1. Factorial Analysis of Variance More than 2 Independent Variables Between-Subjects Designs

2. What is a Factorial At least two independent variablesAt least two independent variables All combinations of each variable (completely crossed)All combinations of each variable (completely crossed) R X C factorialR X C factorial CellsCells

3. Variables and Cells 2 X 2 2 IVs 4 cells2 X 2 2 IVs 4 cells 2 X 3 2 IVs 6 cells2 X 3 2 IVs 6 cells 2 X 3 X 3 3 IVs 18 cells2 X 3 X 3 3 IVs 18 cells 2 X 2 X 2 X 2 4 Ivs 16 cells2 X 2 X 2 X 2 4 Ivs 16 cells

4. Effects Tested 2 X 2A, B, A X B2 X 2A, B, A X B 2 X 2 X 2A, B, C, A X B,2 X 2 X 2A, B, C, A X B, A X C, B X C, A X B X C A X C, B X C, A X B X C 2 X 2 X 2 X 2 A, B, C, D, A X B, A X C, A X D, B X C, B X D, C X D, A X B X C, A X B X D, B X C X D, A X B X C X D2 X 2 X 2 X 2 A, B, C, D, A X B, A X C, A X D, B X C, B X D, C X D, A X B X C, A X B X D, B X C X D, A X B X C X D

5. 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. 2 X 2 Factorial

7. 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. The Data (cell means and standard deviations)

9. Plotting Results

10. 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. 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. Calculations Total sum of squaresTotal sum of squares Main effect sum of squaresMain effect sum of squares Cont.

13. Calculations--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. 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. 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. Calculations--cont. SS genderSS gender Cont.

17. Calculations--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. Calculations--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. Summary Table

20. Conclusions 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. Conclusions--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. 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. Line Graph of Interaction

24. 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. 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. Magnitude of Effect Eta SquaredEta Squared XInterpretation Omega squaredOmega squared XLess biased estimate k = number of levels for the effect in question Cont.

27. 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.