Analysis of Variance ANOVA, Part 2. What We Will Cover in This Section. Factorial ANOVA, Two-way Design

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1 Analysis of Variance ANOVA, Part //007 P33 Analysis of Variance What We Will Cover in This Section Introduction. Overview. Factorial ANOVA Repeated Measures ANOVA. //007 P33 Analysis of Variance Factorial ANOVA, Two-way Design //007 P33 Analysis of Variance 3

2 Definition Experimental design in which there are two or more independent variables and one dependent variable. //007 P33 Analysis of Variance Problem # Effects of Music on Mood A student was researching the influence of music on mood. She hypothesized that tone of the music (aggressive vs. calm) would influence a person s mood but that the type of music (classical vs. popular) would not affect mood. She randomly divided 0 volunteers into one of four groups: classical-aggressive, classical-calm, popularaggressive, or popular-calm. Then she played a sixminute musical selection for the person then had them rate their mood. //007 P33 Analysis of Variance 5 Music Study Descriptive Statistics Music Type Aggressive Calm Classical Popular //007 P33 Analysis of Variance

3 Relationship between music type and mood M ood Classical Popular Calm Aggressive //007 P33 Analysis of Variance 7 Main Effect The independent influence that one independent variable alone has on the dependent variable. //007 P33 Analysis of Variance Job Satisfaction Study Smoker? Smoker Non Smoker Worker Slacker //007 P33 Analysis of Variance

4 Factorial ANOVA: One Main Effect Job Satisfaction Workers Slackers 0 Nonsmokers Smokers //007 P33 Analysis of Variance Teacher Satisfaction Study Final Type Easy Hard A Student F Student //007 P33 Analysis of Variance Factorial ANOVA: Main Effects Teacher Satisfaction F students A students 0 Easy Final Hard Final //007 P33 Analysis of Variance

5 Interaction The combined effects of two or more independent variables on the dependent variable. Is the combined effect of a tranquilizer and alcohol stronger than either taken alone? Is exercise and healthy diet better than either alone? Is distributed studying and studying with a classmate more effective than either one alone. //007 P33 Analysis of Variance 3 Job Satisfaction # Smokers No Yes Smoking Not Allowed 5.5 Smoking Allowed //007 P33 Analysis of Variance Factorial Graphs: Interaction Job Satisfaction Smoking OK No Smoking 0 Nonsmokers Smokers //007 P33 Analysis of Variance 5

6 Another Interaction A B 0 X Y //007 P33 Analysis of Variance Partitioning Sources of Total Factor A (A Main Effect) Between Treatments Factor B (B Main Effect) Within Treatments Interaction //007 P33 Analysis of Variance 7 Problem * Vera Loude was convinced that the volume of commercials would make the commercials more persuasive. She also felt that this effect would be different for males and females. To test her belief Vera recorded an advertisement and played it to a group of male and female volunteers at one of three levels: Soft, Medium, and Loud. After listening to the advertisement the volunteers were asked to rate its persuasiveness. * From Heiman, G. W. (003) Basic Statistics for the Behavioral Sciences. Houghton Mifflin: Boston //007 P33 Analysis of Variance

7 Graph of Vera s Data 0 Soft Medium Loud Male Female //007 P33 Analysis of Variance Vera s Data Soft Medium Loud Mean Male Female Mean (M=.00) (M=.00) (M=.7) (M=.00) (M=.00) (M=.00) M=.00 M=.50 M=.33 M=. M=7.33 M G =. //007 P33 Analysis of Variance 0 Where Do the Numbers Come From? Soft Medium Loud Mean Male Female Mean 7 M=.0 M=.0 M=.0 M=.00 3 M= M=.0 M=.0 M=.7 M=.33 M=. M=7.33 M G =. //007 P33 Analysis of Variance

8 ANOVA Summary Table Source Between Volume Gender Gender x Volume Sum of Squares df MS F Within.7. Total. 7 //007 P33 Analysis of Variance ANOVA Summary Table Source Between Volume Gender Gender x Volume Sum of Squares df MS F Within.7. Total. 7 //007 P33 Analysis of Variance 3 Where do the degrees of freedom come from? Volume Calculation (k volume -) Result 3-= Gender (k gender ) -= Volume x Gender (df volume )(df gender ) x = //007 P33 Analysis of Variance

9 Evaluating the Null Hypothesis F obt F crit Volume Gender Volume x Gender F (,) = 7. F (,) =.3 F (,) = //007 P33 Analysis of Variance 5 Post Hoc Tests For Main effects conduct the regular Tukey HSD test. For Interactions Make comparisons within Each column. Each row. When using Tukey, adjust k when selecting the value for q. //007 P33 Analysis of Variance Effect Size Calculate Eta squared using the regular formula. Volume 7.5 η = =.5. Gender 3.3 η = =.7. Interaction.77 η = =.. //007 P33 Analysis of Variance 7

10 Problem : Chocolate Chip Study The Home for Retired College Professors (HRCP) wants to do a fund raiser using the expertise of its residents as business consultants. After a trial, the clients complained that the advice was too impractical and academic. The director, Gerry Atric, wants to see if feeding these oldsters with chocolate chips would increase the practicality of their recommendations. Atric felt that teaching experience would also have an impact on the treatment effect, so she divided the group into those who taught more than 0 years and those who taught less than 0 years. //007 P33 Analysis of Variance Model Experience Under 0 years Chocolate Chips No Yes n=5 n=5 Over 0 years n=5 n=5 //007 P33 Analysis of Variance Under 0 Over 0 Mean No Chips. Chips 3 M=. M=. M=. M=.0 5. Mean //007 P33 Analysis of Variance 30

11 Chocolate Chip Study Impractical Ideas 0. Under 0 Over 0 No CHIPS Chips //007 P33 Analysis of Variance 3 x Factorial ANOVA Chocolate and experience study summary table Source Between Group Experience Chocolate Chips AXB Within Group Total * p <.0 SS df MS F.05* 5.* 7.3* //007 P33 Analysis of Variance 3 Effect Size η Experience Chips Experience x Chips //007 P33 Analysis of Variance 33

12 Factorial ANOVA: Notation Number of independent variables x 3 x factorial ANOVA Levels of each independent variable. //007 P33 Analysis of Variance 3 Factorial ANOVA Assumptions. The observations within each treatment condition are independent.. The population distribution is relatively normal. 3. The variances within each treatment condition are equal. //007 P33 Analysis of Variance 35 Assignment Homework #3 //007 P33 Analysis of Variance 3

13 Repeated Measures ANOVA //007 P33 Analysis of Variance 37 Sources of Within Group. Measurement error.. Individual differences among the subjects. 3. Random error. //007 P33 Analysis of Variance 3 Sources of Between Group. TREATMENT EFFECT.. Individual differences among the subjects. 3. Measurement Error.. Random error. //007 P33 Analysis of Variance 3

14 Partitioning the Variance Total Between Groups (Conditions) Within Groups Between Subjects Error //007 P33 Analysis of Variance 0 The F-test F (k-)(n-) = MS Treatment MS Error Note that this is n, not N or N-k. Error has random subject error removed. //007 P33 Analysis of Variance Example: Relaxation Therapy Nine migraine sufferers were asked to document the strength of their headaches. There was a two-week baseline period followed by three weeks of relaxation therapy. The therapists wanted to determine if the therapy was effective.. What is the research hypothesis?. What is H o? 3. What is the statistical hypothesis? //007 P33 Analysis of Variance

15 Subject Mean Baseline weeks Treatment Weeks Subject Total //007 P33 Analysis of Variance 3 Mean Headache Strength by Week //007 P33 Analysis of Variance Summary Table Source df SS MS F Between Weeks Within Between Subjects Error Error Total 3.3 //007 P33 Analysis of Variance 5

16 Post hoc Tests Tukey s HSD Replace MS within with Ms error. Replace df within with df error. //007 P33 Analysis of Variance Effect Size SS η = SS between total. η = 3.3 η =.77 //007 P33 Analysis of Variance 7 Assignment Homework # //007 P33 Analysis of Variance

17 //007 P33 Analysis of Variance