Purpose of One-Way Analysis of Variance

THE PURPOSE OF ONE-WAY ANALYSIS OF VARIANCE

The purpose of the one-way analysis of variance (ANOVA) is to determine whether three or more groups have the same mean (i.e., H₀: μ₁ = μ₂ = μ₃, . . . , μ_k). It is a generalization of the t test to three or more groups. But the difference is that for a one-sided t test, when you reject the null hypothesis of equality of means, the alter-native tells you which one of the two means is greater (μ > μ₀, μ < μ₀, or μ₁ > μ₂, μ₁ < μ₂). With the analysis of variance, the corresponding test is an F test. It tells you that the means are different but not necessarily which one is larger than the oth-ers. As a result, if we want to identify specific differences we need to carry out ad-ditional tests, as described in Section 13.5.

The analysis of variance is based on a linear model that says that the response for group j, denoted X_j, satisfies Equation 13.1 for a one-way ANOVA:

X_ij =  μ_j + ε_ij                         (13.1)

where i is the ith observation from the jth group j = 1, 2, . . . , k; j is the group label and we have k ≥ 3 groups; m_j is the mean for group j; and ε_ij is an independent error term assumed to have a normal distribution with mean 0 and variance σ² independent of j.

The test statistic is the ratio of estimates of two sources of variation called the within-group variance and the between-group variance. If the treatment makes a difference, then we expect that the between-group variance will exceed the within-group variance. These variances or sums of squares when normalized have independent chi-square distributions with n_w and n_b degrees of freedom, respectively, when the modeling assumptions in Equation 13.1 hold and the null hypothesis is true. The sums of squares divided by their degrees of freedom are called mean squares.

The ratio of these mean squares is the test statistic for the analysis of variance. When the means are equal, this ratio has an F distribution with n_b degrees of freedom in the numerator and n_w degrees of freedom in the denominator. It is this F distribution that we refer to in order to determine whether or not to reject the null hypothesis. We also can compute a p-value from this F distribution as we have done with other tests. The F distribution is more complicated than the t distribution be-cause it has two degrees of freedom parameters instead of just one.