Download analysis of variance

CHAPTER 12
Analysis of Variance
(ANOVA)
© Copyright McGraw-Hill 2000
12-1
Objectives

Use the one-way ANOVA technique to
determine if there is a significant difference
among three or more means.

Determine which means differ using Scheffe
or Tukey test if the the null hypothesis is
rejected in the ANOVA.

Use the two-way ANOVA technique to
determine if there is a significant difference in
the main effects or interaction.
© Copyright McGraw-Hill 2000
12-2
Introduction

The F-test, used to compare two variances,
can also be used to compare three of more
means.

This technique is called analysis of variance
or ANOVA.

For three groups, the F-test can only show
whether or not a difference exists among the
three means, not where the difference lies.

Other statistical tests, Scheffé test and the
Tukey test, are used to find where the
difference exists.
© Copyright McGraw-Hill 2000
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Analysis of Variance

When an F-test is used to test a hypothesis
concerning the means of three or more
populations, the technique is called analysis
of variance (commonly abbreviated as
ANOVA).

Although the t-test is commonly used to
compare two means, it should not be used to
compare three or more.
© Copyright McGraw-Hill 2000
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Assumptions for the F-test

The following assumptions apply when using
the F-test to compare three or more means.
1.
The populations from which the samples
were obtained must be normally or
approximately normally distributed.
2.
The samples must be independent of each
other.
3.
The variances of the populations must be
equal.
© Copyright McGraw-Hill 2000
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F-Test

In the F-test, two different estimates of the
population variance are made.

The first estimate is called the between-group
variance, and it involves finding the variance
of the means.

The second estimate, the within-group
variance, is made by computing the variance
using all the data and is not affected by
differences in the means.
© Copyright McGraw-Hill 2000
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F-Test (cont’d.)

If there is no difference in the means, the
between-group variance will be approximately
equal to the within-group variance, and the F
test value will be close to 1—the null
hypothesis will not be rejected.

However, when the means differ significantly,
the between-group variance will be much
larger than the within-group variance; the Ftest will be significantly greater than 1—the
null hypothesis will be rejected.
© Copyright McGraw-Hill 2000
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Hypotheses


For a test of the difference among three or more
means, the following hypotheses should be used:
H0: 1  2 
•••
 k, where k is the number of groups

H1: At least one mean is different from the
others.

A significant test value means that there is a high
probability that this difference in the means is not
due to chance.
© Copyright McGraw-Hill 2000
12-8
Degrees of Freedom

The degrees of freedom for this F-test are
d.f.N.  k  1,
where k is the number of groups,

And d.f.D.  N  k
where N is the sum of the sample sizes of the groups,
N  n1 n2 • • •  nk.

The sample sizes do not need to be equal.

The F-test to compare means is always right-tailed.
© Copyright McGraw-Hill 2000
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Procedure for Finding F-Test Value

Step 1
Find the mean and variance of each
sample.

Step 2
Find the grand mean.

Step 3
Find the between-group variance.

Step 4
Find the within-group variance.

Step 5
Find the F-test value.
© Copyright McGraw-Hill 2000
12-10
Analysis of Variance Summary Table
Source
Sum of
squares
d.f.
Mean
Squares
F
Between
SSB
k 1
MSB
Within
SSW
Nk
MSW
MS B
MSW
Total
© Copyright McGraw-Hill 2000
12-11
Sum of Squares Between Groups

The sum of the squares between groups,
denoted SSB, is found using the following
formula:
2
n i  X i  X GM 

2
SB 
k 1

Where the grand mean, denoted by X GM , is
the mean of all values in the samples.
X GM
X


N
© Copyright McGraw-Hill 2000
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Sum of Squares Within Groups

The sum of the squares within groups,
denoted SSW, is found using the following
2
formula:
n i  1 si

2
SW 
 n i  1

Note: This formula finds an overall variance
by calculating a weighted average of the
individual variances. It does not involve using
differences of the means.
© Copyright McGraw-Hill 2000
12-13
The Mean Squares

The mean square values are equal to the sum
of the squares divided by the degrees of
freedom.
SS B
MS B 
k 1
SSW
MSW 
N k
© Copyright McGraw-Hill 2000
12-14
Scheffé Test

In order to conduct the Scheffé test, one must
compare the means two at a time, using all
possible combinations of means.

For example, if there are three means, the
following comparisons must be done:
X 1 versus X 2
X 1 versus X 3
© Copyright McGraw-Hill 2000
X 2 versus X 3
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Formula for Scheffé Test
Xi  X j 

Fs  2
sw 1 n i   1 n j 
2
where X i and X j are the means of the samples
being compared, ni and nj are the respective
sample sizes, and sw2 is the within-group
variance.
© Copyright McGraw-Hill 2000
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F Value for Scheffé Test

To find the critical value F for the Scheffé
test, multiply the critical value for the F-test
by k  1:
F '  k  1C .V .
There is a significant difference between the
two means being compared when Fs is greater
than F.
© Copyright McGraw-Hill 2000
12-17
Tukey Test

The Tukey test can also be used after the
analysis of variance has been completed to
make pair-wise comparisons between means
when the groups have the same sample size.

The symbol for the test value in the Tukey test
is q.
© Copyright McGraw-Hill 2000
12-18
Formula for Tukey Test
Xi  X j
q
sw2 n
where X i and X j are the means of the samples
being compared, n is the size of the sample
and sw2 is the within-group variance.
© Copyright McGraw-Hill 2000
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Tukey Test Results

When the absolute value of q is greater than
the critical value for the Tukey test, there is a
significant difference between the two means
being compared.
© Copyright McGraw-Hill 2000
12-20
Two-Way Analysis of Variance

The two-way analysis of variance is an
extension of the one-way analysis of variance
already discussed; it involves two independent
variables.

The independent variables are also called
factors.
© Copyright McGraw-Hill 2000
12-21
Two-Way Analysis of Variance (cont’d.)

Using the two-way analysis of variance, the
researcher is able to test the effects of two
independent variables or factors on one
dependent variable.

In addition, the interaction effect of the two
variables can be tested.
© Copyright McGraw-Hill 2000
12-22
Two-Way ANOVA Terms

Variables or factors are changed between two
levels (i.e., two different treatments).

The groups for a two-way ANOVA are
sometimes called treatment groups.
© Copyright McGraw-Hill 2000
12-23
Two-Way ANOVA Designs
B1
B2
A1
3  3 design
A2
A3
B1
3  2 design
B2
B3
A1
A2
A3
© Copyright McGraw-Hill 2000
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Two-Way ANOVA Null Hypothesis

A two-way ANOVA has several null
hypotheses.

There is one for each independent variable
and one for the interaction.
© Copyright McGraw-Hill 2000
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Two-Way ANOVA Summary Table
Source
A
Sum of
squares
SSA
B
d.f.
a1
Mean
square
MSA
F
FA
SSB
b1
MSB
FB
AB
SSA  B
(a  1)(b  1)
MSA  B
FA  B
Within
(error)
SSW
ab(n  1)
MSW
Total
© Copyright McGraw-Hill 2000
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Assumptions for the Two-Way ANOVA

The population from which the samples were
obtained must be normally or approximately
normally distributed.

The samples must be independent.

The variances of the population from which
the samples were selected must be equal.

The groups must be equal in sample size.
© Copyright McGraw-Hill 2000
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Graphing Interactions

To interpret the results of a two-way analysis
of variance, researchers suggest drawing a
graph, plotting the means of each group,
analyzing the graph, and interpreting the
results.
© Copyright McGraw-Hill 2000
12-28
Disordinal Interaction


If the graph of the
means has lines that
intersect each other, the
interaction is said to be
disordinal.
y
When there is a
disordinal interaction,
one should not interpret
the main effects without
considering the
interaction effect.
© Copyright McGraw-Hill 2000
x
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Ordinal Interaction

An ordinal interaction is
evident when the lines of
the graph do not cross nor
are they parallel.

If the F-test value for the
interaction is significant
and the lines do not cross
each other, then the
interaction is said to be
ordinal and the main
effects can be interpreted
independently of each
other.
y
x
© Copyright McGraw-Hill 2000
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No Interaction


When there is no
significant interaction
effect, the lines in the
graph will be parallel or
approximately parallel.
When this situation
occurs, the main effects
can be interpreted
independently of each
other because there is
no significant
interaction.
© Copyright McGraw-Hill 2000
y
x
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Summary

The F-test can be used to compare two sample
variances to determine whether they are
equal.

It can also be used to compare three or more
means—this procedure is called an analysis
of variance, or ANOVA.

When there is one independent variable, the
analysis of variance is called a one-way
ANOVA; when there are two independent
variables—a two-way ANOVA.
© Copyright McGraw-Hill 2000
12-32
Summary (cont’d.)

The ANOVA technique uses two estimates of
the population variance.

The between-group variance is the variance of
the sample means; the within-group variance
is the overall variance of all the values.

When there is no significant difference among
the means, the two estimates will be
approximately equal, and the F-test value will
be close to 1.
© Copyright McGraw-Hill 2000
12-33
Summary (cont’d.)

If there is a significant difference among the
means, the between-group variance will be
larger than the within-group and a significant
test value will result.

If there is a significant difference among the
means and the sample sizes are the same, the
Tukey test can be used to find where the
difference lies.

The Scheffé test is more general and can be
used even if the sample sizes are not equal.
© Copyright McGraw-Hill 2000
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Conclusions

The two-way ANOVA enables the researcher to
test the effects of two independent variables
and a possible interactions effect on one
dependent variable.
© Copyright McGraw-Hill 2000
12-35