Page 13.1 (C:\data\StatPrimer\anovab.wpd 8/9/06)
13: Additional ANOVA Topics
Post hoc
Comparisons

ANOVA Assumptions

Assessing Group Variances
When Distributional Assumptions are Severely Violated
 KruskalWallis Test
Post hoc
Comparisons
0
In the prior chapter we used ANOVA to compare means from
k
independent groups. The null hypothesis was
H
: all
i
:
are equal. Moderate
P
values reflect little evidence against the null hypothesis whereas small
P
values indicate
that either the null hypothesis is not true or a rare event had occurred. In rejecting the null declared, we would
0
1
2
declare that at least one population mean differed but did not specify how so. For example, in rejecting
H
:
:
=
:
=
3
4
:
=
:
we were
uncertain whether all four means differed or if there was one “odd man out.” This chapter shows
how to proceed from there.
Illustrative data (Pigmentation study)
.
Data from a study on skin pigmentation is used to illustrate methods and
concepts in this chapter. Data are from four families from the same “racial group.” The dependent variable is a
measure of skin pigmentation. Data are:
Family 1:
36
39
43
38
37
= 38.6
Family 2:
46
47
47
47
43
= 46.0
Family 3:
40
50
44
48
50
= 46.4
Family 4:
45
53
56
52
56
= 52.4
1
2
3
4
There are
k
= 4 groups. Each group has 5 observations(
n
=
n
=
n
=
n
=
n
= 5), so there are
N
= 20 subjects total.
A oneway ANOVA table (below) shows the means to differ significantly (
P
< 0.0005):
Sum of Squares
SS
df
Mean Square
F
Sig.
Between
478.95
3
159.65
12.93
.000
Within
197.60
16
12.35
Total
676.55
19
Sidebyside boxplots (below) reveal a large difference between group 1 and group 4, with intermediate resultsi n
group 2 and group 3.
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Tukey, J. W. (1991). The Philosophy of Multiple Comparisons.
Statistical Science, 6
(1), 100116.
*
Rothman, K. J. (1990). No adjustments are needed for multiple comparisons.
Epidemiology, 1
, 4346.
†
Page 13.2 (C:\data\StatPrimer\anovab.wpd 8/9/06)
The overall oneway ANOVA results are significant, so we concluded the
not
all the population means are equal.
We now compare means two at a time in the form of
post hoc (afterthefact) comparisons
. We conduct the
following six tests:
0
1
2
1
1
2
0
1
3
1
1
3
Test 1:
H
:
:
=
:
vs.
H
:
:
:
Test 2:
H
:
:
=
:
vs.
H
:
:
:
0
1
4
1
1
4
0
2
3
1
2
3
Test 3:
H
:
:
=
:
vs.
H
:
:
:
Test 4:
H
:
:
=
:
vs.
H
:
:
:
0
2
4
1
2
4
0
3
4
1
3
4
Test 5:
H
:
:
=
:
vs.
H
:
:
:
Test 6:
H
:
:
=
:
vs.
H
:
:
:
Conducting multiple
post hoc
comparisons (like these) leads to a problem in interpretation called “The
Problem of
Multiple Comparisons.”
This boils down to identifying too many random differences when many “looks” are taken:
A man or woman who sits and deals out a deck of cards repeatedly will eventually get a very
unusual set of hands. A report of unusualness would be taken quite differently if we knew it was
the only deal ever made, or one of a thousand deals, or one of a million deals.
*
Consider testing 3
true
null hypothesis. In using
a
= 0.05 for each test, the probability of making a correct retention
is 0.95. The probability of making three consecutive correct retentions = 0.95 × 0.95 × 0.95
.
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 Spring '10
 AB
 Addition, Normal Distribution, Standard Deviation, Variance, post hoc comparisons

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