1. Chapter 17 Factorial Analysis of Variance Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell © 2003 Brooks/Cole Publishing Company/ITP

2. 2Chapter 17 Factorial Analysis of Variance Major Points What is a factorial design?What is a factorial design? An exampleAn example Main effectsMain effects InteractionsInteractions Simple effectsSimple effects Cont.

3. 3Chapter 17 Factorial Analysis of Variance Major Points-cont. Unequal sample sizesUnequal sample sizes Magnitude of effectMagnitude of effect Review questionsReview questions

4. 4Chapter 17 Factorial Analysis of Variance What is a Factorial At least two independent variablesAt least two independent variables All combinations of each variableAll combinations of each variable R X C factorialR X C factorial CellsCells

5. 5Chapter 17 Factorial Analysis of Variance Video Violence Bushman studyBushman study XTwo independent variables Two kinds of videosTwo kinds of videos Male and female subjectsMale and female subjects See following diagramSee following diagram

6. 6Chapter 17 Factorial Analysis of Variance 2 X 2 Factorial

7. 7Chapter 17 Factorial Analysis of Variance Bushman’s Study-cont. Dependent variable = number of aggessive associatesDependent variable = number of aggessive associates 50 observations in each cell50 observations in each cell We will work with means and st. dev., instead of raw data.We will work with means and st. dev., instead of raw data. XThis illustrates important concepts.

8. 8Chapter 17 Factorial Analysis of Variance The Data (cell means and standard deviations)

9. 9Chapter 17 Factorial Analysis of Variance Plotting Results

10. 10Chapter 17 Factorial Analysis of Variance Effects to be estimated Differences due to videosDifferences due to videos XViolent appear greater than nonviolent Differences due to genderDifferences due to gender XMales appear higher than females Interaction of video and genderInteraction of video and gender XWhat is an interaction? XDoes violence affect males and females equally? Cont.

11. 11Chapter 17 Factorial Analysis of Variance Estimated Effects--cont. ErrorError Xaverage within-cell variance Sum of squares and mean squaresSum of squares and mean squares XExtension of the same concepts in the one-way

12. 12Chapter 17 Factorial Analysis of VarianceCalculations Total sum of squaresTotal sum of squares Main effect sum of squaresMain effect sum of squares Cont.

13. 13Chapter 17 Factorial Analysis of VarianceCalculations--cont. Interaction sum of squaresInteraction sum of squares XCalculate SS cells and subtract SS V and SS G SS error = SS total - SS cellsSS error = SS total - SS cells Xor, MS error can be found as average of cell variances

14. 14Chapter 17 Factorial Analysis of Variance Degrees of Freedom df for main effects = number of levels - 1df for main effects = number of levels - 1 df for interaction = product of df main effectsdf for interaction = product of df main effects df error = N - ab = N - # cellsdf error = N - ab = N - # cells df total = N - 1df total = N - 1

15. 15Chapter 17 Factorial Analysis of Variance Calculations for Bushman Data SS total requires raw data.SS total requires raw data. XIt is actually = 171.50 SS video SS video Cont.

16. 16Chapter 17 Factorial Analysis of VarianceCalculations--cont. SS genderSS gender Cont.

17. 17Chapter 17 Factorial Analysis of VarianceCalculations--cont. SS cellsSS cells SS VXG = SS cells - SS video - SS gender = 171.375 - 105.125 - 66.125 = 0.125SS VXG = SS cells - SS video - SS gender = 171.375 - 105.125 - 66.125 = 0.125 Cont.

18. 18Chapter 17 Factorial Analysis of VarianceCalculations--cont. MS error = average of cell variances = (4.6 2 + 3.5 2 + 4.2 2 + 2.8 2 )/4 =58.89/4 = 14.723MS error = average of cell variances = (4.6 2 + 3.5 2 + 4.2 2 + 2.8 2 )/4 =58.89/4 = 14.723 Note that this is MS error and not SS errorNote that this is MS error and not SS error

19. 19Chapter 17 Factorial Analysis of Variance Summary Table

20. 20Chapter 17 Factorial Analysis of VarianceConclusions Main effectsMain effects XSignificant difference due to video More aggressive associates following violent videoMore aggressive associates following violent video XSignificant difference due to gender Males have more aggressive associates than females.Males have more aggressive associates than females. Cont.

21. 21Chapter 17 Factorial Analysis of VarianceConclusions--cont. InteractionInteraction XNo interaction between video and gender Difference between violent and nonviolent video is the same for males (1.5) as it is for females (1.4)Difference between violent and nonviolent video is the same for males (1.5) as it is for females (1.4) We could see this in the graph of the data.We could see this in the graph of the data.

22. 22Chapter 17 Factorial Analysis of Variance Elaborate on Interactions Diagrammed on next slide as line graphDiagrammed on next slide as line graph Note parallelism of linesNote parallelism of lines XMeans video differences did not depend on gender A significant interaction would have nonparallel linesA significant interaction would have nonparallel lines XOrdinal and disordinal interactions

23. 23Chapter 17 Factorial Analysis of Variance Line Graph of Interaction

24. 24Chapter 17 Factorial Analysis of Variance Simple Effects Effect of one independent variable at one level of the other.Effect of one independent variable at one level of the other. e.g. Difference between males and females for only violent videoe.g. Difference between males and females for only violent video Difference between males and females for only nonviolent videoDifference between males and females for only nonviolent video

25. 25Chapter 17 Factorial Analysis of Variance Unequal Sample Sizes A serious problem for hand calculationsA serious problem for hand calculations Can be computed easily using computer softwareCan be computed easily using computer software Can make the interpretation difficultCan make the interpretation difficult XDepends, in part, on why the data are missing.

26. 26Chapter 17 Factorial Analysis of Variance Magnitude of Effect Eta SquaredEta Squared XInterpretation Omega squaredOmega squared XLess biased estimate k = number of levels for the effect in question Cont.

27. 27Chapter 17 Factorial Analysis of Variance Effect Size—cont. As with one-way, we can calculate effect size for each kind of effect separately.As with one-way, we can calculate effect size for each kind of effect separately. Most sensible to stick to comparisons of two groups.Most sensible to stick to comparisons of two groups. Same formulae as for t tests.Same formulae as for t tests.

28. 28Chapter 17 Factorial Analysis of Variance Minitab Example Analysis of Variance for AGGASSOCAnalysis of Variance for AGGASSOC Source DF SS MS F PSource DF SS MS F P GENDER 1 66.1 66.1 4.49 0.035GENDER 1 66.1 66.1 4.49 0.035 VIDEO 1 105.1 105.1 7.14 0.008VIDEO 1 105.1 105.1 7.14 0.008 Interaction 1 0.1 0.1 0.01 0.927Interaction 1 0.1 0.1 0.01 0.927 Error 196 2885.6 14.7Error 196 2885.6 14.7 Total 199 3057.0Total 199 3057.0 Cont.

29. 29Chapter 17 Factorial Analysis of VarianceMinitab--cont. Individual 95% CI GENDER Mean --------+---------+---------+---------+--- 1 6.95 (----------*----------) 2 5.80 (----------*----------) --------+---------+---------+---------+--- 5.60 6.30 7.00 7.70 Individual 95% CI VIDEO Mean ---------+---------+---------+---------+-- 1 7.10 (---------*--------) 2 5.65 (---------*--------) ---------+---------+---------+---------+-- 5.60 6.40 7.20 8.00

30. 30Chapter 17 Factorial Analysis of Variance Review Questions What is the definition of a factorial design?What is the definition of a factorial design? How many independent variables can you have in a factorial design?How many independent variables can you have in a factorial design? XHow many levels of an independent variable can you have? Cont.

31. 31Chapter 17 Factorial Analysis of Variance Review Questions--cont. What do all of the calculations for sums of squares have in common?What do all of the calculations for sums of squares have in common? How does a main effect differ from an interaction?How does a main effect differ from an interaction? How does a main effect differ from a simple effect?How does a main effect differ from a simple effect? Cont.

32. 32Chapter 17 Factorial Analysis of Variance Review Questions--cont. Give an example of a situation where you would commonly expect an interaction.Give an example of a situation where you would commonly expect an interaction. What happens to F values when MS error decreases?What happens to F values when MS error decreases? How do eta-squared and omega-squared differ?How do eta-squared and omega-squared differ?