Transcript Chapter 10 Analysis of Variance (Hypothesis Testing III)

Statistics: A Tool For
Social Research
Eighth Edition
Joseph F. Healey
Chapter 10
Hypothesis Testing III :
The Analysis of Variance
Learning Objectives
1. Identify and cite examples of situations in which
ANOVA is appropriate.
2. Explain the logic of hypothesis testing as applied to
ANOVA.
3. Perform the ANOVA test using the five-step model
as a guide, and correctly interpret the results.
4. Define and explain the concepts of population
variance, total sum of squares, sum of squares
between, sum of squares within, mean square
estimates, and post hoc tests.
5. Explain the difference between the statistical
significance and the importance of relationships
between variables.
Chapter Outline
•
•
•
•
•
•
•
Introduction
The Logic of the Analysis of Variance
The Computation of ANOVA
Computational Shortcut
A Computational Example
A Test of Significance for ANOVA
An Additional Example for Computing and Testing the
Analysis of Variance
• The Limitations of the Test
• Interpreting Statistics: Does Sexual Activity Vary by
Marital Status?
In This Presentation
•
•
•
•
The basic logic of analysis of variance (ANOVA)
A sample problem applying ANOVA
The Five Step Model
Limitations of ANOVA
– post hoc techniques
Basic Logic
• ANOVA can be used in situations where the
researcher is interested in the differences in sample
means across three or more categories.
• Examples:
– How do Protestants, Catholics and Jews vary in terms of
number of children?
– How do Republicans, Democrats, and Independents vary in
terms of income?
– How do older, middle-aged, and younger people vary in
terms of frequency of church attendance?
Basic Logic
• Think of ANOVA as extension of t test for more than two
groups.
• ANOVA asks “are the differences between the samples large
enough to reject the null hypothesis and justify the conclusion
that the populations represented by the samples are
different?” (pg. 235)
• The H0 is that the population means are the same:
– H0: μ1= μ2= μ3 = … = μk
Basic Logic
• If the H0 is true, the sample means should be about
the same value.
– If the H0 is true, there will be little difference between
sample means.
• If the H0 is false, there should be substantial
differences between categories, combined with
relatively little difference within categories.
– The sample standard deviations should be low in value.
– If the H0 is false, there will be big difference between
sample means combined with small values for s.
Basic Logic
• The larger the differences between the sample means,
the more likely the H0 is false.-- especially when there is
little difference within categories.
• When we reject the H0, we are saying there are
differences between the populations represented by the
sample.
Basic Logic: Example
• Could there be a relationship between religion and
support for capital punishment? Consider these two
examples.
Steps in Computation of ANOVA
1.
2.
3.
Find total sum of squares (SST) by Formula 10.1.
Find sum of squares between (SSB) by Formula 10.3.
Find sum of squares within (SSW) by subtraction (Formula
10.4).
NX2
Steps in Computation of ANOVA
4.
5.
6.
Calculate the degrees of freedom (Formulas 10.5 and 10.6).
Construct the mean square estimates by dividing SSB and
SSW by their degrees of freedom. (Formulas 10.7 and 10.8).
Find F ratio by Formula 10.9.
Example of Computation of ANOVA
• Problem 10.6 (255)
– Does voter turnout vary by type of election? Data are
presented for local, state, and national elections.
Example of Computation of ANOVA:
Problem 10.6
∑X
∑X2
Group
Mean
Local
State
National
441
559
723
20,213
27,607
45,253
36.75
46.58
60.25
Example of Computation of ANOVA:
Example 10.6
• The difference in the means suggests that
turnout does vary by type of election.
• Turnout seems to increase as the scope of the
election increases.
• Are these differences statistically significant?
Example of Computation of ANOVA:
Example 10.6
• Use Formula 10.1 to find SST.
• Use Formula 10.4 to find SSB
• Find SSW by subtraction
– SSW = SST – SSB
– SSW = 10,612.13 - 3,342.99
– SSW= 7269.14
SST   X 2  NX 2
SST  93073   36  47.86 
SST  93073  (82460.87)
SST  10612.13
• Use Formulas 10.5 and 10.6 to calculate
degrees of freedom.
2
Example of Computation of ANOVA:
Example 10.6
• Use Formulas 10.7 and 10.8 to find the Mean
Square Estimates:
– MSW = SSW/dfw
– MSW =7269.14/33
– MSW = 220.28
– MSB = SSB/dfb
– MSB = 3342.99/2
– MSB = 1671.50
Example of Computation of ANOVA:
Example 10.6
• Find the F ratio by Formula 10.9:
– F = MSB/MSW
– F = 1671.95/220.28
– F = 7.59
Step 1: Make Assumptions and Meet Test
Requirements
• Independent Random Samples
• Level of Measurement is Interval-Ratio
– The dependent variable (e.g., voter turnout) should be I-R to justify
computation of the mean. ANOVA is often used with ordinal variables with
wide ranges.
• Populations are normally distributed.
• Population variances are equal.
Step 2: State the Null Hypothesis
• H0: μ1 = μ2= μ3
– The H0 states that the population means are the
same.
• H1: At least one population mean is different.
– If we reject the H0, the test does not specify which
population mean is different from the others.
Step 3: Select the Sampling Distribution
and Determine the Critical Region
•
•
•
•
•
Sampling Distribution = F distribution
Alpha = 0.05
dfw = (N – k) = 33
dfb = k – 1 = 2
F(critical) = 3.32
– The exact dfw (33) is not in the table but dfw = 30
and dfw = 40 are. Choose the larger F ratio as F
critical.
Step 4
Calculate the Test Statistic
• F (obtained) = 7.59
Step 5 Making a Decision and Interpreting
the Test Results
• F (obtained) = 7.59
• F (critical) = 3.32
– The test statistic is in the critical region. Reject the
H0.
– Voter turnout varies significantly by type of
election.
Suggestion
• Go carefully through the examples in the book
to be sure you understand and can apply
ANOVA.
– Support for capital punishment example: Section
10.3.
– Efficiency of three social service agencies: Section
10.6
Limitations of ANOVA
1. Requires interval-ratio level measurement of the
dependent variable and roughly equal numbers of
cases in the categories of the independent variable.
2. Statistically significant differences are not
necessarily important.
3. The alternative (research) hypothesis is not
specific. Asserts that at least one of the population
means differs from the others.
–
–
Use post hoc techniques for more specific differences.
See example in Section 10.8 for post hoc technique.