10/28/2006 P331 Analysis of Variance Introduction 1 Analysis of Variance (ANOVA) 10/28/2006 P331 Analysis of Variance Introduction 2 What We Will Cover in This Section • Introduction. • Overview. • Simple ANOVA. – Theory – Practice 10/28/2006 P331 Analysis of Variance Introduction 3 Situation The management of Saul T. Dogg’s restaurant wanted to see if the saltiness of appetizers would influence the number of drinks people purchased. Three sections of the club are targeted to receive appetizers that have either low, medium, or high saltiness. The dependent variable is the number of drinks ordered.
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10/28/2006 P331 Analysis of Variance Introduction
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Analysis of Variance(ANOVA)
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What We Will Cover in This Section
• Introduction.• Overview.• Simple ANOVA.
– Theory– Practice
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Situation
The management of Saul T. Dogg’srestaurant wanted to see if the saltiness of appetizers would influence the number of drinks people purchased. Three sections of the club are targeted to receive appetizers that have either low, medium, or high saltiness. The dependent variable is the number of drinks ordered.
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Appetizer saltiness and number or drinks ordered.
M = 1.80M = 3.90M = 2.00244142222131142362251241143332
Group 3 High SaltGroup 2 Medium SaltGroup 1 Low Salt
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Issue
How to determine if one mean is significantly different from the other means while minimizing the probability of committing a Type I error.
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Analysis of Variance:Background and Theory
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t-test Logic
1X 2X
Distance Between Means
Treatment + Random Error
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ANOVA Logic
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Treatment Effects
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Total Variability
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Between Groups Variability
:t1 :t2 :t3
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Within Groups Variability
:t1 :t2 :t3
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Partitioning the Variance
TotalVariability
Between GroupsVariability
Within GroupsVariability
Treatment Effect Random Error Random Error
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Where We Are Going
( )Random Erro) rR
Treatmandom
ent + (Error
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ANOVA Model
:t1 :t2 :t3
:G
+= RandomErrorTreatment EffectTotal
Effect
+= Within GroupsVariability
Between Groups
Variability
TotalVariability
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Partitioning the Variance
+= Within GroupsVariability
Between Groups
Variability
TotalVariability
BetweenGroups WithinGroupsTotal
Total Between Within
SS SSSSdf df df
= +
(N - 1) (k – 1) (N – k)
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The F ratio
:t1 :t2 :t3
:G
F(k-1,N-k) MSW
MSB=
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F-test Compared to t-test
F(k-1,N-k)MSW
MSB=t(df)
σX
X1 - X2=
7
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What Does the F Mean?
Random Error
Treatment Variability+
Random Error=F
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The Critical Value of F
• See page 485 in text book.• Notice
– Need df between (numerator) for columns.– Need df within (denominator) for rows.– As df increases the critical values get
smaller.
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The ProblemReturns
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The Situation (in case you forgot)
The management of Sal T. Dogg’s restaurant wanted to see if the saltiness of appetizers would influence the number of drinks people purchased. Three sections of the club are targeted to receive appetizers that have either low, medium, or high saltiness. The dependent variable is the number of drinks ordered.
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Hypotheses
Research Hypothesis.Saltiness of the appetizers will influence the number of drinks that people buy.
Null Hypothesis. Saltiness will not influence the number of drinks that people buy.
Statistical Hypothesis.µ1 ≠ µ2 ≠ µ3
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Appetizer saltiness and number or drinks ordered.
X = 1.80X = 3.90X = 2.00244142222131142362251241143332
Group 3 High SaltGroup 2 Medium SaltGroup 1 Low Salt
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Graph of Saltiness Ratings
GROUP
High SaltinessMedium SaltinessLow saltiness
Mea
n of
SAL
TINE
S
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
.5
0.0
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ANOVA Summary Table
2951.37Total
.912724.50Within Groups
14.7713.435226.87Between Groups
F(crit=3.35)MSdfSSSource
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How to Express F
F (2,27) = 14.77, p<.05
Calculated FDegrees of freedom(between, within) Alpha
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Post Hoc Tests
When F is significant, how do you determine which of the means differs from the others?
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Tukey Honestly Significant Difference Test (HSD)
( , , )within
withindf k
MSHSD qnα=
q = Value from table, page 488α = desired significance leveldfwithin = within groups dfk = Number of groups.
(.05,27,3).9110
HSD q=
3.53 .091HSD x=
1.06HSD =
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2.00
3.90
1.80
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
How To Use Tukey (HSD = 1.06)
1.902.10
.20
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Effect Size: Eta Squared (η2)
2 Between
Total
SSSS
η =2 26.87
51.37η =
2 .523η =
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Assumptions
1. The observations within each sample are independent.
2. The population from which the samples are selected is normally distributed.
3. The population from which the samples are selected have equal variances (homogeneity of variance)
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Another ANOVA Example
Sal O. Gysm felt that the perceived difficulty of logic problems would influence performance on these problems. Sal developed a set of problems and gave them to three groups. One group was told that the problems was easy, another was told that they were moderately difficult, and the third was told that they were difficult. The dependent variable was the number of problems solved.
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ANOVA: Example 2
M = 3.0M = 6.0M = 8.0
13452
463210
912487
DifficultModerateEasy
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Logic Problem Results
012345678
No of problems solved
Easy Moderate Difficult
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ANOVA: Summary Table
4.5231.677.00
21214
63.3384.00
147.33
BetweenWithinTotal
FMSdfSSSource
Is this value significant?
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Post hoc Analysis: Tukey HSD4.46HSD =
8
6
3
0123456789
Easy Moderate Difficult
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Effect Size: Eta2 2( )η
2 between
total
SSSS
η =2 63.33
147.33η =
2 .428η =
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Key Learning Points
1. The F test is used for comparing three or more means.
2. The Total Variance is broken down into two components
- Between Group Variance (Treatment plus error)- Within Group Variance (Error)
3. The F ratio is computed by dividing Mean Square Between by the Mean Square Within.
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Key Learning Points
4. The F ratio tells you ONLY if all of the means together are different.
5. The post hoc tests tell you which pairs of means differ.
6. Eta2 tells you the amount of variability accounted for in the treatment.
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Tess Tosterone is studying aggression among adolescent girls. She believes that there is a relationship between the level of interaction a girl has with her mother and
the girl’s level of aggression. She has identified fifteen girls who fall into one of
three maternal interaction levels (low, medium, and high) and has measured their
aggression scores.
The scores are shown on the next slide.
Another Practice Problem
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Data Summary Table
M = 1.00M = 5.00M = 6.00
026
144
059
485
066
High InteractionModerate InteractionLow interaction
