07. Repeated-Measures and Two-Factor Analysis of Variance.pdf
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The document provides an overview of repeated-measures and two-factor ANOVA as part of a statistics lecture. It outlines learning outcomes, hypotheses, structure, computations, and interpretations relevant to these statistical methods, emphasizing the control of individual differences and the significance of interaction effects. Key aspects include advantages and disadvantages of designs, assumptions for validity, and recommendations for presenting results.
07. Repeated-Measures and Two-Factor Analysis of Variance.pdf
- 1. Repeated-Measures and Two-Factor Analysis of Variance PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau
- 2. Learning Outcomes • Understand logic of repeated-measures ANOVA study 1 • Compute repeated-measures ANOVA to evaluate mean differences for single-factor repeated-measures study 2 • Measure effect size, perform post hoc tests and evaluate assumptions required for single-factor repeated-measures ANOVA 3
- 3. • Measure effect size, interpret results and articulate assumptions for two-factor ANOVA Learning Outcomes (continued) • Understand logic of two-factor study and matrix of group means 4 • Describe main effects and interactions from pattern of group means in two-factor ANOVA 5 • Compute two-factor ANOVA to evaluate means for two-factor independent-measures study 6 7
- 4. 7.1 Overview • Analysis of Variance – Evaluated mean differences for two or more groups – Limited to one independent variable (IV) • Complex Analysis of Variance – Samples are related; not independent (Repeated-measures ANOVA) – Two independent variables are manipulated (Factorial ANOVA; only Two-Factor in this text)
- 5. 7.2 Repeated-Measures ANOVA • Independent-measures ANOVA uses multiple participant samples to test the treatments • Participant samples may not be identical • If groups are different, what was responsible? – Treatment differences? – Participant group differences? • Repeated-measures solves this problem by testing all treatments using one sample of participants
- 6. Repeated-Measures ANOVA • Repeated-Measures ANOVA used to evaluate mean differences in two general situations – In an experiment, compare two or more manipulated treatment conditions using the same participants in all conditions – In a nonexperimental study, compare a group of participants at two or more different times • Before therapy; After therapy; 6-month follow-up • Compare vocabulary at age 3, 4 and 5
- 8. Repeated-Measures ANOVA Hypotheses • Null hypothesis: in the population there are no mean differences among the treatment groups • Alternate hypothesis: there is one (or more) mean differences among the treatment groups ... : 3 2 1 0 H H1: At least one treatment mean μ differs from another
- 9. General structure of the ANOVA F-Ratio • F ratio based on variances – Numerator measures treatment mean differences – Denominator measures treatment mean differences when there is no treatment effect – Large F-ratio greater treatment differences than would be expected with no treatment effects effect treatment no with expected es) (differenc variance treatments between es) (differenc variance F
- 10. Individual differences • Participant characteristics may vary considerably from one person to another • Participant characteristics can influence measurements (Dependent Variable) • Repeated measures design allows control of the effects of participant characteristics – Eliminated from the numerator by the research design – Must be removed from the denominator statistically
- 11. Structure of the F-Ratio for Repeated-Measures ANOVA ally) mathematic removed s difference l (individua effect treatment no with expected es) (differenc variance s) difference individual (without eatments between tr es) (differenc variance F The biggest change between independent- measures ANOVA and repeated-measures ANOVA is the addition of a process to mathematically remove the individual differences variance component from the denominator of the F-ratio
- 12. Repeated-Measures ANOVA Logic • Numerator of the F ratio includes – Systematic differences caused by treatments – Unsystematic differences caused by random factors are reduced because the same individuals are in all treatments • Denominator estimates variance reasonable to expect from unsystematic factors – Effect of individual differences is removed – Residual (error) variance remains
- 13. Figure 7.1 Structure of the Repeated-Measures ANOVA
- 14. Repeated-Measures ANOVA Stage One Equations N G X SStotal 2 2 treatment each inside treatments within SS SS N G n T SS treatments between 2 2
- 15. Two Stages of the Repeated- Measures ANOVA • First stage – Identical to independent samples ANOVA – Compute SStotal, SSbetween treatments and SSwithin treatments • Second stage – Done to remove the individual differences from the denominator – Compute SSbetween subjects and subtract it from SSwithin treatments to find SSerror (also called residual)
- 16. Repeated-Measures ANOVA Stage Two Equations N G k P SS subjects between 2 2 _ bjects between_su atments within tre SS SS SSerror
- 17. Degrees of freedom for Repeated-Measures ANOVA dftotal = N – 1 dfwithin treatments = Σdfinside each treatment dfbetween treatments = k – 1 dfbetween subjects = n – 1 dferror = dfwithin treatments – dfbetween subjects
- 18. Mean squares and F-ratio for Repeated-Measures ANOVA error error error df SS MS treatments between treatments between treatments between df SS MS _ _ _ error ments treat between MS MS F
- 19. F-Ratio General Structure for Repeated-Measures ANOVA ) ( ) ( s difference individual without s difference ic unsystemat s difference individual without s difference ic unsystemat effects treatment F
- 20. Effect size for the Repeated-Measures ANOVA • Percentage of variance explained by the treatment differences • Partial η2 is percentage of variability that has not already been explained by other factors or subjects between total eatments between tr 2 SS SS SS error SS SS SS eatments between tr eatments between tr 2
- 21. In the Literature • Report a summary of descriptive statistics (at least means and standard deviations) • Report a concise statement of the ANOVA results – E.g., F (3, 18) = 16.72, p<.01, η2 = .859
- 22. Repeated Measures ANOVA post hoc tests (posttests) • Significant F indicates that H0 (“all populations means are equal”) is wrong in some way • Use post hoc test to determine exactly where significant differences exist among more than two treatment means – Tukey’s HSD and Scheffé can be used – Substitute SSerror and dferror in the formulas
- 23. Repeated-Measures ANOVA Assumptions • The observations within each treatment condition must be independent • The population distribution within each treatment must be normal • The variances of the population distribution for each treatment should be equivalent
- 24. Repeated-Measures ANOVA Advantages and Disadvantages • Advantages of repeated-measures designs – Individual differences among participants do not influence outcomes – Smaller number of participants needed to test all the treatments • Disadvantages of repeated-measures designs – Some (unknown) factor other than the treatment may cause participant’s scores to change – Practice or experience may affect scores independently of the actual treatment effect
- 25. 7.3 Two-Factor ANOVA • Both independent variables and quasi- independent variables may be employed as factors in Two-Factor ANOVA • An independent variable (factor) is manipulated in an experiment • A quasi-independent variable (factor) is not manipulated but defines the groups of scores in a nonexperimental study
- 26. 7.3 Two-Factor ANOVA • Factorial designs – Consider more than one factor • We will study two-factor designs only • Also limited to situations with equal n’s in each group – Joint impact of factors is considered • Three hypotheses tested by three F-ratios – Each tested with same basic F-ratio structure effect treatment no with expected es) (differenc variance treatments between es) (differenc variance F
- 28. Main Effects • Mean differences among levels of one factor – Differences are tested for statistical significance – Each factor is evaluated independently of the other factor(s) in the study 2 1 2 1 : : 1 0 A A A A H H 2 1 2 1 : : 1 0 B B B B H H
- 29. Interactions Between Factors • The mean differences between individuals treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors • H0: There is no interaction between Factors A and B • H1: There is an interaction between Factors A and B
- 30. Interpreting Interactions • Dependence of factors – The effect of one factor depends on the level or value of the other – Sometimes called “non-additive” effects because the main effects do not “add” together predictably • Non-parallel lines (cross, converge or diverge) in a graph indicate interaction is occurring • Typically called the A x B interaction
- 31. Figure 13.2 Group Means Graphed without (a) and with (b) Interaction
- 32. Combination of significant and/or nonsignificant main effects and interactions
- 33. Independence of Main Effects and Interactions
- 34. Independence of Main Effects and Interactions
- 35. Structure of the Two-Factor Analysis of Variance • Three distinct tests – Main effect of Factor A – Main effect of Factor B – Interaction of A and B • A separate F test is conducted for each • Results of one are independent of the others effect treatment no is there if expected s difference mean variance treatments between s difference mean variance F ) ( ) (
- 37. Two Stages of the Two-Factor Analysis of Variance • First stage – Identical to independent samples ANOVA – Compute SStotal, SSbetween treatments and SSwithin treatments • Second stage – Partition the SSbetween treatments into three separate components: differences attributable to Factor A; to Factor B; and to the AxB interaction
- 38. Figure 13.3 Structure of the Two-Factor Analysis of Variance
- 39. Stage One of the Two-Factor Analysis of Variance N G X SStotal 2 2 ment each treat inside SS SS treatments within N G n T SS treatments between 2 2
- 40. Stage Two of the Two Factor Analysis of Variance • This stage determines the numerators for the three F-ratios by partitioning SSbetween treatments N G n T SS row row A 2 2 N G n T SS col col B 2 2 B A treatments between AxB SS SS SS SS
- 41. Degrees of freedom for Two-Factor ANOVA dftotal = N – 1 dfwithin treatments = Σdfinside each treatment dfbetween treatments = k – 1 dfA = (number of rows) – 1 dfB = (number of columns)– 1 dfAxB = dfbetween treatments – dfA – dfB
- 42. Mean squares and F-ratios for the Two-Factor ANOVA reatments t within reatments t within reatments t within df SS MS AxB AxB AxB B B B A A A df SS MS df SS MS df SS MS within AxB AxB within B B within A A MS MS F MS MS F MS MS F
- 43. Two-Factor ANOVA Summary Table Example Source SS df MS F Between treatments 200 3 Factor A 40 1 40 4 Factor B 60 1 60 *6 A x B 100 1 100 **10 Within Treatments 300 20 10 Total 500 23 F.05 (1, 20) = 4.35* F.01 (1, 20) = 8.10** (N = 24; n = 6)
- 44. Two-Factor ANOVA Effect Size • η2, is computed to show the percentage of variability not explained by other factors treatments within A A AxB B total A A SS SS SS SS SS SS SS 2 treatments within B B AxB A total B B SS SS SS SS SS SS SS _ 2 treatments within AxB AxB B A total AxB AxB SS SS SS SS SS SS SS 2
- 45. In the Literature • Report mean and standard deviations (usually in a table or graph due to the complexity of the design) • Report results of hypothesis test for all three terms (A & B main effects; A x B interaction) • For each term include F, df, p-value & η2 • E.g., F (1, 20) = 6.33, p<.05, η2 = .478
- 46. Interpreting the Results • Focus on the overall pattern of results • Significant interactions require particular attention because even if you understand the main effects, interactions go beyond what main effects alone can explain. • Extensive practice is typically required to be able to clearly articulate results which include a significant interaction
- 47. Figure 7.4 Sample means for Example 7.4
- 48. Two-Factor ANOVA Assumptions • The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests – The observations within each sample must be independent of each other – The populations from which the samples are selected must be normally distributed – The populations from which the samples are selected must have equal variances (homogeneity of variance)
- 49. Figure 7.5 Independent- Measures Two-Factor Formulas
- 50. Figure 7.6 Example 7.1 SPSS Output for Repeated-Measures
- 51. Figure 7.7 Example 7.4 SPSS Output for Two-Factor ANOVA