1. MM570Sec02 Zrotowski Unit 8, Unit 8, Chapter 9 Chapter 9 1 ANOVA 1 ANOVA

2. Chapter 9: Introduction to the Analysis of Variance Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

3. ANOVA Research Design

4. 1 ANOVA EXAMPLE Research Question Is there a difference in the study time of Kaplan students in Business, Psychology, Nursing and IT? Is there a difference in the study time of Kaplan students in Business, Psychology, Nursing and IT?

5. Analysis of Variance Testing ‘variation among the means’ of several (k=NG) groups, Testing ‘variation among the means’ of several (k=NG) groups, due to ‘chance’ or significant due to ‘chance’ or significant ANOVA ANOVA One-way analysis of variance One-way analysis of variance Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

6. Basic Logic of ANOVA Null hypothesis Ho: μ1=…=μk Null hypothesis Ho: μ1=…=μk Several populations all have same mean Several populations all have same mean Do the means of the samples differ more than ‘expected’? (if the null hypothesis were true) (expected relates to confidence level) Do the means of the samples differ more than ‘expected’? (if the null hypothesis were true) (expected relates to confidence level) Analyze variances – hence ANOVA Analyze variances – hence ANOVA Two different ways of estimating population variance Two different ways of estimating population variance Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

7. Basic Logic of ANOVA Estimating population variance from variation from WITHIN each sample Estimating population variance from variation from WITHIN each sample Within-groups estimate of the population variance, k-sum Within-groups estimate of the population variance, k-sum Not affected by whether the null hypothesis is true Not affected by whether the null hypothesis is true SSW= SS+…+SS=∑(X-M)+…+ ∑(X-M) SSW= SS+…+SS=∑(X-M) 2 +…+ ∑(X-M) 2 Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

8. Basic Logic of ANOVA Estimating population variance from variation BETWEEN the means of the samples Estimating population variance from variation BETWEEN the means of the samples Between-groups estimate of the population variance Between-groups estimate of the population variance When the null hypothesis is true When the null hypothesis is true When the null hypothesis is not true When the null hypothesis is not true Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

9. SSB= ∑n(M-GM), k-sum SSB= ∑n(M-GM) 2, k-sum M sample average (for each group there is one M) M sample average (for each group there is one M) GM grand average GM grand average n sample size (for each sample there is one n) n sample size (for each sample there is one n) GN=n+…+n, grand sample size GN=n+…+n, grand sample size

10. Basic Logic of ANOVA Sources of variation in within-groups and between-groups variance estimates Sources of variation in within-groups and between-groups variance estimates Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

11. Basic Logic of ANOVA The F ratio The F ratio Ratio of the between-groups population variance estimate (after df normalization) to the within-groups population variance estimate (after df normalization) Ratio of the between-groups population variance estimate (after df normalization) to the within-groups population variance estimate (after df normalization) The F distribution The F distribution The F table The F table Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

12. Carrying out an ANOVA Estimating the population variance from the variation of scores WITHIN each group Estimating the population variance from the variation of scores WITHIN each groupMSW=SSW/dfW, dfW=GN-k=df+…+df, df=n-1 Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

13. Carrying out an ANOVA Estimating the population variance from the differences BETWEEN group means Estimating the population variance from the differences BETWEEN group means MSB=SSB/dfB MSB=SSB/dfB Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

14. Carrying out an ANOVA Estimating the population variance from the differences between group means Estimating the population variance from the differences between group means Estimate the variance of the population of individual scores Estimate the variance of the population of individual scores Figuring the F ratio Figuring the F ratio Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

15. Carrying out an ANOVA An F distribution An F distribution Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

16. Carrying out an ANOVA The F table The F table Between-groups degrees of freedom Between-groups degrees of freedom Within-groups degrees of freedom Within-groups degrees of freedom Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

17. Assumptions in ANOVA Populations follow a normal curve Populations follow a normal curve Populations have equal variances Populations have equal variances Samples: random and independent Samples: random and independent (same as two independent samples t-test) (same as two independent samples t-test) Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

18. Planned Contrasts Reject null hypothesis (if F>f(α,dfT,dfB) Reject null hypothesis (if F>f(α,dfT,dfB) Population means are significantly not all the same Population means are significantly not all the same Planned contrasts Planned contrasts Within-groups population variance estimate Within-groups population variance estimate Between-groups population variance estimate Between-groups population variance estimate Use the two means of interest Use the two means of interest Figure F in the usual way Figure F in the usual way Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

19. Planned Contrasts Bonferroni procedure Bonferroni procedure Provides a more stringent significance level for each comparison Provides a more stringent significance level for each comparison Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

20. Post-Hoc Comparisons Exploratory approach Exploratory approach Scheffé test Scheffé test Figure the F in the usual way Figure the F in the usual way Divide the F by the overall study’s df Between Divide the F by the overall study’s df Between Compare this to the overall study’s F cutoff Compare this to the overall study’s F cutoff Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

21. Effect Size for ANOVA Proportion of variance accounted for (R 2 ) Proportion of variance accounted for (R 2 ) Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

22. Effect Size for ANOVA R 2 also known as η 2 (eta squared) R 2 also known as η 2 (eta squared) small R 2 =.01 small R 2 =.01 medium R 2 =.06 medium R 2 =.06 large R 2 =.14 large R 2 =.14 Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

23. Power for ANOVA (.05 significance level) Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

24. Approximate Sample Size Needed in Each Group for 80% Power (.05 significance level) Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

25. ANOVA in Research Articles F(3, 68) = 5.81, p <.01 F(3, 68) = 5.81, p <.01 Means given in a table or in the text Means given in a table or in the text Follow-up analyses Follow-up analyses Planned comparisons Planned comparisons Using t tests Using t tests Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

26. Controversies and Limitations Omnibus test versus planned contrasts Omnibus test versus planned contrasts Conduct specific planned contrasts to examine Conduct specific planned contrasts to examine Theoretical questions Theoretical questions Practical questions Practical questions Controversial approach Controversial approach Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

27. Reporting in Research Articles Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

28. The Structural Model Flexible way of figuring the two population variance estimates Flexible way of figuring the two population variance estimates Handles situation when sample sizes in each group are not equal Handles situation when sample sizes in each group are not equal Insight into underlying logic of ANOVA Insight into underlying logic of ANOVA Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

29. Principles of the Structural Model Dividing up the deviations Dividing up the deviations Deviation of a score from the grand mean Deviation of a score from the grand mean Deviation of the score from the mean of its group Deviation of the score from the mean of its group Deviation of the mean of its group from the grand mean Deviation of the mean of its group from the grand mean Summing the squared deviations Summing the squared deviations Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

30. Principles of the Structural Model From the sums of squared deviations to the population variance estimates From the sums of squared deviations to the population variance estimates Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

31. Principles of the Structural Model Relation of the structural model approach to the previous approach Relation of the structural model approach to the previous approach Within-groups variance estimate Within-groups variance estimate Never figure the variance estimate for each group and average them Never figure the variance estimate for each group and average them Between-groups variance estimate Between-groups variance estimate Never multiply anything by the number of scores in each sample Never multiply anything by the number of scores in each sample Same ingredients for the F ratio Same ingredients for the F ratio Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

32. Principles of the Structural Model Relation of the structural model approach to the previous approach Relation of the structural model approach to the previous approach Previous approach Previous approach Emphasizes entire groups Emphasizes entire groups Focuses directly on what contributes to the overall population variance estimates Focuses directly on what contributes to the overall population variance estimates Structural model Structural model Emphasizes individual scores Emphasizes individual scores Focuses directly on what contributes to the divisions of the deviations of scores from the grand mean Focuses directly on what contributes to the divisions of the deviations of scores from the grand mean Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

33. Using the Structural Model to Figure an ANOVA Example analysis of variance table Example analysis of variance table Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

34. Analysis of Variance Table Copyright © 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458. All rights reserved.

35. SPSS Steps p365 Steps p365 Tables: descriptives, ANOVA basic p366 Tables: descriptives, ANOVA basic p366 Post hoc: multicomparisons p368 Scheffe Post hoc: multicomparisons p368 Scheffe