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Unformatted text preview: ltiple Comparisons
The KruskalWallis Test
TwoWay ANOVA An Example Cont’d There were four replication of each treatment . The reduction in
blood pressure (mm Hg) over a period of four weeks observed for
the experimental subjects were as tabulated below.
Treatment Response (mm Hg) Mean (¯i . )
y
1
27, 26, 21, 26
25.00
2
19, 13, 15, 16
15.75
3
15, 10, 10, 11
11.50
4
22, 15, 21, 18
19.00
5
20, 18, 17, 16
17.75
2
Note SSB = 388.7 and sW = 6.433. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA An Example Cont’d
The following are questions the investigator asked about the
treatment eﬀects.
1 Is there a diﬀerence between the eﬀects of the low and high
doses of A? 2 Is there a diﬀerence between the eﬀects of the low and high
doses of B? 3 Is there a diﬀerence between the average of the expected
responses for the two doses of A and the average of the
expected responses for the two doses of B? Express the above questions in terms of contrasts, compute the
sample contrasts and estimated variances. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Error Rates
PerComparison (ComparisonWise) Error Rate : The
expected proportion of contrasts that will be falsely declared
positive. We denote this error rate by αC .
Experimentwise (Familywise) Error Rate:The probability
of at least one false positive. We denote this error rate by αE .
Suppose we would like to test m contrasts. For a given
multiple comparison procedure, we have the scenarios
described in the following table
Number
Number
Number of
not rejected rejected Total
True Null Hypothesis
U
V
m0
T
S
m1
False Null Hypotheses
Total
m−R
R
m
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Error Rates Cont’d...
Notice that
αC = E (V )
.
m Also that
αE = P (V ≥ 1).
A multiple comparison procedure is said to control a particular
error rate at a level α, if this error rate is less than or equal to
α.
In general, we have the relationship
αC ≤ αE
That is, a multiple comparison procedure that controls EWER
will automatically control PCER.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Fisher’s Method
Suppose we are interested in testing the signiﬁcance of ˆ1 , ˆ2 ,
. . ., ˆm . The Fisher’s MC procedure declares ˆi signiﬁcant if
 ˆi  ≥ t1−α/2,n−k ˆ
V ( ˆi ). The Fisher’s MC procedure controls PCER at a level α.
A 100(1 − α)% conﬁdence interval for i is
ˆi ± t1−α/2,n−k ˆ
V ( ˆi ) Note: if a MC procedure controls only PCER then expected
number of false positives may increases with m.
The m contrasts must be preplanned comparisons for Fisher’s
procedure to have any use.
Chapter 12: Multisample Inference Stat 491...
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 Fall '12
 SolomonHarrar
 Statistics, Biostatistics

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