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Statistics: A Tool For Social Research

Statistics: A Tool For Social Research. Eighth Edition Joseph F. Healey. Chapter 10. Hypothesis Testing III : The Analysis of Variance. Learning Objectives. Identify and cite examples of situations in which ANOVA is appropriate. Explain the logic of hypothesis testing as applied to ANOVA.

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Statistics: A Tool For Social Research

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  1. Statistics: A Tool ForSocial Research Eighth Edition Joseph F. Healey

  2. Chapter 10 Hypothesis Testing III : The Analysis of Variance

  3. Learning Objectives • Identify and cite examples of situations in which ANOVA is appropriate. • Explain the logic of hypothesis testing as applied to ANOVA. • Perform the ANOVA test using the five-step model as a guide, and correctly interpret the results. • 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. • Explain the difference between the statistical significance and the importance of relationships between variables.

  4. 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?

  5. 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

  6. 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?

  7. 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

  8. 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.

  9. Basic Logic • The larger the differences between the sample means, the more likely the H0is 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.

  10. Basic Logic: Example • Could there be a relationship between religion and support for capital punishment? Consider these two examples.

  11. Steps in Computation of ANOVA • 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

  12. Steps in Computation of ANOVA • 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.

  13. 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.

  14. Example of Computation of ANOVA:Problem 10.6

  15. 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?

  16. 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 Use Formulas 10.5 and 10.6 to calculate degrees of freedom. Example of Computation of ANOVA:Example 10.6

  17. 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

  18. 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

  19. 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.

  20. 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.

  21. 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 largerF ratio as F critical.

  22. Step 4 Calculate the Test Statistic • F (obtained) = 7.59

  23. 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.

  24. 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

  25. Limitations of ANOVA • Requires interval-ratio level measurement of the dependent variable and roughly equal numbers of cases in the categories of the independent variable. • Statistically significant differences are not necessarily important. • 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.

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