PPA 415

1
PPA 415 Research Methods in Public
Administration

• Lecture 7 Analysis of Variance

2
Introduction

• Analysis of variance (ANOVA) can be considered an
  extension of the t-test.
• The t-test assumes that the independent variable
  has only two categories.
• ANOVA assumes that the nominal or ordinal
  independent variable has two or more categories.

3
Introduction

• The null hypothesis is that the populations from
  which the each of samples (categories) are drawn
  are equal on the characteristic measured (usually
  a mean or proportion).

4
Introduction

• If the null hypothesis is correct, the means for
  the dependent variable within each category of
  the independent variable should be roughly equal.
• ANOVA proceeds by making comparisons across the
  categories of the independent variable.

5
Computation of ANOVA

• The computation of ANOVA compares the amount of
  variation within each category (SSW) to the
  amount of variation between categories (SSB).
• Total sum of squares.

6
Computation of ANOVA

• Sum of squares within (variation within
  categories).
• Sum of squares between (variation between
  categories).

7
Computation of ANOVA

• Degrees of freedom.

8
Computation of ANOVA

• Mean square estimates.

9
Computation of ANOVA

• Computational steps for shortcut.
• Find SST using computation formula.
• Find SSB.
• Find SSW by subtraction.
• Calculate degrees of freedom.
• Construct the mean square estimates.
• Compute the F-ratio.

10
Five-Step Hypothesis Test for ANOVA.

• Step 1. Making assumptions.
• Independent random samples.
• Interval ratio measurement.
• Normally distributed populations.
• Equal population variances.
• Step 2. Stating the null hypothesis.

11
Five-Step Hypothesis Test for ANOVA.

• Step 3. Selecting the sampling distribution and
  establishing the critical region.
• Sampling distribution F distribution.
• Alpha .05 (or .01 or . . .).
• Degrees of freedom within N k.
• Degrees of freedom between k 1.
• F-criticalUse Appendix D, p. 499-500.
• Step 4. Computing the test statistic.
• Use the procedure outlined above.

12
Five-Step Hypothesis Test for ANOVA.

• Step 5. Making a decision.
• If F(obtained) is greater than F(critical),
  reject the null hypothesis of no difference. At
  least one population mean is different from the
  others.

13
ANOVA Example 1 JCHA 2000
What impact does marital status have on
respondents rating Of JCHA services? Sum of
Rating Squared is 615
14
ANOVA Example 1 JCHA 2000

• Step 1. Making assumptions.
• Independent random samples.
• Interval ratio measurement.
• Normally distributed populations.
• Equal population variances.
• Step 2. Stating the null hypothesis.

15
ANOVA Example 1 JCHA 2000

• Step 3. Selecting the sampling distribution and
  establishing the critical region.
• Sampling distribution F distribution.
• Alpha .05.
• Degrees of freedom within N k 38 5 33.
• Degrees of freedom between k 1 5 1 4.
• F-critical2.69.

16
ANOVA Example 1 JCHA 2000

• Step 4. Computing the test statistic.

17
ANOVA Example 1 JCHA 2000
18
ANOVA Example 1 JCHA 2000
19
ANOVA Example 1 JCHA 2000.

• Step 5. Making a decision.
• F(obtained) is 1.93. F(critical) is 2.69.
  F(obtained) lt F(critical). Therefore, we fail to
  reject the null hypothesis of no difference.
  Approval of JCHA services does not vary
  significantly by marital status.

20
ANOVA Example 2 Ford-Carter Disaster Data Set
What impact does Presidential administration have
on the presidents recommendation of disaster
assistance?
21
ANOVA Example 2 Ford-Carter Disaster Data Set

• Step 1. Making assumptions.
• Independent random samples.
• Interval ratio measurement.
• Normally distributed populations.
• Equal population variances.
• Step 2. Stating the null hypothesis.

22
ANOVA Example 2 Ford-Carter Disaster Data Set

• Step 3. Selecting the sampling distribution and
  establishing the critical region.
• Sampling distribution F distribution.
• Alpha .05.
• Degrees of freedom within N k 371 2
  369.
• Degrees of freedom between k 1 2 1 1.
• F-critical3.84.

23
ANOVA Example 2 Ford-Carter Disaster Data Set

• Step 4. Computing the test statistic.

24
ANOVA Example 2 Ford-Carter Disaster Data Set

• Step 5. Making a decision.
• F(obtained) is 5.288. F(critical) is 3.84.
  F(obtained) gt F(critical). Therefore, we can
  reject the null hypothesis of no difference.
  Approval of federal disaster assistance does vary
  by presidential administration.