ivmultiple comparisons a.contrast among population means ( i )
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IV Multiple Comparisons
A. Contrast Among Population Means (i)
1. A contrast among population means is a
difference among the means with appropriate
algebraic sign.
pairwise contrast:
nonpairwise contrast:
i =μ j −μ ′j
i =μ j −(μ ′j + μ ′′j ) / 2
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2. Contrasts are defined by a set of underlying
coefficients (cj ) with the following characteristics:
The sum of the coefficients must equal zero,
c j =c1 + c2 +L + cp =0j=1
p∑ .
cj ≠ 0 for some j
For convenience (to put all contrasts on the
same measurement scale), coefficients are
chosen so that c jj=1
p∑ =2.
3
3. Pairwise contrast for means 1 and 2, where c1 = 1
and c2 = –1
1 =(1)μ1 + (−1)μ2
4. Nonpairwise contrast for means 1, 2, and 3, where
c1 = 1, c2 = –1/2, and c3 = –1/2
2 =(1)μ1 + (−1 / 2)μ2 + (−1 / 2)μ3
=μ1 −
μ2 + μ32
=μ1−μ2
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5. Pairwise contrast: all of the coefficients except
two are equal to 0.
6. Nonpairwise contrast: at least three coefficients
are not equal to 0.
7. A contrast among sample means, denoted by
is a difference among the sample means with
appropriate algebraic sign.
i ,
Pairwise contrast 1 =(1)X1 + (−1)X2
Nonpairwise contrast
2 =(1)X1 + (−1/ 2)X2 + (−1/ 2)X3
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V Fisher-Hayter Multiple Comparison Test
A. Characteristics of the Test
1. The test uses a two-step procedure. The
first steps consists of testing the omnibus
null hypothesis using an F statistic.
2. If the omnibus test is significant, the
Fisher-Hayter statistic is used to test all
pairwise contrasts among the p means.
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B. Fisher-Hayter Test Statistic
qFH =X. j −X. ′j
MSWG2
1nj
+1
n ′j
⎛
⎝⎜
⎞
⎠⎟
where and are means of random samples from
normal populations, MSWG is the denominator of the
F statistic from an ANOVA, and nj and nj′ are the
sizes of the samples used to compute the sample
means.
X. j
X. ′j
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1. Reject H0: μj = μj′ if |qFH| statistic exceeds or
equals the critical value, , from the
Studentized range table (Appendix Table D.10).
C. Computational Example Using the Weight- Loss Data
1. Step 1. Test the omnibus null hypothesis
qα; p−1,ν
F =
MSBGMSWG
=43.3345.037
=8.60*
2. Step 2. Because F is significant, test all pairwise
contrasts using qFH.
*p < .02
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qFH =X. j −X. ′j
MSWG2
1nj
+1
n ′j
⎛
⎝⎜
⎞
⎠⎟
qFH =8.00 −9.00
5.0372
110
+110
⎛⎝⎜
⎞⎠⎟
=−1.41
qFH =8.00 −12.00
5.0372
110
+110
⎛⎝⎜
⎞⎠⎟
=−5.64*
qFH =9.00 −12.00
5.0372
110
+110
⎛⎝⎜
⎞⎠⎟
=−4.23*
1 =X.1 −X.2
2 =X.1 −X.3
3 =X.2 −X.3
q.05; 3−1, 27 ≅2.90
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D. Assumptions of the Fisher-Hayter Test
1. Random sampling or random assignment
of participants to the treatment levels
2. The j = 1, . . . , p populations are normally
distributed.
3. The variances of the j = 1, . . . , p populations are
equal.
10
VI Scheffé Multiple Comparison Test and Confidence Interval
A. Characteristics of the Test
1. The test does not require a significant
omnibus test.
2. Can test both pairwise and nonpairwise contrasts
and construct confidence intervals.
3. The test is less powerful than the Fisher-Hayter
test for pairwise contrasts.
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B. Scheffé Test Statistic
FS =(c1X.1 + c2X.2 +L + cpX.p)
2
MSWGc12
n1+
c22
n2+L +
cp2
np
⎛
⎝⎜
⎞
⎠⎟
where c1, c2, . . . , cp are coefficient that define a
contrast, , . . . , are sample means, MSWG
is the denominator of the ANOVA F statistic, and n1,
n2, . . . , np are the sizes of the samples used to
compute the sample means.
X.1
X.p
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1. Reject a null hypothesis for a contrast if the FS
statistic exceeds or equals the critical value,
is obtained from the
F table (Appendix Table D.5).
C. Computational Example Using the Weight- Loss Data
1. Critical value is
( p −1)Fα;ν1, ν2
. Fα;ν1, ν2
(3−1)F.05; 2, 27 =(2)(3.35) =6.70.
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FS =(c1X.1 + c2X.2 +L + cpX.p)
2
MSWGc12
n1+
c22
n2+L +
cp2
np
⎛
⎝⎜
⎞
⎠⎟
FS =[(1)8.00 + (0)9.00 + (−1)12.00]2
5.037(1)2
10+(0)2
10+(−1)2
10
⎛
⎝⎜
⎞
⎠⎟
=15.88* 1 =X.1 −X.3
FS =[(0)8.00 + (1)9.00 + (−1)12.00]2
5.037(0)2
10+
(1)2
10+
(−1)2
10
⎛
⎝⎜
⎞
⎠⎟
= 8.93* 2 =X.2 −X.3
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FS =[(12)8.00 + (1
2)9.00 + (−1)12.00]2
5.037(12)2
10+(12)2
10+(−1)2
10
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=16.21* 3 =
X.1−X.2
2−X.3
D. Two-Sided Confidence Interval
i − (p−1)Fα;ν1,ν2 MSWGcj2
njj=1
p∑ < i
< ψ i + (p −1)Fα ;ν1,ν 2MSWG
c j2
n jj =1
p∑
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1. Computational example for =( 12 )μ1 + ( 12 )μ2 −μ3
(12)8.0 + (12)9.0 + (−1)12.0⎡⎣ ⎤⎦− (2)(3.35) (5.037)
(12)2 + (12)
2 + (−1)2
10 +10 +10⎡
⎣⎢
⎤
⎦⎥
< (12)8.0 + (1
2)9.0 + (−1)12.0⎡⎣ ⎤⎦+ (2)(3.35) (5.037)(1
2)2+ (1
2)2+ (−1)2
10 + 10 + 10
⎡
⎣⎢
⎤
⎦⎥
<
−5.45 < < −1.55
L
1 = – 5 . 4 5 L
2 = – 1 . 5 5
– 5 – 4 – 3 – 2 – 1 0– 6
( 1
2)μ1 + ( 12 )μ2 −μ3
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E. Assumptions of the Scheffé Test and Confidence Interval
1. Random sampling or random assignment
of participants to the treatment levels
2. The j = 1, . . . , p populations are normally
distributed.
3. The variances of the j = 1, . . . , p populations are
equal.
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F. Comparison of Fisher-Hayter and SchefféTests
1. The Fisher-Hayter test controls the Type I error
at α for the collection of all pairwise contrasts.
2. The Scheffé test controls the Type I error at α for
the collection of all pairwise and nonpairwise
contrasts.
3. The Scheffé statistic can be used to construct
confidence intervals.
18
VII Practical Significance
A. Omega Squared
1. Omega squared estimates the proportion of the
population variance in the dependent variable that
is accounted for by the p treatments levels.
2. Computational formula
(ω2 )
ω 2 =(p−1)(F −1)
(p−1)(F −1)+np
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3. Cohen’s guidelines for interpreting omega squared
ω2 =.010 is a small association
ω2 =.059 is a medium association
ω2 =.138 is a large association
4. Computational example for the weight-loss data
ω 2 =
(p−1)(F −1)(p−1)(F −1) + np
=(3−1)(8.60−1)
(3−1)(8.60−1) + (10)(3)=.34
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B. Hedges’s g Statistic
1. g is used to assess the effect size of contrasts
g =X. j −X. ′j
σPooled
σPooled = MSWG
2. Computational example for the weight-loss data
σPooled = 5.037 =2.244
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g =|8 −12|2.244
=1.78 2 =X.1 −X.3
g =|9 −12|2.244
=1.34 3 =X.2 −X.3
g =|8 −9|2.244
=0.45 1 =X.1 −X.2
g =.2 is a small effect
g =.5 is a medium effect
g =.8 is a large effect
3. Guidelines for interpreting Hedges’s g statistic
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