Chapter 15 Introduction to the Analysis of Variance IThe 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.
Introduction to the Analysis of Variance
IThe Omnibus Null Hypothesis
H0: 1 = 2 = . . . = p
H1: j = j’
A.Answering General Versus Specific Research
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.
IIBasic Concepts In ANOVA
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
B.Composite Nature of a Score
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
2.Sample model equation for a score
3.The statistics estimate parameters of the model
equation as follows
4.Illustration of the sample model equation using the
weight-loss data in Table 1.
Table 1. One-Month Weight Losses for Three Diets
a1 a2 a3
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).
C.Partition of the Total Sum of Squares (SSTO)
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.
IIICompletely Randomized Design (CR-p Design)
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
C.Descriptive Statistics for Weight-Loss Data
In Table 1
Table 2. Means and Standard Deviations for Weight-Loss Data
Figure 1. Stacked box plots for the weight-loss data. The
distributions are relatively symmetrical and have similar
Table 3. Computational Procedures for CR-3 Design
a1 a2 a3
D.Sum of Squares Formulas for CR-3 Design
Table 4. ANOVA Table for Weight-Loss Data
SourceSS df MS F
1.Between86.667p – 1 = 243.3348.60*groups (BG)
2.Within 136.000p(n – 1) = 275.037
3. Total222.667np – 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