University of Missouri Kansas City Statistics Worksheet

665 – Assignment 6.2One-Way ANOVA
A researcher wants to examine the production capability of three manufacturing plants that
utilize different production methods of the same part in order to select a plant as a supplier for a
company. The effectiveness of each plant will be measured in the number of parts it can
produce in an hour. Representative hourly production amounts are recorded from each plant
over a period of 12 hours and provided to the researcher.
The results of each of the three plants are as follows:
Plant A
131
111
165
188
175
173
188
186
145
132
128
123
Plant B
141
165
174
185
172
188
145
177
162
151
147
133
Plant C
108
185
190
206
175
197
186
221
214
211
214
208
1. Use the five-step hypothesis testing process and StatCrunch to evaluate this scenario.
Perform a complete analysis and interpretation of the results. If appropriate, use
diagrams or graphs. Ensure your answer is detailed in all aspects. Be sure to comment
on the assumptions of the ANOVA and if there are any violations. Which manufacturing
plant, if any, would you recommend as the supplier? Why? Write up your results in APA
format.
Null hypothesis – fatality rates in aircraft = fatality rates in automobiles.
Each year is one data set.
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Statistics for the Behavioral and Social
Sciences: A Brief Course
Sixth Edition
Chapter 10
Introduction to Analysis of
Variance
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Learning Objectives (1 of 2)
10.1 Describe how ANOVA is used to compare two
estimates of population variance
10.2 Explain the F statistic in an analysis of variance
10.3 Conduct an analysis of variance
10.4 Summarize the steps of an analysis of variance
10.5 Outline the assumptions for an analysis of variance
10.6 Explain why researchers continue analysis after
getting a significant result from an ANOVA
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Learning Objectives (2 of 2)
10.7 Determine the appropriate sample size required to
ensure high power for an ANOVA
10.8 Interpret results of factorial research designs
10.9 Summarize how results of ANOVA are reported in
research articles
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Analysis of Variance
• Very often when doing research, you will have more than
one experimental group and one control group.
– e.g., one group receives treatment A, one group
receives treatment B, and one group does not receive
any treatment
• The analysis of variance (ANOVA) procedure is a
statistical procedure that tests the variation among the
means of more than two groups.
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Basic Logic of the Analysis of Variance
(ANOVA): Table 10-1
• Objective: Describe How ANOVA is Used to Compare Two Estimates of
Population Variance
• Question Used in Hazan and Shaver (Hazan, Shaver, 1987) Newspaper
Survey
Which of the following best describes your feelings? [Check One]
[ ] I find it relatively easy to get close to others and am comfortable depending
on them and having them depend on me. I don’t often worry about being
abandoned or about someone getting too close to me.
[ ] I am somewhat uncomfortable being close to others; I find it difficult to trust
them completely, difficult to allow myself to depend on them. I am nervous when
anyone gets too close, and often, love partners want me to be more intimate than
I feel comfortable being.
[ ] I find that others are reluctant to get as close as I would like. I often worry that
my partner doesn’t really love me or won’t want to stay with me. I want to merge
completely with another person, and this desire sometimes scares people away.
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Hypothesis Testing with the Analysis of
Variance
• The null hypothesis when using the ANOVA is that the
populations being compared all have the same mean.
• The goal of hypothesis testing with ANOVA is to
determine whether the means of the sample differ more
than you would expect if the null hypothesis were true.
• With ANOVA, we are examining the variation among
means.
• The ANOVA is about a comparison of the results of two
different ways of estimating population variances.
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Estimating Population Variance from
Variation within Each Sample
• With ANOVA, you estimate the population variance.
• You assume that all populations have the same variance.
• You pool the estimated population variances.
– This is called the within-groups estimate of the
population variance.
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Estimating the Population Variance from
Variation between the Means of the Samples
• Variation among the means is another way to estimate
the variance in the populations from which the samples
are chosen.
• If the null hypothesis is true, all populations are identical
and have the same mean variance and shape.
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The Effect of Variation in Population
Scores
• The more variance in each of several identical
populations, the more variance there will be among the
means of samples when you take a random sample from
each population
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Figure 10-01
Means of samples from identical populations will not be identical. (a) Sample
means from populations with less variation will vary less. (b) Sample means
from populations with more variation will vary more. Population means are
indicated by a triangle, sample means by an X.)
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When the Null Hypothesis is Not True
• If the null hypothesis is not true (and thus the research
hypothesis is true), the populations themselves do not
have the same mean.
• When the research hypothesis is true, the means of the
sample are spread out due to:
– Variation within each of the populations
▪ overall population variance
– Because of treatment effects
▪ variation among the means of the populations
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Figure 10-02
Means of samples from populations whose means differ (b) will vary more than
sample means taken from populations whose means are the same (a).
(Population means are indicated by a triangle, sample means by an X.)
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Summary of the Between-Groups Estimate
of the Population Variance
• Figured based on the variation among the means of the samples.
• If the null hypothesis is true
– this estimate gives an accurate indication of the variation within each of
the populations
• If the null hypothesis is false
– this method of estimating the population variance is influenced by
▪ the variation within each of the different populations and
▪ the variation among the different population’s means
– Variation due to treatment effect
– the between group estimate will not be an accurate estimate of the
variation within each of the populations due to the impact of the
variation among the populations
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Table 10-2
Sources of Variation in Within-Groups and BetweenGroups Variance Estimates
Blank
Variation Within
Populations
Variation Between Populations (Due to a
Treatment Effect Creating Differences
Between Populations)
Null hypothesis is true
Within-groups estimate
reflects
blank
Null hypothesis is true
Between-groups estimate
reflects
blank
Research hypothesis is true
Within-groups estimate
reflects
blank
Research hypothesis is true
Between-groups estimate
reflects
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The F Ratio
Objective: Explain the F statistic in an analysis of variance
• The ratio of the between-groups to the within-groups population
variance
• F Distribution
– distribution of F ratios when the null hypothesis is true
– mathematically defined curve that is the comparison distribution
• F Table
– table of cutoff scores on F distribution for various degrees of
freedom and significance levels
▪ You look up in the F table how extreme your F ratio needs to
be to reject the null hypothesis.
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An Analogy for the Analysis of Variance
• “Signal-to-noise ratio”
– your ability to make out the words in a staticky cell-phone
conversation depends on the strength of the signal versus
the amount of random noise.
– F ratio
▪ difference among the means of the samples is like the
signal
– information of interest
▪ variation within the samples is like the noise
▪ if the variation among the samples is sufficiently great
in comparison to the variation within the samples
– conclude that there is a significant effect
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How Are You Doing? (1 of 2)
• When do you use ANOVA?
• What are two sources of variation that can contribute to
the between-groups population variance estimate?
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How Are You Doing? (2 of 2)
• You use the analysis of variance when you are
comparing means of samples from more than two
populations
• Two sources of variation that can contribute to the
between-groups population variance estimate
– variation among the scores within each of the
populations
– variation among the means of the populations (that is,
variation due to a treatment effect making the
populations each have different means from each
other).
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Example of Carrying out an Analysis of
Variance (1 of 2)
• 15 volunteers who have been selected for jury duty (but have
not yet served at a trial)
– Criminal Record Group
▪ Last section says that the woman has been convicted
several times before for passing bad checks
– Clean Record Group
▪ Last section says the woman has a completely clean
criminal record
– No Information Group
▪ Sheet does not mention anything about criminal record
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Table 10-3 (1 of 3)
Results of the Criminal Record Study (Fictional Data)
Criminal Record Group
Rating
Deviation from
Mean
Squared Deviation
from Mean
10
2
4
7
−1
1
5
−3
9
10
2
4
8
0
0
0
18
summation: 40; capital M

: 40
equals 40 by 5 equals 8;
capital S squared equals
M
=440
54.5
=8
18 by
equals
S 2 = 18 4 = 4.5
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Table 10-3 (2 of 3)
Clean Record Group
Rating
Deviation
from Mean
Squared Deviation
from Mean
5
1
1
1
−3
9
3
−1
1
7
3
9
4
0
0
20
0
20
capital M equals 20 by 5 equals 4; capital S
squared equals 20 by 4 equals 5.0
M = 20 5 = 4
S 2 = 20 4 = 5.0
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Table 10-3 (3 of 3)
No Information Group
Rating
Deviation
from Mean
Squared Deviation
from Mean
4
−1
1
6
1
1
9
4
16
3
−2
4
3
−2
4
25
M = 25 5 = 5
0
26
capital M equals 25 by 5 equals 5; capital S squared
equals 20 by 4 equals 5.0
S 2 = 26 4 = 6.5
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Example of Carrying out an Analysis of
Variance (2 of 2)
Objective: Conduct an analysis of variance
• You need to figure the following numbers to test the
hypothesis that the three populations are different:
– within-groups estimate of the population variance
– population variance estimates based on each group’s
scores.
– the ratio of the two
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Figuring the Within-Groups Estimate of
the Population Variance
• Figure the population variance estimates based on each
group’s scores
– Calculate the sum of the squared deviation scores.
▪ Take the deviation of each score from its group’s
mean.
▪ Square that deviation score.
▪ Divide the sum of the squared deviation scores by
that group’s degrees of freedom (df = N – 1).
• Average these variance estimates
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Formula for the Within-Groups Estimate of the
Population Variance (When Sample Sizes Are Equal)
• From Table 10-3 the estimated population variances are:
– Criminal Record Group = 4.5
– Clean Record Group = 5.0
– No Information Group = 6.5
• Pool these variance estimates by straight averaging:
2
Within
S
2
SWithin
=
2
(S12 + S22 ……. + SLast
)
=
NGroups
(4.5 + 5.0 + 6.5.) 16
=
= 5.33
3
3
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Figuring the Between-Groups Estimate of
the Population Variance
• Step A: Estimate the variance of the distribution of
means.
– from the means of your samples
• Step B: Figure the estimated variance of the population of
individuals scores.
– based on the variance of the distribution of means
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Step A: Estimate the Variance of the
Distribution of Means
• Add up the sample means’ squared deviations from the
mean of all the scores (overall mean) and divide this by
the number of means minus 1.
– Find the mean of your sample means.
– Calculate the deviation of each sample mean from the
overall mean.
– Square each of these deviations.
– Sum the squared deviations.
– Divide the sum of the squared deviation by the
degrees of freedom (number of means minus 1).
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Formula for Estimating the Variance of the
Distribution of Means
• You are treating the means of your samples as the scores and the
distribution of means as the population from which the scores are
taken.
SM2 =
2
(
M
−
GM
)

dfBetween
– SM2 = estimated variance of the distribution of means
– M = mean of each of the samples
– GM = Grand Mean or overall mean of all your scores
– dfBetween = number of groups minus 1(NGroups − 1)
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Table 10-4
Estimated Variance of the Distribution of Means Based on Means of the Three
Experimental Groups in the Criminal Record Study (Fictional Data)
Sample Means
Deviation from Grand Mean
Squared Deviation
from Grand Mean
(M)
(M − GM)
( M − GM )
4
−1.67
2.79
8
2.33
5.43
5
−.67
.45
 17
−0.01
8.67
summation 17
capital M minus capital G capital M square
2
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Step B: Figure the Estimated Variance of
the Population of Individuals Scores
• Multiply the variance of the distribution of means by the number of
scores in each group.
2
SBetween
= ( SM2 ) ( n ) = ( 4.34 )( 5 ) = 21.70
2
= estimate of the population variance based on
– SBetween
variation among the means
– n = the number of participants in each sample
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Table 10-5
Comparison of Figuring the Variance of a Distribution of Means from
the Variance of a Distribution of Individuals, and the Reverse
• From distribution of individuals to distribution of means: SM2 = S 2 / n
• From distribution of means to distribution of individuals: S 2 = (SM2 ) ( n )
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Figuring the F Ratio
• Ratio of the between-groups estimate of the population
variance to the within-groups estimate of the population
variance
2
SBetween
21.70
F= 2
=
= 4.07
SWithin
5.33
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The F Distribution
• The F distribution is a distribution of all possible F ratios
calculated from random samples from identical populations.
• There are many different F distributions, the shapes of which
are determined by how many samples are taken each time
and how many scores are in each sample.
• The general shape is non-symmetrical and positively skewed
(long tail to the right).
• The F distribution cannot be lower than 0 and can rise quite
high.
– Most F ratios pile up near 1 and spread out more on the
positive side.
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Creating an F Distribution
• Start with three identical populations.
• Randomly select five scores from each.
• Figure the F ratio on the basis of these three samples (of five
scores each)
• Select three new random samples of five scores each and
figure the F ratio using these three samples.
• Do this whole process many, many times, and you will
eventually get a lot of F ratios.
• The distribution of all possible F ratios figured in this way (from
random samples from identical populations) is called the F
distribution.
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Figure 10-3
An F distribution
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The F Table
• To look up a cutoff score, you need to know
– between-groups degrees of freedom
▪ number of groups minus
dfBetween = NGroups − 1= 3 − 1= 2
– within-groups degrees of freedom
▪ sum of the degrees of freedom from each sample
you use when figuring out the within-groups variance
estimate
dfWithin = df1 + df2 +…..+ dfLast = 4 + 4 + 4 = 12
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Table 10-06
Selected Cutoffs for the F Distribution (with Values Highlighted for the Criminal Record
Study)
Denominat
or Degrees
of Freedom
Significance
Level
Numerator
Degrees of
Freedom 1
Numerator
Degrees of
Freedom 2
Numerator
Degrees of
Freedom 3
Numerator
Degrees of
Freedom 4
Numerator
Degrees of
Freedom 5
Numerator
Degrees of
Freedom 6
10
.01
10.05
7.56
6.55
6.00
5.64
5.39
10
.05
4.97
4.10
3.71
3.48
3.33
3.22
10
.10
3.29
2.93
2.73
2.61
2.52
2.46
11
.01
9.65
7.21
6.22
5.67
5.32
5.07
11
.05
4.85
3.98
3.59
3.36
3.20
3.10
11
.10
3.23
2.86
2.66
2.54
2.45
2.39
12
.01
9.33
6.93
5.95
5.41
5.07
4.82
12
.05
4.75
3.89
3.49
3.26
3.11
3.00
12
.10
3.14
2.76
2.56
2.43
2.35
2.28
13
.01
9.07
6.70
5.74
5.21
4.86
4.62
13
.05
4.67
3.81
3.41
3.18
3.03
2.92
13
.10
3.14
2.76
2.56
2.43
2.35
2.28
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Hypothesis Testing with the Analysis of
Variance: Table 10-07 (1 of 3)
Objective: Summarize the Steps of an Analysis of Variance
Steps for the Analysis of Variance
• Restate the question as a research hypothesis and a null
hypothesis about the populations.
• Determine the characteristics of the comparison distribution.
– The comparison distribution is an F distribution.
– The between-groups degrees of freedom is the number of
groups minus 1: dfBetween = NGroups − 1.
– The within-groups degrees of freedom is the sum of the degrees
of freedom in each group (the number in the group minus 1):
dfWithin = df1 + df2 +
+ dfLast .
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Hypothesis Testing with the Analysis of
Variance: Table 10-07 (2 of 3)
• Determine the cutoff sample score on the comparison
distribution at which the null hypothesis should be rejected.
– Decide the significance level.
– Look up the appropriate cutoff in an F table, using the degrees of
freedom from Step 2.
• Determine your sample’s score on the comparison distribution.
This will be an F ratio.
2
– Figure the between-groups population variance estimate SBetween .
(
)
▪ Figure the mean of each group
– Estimate the variance of the distribution of means:
2
SM2 =   ( M − GM )  dfBetween .


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Hypothesis Testing with the Analysis of
Variance: Table 10-07 (3 of 3)
▪ Figure the estimated variance of the population of
individual scores:
2
2 )(n)
SWithin
=(SM
(
2
)
– Figure the within-groups population variance estimate SBetween .
▪ Figure population variance estimates based on each group’s
2
scores: For each group, S 2 =  ( X − M )  ( n − 1)
– Average these variance estimates:
2
2
SWithin
or MSWithin = ( S12 + S22 + + SLast
) NGroups .
– Figure the F ratio: F = SBetween SWithin .
2
2
• Decide whether to reject the null hypothesis: Compare the scores
from Steps 3 and 4.
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Example of Hypothesis Testing with the
Analysis of Variance: Step 1
• Restate the question as a research hypothesis and a null
hypothesis about the populations.
– Population 1: jurors told that the defendant has a criminal
record
– Population 2: jurors told that the defendant has a clean
record
– Population 3: jurors given no information concerning the
defendant’s record
– The null hypothesis is that these three groups have the
same mean.
– The research hypothesis is that the population means are
not the same.
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Example of Hypothesis Testing with the
Analysis of Variance: Step 2 & 3
• Determine the characteristics of the comparison
distribution.
– The comparison distribution is an F distribution with 2
and 12 degrees of freedom.
• Determine the cutoff sample score on the comparison
distribution at which the null hypothesis should be
rejected.
– This is done using the F table.
▪ For a .05 significance level, the cutoff F ratio is
3.89.
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Example of Hypothesis Testing with the
Analysis of Variance: Step 4
• Determine your sample’s score on the comparison
distribution.
2
SBetween
F= 2
SWithin
(4.5 + 5.0 + 6.5) 16
=
= 5.33
3
3
2
SBetween
= SM2 (n ) = 4.34(5) = 21.70
2
SWithin
=
F=
21.70
= 4.07
5.33
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Example of Hypothesis Testing with the
Analysis of Variance: Step 5
• Decide whether to accept or reject the null
– The sample’s F ratio of 4.07 is more extreme than the
.05 significance level cutoff of 3.89.
– The researcher would reject the null hypothesis that
the three groups come from populations with the
same means.
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Assumptions in the Analysis of Variance
Objective: Outline the assumptions for an analysis of
variance
• The populations follow a normal curve.
• The populations’ variances are equal.
• If the variance estimate of the group with the largest
estimate is not more than four or five times that of the
smallest and the samples sizes are equal, the
conclusions using the F distribution should be reasonably
accurate.
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Comparing Each Group to Each Other
Group (1 of 2)
Objective: Explain why researchers continue analysis after
getting a significant result from an ANOVA
• After concluding that an F ratio is considered statistically
significant, the next step is to compare each population to
each other population.
– group 1 compared to group 2
– group 1 compared to group 3
– group 2 compared to group 3
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Comparing Each Group to Each Other
Group (2 of 2)
• Protected t tests
– t tests among pairs of means after finding that the F
for the overall difference among the means is
significant.
• Other tests such as Tukey’s HSD test are also used.
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Effect Size for the Analysis of Variance
Objective: Determine the appropriate sample size required to
ensure high power for an ANOVA
• Proportion of variance accounted for (R2 )
– measure of the effect size for the analysis of variance
2
▪ R is the proportion of the total variation of scores from
the grand mean that is accounted for by the variation
between the means of the groups.
– How much of the variance in the measured variable
is accounted for by the variable that divides up the
groups?
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2
Rcapital R square
2
R =
(S
2
Between
(S
) ( df
2
Between
)
) ( df
) + (S ) ( df
Between
Between
2
Within
Within
)
• R2 has a minimum of 0 and a maximum of 1.
• Cohen’s conventions for R2 :
– small effect size = .01
– medium effect size =.06
– large effect size = .14
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Power for the Analysis of Variance:
Table 10-8 (1 of 3)
Approximate Power for Studies Using the Analysis of Variance Testing
Hypotheses at the .05 Significance Level
Three groups ( dfBetween = 2 )
blank
Effect Size
Small ( R 2 = .01)
Medium ( R 2 = .06 ) Large ( R 2 = .14 )
capital R 2 equals .14
capital R 2 equals .06
capital R 2 equals .01
10
.07
.20
.45
20
.09
.38
.78
30
.12
.55
.93
40
.15
.68
.98
50
.18
.79
.99
100
.32
.98
*
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Power for the Analysis of Variance:
Table 10-8 (2 of 3)
Four groups ( dfBetween = 3 )
blank
Effect Size
Small ( R 2 = .01)
Medium ( R 2 = .06 ) Large ( R 2 = .14 )
capital R 2 equals .14
capital R 2 equals .06
capital R 2 equals .01
10
.07
.21
.51
20
.10
.43
.85
30
.13
.61
.96
40
.16
.76
.99
50
.19
.85
*
100
.36
.99
*
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Power for the Analysis of Variance:
Table 10-8 (3 of 3)
Five groups ( dfBetween = 4 )
blank
Effect Size
Small ( R 2 = .01)
Medium ( R 2 = .06 ) Large ( R 2 = .14 )
capital R 2 equals .14
capital R 2 equals .06
capital R 2 equals .01
10
.07
.23
.56
20
.10
.47
.90
30
.13
.67
.98
40
.17
.81
*
50
.21
.90
*
100
.40
*
*
*Nearly 1.
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Planning Sample Size: Table 10-9
Approximate Number of Participants Needed in Each Group (Assuming
Equal Sample Sizes) for 80% Power for the One-Way Analysis of
Variance Testing Hypotheses at the .05 Significance Level
blank
Effect Size
Small ( R 2 = .01)
capital R 2 equals .01
Effect Size
Effect Size
2
Medium ( R = .06 ) Large ( R 2 = .14 )
capital R 2
capital R 2 equals .14
equals .06
Three groups
df
322
52
21
d f sub
274
45
18
d f sub
240
39
16
( dfBetween = 2 )
sub Between equals 2
Four groups
( dfBetween = 3 )
Between equals 3
Five groups
( dfBetween = 4 )
Between equals 4
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Factorial Analysis of Variance
Objective: Interpret results of factorial research designs
• Allows for the analysis of data from studies looking at the
effect of two different factors on a variable of interest
• Factorial Research Design
– a study in which the influence of two or more
variables is studied at once by setting up the situation
so that a different group of people are tested for each
combination of the variables
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The Effects of Sensitivity and Test
Difficulty: Table 10-11
Factorial Research Design Employed by E. Aron and Colleagues (Aron,
Davies,2005)
Sensitivity
Test Difficulty
Easy
Test Difficulty
Hard
Not High
a
c
High
b
d
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Interaction Effects
• Situation in a factorial analysis of variance in which the
combination of variables has an effect that could not be
predicted from the effect of the two variables individually
– The effect of one variable on the measured variable is
different across the levels of the other variable that
divides the group.
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Example of an Interaction Effect: Table
10-12
Mean Levels of Negative Mood in the E. Aron and Colleagues (Aron,
Aron, Davies,2005) Study
Sensitivity
Test Difficulty
Easy
Test Difficulty
Hard
Not High
2.43
2.58
High
2.19
3.01
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Some Terminology
• Two-Way Factorial Research Design
– factorial design with two variables that each divide the
groups
• Two-Way Analysis of Variance
– analysis of variance for a two-way factorial research
design
• Grouping Variable
– variable that separates groups in an analysis of variance
• Independent Variable
– variable considered to be a cause, such as what group a
person is part of, for an analysis of variance
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More Terminology (1 of 2)
• One-Way Analysis of Variance
– analysis of variance in which there is only one
grouping variable
• Main Effect
– when the result for a variable averaged across the
other grouping variables is significant
• Cell
– each grouping in a factorial design
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More Terminology (2 of 2)
• Cell Mean
– the mean of the scores in each grouping combination
• Marginal Mean
– means of one grouping variable alone
• Dependent Variable
– a variable that represents the effect of the
experimental procedure
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Recognizing and Interpreting Interaction
Effects
• Can be described in words, using a graph, or using numbers
– Interaction
▪ effect of one grouping variable on the dependent
variable varies according to the level of another
grouping variable.
– Numerically
▪ pattern of difference in cell means across one row is
not be the same as the patterns of differences in cell
means across another row.
– Graphically
▪ pattern of bars in one section of the graph is different
from the pattern on the other section of the graph.
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Table 10-12
Mean Levels of Negative Mood in the E. Aron and Colleagues
Sensitivity
Test Difficulty
Easy
Test Difficulty
Hard
Not High
2.43
2.58
High
2.19
3.01
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Figure 10-04
Bar graph for level of negative mood as a function of sensitivity (not high v s
high) and test difficulty (easy v s hard).
ersu
ersu
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Relation of Interaction and Main Effects (1 of 2)
• There can be any combination of main and interaction
effects.
– All main effects and the interactions may be
significant.
– main effects for all variables, but no interaction effects
– main effect for one of the variables but no other
variables and an interaction effect
– no significant main or interaction effects
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Relation of Interaction and Main Effects (2 of 2)
• When there is no interaction, the main effect has a
straightforward meaning.
• When there is an interaction and main effect, the main
effects have to be interpreted cautiously because the
main effect may be created by just one of the levels of
the variable—but sometimes the main effect holds up
over and above the interaction.
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Extensions and Special Cases of the
Factorial Analysis of Variance
• Factorial analyses of variance can be extended into
three-way or higher designs.
• Repeated-measures analysis of variance
– used in cases when the same participants are tested
more than twice
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Analysis of Variance in Research Articles
Objective: Summarize how results of ANOVA are reported in
research articles
• ANOVA results are reported by giving the F, degrees of
freedom, and significance level.
– F ( dfBetween ,dfWithin ) = x.xx, p < .01 • Group means are usually given in a table. • For a factorial analysis of variance, the F ratios for any main effects and interaction effects are reported in the text and the cell means, and sometimes cell means and marginal means are reported in a table. Copyright © 2019, 2011, 2008 Pearson Education, Inc. All Rights Reserved Summary (1 of 3) • ANOVA compares the between-groups estimate of the population variance to the within-groups estimate of the population variance. • The F Ratio is the between-groups estimate divided by the within-groups estimate. • If the null hypothesis is true, the F ratio should be about 1. If the research hypothesis is true, the F ratio is larger than 1. • When the samples are of equal size, the within-groups population variance estimate is the average of the estimates of the population variance figured from each sample. Copyright © 2019, 2011, 2008 Pearson Education, Inc. All Rights Reserved Summary (2 of 3) • The distribution of F ratios when the null hypothesis is true is skewed to the right. • Significant cutoffs are determined using an F table according to the degrees of freedom for each population variance estimate. • The assumptions for the ANOVA are that the populations are normally distributed and the variances are equal. • When the null hypothesis is rejected in an analysis of variance, it suggests that the population means are not all the same. • Researchers often go on to use statistics to compare each population to each other population 2 • R is the proportion of the variance accounted for, and it is the measure of effect size for the ANOVA. Copyright © 2019, 2011, 2008 Pearson Education, Inc. All Rights Reserved Summary (3 of 3) • In a factorial design, you can study the effects of two independent variables on a dependent variable without needing twice as many participants, and it allows you to study the combination of the groupings. • An interaction effect is when the impact of one grouping varies according to the level of the other grouping variable. • A main effect is the impact of one grouping variable, ignoring the effect of the other grouping variables. • Analysis of variance results are reported in the following format: F ( dfBetween ,dfWithin ) = x.xx, p < .01 • A factorial analysis of variance, is described in the text as well as a table. • A graph may also be used to show interactions. Copyright © 2019, 2011, 2008 Pearson Education, Inc. All Rights Reserved Copyright Copyright © 2019, 2011, 2008 Pearson Education, Inc. All Rights Reserved