Post Hoc Tests on One-Way ANOVA Lesson 13 - 2 Post Hoc Tests on One-Way ANOVA

Objectives Perform the Tukey Test

Vocabulary Post Hoc – Latin for after this Multiple comparison methods – to determine which means differ significantly Studentized Range Distribution – q-test statistic distribution (Table 8) Experiment-wise error rate – also known as the family-wise error rate is also called the level of significance, α Comparison-wise error rate – is the probability of making a Type I error when comparing two means

After a One-way ANOVA We have performed a one-way ANOVA test If we do not reject the null hypothesis, then There is not enough evidence that the population means are different There is not much else to do If we do reject the null hypothesis, then There is at least two population means that are different Which ones are they?

Tukey Test The Tukey Test analyzes all pairs of population means H0: μi = μj for all i and j where i ≠ j That is to say, the Tukey Test analyzes each of H0: μ1 = μ2 H0: μ1 = μ3 H0: μ1 = μ4 H0: μ2 = μ3 Etc

Tukey’s Test Computing the Q-test Statistic Arrange the sample means in ascending order. Compute the pair-wise differences, xi – xj, where xi > xj Compute the test statistic, q, for each pair-wise difference (listed below) Determine the critical value, qα,n-k,k If q ≥ qα,n-k,k , reject H0: μi = μj and conclude that the means are significantly different Compare all pair-wise differences to identify which means are considered equal x2 – x1 q = -------------------------- s2 1 1 --- · --- + --- 2 n1 n2 where x2 > x1, n1 is sample size from population 1, n2 is the sample size from population 2 and s2 is the MSE estimate of σ2 from ANOVA

Critical Value for Tukey qα,v,k The critical values for the test statistic q depend on three parameters The significance level, α, as usual The error degrees of freedom (from the ANOVA output), ν, which is n - k The total number of means compared, k These critical values are found in Table VIII

ANOVA – Analysis of Variance Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-test Statistic F Critical Value Treatment Σ ni(xi – x)2 k - 1 MST MST/MSE F α, k-1, n-k Error Σ (ni – 1)si2 n - k MSE Total SST + SSE n - 1 Use this as the estimate for s2

Excel ANOVA Output For Test Statistic: need the means, the ni’s and MSE For the Critical Value: α, df (error) and k

See Example 2 on page 694

Summary and Homework Summary Homework When a One-Way ANOVA indicates that not all the means are equal, we would want to identify which ones are equal and which ones are not The Tukey Test provides a method for that After performing The Tukey Test, we can conclude whether each mean is significantly different from, or can be considered equal to, each other mean Homework pg 696 – 699: 1, 2, 3, 5, 6, 15, 18