Unformatted text preview: her the means that are not signiﬁcantly diﬀerent.
The groups may overlap. They are identiﬁed by underlining
the ordered means belonging to the same groups.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Methods
1. Fisher (only PCER)
LSDij (F ) = t1−α/2,n−k 1
1
+
n(i ) n(j ) 2
sW 2. Bonferroni (EWER)
LSDij (B ) = t1− k (kα 1) ,n−k
− 1
1
+
n(i ) n(j ) 2
sW 3. Scheﬀe (EWER)
LSDij (S ) = (k − 1)F1−α,k −1,n−k Chapter 12: Multisample Inference 1
1
+
n(i ) n(j ) Stat 491: Biostatistics 2
sW Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Example: The BP Data apc<glht(A,linfct=mcp(f="Tukey"))
summary(apc,test=adjusted(type=c("bonferroni")))
confint(apc,c_alpha=(10.05/20)) #k(k1)=20 Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Example: Insectology
In a research to investigate the attractiveness of ﬁve diﬀerent
colors to insects, the number of cereal leaf beetles trapped when
six boards of each of the ﬁve colors were placed in a ﬁeld of oats
were recorded. The following summary statistics were computed.
Color
ni
yi .
¯
Yellow 6 49.2
Orange 6 34.9
Red
6 26.4
Blue
6 17.2
White
6 15.0
2 = 63.298.
sW Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Example Cont’d ...
Pair
Y vs W
Y vs B
Y vs R
Y vs O
O vs W
B vs O
O vs R
R vs W
R vs B
B vs W Diﬀ
34.2
32
22.8
14.3
19.9
17.7
8.5
11.4
9.2
2.2 F
9.46
9.46
9.46
9.46
9.46
9.46
9.46
9.46
9.46
9.46 B
14.139
14.139
14.139
14.139
14.139
14.139
14.139
14.139
14.139
14.139 Chapter 12: Multisample Inference S
15.259
15.259
15.259
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15.259
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15.259
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15.259 Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Remarks The cut oﬀ points for diﬀerent pairwise comparisons would be
diﬀerent for each of the method if the sample sizes were not
equal.
We want a procedure that, ideally, not only controls EWER
but also has high power to detect true signiﬁcance (has the
smallest LSD).
Which of Bonferroni or Scheﬀe method should we use?
Fisher is not generally recommended because it does not
control EWER. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA False Discovery Rate (FDR)
Bonferroni’s and Schefe procedures control EWER but they
tend to be conservative when m and k are large.
Controlling EWER, in general, may not be practical for
largescale inference (large k and m).
For large scale inference, EWER results in conservative
inferential procedures.
An error rate which has gained popularity in genetic studies is
the socalled False Discovery Rate.
FDR is deﬁned as the expected proportion of falsely declared
signiﬁcant among all declared signiﬁcant,
FDR = E V
R . In general,
FDR ≤ αE .
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA BenjaminiHochberg Procedure
We ﬁrst compute p value for each comparison using a method
that controls (PCER) such as Fisher’s procedure.
Order the p values from the smallest to the largest...
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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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