Question

Application Exercise:
A team of researchers recruited a random sample of migraine sufferers. Before the study began at baseline, participants were asked to record the duration of headaches they experienced in a week. A few days later, all participants were asked to practice a week-long relaxation technique designed to reduce migraine headaches. A two-week off period in which the participants practiced nothing was given between the relaxation technique. Throughout the relaxation technique participants were asked to record the duration of headaches. What can be concluded with an α of 0.05?

Baseline	Week 1	Week 3
9 7 12 8 12 7 11 7 10	9 7 6 6 9 5 6 5 9	8 9 6 4 7 4 6 5 8

e) Regardless of the H₀ decision in b), conduct Tukey's post hoc test for the following comparisons:
1 vs. 3: difference =_______________ ; significant: ---Select---Yes OR No
1 vs. 2: difference =_______________ ; significant: ---Select---Yes OR No

f) Regardless of the H₀ decision in b), conduct Scheffe's post hoc test for the following comparisons:
1 vs. 3: test statistic = ______________ ; significant: ---Select---Yes OR No
2 vs. 3: test statistic = ______________ ; significant: ---Select---Yes OR No

Answer 1

We are given the following data and are asked to conduct Tukey's post hoc test and Scheffe's post hoc for the following comparisons

Baseline	Week 1	Week 3
9	9	8
7	7	9
12	6	6
8	6	4
12	9	7
7	5	4
11	6	6
7	5	5
10	9	8

e) Tukey's post hoc analysis for the following comparisons:

1 vs 3 and 1 vs 2

To conduct Tukey's post hoc analysis we have to conduct ANOVA test so that we can get MS_within value and the q statistic for the Tukey's post hoc analysis

Baseline	Week 1	Week 3	Ti.
9	9	8	26	225.33	226
7	7	9	23	176.33	179
12	6	6	24	192	216
8	6	4	18	108	116
12	9	7	28	261.33	274
7	5	4	16	85.33	90
11	6	6	23	176.33	193
7	5	5	17	96.33	99
10	9	8	27	243	245
		Totals

Correction Factor

ANOVA Table

Source of Variation	S.S	d.f.	M.SS	F-ratio
Between Process(Treatments)	52.72	(k-1=3-1) 2
Within Process (error)	74.02	(N-k=27-3) 24
Total	126.74	(N-1) 26

Now that we have constructed the ANOVA table, we can proceed with Tukey's post hoc calculation.

For that we have to calculate absolute difference of means of 1 and 3, 1 and 2 groups.

Baseline	Week 1	Week 3
9	9	8
7	7	9
12	6	6
8	6	4
12	9	7
7	5	4
11	6	6
7	5	5
10	9	8
Sum=83	Sum=62	Sum=57

, ,

1 vs 3. difference=9.22-6.33=2.89

1 vs 2 difference=9.22-6.89=2.33

Here n is the group size not the overall size

Since 1 vs 3 difference is larger than HSD value, we say that the difference is significant

Since 1 vs 2 difference is larger than HSD value, we say that the difference is significant

Hence we say that 1 vs 3 difference is significant and 1 vs 2 difference is also significant

f) We are asked to conduct Scheffe's post hoc test for the following comparisons

1 vs 3 and 2 vs 3.

First we find the absolute mean difference between 1 vs and between 2 vs 3.

Since we have already calculated absolute mean difference of 1 v3, we only need to calculate absolute mean difference of 2 vs 3.

2 vs 3 difference=6.89-6.33=0.56

We use the following formula to get Scheffe formula values.

=

Since the sample size is the same for all the three groups, one calculation applies to all other comparisons.

Comparing Scheffe formula value with 1 vs 3: test statistic is not significant because 1 vs 3 difference(2.89) is less than Scheffe formula(3.42).

Comparing Scheefe formula value with 2 vs 3: test statistic is not significant because 2 vs 3 difference(0.56) is less than Scheffe formula(3.42)

Larger values are statistically significant at the chosen significant level(0.05 in our case).

Hence 1 vs 3: test statistic is not significant and 2 vs 3: test statistic is not significant.