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Unformatted text preview: ysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Introduction Simultaneously testing more than one hypothesis among the
treatment means.
For example, a ﬁnal answer to a research question depends on
the results from more than one comparison.
We need to understand how the error rate of the individual
comparisons inﬂuence the error rate applicable to the overall
conclusions
The comparisons could be preplanned or posthoc.
The assumptions needed are the same as before. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA PostHoc Comparisons Hypothesis generated after looking at the data.
If H0 : µ1 = µ2 = · · · = µk is rejected, we would like to know
in what way the means diﬀer.
which treatment is the best, second best,...
if there is an increasing trend in the mean. We can do posthoc comparisons if we
account for the after the fact nature of the hypothesis.
generate hypothesis but no inference with the current data. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Contrasts among Treatment Means
The quantity
= a1 µ1 + a2 µ2 + . . . + ak µk = ai µi with the coeﬃcients satisfying
ai = 0 is called a linear
contrast between treatment means.
Examples: Suppose we have k = 5 treatments with mean
responses µ1 ,µ2 ,µ3 ,µ4 and µ5 .
(a)
(b)
(c)
(d) 1
2
3
4 = µ1 − µ2
= 1 (µ2 + µ4 ) − 1 (µ3 + µ5 )
2
2
= 1 (µ1 + µ2 ) − 1 (µ3 + µ4 + µ5 )
2
3
= 3µ1 + 3µ2 + (−2)µ3 + (−2)µ4 + (−2)µ5 Contrasts 3 and 4 are equivalent.
Comparisons can be stated as linear contrasts.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Contrasts among Sample Means
Contrast among sample means
ˆ= ai yi .
¯ are used to estimate the corresponding contrast among
treatment means
=
ai µ i .
The estimated variance for a contrast among sample means is
2
ˆ
V ( ˆ) = sW Chapter 12: Multisample Inference ai2 /ni . Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA An Example
The reduction in systolic blood pressure (BP) after a drug for
hypertension is administered is one of key indicators of how well the
patient is responding to the drug. When treating for hypertension,
the side eﬀects associated with the drug are of particular concern.
In a study, two drugs A and B for reducing the side eﬀects of a
standard hypertension drug S were evaluated. Drugs A and B were
administered concurrently with drug S. The study was conducted
using a completely randomized design, with ﬁve treatments, as
Treatment
Drug
1
Standard (S)
2
S combined with low dose of A (S+AL)
follows:
3
S combined with high dose of A (S+AH)
4
S combined with low dose of B (S+BL)
5
S combined with high dose of B (S+BH)
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
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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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