Multiple Comparisons for the
Single Factor ANOVA
© 2010 Pearson Prentice Hall. All rights reserved
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.
13-2
Tukey Intervals for Multiple Comparisons
After rejecting the null hypothesis H0: 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.
13-3
Tukey confidence intervals
The Tukey method of constructing these
intervals uses the following formula:
 MS  1 1  


*
E 

( xi  x j )  q ,n k ,k 




 2  ni n j  
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.
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 A
6.06
8.62
10.3
9.7
8.85
Diet B
14.84
14.75
14.72
13.73
15.65
Diet C
9.37
11.43
8.58
6.05
11.05
IS there evidence of a difference between diet
methods and if so what is the difference?
Solution
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
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Diet A subtracted from:
Diet B
Diet C
Lower
3.318
-2.124
Center
6.032
0.590
Upper
8.746
3.304
------+---------+---------+---------+--(----*----)
(----*-----)
------+---------+---------+---------+---5.0
0.0
5.0
10.0
Diet B subtracted from:
Diet C
Lower
-8.156
Center
-5.442
Upper
-2.728
------+---------+---------+---------+--(----*-----)
------+---------+---------+---------+---5.0
0.0
5.0
10.0
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:
H 0 : i   j For i  j
H a : i   j For i  j
Solution
TukeySimultaneous Tests
Response Variable Weight change
All Pairwise Comparisons among Levels of Diet
Diet = Diet A subtracted from:
Diet
Diet B
Diet C
Difference
of Means
6.0320
0.5900
Diet = Diet B
Diet
Diet C
SE of
Difference
1.018
1.018
T-Value
5.9242
0.5795
Adjusted
P-Value
0.0002
0.8335
T-Value
-5.345
Adjusted
P-Value
0.0005
subtracted from:
Difference
of Means
-5.442
SE of
Difference
1.018
Solution
A
C
B
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.