Multiple Comparisons
Overall Risk of Type I Error in Using
Repeated t Tests at  = 0.05
ANOVA: Graphical
Example: ANOVA terms
1
2
3
4
Treatment 1
y11 = 48
y12 = 39
y13 = 42
y14 = 43
Treatment 2
y21 = 40
y22 = 48
y23 = 44
Treatment 3
y31 = 39
y32 = 30
y33 = 32
y34 = 35
Overall
n1 = 4
n2 = 3
n3 = 4
11
ȳ1 = 43
ȳ2 = 44
ȳ3 = 34
40
s1 = 3.74
s2 = 4
s3 = 3.92
ANOVA Table
Source
Between
Within
Total
df
2
8
10
SS
228
120
348
MS
114
15
ANOVA Table: Formulas
Source
df
SS
(Sum of Squares)
MS
(Mean Square)
I
Between
I–1
2
n
(y

y)
i i
i 1
SS/df
I
Within
 (n  1)s
n• – I
i 1
I
Total
n• – 1
ni
i
2
i
2
(y

y)
 ij
i1 j1
F distribution
http://www.vosesoftware.com/ModelRiskHelp/index.htm#Distributions/Continuous_distributions/F_distribution.htm
F Table
Scientific Conclusion for F test
This study (does not) provide(s) evidence [(P = )]
at the  significance level that there is a
difference in ____ among the ____ groups.
Example: ANOVA
A random sample of 15 healthy young men are
split randomly into 3 groups of 5. They receive
0, 20, and 40 mg of the drug Paxil for one
week. Then their serotonin levels are
measured to determine whether Paxil affects
serotonin levels.
Example: ANOVA (cont).
Dose
ni
ȳi
si
(ni-1)si2
ni(ȳi - y̅̅)̅̅ 2
0 mg
48.62
49.85
64.22
62.81
62.51
5
57.60
7.678
235.78
492.56
20 mg
58.60
72.52
66.72
80.12
68.44
5
69.28
7.895
249.32
15.36
40 mg
68.59
78.28
82.77
76.53
72.33
5
75.70
5.460
119.24
333.96
overall
15
67.53
604.34
841.88
Example: ANOVA (cont)
Source
Between
Within
Total
df
2
12
14
SS
841.88
604.34
1446.23
MS
420.94
50.36
Example: ANOVA (cont)
Does Paxil affect serotonin levels in healthy
young men?
Let 1 be the mean serotonin level for men
receiving 0 mg of Paxil.
Let 2 be the mean serotonin level for men
receiving 20 mg of Paxil.
Let 3 be the mean serotonin level for men
receiving 40 mg of Paxil.
Example: ANOVA (cont)
H0: 1 = 2 = 3; mean serotonin levels are the
same at all 3 dosage levels [or, mean serotonin
levels are unaffected by Paxil dose]
HA: The mean serotonin levels of the three
groups are not all equal. [or, serotonin levels
are affected by Paxil does]
Example: ANOVA (cont)
Source
Between
Within
Total
df
2
12
14
SS
841.88
604.34
1446.23
MS
420.94
50.36
Example: ANOVA (cont)
Source
Between
Within
Total
df
2
12
14
SS
MS
F-Ratio
841.88 420.94 8.36
604.34 50.36
1446.23
P-Value
0.0053
This study provides evidence (P = 0.0053) at the 0.05
significance level that there is a difference in serotonin
levels among the groups of men taking 0, 20, and 40 mg
of Paxil.
This study provides evidence (P = 0.0053) at the 0.05
significance level that Paxil intake affects serotonin
levels in young men.
Verification of Conditions
Example 11.6.1: Randomized Block
Procedure
Researchers are interested in the effect that acid
has on growth rate of alfalfa plants. To control
sunlight, the randomized block procedure is
used.
Example 11.6.9: F test
Example
11.7.3: TwoWay ANOVA
Example 11.7.4: Two-Way ANOVA
Bonferroni t Table
Example: ANOVA
A random sample of 15 healthy young men are
split randomly into 3 groups of 5. They receive
0, 20, and 40 mg of the drug Paxil for one
week. Then their serotonin levels are
measured to determine whether Paxil affects
serotonin levels.
Example: Bonferroni Adjustment
Dose
ni
y̅̅i
SSi
Source
Between
Within
Total
0 mg
5
57.60
235.78
df
2
12
14
20 mg
5
69.28
249.32
40 mg
5
75.70
119.24
SS
MS
F-Ratio
841.88 420.94 8.36
604.34 50.36
1446.23
overall
15
67.53
604.34
P-Value
0.0053
Example: Paxil, Graphical
Representation
0 mg
20 mg
40 mg