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Unformatted text preview: 3(n + 1)
n(n + 1)
ni
i =1 where Ri is the sum of ranks of observations in the i th group.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA KruskalWallis Test
If there are no ties we reject H0 if H > χ2 −1,1−α and the
k
pvalue is computed as
p−value = P (χ2 −1 > Hcomputed ).
k
When there are ties, we deﬁne
H∗ = H/ 1 − g
3
i =1 (ti − ti )
n3 − n . Then we reject H0 if H ∗ > χ2 −1 and the pvalue is computed
k
as
∗
p−value = P (χ2 −1 > Hcomputed ).
k
The KruskalWallis test should be used only if the smallest
sample size is at least 5.
In R, the KruskalWallis test may be done using
kruskal.test(formula, data)
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Example:Opthalmology
Arachidonic acid is well known to have an eﬀect on ocular metabolism. In
particular, topical application of arachidonic acid has caused lid closure,
itching and ocular discharge, among other eﬀects. A study was conducted
to compare the antiinﬂammatory eﬀects of four diﬀerent drugs in albino
rabbits after administration of arachidonic acid. Six rabbits were studied
in each group. Diﬀerent rabbits were used in each of the four groups. For
each animal in a group, one of the four drugs was administered to one
eye and a saline solution was administered to the other eye. Ten minutes
later arachidonic acid (Sodium arachidonate) was administered to both
eyes. Both eyes were evaluated every 15 minutes thereafter for lid
closure. At each assessment the lids of both eyes were examined and a
lidclosure score from 0 to 3 was determined, where 0 = eye completely
open, 3 = eye completely closed, and 1,2= intermediate states.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA The measure of eﬀectiveness (X) is the change in lidclosure (from
baseline to followup) in the treated eye minus the change in
lidclosure score in the saline eye. A high value for X is indicative
of an eﬀective drug. The results, after 15 minutes of followup, are
presented in Table below.
Indomethacin
+2
+3
+3
+3
+3
0 Aspirin
+1
+3
+1
+2
+2
+3 Chapter 12: Multisample Inference Piroxicam
+3
+1
+2
+1
+3
+3 BW755C
+1
0
0
0
0
1 Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA AllPairwise Comparisons
The i th and j th treatments are declared diﬀerent if,
reject H0 if Ri − Rj  > z1− k (α )
k
1 n(n + 1)
12 1
1
+
.
ni
nj The above procedure does Bonferroni adjustment for
multiplicity.
Example: Do all pairwise comparisons for the Ophthalmology
example.
In R,
library(pgirmess) #needs to be installed
kruskalmc(x,g,probs=0.05)
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Assignment of Treatments to Experimental Units
The ANOVA we considered previously is know spe...
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This note was uploaded on 02/03/2014 for the course STAT 491 taught by Professor Solomonharrar during the Fall '12 term at Montana.
 Fall '12
 SolomonHarrar
 Statistics, Biostatistics

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