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-critical=Use 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 respondent’s 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-critical=2.69.

16. ANOVA – Example 1 – JCHA 2000 Step 4. Computing the test statistic.

17. 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) < 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 president’s 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-critical=3.84.

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) > F(critical). Therefore, we can reject the null hypothesis of no difference. Approval of federal disaster assistance does vary by presidential administration.