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PS 225
Lecture 15
Analysis of Variance
ANOVA Tables
Analysis of Variance
 Compare the values of a variable when
grouped according to the values of
another variable
 Grouping variable called a ‘factor’
Differences Between Groups
 Observed sample differences caused by:


True population differences
Variation
 Different Types of Variance:


Variation within groups
Variation between group means
Two Types of Variability
 With-in group variability


Similar in concept to standard deviation
Variability between the data in a sample
 Between-group variability


Similar to the standard deviation of sample
means
Variation between sample means
4
6
8
Hours Spent at School Per Day
10
12
Box Plot
Middle
High School
College
Assumptions Needed for
ANOVA
 Independent random samples taken
from each population
 Normality
 Equality of Variance


Levine Test
Visual Examination of Box plot
Comparing the Variation
Types
Between-Groups Mean
Square Error
Between-Groups Variation
F =
=
Within-Groups Variation
Division creates a ratio
of the different types of
variation
Within-Groups Mean Square
Error
Mean Square Error
 A measure of variation
 Takes into account number of samples
Within-Groups Sum of
Squares
 Take standard deviation for all groups
and square them to obtain variance
 Multiply each variance by the degree of
freedom for each group (n-1)
Between-Groups Sum of
Squares
 Subtract the overall mean from each
group mean and square the difference.
Multiply the square by the number of
observations in the group. Add all of the
results to get the mean square
 Calculate the degrees of freedom- the
number of groups minus 1
 Divide the mean square by the degrees
of freedom
F-distribution
 Probability Distribution of the ratio of
mean squares
 Small significance means reject Ho
 Ho: The mean is the same for all groups
Why The F-Distribution?
 The more individual mean difference
tests conducted, the greater the
probability of observing a mean
difference when there is none
 Conduct f-test to determine if there are
differences
 Conduct Bonferroni multiple
comparisons test to determine which
means are different
SPSS Anova Table
Salary divided by Education Level
SPSS Bonferroni Comparison
Multiple Comparisons
Dependent Variable: Current Salary
Bonferroni
(J)
Employme
nt
(I) Employment Category Category
Clerical
Cus todial
Manager
Cus todial
Clerical
Manager
Manager
Clerical
Cus todial
Mean
Difference
(I-J)
-$3,100.35
-$36,139.26*
$3,100.35
-$33,038.91*
$36,139.26*
$33,038.91*
*. The mean difference is s ignificant at the .05 level.
Std. Error
$2,023.760
$1,228.352
$2,023.760
$2,244.409
$1,228.352
$2,244.409
Sig.
.379
.000
.379
.000
.000
.000
95% Confidence Interval
Lower Bound Upper Bound
-$7,962.56
$1,761.86
-$39,090.45
-$33,188.07
-$1,761.86
$7,962.56
-$38,431.24
-$27,646.58
$33,188.07
$39,090.45
$27,646.58
$38,431.24
Assignment
 What determines the age at which an
individual is first married? Sex?
Education (highest degree)?



Determine if there are mean differences
using an Anova table or Independent
Sample T-test
Determine which means are significantly
different using the Boniferri Comparison
when applicable
Explain the relationship between all
variable pairs, can you determine a
specific cause of early marriage?