1. © 2010 Pearson Prentice Hall. All rights reserved Multiple Comparisons for the Single Factor ANOVA

2. 13-2 When the results from a one-way ANOVA lead us to conclude that at least one population mean is different from the others, we can make additional comparisons between the means to determine which means differ significantly. The procedures for making these comparisons are called multiple comparison methods.

3. 13-3 After rejecting the null hypothesis H 0 : 1 = 2 = ··· = k the following steps can be used to compare pairs of means for significant differences, provided that 1.There are k simple random samples from k populations. 2.The k samples are independent of each other. 3.The populations are normally distributed. 4.The populations have the same variance. Step 1: Arrange the sample means in ascending order. Tukey Intervals for Multiple Comparisons

4. Tukey confidence intervals The Tukey method of constructing these intervals uses the following formula:

5. From the constructed intervals we can determine if two means are statistically different if the confidence interval for the difference does not contain the value zero.

6. Parallel Example 2: Tukey Intervals Using Technology Suppose we have the measurements of body weight change for three popular diet plans after 10 weeks on the diet. The results are tabulated below: Diet ADiet BDiet C 6.0614.849.37 8.6214.7511.43 10.314.728.58 9.713.736.05 8.8515.6511.05 IS there evidence of a difference between diet methods and if so what is the difference?

7. Using technology we have the results of the ANOVA below: One-way ANOVA: Diet A, Diet B, Diet C Source DF SS MS F P Factor 2 110.58 55.29 21.33 0.000 Error 12 31.10 2.59 Total 14 141.68 S = 1.610 R-Sq = 78.05% R-Sq(adj) = 74.39% We have sufficient evidence at the 5% level of significance to support the claim that at least two of the diets differ with respect to the mean weight loss after 10 weeks. Solution

8. Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons Diet A subtracted from: Lower Center Upper ------+---------+---------+---------+--- Diet B 3.318 6.032 8.746 (----*----) Diet C -2.124 0.590 3.304 (----*-----) ------+---------+---------+---------+--- -5.0 0.0 5.0 10.0 Diet B subtracted from: Lower Center Upper ------+---------+---------+---------+--- Diet C -8.156 -5.442 -2.728 (----*-----) ------+---------+---------+---------+--- -5.0 0.0 5.0 10.0

9. Solution An alternative to the Tukey interval comes from the Tukey tests where small test P-values signify a difference between means where we are testing the hypotheses:

10. Solution TukeySimultaneous Tests Response Variable Weight change All Pairwise Comparisons among Levels of Diet Diet = Diet A subtracted from: Difference SE of Adjusted Diet of Means Difference T-Value P-Value Diet B 6.0320 1.018 5.9242 0.0002 Diet C 0.5900 1.018 0.5795 0.8335 Diet = Diet B subtracted from: Difference SE of Adjusted Diet of Means Difference T-Value P-Value Diet C -5.442 1.018 -5.345 0.0005

11. Solution Interpretation: The mean change in body weight for diets A and C are not significantly different. The mean change in body weight for diet B is significantly different than diets A and C. It seems that diet B results in the highest mean weight loss after 10 weeks.