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Chapter Thirteen

Chapter Thirteen. The One-Way Analysis of Variance. New Statistical Notation. Analysis of variance is abbreviated as ANOVA An independent variable is called a factor

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Chapter Thirteen

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  1. Chapter Thirteen The One-Way Analysis of Variance

  2. New 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 Chapter 13 - 2

  3. An Overview of ANOVA Chapter 13 - 3

  4. 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 Chapter 13 - 4

  5. One-Way ANOVA A one-way ANOVA is performed when only one independent variable is tested in the experiment Chapter 13 - 5

  6. Between Subjects When a factor 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 Chapter 13 - 6

  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 Chapter 13 - 7

  8. Diagram of a Study Having ThreeLevels of One Factor Chapter 13 - 8

  9. Experiment-Wise Error The overall probability of making a Type I error somewhere in an experiment is called 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 Chapter 13 - 9

  10. Comparing Means When there are more than two means in an experiment, using multiple t-tests result in an experiment-wise error rate 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 a Chapter 13 - 10

  11. Assumptions of the One-WayBetween-Subjects ANOVA The experiment has only one independent variable and all conditions contain independent samples The dependent variable measures normally distributed interval or ratio scores The variances of all populations represented are homogeneous Chapter 13 - 11

  12. Statistical Hypotheses Chapter 13 - 12

  13. The F-Test The statistic for the ANOVA is F When Fobt is significant, it indicates 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 Chapter 13 - 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 Chapter 13 - 14

  15. Components of ANOVA Chapter 13 - 15

  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 Chapter 13 - 16

  17. Mean Squares The mean square within groups describes the variability of scores within the conditions of an experiment The mean square between groups describes the variability between the means of our levels Chapter 13 - 17

  18. Performing the ANOVA Chapter 13 - 18

  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 Chapter 13 - 19

  20. Summary Table of a One-way ANOVA Source Sum of df Mean F Squares Squares Between SSbndfbnMSbnFobt Within SSwndfwnMSwn Total SStotdftot Chapter 13 - 20

  21. Computing Fobt Compute the total sum of squares (SStot) Chapter 13 - 21

  22. Computing Fobt Compute the sum of squares between groups (SSbn) Chapter 13 - 22

  23. Computing Fobt Compute the sum of squares within groups (SSwn) SSwn = SStot - SSbn Chapter 13 - 23

  24. 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 Chapter 13 - 24

  25. Compute the mean squares Computing Fobt Chapter 13 - 25

  26. Computing Fobt Compute Fobt Chapter 13 - 26

  27. 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 Chapter 13 - 27

  28. Sampling Distribution of F When H0 Is True Chapter 13 - 28

  29. Critical F Value 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 one-tailed test Chapter 13 - 29

  30. Performing Post Hoc Comparisons Chapter 13 - 30

  31. Fisher’s Protected t-Test When the ns in the levels of the factor are not equal, use Fisher’s protected t-test Chapter 13 - 31

  32. 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 Tukey’s HSD Test Chapter 13 - 32

  33. Additional Procedures in the One-Way ANOVA Chapter 13 - 33

  34. Confidence Interval The computational formula for the confidence interval for a single m is Chapter 13 - 34

  35. Graphing the Results in ANOVA A graph showing means from three conditions of an independent variable Chapter 13 - 35

  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 Chapter 13 - 36

  37. Example Using the following data set, conduct a one-way ANOVA. Use a = 0.05 Chapter 13 - 37

  38. Example Chapter 13 - 38

  39. Example dfbn = k - 1 = 3 - 1 = 2 dfwn = N - k = 18 - 3 = 15 dftot = N - 1 = 18 - 1 = 17 Chapter 13 - 39

  40. Example Chapter 13 - 40

  41. 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 Chapter 13 - 41

  42. Example The mean of sample 3 is significantly different from the mean of sample 2 Chapter 13 - 42

  43. Key Terms • factor • F-distribution • Fisher’s protected t-test • F-ratio • level • mean square between groups • mean square within groups analysis of variance ANOVA between-subjects ANOVA between-subjects factor error variance eta squared experiment-wise error rate Chapter 13 - 43

  44. Key Terms (Cont’d) • treatment variance • Tukey’s HSD multiple comparisons test • univariate statistics • within-subjects ANOVA multivariate statistics one-way ANOVA post hoc comparisons sum of squares treatment treatment effect Chapter 13 - 44

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