13.1 One-Way ANOVA

The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test uses variances to help determine if the means are equal or not. To perform a one-way ANOVA test, there are five basic assumptions to be fulfilled:

• Each population from which a sample is taken is assumed to be normal.
• All samples are randomly selected and independent.
• The populations are assumed to have equal standard deviations (or variances).
• The factor is a categorical variable.
• The response is a numerical variable.

The Null and Alternative Hypotheses

The null hypothesis is that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are k groups

H₀: μ₁ = μ₂ = μ₃ = ... = μ_k

H_a: At least two of the group means μ₁, μ₂, μ₃, ..., μ_k are not equal. That is, μ_i ≠ μ_j for some i ≠ j.

The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), H₀: μ₁ = μ₂ = μ₃ and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations.

If the null hypothesis is false, then the variance of the combined data is larger, which is caused by the different means as shown in the second graph (green box plots).

Figure 13.2 (a) We fail to reject H₀ as it may be true. All the means are about the same; the differences may be due to random variation. (b) We reject H₀ as all the means are not the same; the differences are too large to be due to random variation.