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Chapter Thirteen
The One-Way Analysis of Variance
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 2
New Statistical Notation
1.Analysis of variance is abbreviated as ANOVA
2.An independent variable is called a factor
3.Each condition of the independent variable is also called a level or a treatment, and differences produced by the independent variable are a treatment effect
4.The symbol for the number of levels in a factor is k
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 3
An Overview of ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 4
One-Way ANOVA
A one-way ANOVA is performed when
only one independent variable is tested in
the experiment
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 5
Between Subjects
• When an independent variable is studied using independent samples in all conditions, it is called a between-subjects factor
• A between-subjects factor involves using the formulas for a between-subjects ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 6
Within Subjects Factor
• When a factor is studied using related (dependent) samples in all levels, it is called a within-subjects factor
• This involves a set of formulas called a within-subjects ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 7
Analysis of Variance
• The analysis of variance is the parametric procedure for determining whether significant differences occur in an experiment containing two or more sample means
• In an experiment involving only two conditions of the independent variable, you may use either a t-test or the ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 8
Diagram of a Study Having ThreeLevels of One Factor
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 9
Experiment-Wise Error
• The overall probability of making a Type I error somewhere in an experiment is call the experiment-wise error rate
• When we use a t-test to compare only two means in an experiment, the experiment-wise error rate equals
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 10
Comparing Means
• When there are more than two means in an experiment, the multiple t-tests result in an experiment-wise error rate that is much larger than the one we have selected
• Using the ANOVA allows us to compare the means from all levels of the factor and keep the experiment-wise-error rate equal to
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 11
Assumptions of the ANOVA
1.The experiment has only one independent variable and all conditions contain independent samples
2.The dependent variable measures interval or ratio scores
3.The population represented by each condition forms a normal distribution
4.The variances of all populations represented are homogeneous
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 12
kH 210 :
equalaresallnot:a H
Statistical Hypotheses
•
•
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 13
The F-Test
• The statistic for the ANOVA is F
• When Fobt is significant, it indicates only that somewhere among the means at least two of them differ significantly
• It does not indicate which specific means differ significantly
• When the F-test is significant, we perform post hoc comparisons
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 14
Post Hoc Comparisons
• Post hoc comparisons are like t-tests
• We compare all possible pairs of means from a factor, one pair at a time, to determine which means differ significantly
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 15
Components of ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 16
Sources of Variance
• There are two potential sources of variance
• Scores may differ from each other even when participants are in the same condition. This is called variance within groups
• Scores may differ from each other because they are from different conditions. This is called the variance between groups
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 17
Mean Squares
• The mean square within groups is an estimate of the variability in scores as measured by differences within the conditions of an experiment
• The mean square between groups is an estimate of the differences in scores that occurs between the levels in a factor
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 18
Performing the ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 19
Sum of Squares
• The computations for the ANOVA require the use of several sums of squared deviations
• Each of these terms is called the sum of squares and is symbolized by SS
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 20
Source Sum of df Mean FSquares Squares
Between SSbn dfbn MSbn Fobt
Within SSwn dfwn MSwn
Total SStot dftot
Summary Table of a One-way ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 21
N
XXSS
2tot2
tottot
)(
Computing Fobt
1.Compute the total sum of squares (SStot)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 22
N
X
nSS
2tot
2
bn
)(
columntheinscoresof
)columntheinscoresofsum(
Computing Fobt
2.Compute the sum of squares between groups (SSbn)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 23
Computing Fobt
3.Compute the sum of squares within groups (SSwn)
SSwn = SStot - SSbn
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 24
Computing Fobt
4. Compute the degrees of freedom1. The degrees of freedom between groups
equals k - 1
2. The degrees of freedom within groups equals N - k
3. The degrees of freedom total equals N - 1
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 25
5.Compute the mean squares
bn
bnbn df
SSMS
wn
wnwn df
SSMS
Computing Fobt
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 26
wn
bnobt MS
MSF
Computing Fobt
6.Compute Fobt
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 27
The F-Distribution
The F-distribution is the sampling
distribution showing the various values of
F that occur when H0 is true and all
conditions represent one population
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 28
Sampling Distribution of F When H0 Is True
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 29
Critical F Value
• The critical value of F (Fcrit) depends on
– The degrees of freedom (both the dfbn = k - 1 and the dfwn = N - k)
– The selected
– The F-test is always a one-tailed test
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 30
Performing Post Hoc Comparisons
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 31
21wn
21obt
11nn
MS
XXt
Fisher’s Protected t-Test
• When the ns in the levels of the factor
are not equal, use Fisher’s protected
t-test
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 32
• When the ns in all levels of the factor are equal, use the Tukey HSD multiple comparisons test
where qk is found using the appropriate
table
n
MSqHSD k
wn)(
Tukey’s HSD Test
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 33
Additional Procedures in the One-Way ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 34
Xtn
MSXt
n
MS
)()( crit
wncrit
wn
Confidence Interval
• The computational formula for the confidence interval for a single is
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 35
A graph showing means from three conditions of an independent variable.
Graphing the Results in ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 36
• Eta squared indicates the proportion
of variance in the dependent variable
that is accounted for by changing the
levels of a factor
2
tot
bn2
SS
SS
Proportion of Variance Accounted For
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 37
Group 1 Group 2 Group 3
14 14 10 13 11 15
13 10 12 11 14 13
14 15 11 10 14 15
Example
• Using the following data set, conduct a one-way ANOVA. Use = 0.05
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 38
611.5518
2292969
)( 22tot2
tottot
N
XXSS
111.2218
229
6
82
6
67
6
80
)(
columntheinscoresof
)columntheinscoresofsum(
2222
2tot
2
bn
N
X
nSS
50.33111.22611.55bntotwn SSSSSS
Example
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 39
Example
• dfbn = k - 1 = 3 - 1 = 2
• dfwn = N - k = 18 - 3 = 15
• dftot = N - 1 = 18 - 1 = 17
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 40
055.112
111.22
bn
bnbn
df
SSMS
233.215
50.33
wn
wnwn
df
SSMS
951.4233.2
055.11
wn
bnobt
MS
MSF
Example
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 41
Example
• Fcrit for 2 and 15 degrees of freedom and = 0.05 is 3.68
• Since Fobt = 4.951, the ANOVA is significant
• A post hoc test must now be performed
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 42
242.26
233.2675.3)( wn
n
MSqHSD k
334.0333.13667.13
500.2167.11667.13
166.2167.11333.13
13
23
21
XX
XX
XX
Example
• The mean of sample 3 is significantly different from the mean of sample 2