Question

10. Tukey’s HSD test

Sleep apnea is a disorder characterized by pauses in breathing during sleep. Children who suffer from untreated sleep apnea often have behavior problems, including hyperactivity, inattention, and aggression. A common treatment for pediatric sleep apnea is the surgical removal of enlarged tonsils and adenoids that are obstructing the airways.

Suppose researchers at a sleep clinic are interested in the effect of surgical treatment for pediatric sleep apnea on aggressive behavior. They study 11 children without sleep apnea, 11 children with untreated sleep apnea, and 11 children who have been surgically treated for sleep apnea. Aggression is measured using the Conners Rating Scales.

The sample means and sums of squares of the scores for each of the three groups are presented in the following table.

Group	Sample Mean	Sum of Squares
No Sleep Apnea	0.59	0.3240
Untreated Sleep Apnea	0.45	0.4410
Treated Sleep Apnea	0.31	0.2250

The researchers perform an analysis of variance (ANOVA) at α = 0.05 to test the hypothesis that the treatment means are equal. The results are presented in the following ANOVA table.

ANOVA Table

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F
Between Treatments	0.4312	2	0.2156	6.53
Within Treatments	0.9900	30	0.0330
Total	1.4212	32

The ANOVA yielded a significant F statistic, so the null hypothesis is rejected. Since there are more than two groups, the researchers are interested in determining which groups are different. The Tukey’s Honestly Significant Difference (HSD) test will be used to evaluate the pairs. First, use the table given below to determine the appropriate value of q at α = 0.05. The q value for this problem is   .

The Studentized Range Statistic (q)

df for Error Term	2	3
20	2.95	3.58
	4.02	4.64
24	2.92	3.53
	3.96	4.55
30	2.89	3.49
	3.89	4.45
40	2.86	3.44
	3.82	4.37
60	2.83	3.40
	3.76	4.28

The top value is α = .05; the bottom (bold) value is α = .01. The number of treatments is listed across. The df for the error term is in the left column, where the “error term” is another name for the within-treatments variance.

Now, use the q value to calculate Tukey’s HSD. Tukey’s HSD is   . Thus, the mean difference between any two samples must be at least   to be significant.

The researchers   conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ.

They   conclude that the population means for children without sleep apnea and children with treated sleep apnea differ.

They   conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ.

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Answer 1

The q value for this problem is 3.49

Tukey's HSD = q*√(MSW/n) = 3.49*√(0.033/11) = 0.19

Thus, the mean difference between any two samples must be at least 0.19 to be significant.

Comparison	Difference
x̅1 - x̅2	0.14
x̅1 - x̅3	0.28
x̅2 - x̅3	0.14

The researchers cannot conclude that the population means for children without sleep apnea and children with untreated sleep apnea differ.

They can conclude that the population means for children without sleep apnea and children with treated sleep apnea differ.

They cannot conclude that the population means for children with untreated sleep apnea and children with treated sleep apnea differ.