of another factor*.

## One-Way ANOVA

There is a real paradox about analysis of variance, which often stands in the way of a clear understanding of exactly what is going on. The idea of ANOVA is to compare two or more means. But it does this by comparing variances. How can that work?

The best way to see what is happening is to work through a graphical example. To keep things as simple as possible, we shall use a factor with just two levels at this stage, but the argument extends to any number of levels. Suppose that we have atmospheric ozone concentrations measured in parts per hundred million (pphm) in ...

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