Chapter 15 Introduction to the Analysis of Variance I The Omnibus Null Hypothesis H 0 : 1 = 2 = . . . = p H 1 : j = j ’. A. Answering General Versus Specific Research Questions 1. Population contrast, i , and sample contrast.
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Introduction to the Analysis of Variance
I The Omnibus Null Hypothesis
H0: 1 = 2 = . . . = p
H1: j = j’
1. Population contrast, i, and sample contrast
2. Pairwise and nonpairwise contrasts
B. Analysis of Variance Versus Multiple t Tests
1. Number of pairwise contrasts among p means
is given by p(p– 1)/2
p = 3 3(3 – 1)/2 = 3
p = 4 4(4 – 1)/2 = 6
p = 5 5(5 – 1)/2 = 10
2. If C = 3 contrasts among p = 3 means are tested
using a t statistic at = .05, the probability of
one or more type I errors is less than
3. As C increases, the probability of making one or
more Type I errors using a t statistic increases
4.Analysis of variance tests the omnibus null
hypothesis, H0: 1 = 2 = . . . = p , and controls
probability of making a Type I error at, say,
= .05 for any number of means.
5. Rejection of the null hypothesis makes the
alternative hypothesis, H1: j ≠ j’, tenable.
1. Two subscripts are used to denote a score, Xij.
The i subscript denotes one of the i = 1, . . . , n
participants in a treatment level. The j subscript
denotes one of the j = 1, . . . , p treatment levels.
2. The jth level of treatment A is denoted by aj.
a1 a2 a3 a4
X11 X12 X13 X14
X21 X22 X23 X24
Xn1 Xn2 Xn3 Xn4
1. A score reflects the effects of four variables:
characteristics of the participants in the
chance fluctuations in the participant’s
environmental and other uncontrolled
3. The statistics estimate parameters of the model
equation as follows
weight-loss data in Table 1.
Table 1. One-Month Weight Losses for Three Diets
a1 a2 a3
7 10 12
9 13 11
8 9 15
6 7 14
5. Let X11 = 7 denote Joan’s weight loss. She used
diet a1. Her score is a composite that tells a story.
6. Joan used a less effective diet than other girls
(8 – 9.67 = –1.67), and she lost less weight than
other girlson the same diet (8 – 9 = –1).
1. The total variability among scores in the diet
also is a composite that can be decomposed into
between-groups sum of squares (SSBG)
within-groups sum of squares (SSWG)
D. Degrees of Freedom for SSTO, SSBG, and
1. dfTO = np – 1
2. dfBG = p – 1
3. dfWG = p(n – 1)
E. Mean Squares, MS, and F Statistic
F. Nature of MSBG and MSWG
1. Expected value of MSBG and MSWG when the
null hypothesis is true.
2. Expected value of MSBG and MSWG when the
null hypothesis is false.
3. MSBG represents variation among participants
who have been treated differently—received
different treatment levels.
4. MSWG represents variation among participants
who have been treated the same—received
the same treatment level.
5. F = MSBG/MSWG values close to 1 suggest that
the treatment levels did not affect the dependent
variable; large values suggest that the treatment
levels had an effect.
A. Characteristics of a CR-p Design
1. Design has one treatment, treatment A, with p
2. N = n1 + n2 + . . . + np participants are randomly
assigned to the p treatment levels.
3. It is desirable, but not necessary, to have the same
number of participants in each treatment level.
B. Comparison of layouts for a t-test design for
independent samples and a CR-3 design
Participant1 a1Participant1 a1
Participant2 a1 Participant2 a1
Participant10 a1 Participant10 a1
Participant11 a2 Participant11 a2
Participant12 a2 Participant12 a2
Participant20 a2 Participant20 a2
In Table 1
Table 2. Means and Standard Deviations for Weight-Loss Data
a1 a2 a3
2.21 2.21 2.31
distributions are relatively symmetrical and have similar
a1 a2 a3
7 10 12
9 13 11
8 9 15
6 7 14
Source SS df MS F
1. Between 86.667 p – 1 = 2 43.334 8.60* groups (BG)
2. Within 136.000 p(n – 1) = 27 5.037
3. Total 222.667 np – 1 = 29
*p < .002
E. Assumptions for CR-p Design
1. The model equation,
reflects all of the sources of variation that affect
2. Random sampling or random assignment
3. The j = 1, . . . , p populations are normally
4. Variances of the j = 1, . . . , p populations are