Two-Way ANOVAs

Two-Way ANOVAs ______________________________________ 1) Describe the difference between one- and multi-factor ANOVAS. 2) Discuss the three kinds of potential effects in two-way ANOVA designs: • omnibus effects • main effects • interaction effects 3) Describe the proper way to interpret the results of a two-way ANOVA: • omnibus • interaction • main effects 4) Conduct the voluminous calculations required to fully analyze a two-way ANOVA. • MStotal, MSmodel, MSa, MSb, MSe, etc. 5) Practice filling in the blanks for the ANOVA table. Factorial Experiments Two-Way ANOVAs Two-Way ANOVA – an experimental design in which two or more factors are used to evaluate the DV. ________________________________________ Dog Breed X Collie Retriever Mutt Gender Male Female ________________________________________ Gender Male Female x Weight Under Normal Over x SES Low Med Hight Types of Effects in Factorial Designs ________________________________________ Main Effects ________________________________________ Interactions ________________________________________ Caffeine, Studying, and Exam Performance No Coffee 2 Cups > 2 Cups Studied 82 95 80 Did not Study 74 73 60 Two-Way ANOVA: Examining the possibilities ________________________________________ Let’s look at the possible outcomes of a 2 x 2 experiment meant to assess the Males’ and Females’ preferences for ‘Grey’s Anatomy’ and Monday Night Football. There are two factors in the experiment. The first factor is gender: Male vs. Female. The second factor is television show: GA vs. MNF. Thus, there are a total of 4 treatments in the experiment. Grey’s Anatomy Women Men Monday Night Football Women’s Ratings of Women’s Ratings of John & Kate + 8 Monday Night Football Men’s Ratings of John & Kate + 8 Men’s Ratings of Monday Night Football Interaction takes precedence ________________________________________ Why? • Because interaction tells us more than main effects alone ________________________________________ Stat State Math SE Verbal U. 500 800 650 Verbal 800 500 650 650 650 Is there a main effect of School? Is there a main effect of Test Type? Is there an interaction effect? What does the interaction tell us? More on the precedence of the Interaction ________________________________________ Low Salt Exercises Regularly 100 Couch Potato 200 150 Normal 200 200 200 150 200 Is there a main effect of Exercise? Is there a main effect of Sodium? Is there an interaction? What does the interaction tell us? Factorial Experiments (n-way ANOVA) ________________________________________ Main Effects • Collies are smarter than Labs • Attending Class improves your grade • People like Grey’s Anatomy better than MNF Interaction • Female Collies are smarter than males but • Attending class only improves your grade IF • Females like JK better than MNF, but Assumptions for Factorial Designs (n-way ANOVA) ________________________________________ 1) Distribution for each treatment 2) Homogeneity of variance 3) Each treatment consists of random, independent samples. 4) Balanced design: Equal n in all treatments Steps for n-way Factorial (ANOVA) __________________________________________ Study No study No Coffee 82 74 2 Cups 95 73 > 2 Cups 80 60 1) Partition the variance into treatment and error Treatment df Main Effect A Main Effect B A X B Interaction Model SS(A) SS(B) SS(AB) SS(M) Error Error SSE Total Total SS(Total) Multiple Tests ________________________________________ Omnibus test (test of the overall model) • Ho: • Ha: F = df = ________________________________________ Main Effects • Ho: No differences b/t • Ha: At least two levels F = df = or ________________________________________ Interaction Effects • Ho: Factors A and B • Ha: Factors A and B F = df = Formulae ________________________________________ SST SSM SSE (x2) - G2/N [(T2/n)] - G2/N SS within each sample (SST - SSM) [(x2) – (T2/n)] __________________________________________ However, now three sources contribute to SSM SSA = SSB = SSAxB = Ta2  G 2   − N  n  Tb2  G 2   − N  n  SSM - (SSA + SSB) PlayStation __________________________________________ PlayStation, Inc. is test-marketing a new commercial for it's Super-Duper Game Thingy. They imbed the commercial into a news program and show it to two groups of subjects, college students and senior citizens. The commercial either appears once, or is repeated three times. After the show, the two groups were asked to rate how likely they would be to purchase a PlayStation for Christmas this year. Higher numbers represent a greater likelihood to buy. Older Adults Younger Adults x M x x2 SS 2, 4, 5, 3, 6 7, 7, 5, 5, 6 x M x x2 SS 3, 1, 2, 3, 1 10, 8, 7, 7, 8 Once Repeated Calculations __________________________________________ = x2 - G2/N = (90 + 184 + 24 + 326) - 1002/20 = 624 - 10000/20 = 624 - 500 = 124 __________________________________________ SST = [(T2/n)] - G2/N = (202/5) + (302/5) + (102/5) + (402/5) - 500 = 80 + 180 + 20 + 320 - 500 = 600 - 500 = 100 __________________________________________ SSM SSE = = SST - SSM 124 - 100 = 24 Partitioning SSM __________________________________________ = [(TA2/n)] - G2/N = (20+10)2/10 + (30+40)2/10 - 500 = (302/10) + (702/10) - 500 = 90 + 490 - 500 = 580 - 500 = 80 __________________________________________ SSAge = [(TR2/n)] - G2/N = (20+30)2/10 + (10+40)2/10 - 500 = (502/10) + (502/10) - 500 = 250 + 250 - 500 = 500 - 500 = 0 __________________________________________ SSRep SSAxR = = = SSM - (SSA + SSB) 100 - (80 + 0) 100 - 80 = 20 X-box ANOVA table __________________________________________ Source df SS MS F df SS MS F Model Error Total Source Age Rep AxR __________________________________________ Rep Once Repeated Age Older Younger 4 6 2 8 3 7 5 5 Reporting the results of a 2-Way ANOVA _______________________________________ The analysis indicated that there was an overall main effect (F (3, 16) = 22.22, MSE = 1.5, p < .05). This indicates that our independent variables influenced purchase intent. The results indicated a significant main effect of Age (F (1, 16) = 53.33, MSE = 1.5, p < .05), with younger adults more likely to purchase the product than older adults. The main effect of ad repetition was not significant (F (1, 16) = 0, MSE = 1.5, p = 1). Repeating the ads did not alter purchase intent. Finally, the interaction effect was significant (F (1, 16) = 13.33, MSE = 1.5, p < .05). This result occurred because, whereas repetition increased purchase intent for the younger adults, it decreased purchase intent for the older adults. Filling in the Blanks ________________________________________ Source Model Error Total Source Df SS MS 11 F P 10.00 .0046 F P 10.00 119 2180.00 Df SS MS A 3 45.00 1.50 .2493 B 2 98.00 .0371 AxB 957.00 .0011 ________________________________________ 1) How many levels for Factor A? 2) How many levels for Factor B? 3) How many total units in the experiment? 4) How many units per treatment? 5) Which factors are significant? Social support in Stressful Situations ______________________________________ The subjects in an experiment were told that they were going to experience a mildly painful medical procedure. They were asked to rate how much they would like to wait with another subject, who was a confederate of the experimenter. The two independent variables were the gender of the subject, and the gender of the confederate. The data are presented in the table below. Conduct a 2way ANOVA to examine the effect of these two variables on the subjects' desire to wait with another person ( = .05). Subject Male Female Confederate Male Female M=4 M=4 x = 20 x = 20 x2 = 88 x2 = 86 M=5 M=8 x = 25 x = 40 x2 = 129 x2 = 324 Social support Solution ______________________________________ SST = = x2 - G2/N __________________________________________ SSM = (T2/n) - G2/N __________________________________________ SSE = SST - SSM Partitioning SSM __________________________________________ SSsubj = [(TS2/n)] - G2/N __________________________________________ SSconfed = [(TC2/n)] - G2/N __________________________________________ SSAxR = SSM - (SSA + SSB) Social Support ANOVA table __________________________________________ Source df SS MS F df SS MS F Model Error Total Source Subject Confed SxC __________________________________________ Subject Male Female Conclusions: Confederate Male Female 4 4 5 8 4.5 6 4 6.5