1. Post Hoc Tests on One-Way ANOVA
Lesson
Post Hoc Tests on One-Way ANOVA

2. Objectives
Perform the Tukey Test

3. 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

4. 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?

5. 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

6. 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 – x q = s
--- ·
n 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

7. 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

8. 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

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

10. See Example 2 on page 694

11. 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