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Lecture 13: Factorial ANOVA 1

Lecture 13: Factorial ANOVA 1. Laura McAvinue School of Psychology Trinity College Dublin. Analysis of Variance. One way ANOVA. Factorial ANOVA. More than One Independent Variable. One Independent Variable. Between subjects. Repeated measures / Within subjects. Two way.

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Lecture 13: Factorial ANOVA 1

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

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

  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 • Ho: There is no effect of gender • Mean of males = Mean of females • Halt: 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 • Ho: There is no significant effect of therapy • Mean CBT = Mean Psychoanalytic = Mean Drug • Halt: 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 IV1 & IV2 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 • Ho: There is no interaction between gender & therapy • All mean differences are due only to main effects

  15. Type of Therapy Gender

  16. Is there a main effect of Gender?

  17. Is there a main effect of Therapy?

  18. Is there an Interaction between Gender & Therapy? Examine the pattern of means…

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

  20. Calculations Total Variance Variance due to the interaction between IV1 & IV2 Gender x Therapy Variance Due to random error Variance due to IV1 Gender Variance due to IV2 Therapy

  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. Total Variance • ∑ (xij - Grand Mean )2 SStotal 1000

  23. Variance due to Gender • ngender∑ (Mean for each level of gender - Grand Mean )2 SSgender

  24. Variance due to Gender • 9∑ (17.33 – 16.67 )2 + (16 – 16.67)2 SSgender 8

  25. Variance due to Therapy • ntherapy∑ (Mean for each level of therapy - Grand Mean )2 SStherapy

  26. Variance due to Therapy • 6∑ (14 – 16.67 )2 + (12 – 16.67)2 + (24 – 16.67)2 SStherapy 496

  27. Variance due to the interaction Each cell mean is a combination of a level of each independent variable

  28. Variance due to the Interaction • SScells • 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 • SScells = SSgender + SStherapy + SSgenderx therapy • SSgenderx therapy = SScells – SSgender – SStherapy

  30. Variance due to the interaction • No. of participants in each cell ∑ (Each cell mean - Grand Mean )2 SScells

  31. 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 SScells

  32. Variance due to the Interaction • SSgenderx therapy = SScells – SSgender – SStherapy • SSgenderx therapy = 952 – 8 – 496 • SSgenderx therapy = 448

  33. Variance due to Random Error • Two Methods… • Directly • SSerror = ∑(each score in each cell – mean of that cell)2 • 48 • Indirectly • SStotal= [SSgender + SStherapy +SSgenderx therapy ]+SSerror • SStotal= [SScells ]+SSerror • SSerror = SStotal – SScells = 1000 – 952 = 48

  34. ANOVA table

  35. ANOVA table

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