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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.
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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.
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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
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⎛
⎝⎜
⎞
⎠⎟
=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.
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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