1. Lecture 13: Factorial ANOVA 1 Laura McAvinue School of Psychology Trinity College Dublin

2. Analysis of Variance One way ANOVAFactorial ANOVA One Independent Variable More than One Independent Variable Two way Three way Four way Between subjects Repeated measures / Within subjects Different participants Same participants

3. Factorial ANOVA Factor –Another word for an independent variable in ANOVA Factorial Design –Design in which there are two or more independent variables or factors

4. Labelling Number of independent variables / factors –One independent variable – One way ANOVA –Two independent variables – Two way ANOVA –Three independent variables – Three way ANOVA Number of levels of each variable / factor –Comparing men and women’s performance on an attention task under three conditions of noise –Two independent variables Gender (2 levels: male & female) Noise (3 levels: none, white noise, random tones) –2 x 3 factorial ANOVA

5. Factorial ANOVA Allows you to examine two things… –The main effect of each independent variable, when controlling for the other variable –The interaction between the two variables

6. Research Example We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women What is our dependent variable? –Number of depressive symptoms How many independent variables do we have? –2–2

7. Research Example We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women What are the independent variables? –Gender & Therapy How many levels do they have? –Gender: 2 levels (Male/Female) –Therapy: 3 levels (CBT, Psychoanalytic, Drug)

8. Research Example We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women Label this experiment in two ways –Two Way Factorial ANOVA –2 x 3 Factorial ANOVA

9. Factorial ANOVA This design will enable us to investigate three things –Main Effect of Gender –Main Effect of Therapy –Interaction between Gender and Therapy

10. Main Effect The effect of one independent variable averaged across the levels of the other independent variable The effect of one independent variable ignoring the other variable

11. Main Effect of Gender There is a significant difference between men and women’s no. of depressive symptoms across all therapy groups –Men and women’s depressive symptoms differ, irrespective of the type of therapy they got –The type of therapy does not influence the effect of gender E.g. Men have a significantly lower number of depressive symptoms than women overall, across all three therapy conditions –H o : There is no effect of gender Mean of males = Mean of females –H alt : There is a main effect of gender Mean of males ≠ Mean of females

12. Main Effect of Therapy The kind of therapy administered significantly affected the number of depressive symptoms, irrespective of the gender of the client H o :There is no significant effect of therapy –Mean CBT = Mean Psychoanalytic = Mean Drug H alt : At least one mean for therapy is different from the other two E.g. CBT significantly reduced the number of depressive symptoms for both men and women

13. Interaction Factorial Design –Enables you to pair each level of each variable with each level of the other variable / variables Interaction –Combined effect IV 1 & IV 2 on the DV –Means that the effects of one independent variable depend on the level of the other independent variable Simple Effect –The effect of one independent variable at one level of another variable

14. Interaction between Gender & Therapy One therapy is more effective for one type of client Men & women benefit equally from CBT and drugs but women respond better to psychoanalysis H o : There is no interaction between gender & therapy –All mean differences are due only to main effects

15. CBTPsychoanaly tic Drug Males101624 81826 62028 Females22620 422 18824 Type of Therapy Gender

16. CBTPsychoanaly tic Drug Males101624 81826 62028 81826 17.33 Females22620 422 18824 20622 16 16.67 Is there a main effect of Gender?

17. CBTPsychoanaly tic Drug Males101624 81826 62028 Females22620 422 18824 14122416.67 Is there a main effect of Therapy?

18. CBTPsychoanaly tic Drug Males101624 81826 62028 81826 Females22620 422 18824 20622 16.67 Is there an Interaction between Gender & Therapy? Examine the pattern of means…

19. Line graph of the six cell means 0 5 10 15 20 25 30 CBT Psycho- analytic Drug male female

20. Calculations Total Variance Variance due to IV 1 Gender Variance due to IV 2 Therapy Variance due to the interaction between IV 1 & IV 2 Gender x Therapy Variance Due to random error

21. Three F Ratios Compare the variance due to the main effects and the interaction to the variance due to random error Variance due to Gender Variance due to Random Error Variance due to Therapy Variance due to Random Error Variance due to Gender x Therapy Variance due to Random Error

22. CBTPsychoanaly tic Drug Males101624 81826 62028 Females22620 422 18824 16.67 Total Variance SS total ∑ (x ij - Grand Mean ) 2 1000

23. CBTPsychoanaly tic Drug Males101624 81826 62028 17.33 Females22620 422 18824 16 16.67 Variance due to Gender SS gender n gender ∑ (Mean for each level of gender - Grand Mean ) 2

24. CBTPsychoanaly tic Drug Males101624 81826 62028 17.33 Females22620 422 18824 16 16.67 Variance due to Gender SS gender 9 ∑ (17.33 – 16.67 ) 2 + (16 – 16.67) 2 8

25. CBTPsychoanaly tic Drug Males101624 81826 62028 Females22620 422 18824 14122416.67 Variance due to Therapy SS therapy n therapy ∑ (Mean for each level of therapy - Grand Mean ) 2

26. CBTPsychoanaly tic Drug Males101624 81826 62028 Females22620 422 18824 14122416.67 Variance due to Therapy SS therapy 6 ∑ (14 – 16.67 ) 2 + (12 – 16.67) 2 + (24 – 16.67) 2 496

27. CBTPsychoanaly tic Drug Males101624 81826 62028 81826 Females22620 422 18824 2062216.67 Variance due to the interaction Each cell mean is a combination of a level of each independent variable

28. Variance due to the Interaction SS cells –The sum of squared deviations of each cell mean from the grand mean –The variance of the cell means –A measure of how much the cell means differ Cell means can differ due to… –Level of Gender –Level of Therapy –Interaction between Gender & Therapy

29. Variance due to the Interaction SS cells = SS gender + SS therapy + SS gender x therapy SS gender x therapy = SS cells – SS gender – SS therapy

30. CBTPsychoanaly tic Drug Males101624 81826 62028 81826 Females22620 422 18824 2062216.67 Variance due to the interaction No. of participants in each cell ∑ (Each cell mean - Grand Mean ) 2 SS cells

31. CBTPsychoanaly tic Drug Males101624 81826 62028 81826 Females22620 422 18824 2062216.67 Variance due to the interaction 3 ∑ (8 – 16.67 ) 2 + (18 – 16.67) 2 + (26 – 16.67) 2 + (20 – 16.67) 2 + (6 – 16.67) 2 + (22 – 16.67) 2 952 SS cells

32. Variance due to the Interaction SS gender x therapy = SS cells – SS gender – SS therapy SS gender x therapy = 952 – 8 – 496 SS gender x therapy = 448

33. Variance due to Random Error Two Methods… Directly –SS error = ∑(each score in each cell – mean of that cell) 2 –48 Indirectly –SS total = [SS gender + SS therapy + SS gender x therapy ] + SS error –SS total = [SS cells ] + SS error –SS error = SS total – SS cells = 1000 – 952 = 48

34. ANOVA table Source of variation SSdfMSFp GenderSS gender k gender – 1SS gender / df gender MS gender MS error P value TherapySS therapy k therapy – 1 SS therapy / df therapy MS therapy MS error P value Gender* Therapy SS gender*therapy Df gender * Df therapy SS gender*therapy / df gender*therapy MS gender*therapy MS error P value ErrorSS error k gender * k therapy (n-1) SS error / df error TotalSS total N – 1

35. ANOVA table Source of variation SSdfMSFp Gender8182.183 Therapy496224862.000 Gender* Therapy 448222456.000 Error48124 Total100017