1. Chapter Thirteen The One-Way Analysis of Variance

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 2 New Statistical Notation 1.Analysis of variance is abbreviated as ANOVA 2.An independent variable is called a factor 3.Each condition of the independent variable is also called a level or a treatment, and differences produced by the independent variable are a treatment effect 4.The symbol for the number of levels in a factor is k

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 3 An Overview of ANOVA

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 4 One-Way ANOVA A one-way ANOVA is performed when only one independent variable is tested in the experiment

5. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 5 Between Subjects When an independent variable is studied using independent samples in all conditions, it is called a between- subjects factor A between-subjects factor involves using the formulas for a between- subjects ANOVA

6. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 6 Within Subjects Factor When a factor is studied using related (dependent) samples in all levels, it is called a within-subjects factor This involves a set of formulas called a within-subjects ANOVA

7. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 7 Analysis of Variance The analysis of variance is the parametric procedure for determining whether significant differences occur in an experiment containing two or more sample means In an experiment involving only two conditions of the independent variable, you may use either a t-test or the ANOVA

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 8 Diagram of a Study Having Three Levels of One Factor

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 9 Experiment-Wise Error The overall probability of making a Type I error somewhere in an experiment is call the experiment-wise error rate When we use a t-test to compare only two means in an experiment, the experiment-wise error rate equals 

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 10 Comparing Means When there are more than two means in an experiment, the multiple t-tests result in an experiment-wise error rate that is much larger than the one we have selected Using the ANOVA allows us to compare the means from all levels of the factor and keep the experiment-wise-error rate equal to 

11. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 11 Assumptions of the ANOVA 1.The experiment has only one independent variable and all conditions contain independent samples 2.The dependent variable measures interval or ratio scores 3.The population represented by each condition forms a normal distribution 4.The variances of all populations represented are homogeneous

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 12 Statistical Hypotheses

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 13 The F-Test The statistic for the ANOVA is F When F obt is significant, it indicates only that somewhere among the means at least two of them differ significantly It does not indicate which specific means differ significantly When the F-test is significant, we perform post hoc comparisons

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 14 Post Hoc Comparisons Post hoc comparisons are like t-tests We compare all possible pairs of means from a factor, one pair at a time, to determine which means differ significantly

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 15 Components of ANOVA

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 16 Sources of Variance There are two potential sources of variance Scores may differ from each other even when participants are in the same condition. This is called variance within groups Scores may differ from each other because they are from different conditions. This is called the variance between groups

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 17 Mean Squares The mean square within groups is an estimate of the variability in scores as measured by differences within the conditions of an experiment The mean square between groups is an estimate of the differences in scores that occurs between the levels in a factor

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 18 Performing the ANOVA

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 19 Sum of Squares The computations for the ANOVA require the use of several sums of squared deviations Each of these terms is called the sum of squares and is symbolized by SS

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 20 SourceSum ofdf MeanFSquares BetweenSS bn df bn MS bn F obt WithinSS wn df wn MS wn TotalSS tot df tot Summary Table of a One-way ANOVA

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 21 Computing F obt 1.Compute the total sum of squares (SS tot )

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 22 Computing F obt 2.Compute the sum of squares between groups (SS bn )

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 23 Computing F obt 3.Compute the sum of squares within groups (SS wn ) SS wn = SS tot - SS bn

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 24 Computing F obt 4.Compute the degrees of freedom 1.The degrees of freedom between groups equals k - 1 2.The degrees of freedom within groups equals N - k 3.The degrees of freedom total equals N - 1

25. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 25 5.Compute the mean squares Computing F obt

26. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 26 Computing F obt 6.Compute F obt

27. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 27 The F-Distribution The F-distribution is the sampling distribution showing the various values of F that occur when H 0 is true and all conditions represent one population

28. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 28 Sampling Distribution of F When H 0 Is True

29. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 29 Critical F Value The critical value of F (F crit ) depends on –The degrees of freedom (both the df bn = k - 1 and the df wn = N - k ) –The  selected –The F-test is always a one-tailed test

30. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 30 Performing Post Hoc Comparisons

31. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 31 Fisher’s Protected t-Test When the n s in the levels of the factor are not equal, use Fisher’s protected t-test

32. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 32 When the n s in all levels of the factor are equal, use the Tukey HSD multiple comparisons test where q k is found using the appropriate table Tukey’s HSD Test

33. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 33 Additional Procedures in the One-Way ANOVA

34. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 34 Confidence Interval The computational formula for the confidence interval for a single  is

35. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 35 A graph showing means from three conditions of an independent variable. Graphing the Results in ANOVA

36. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 36 Eta squared indicates the proportion of variance in the dependent variable that is accounted for by changing the levels of a factor Proportion of Variance Accounted For

37. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 37 Group 1 Group 2 Group 3 14 10131115 131012111413 141511101415 Example Using the following data set, conduct a one-way ANOVA. Use  = 0.05

38. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 38 Example

39. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 39 Example df bn = k - 1 = 3 - 1 = 2 df wn = N - k = 18 - 3 = 15 df tot = N - 1 = 18 - 1 = 17

40. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 40 Example

41. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 41 Example F crit for 2 and 15 degrees of freedom and  = 0.05 is 3.68 Since F obt = 4.951, the ANOVA is significant A post hoc test must now be performed

42. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 13 - 42 Example The mean of sample 3 is significantly different from the mean of sample 2