Chapter Thirteen

2. Chapter Thirteen Hypothesis Testing for Two or More Means: The One-Way Analysis of Variance

3. More Statistical Notation • Analysis of variance is abbreviated as ANOVA • An independent variable is called a factor • 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 • The symbol for the number of levels in a factor is k

4. An Overview of ANOVA

5. One-Way ANOVA A one-way ANOVA is performed when only one independent variable is tested in the experiment

6. 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

7. 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

8. 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

9. Diagram of a Study Having ThreeLevels of One Factor

10. 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 a

11. Comparing Means • When there are more than two means in an experiment, the multiple t-tests result in an experiment-wise error rate that this much larger than the a 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 a

12. Assumptions of the ANOVA • Each condition contains a random sample of interval or ratio scores • The population represented in each condition forms a normal distribution • The variances of all populations represented are homogeneous

13. Statistical Hypotheses

14. The F-Test • The statistic for the ANOVA is F • When Fobt is significant, 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

15. Post Hoc Comparisons • Post hoc comparisons are like t-tests • We compare all possible pairs of level means from a factor, one pair at a time

16. Components of ANOVA

17. 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

18. 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

19. The F-Distribution The F-distribution is the sampling distribution showing the various values of F that occur when H0 is true and all conditions represent one population

20. Sampling Distribution of F When H0 Is True

21. Degrees of Freedom • The critical value of F (Fcrit) depends on • The degrees of freedom (both the dfbn = k - 1 and the dfwn = N - k) • The a selected • The F-test is always a two-tailed test

22. Computing the F-Ratio

23. 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

24. Summary Table of a One-way ANOVA Source Sum of df Mean F Squares Squares Between SSbndfbnMSbnFobt Within SSwndfwnMSwn Total SStotdftot

25. Computing Fobt • Compute the total sum of squares (SStot)

26. Computing Fobt • Compute the sum of squares between groups (SSbn)

27. Computing Fobt • Compute the sum of squares within groups (SSwn) • SSwn = SStot - SSbn

28. Computing Fobt • Compute the degrees of freedom • The degrees of freedom between groups equals k - 1 • The degrees of freedom within groups equals N - k • The degrees of freedom total equals N - 1

29. Computing Fobt • Compute the mean squares

30. Computing Fobt • Compute Fobt

31. Performing Post Hoc Comparisons

32. Fisher’s Protected t-Test • When the ns in the levels of the factor are not equal, use Fisher’s protected t-test

33. Tukey’s HSD Test • When the ns in all levels of the factor are equal, use the Tukey HSD multiple comparisons test where qk is found using the appropriate table

34. Describing the Relationshipin a One-Way ANOVA

35. Confidence Interval • The computational formula for the confidence interval for a single m is

36. Graphing the Results in ANOVA A graph showing means from three conditions of an independent variable.

37. Proportion of Variance Accounted For • Eta squared indicates the proportion of variance in the dependent variable that is accounted for by changing the levels of a factor

38. Omega Squared • In some instances, the effect size is reported using the measurement omega squared • is an estimate of the proportion of the variance in the population that would be accounted for by the relationship

39. Example • Using the following data set, conduct a one-way ANOVA. Use a = 0.05

40. Example

41. Example • dfbn = k - 1 = 3 - 1 = 2 • dfwn = N - k = 18 - 3 = 15 • dftot = N - 1 = 18 - 1 = 17

42. Example

43. Example • Fcrit for 2 and 15 degrees of freedom and a = 0.05 is 3.68 • Since Fobt = 4.951, the ANOVA is significant • A post hoc test must now be performed

44. Example • The mean of sample 3 is significantly different from the mean of sample 2