 Analysis of Variance (ANOVA)

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Definition

A statistical test that allows for comparisons of multiple sources of variation, or effects, to determine if any of these sources significantly effect the variability of the outcome being studied. The null hypothesis is that the treatment effects are all the same and the alternative is that they're not all the same (at least one of them differs significantly from the others).

Examples A moving company has three different types of trucks and wants to determine whether they differ in mileage efficiency. The data on mileage were collected and an analysis of variance performed at a 5% level of significance. (See Additional Images for the data)

The typical Anova output is shown -it consists of a table showing the sources of variation, degrees of freedom, sum of squares (SS), mean SS, F-statistic and p-value associated with it.

The p-value corresponding to the F-statistic of 1.08 with 2 and 12 degrees of freedom is greater than the pre-determined level of significance (0.05), so the null hypothesis of ‘No Difference’ cannot be rejected.

Application

Even though the hypotheses are statements about the mean treatment effects, the test is based on the F-test of two variances, comparing the within treatment variance (or error variance) with the between-treatment variance. If the ratio of the between- to within treatment variance (the F statistic) is greater than a pre-determined cut-off value based on the choice of significance level, we reject the null hypothesis of no difference. Alternatively, if the p-value associated with the F-statistic is less than the pre-specified significance level, the hypothesis of no difference is rejected. 