Analysis of Variance ANOVA 8//007 P7 Analysis of Variance What We Will Cover in This Section Introduction. Overview. Simple ANOVA. Repeated Measures ANOVA. Factorial ANOVA 8//007 P7 Analysis of Variance Situation The management of Sal T. Dogg s restaurant wanted to see if the saltiness of appetizers would influence the number of drinks people purchased. Three sections of the club are targeted to receive appetizers that have either low, medium, or high saltiness. The dependent variable is the number of drinks ordered.. What is the research hypothesis?. What is H o?. What is the statistical hypothesis? 8//007 P7 Analysis of Variance
Appetizer saltiness and number or drinks ordered. Group Low Salt Group Medium Salt Group High Salt 5 M =.00 M =.90 M =.80 8//007 P7 Analysis of Variance Issue How to determine if one mean is significantly different from the other means while minimizing the probability of committing a Type I error. 8//007 P7 Analysis of Variance 5 Analysis of Variance: Background and Theory 8//007 P7 Analysis of Variance
Logic 8//007 P7 Analysis of Variance 7 Treatment Effects 8//007 P7 Analysis of Variance 8 : G : t : t : t 8//007 P7 Analysis of Variance 9
: t : t : t Between Groups 8//007 P7 Analysis of Variance 0 Sources of Between Group TREATMENT effect Random error from... Subjects. Measurement. Random. 8//007 P7 Analysis of Variance Within Groups : t : t : t 8//007 P7 Analysis of Variance
Sources of Within Group Random Error from Subjects Measurement. Random. 8//007 P7 Analysis of Variance Partitioning the Variance Between Groups Within Groups 8//007 P7 Analysis of Variance Partitioning the Variance Between Groups Within Groups Treatment Effect Random Error Random Error 8//007 P7 Analysis of Variance 5
Where We Are Going ( Treatment ) + ( Random Error) Random Error 8//007 P7 Analysis of Variance F-test Compared to t-test t (df) = X -X σ X F (k-,n-k) = MS B MS W 8//007 P7 Analysis of Variance 7 Partitioning the : G : t : t : t ( X µ ) G ( µ ) ( X µ ) T µ G T N = k = Between + + Σ N k Within 8//007 P7 Analysis of Variance 8
ANOVA Model : G Effect : t : t : t = Treatment Effect + Random Error = Between Groups 8//007 P7 Analysis of Variance 9 + Within Groups Partitioning the Variance = Between Groups + Within Groups df (N - ) BetweenGroups = + df df Between (k ) WithinGroups Within (N k) 8//007 P7 Analysis of Variance 0 F-test Model Between Groups Within Groups TREATMENT EFFECT Subject Error Measurement Error Random Error Subject Error Measurement Error Random Error 8//007 P7 Analysis of Variance
The F ratio : G : t : t : t F (k-,n-k) = MS B MS W = Between Groups Within Groups 8//007 P7 Analysis of Variance The Critical Value of F See page 95 in old text book, 9 in new text book. Notice Need df between (numerator) for columns. Need df within (denominator) for rows. As df increases the critical values get smaller. 8//007 P7 Analysis of Variance The Problem Returns 8//007 P7 Analysis of Variance
The Situation (in case you forgot) The management of Sal T. Dogg s restaurant wanted to see if the saltiness of appetizers would influence the number of drinks people purchased. Three sections of the club are targeted to receive appetizers that have either low, medium, or high saltiness. The dependent variable is the number of drinks ordered. 8//007 P7 Analysis of Variance 5 Hypotheses Research Hypothesis. Saltiness of the appetizers will influence the number of drinks that people buy. Null Hypothesis. Saltiness will not influence the number of drinks that people buy. Statistical Hypothesis. µ µ µ 8//007 P7 Analysis of Variance Appetizer saltiness and number or drinks ordered. Group Low Salt Group Medium Salt Group High Salt 5 M =.00 M =.90 M =.80 8//007 P7 Analysis of Variance 7
Graph of Saltiness Ratings.5.0.5.0.5.0 Mean of SALTINES.5.0.5 0.0 Low saltiness Medium Saltiness High Saltiness GROUP 8//007 P7 Analysis of Variance 8 ANOVA Summary Table Source df MS F (crit=.5) Between Groups.87.5.77 Within Groups.50 7.9 5.7 9 8//007 P7 Analysis of Variance 9 How to Express F F (,7) =.77, p<.05 Degrees of freedom (between, within) Calculated F Alpha 8//007 P7 Analysis of Variance 0
Post Hoc Tests When F is significant, how do you determine which of the means differs from the others? 8//007 P7 Analysis of Variance Tukey Honestly Significant Difference Test (HSD) HSD = q ( α, dfwithin, k) MS n within HSD = q (.05,7,).9 0 q = Value from table α = desired significance level df within = within groups df k = Number of groups. HSD =.5 x.09 HSD =.0 8//007 P7 Analysis of Variance How To Use Tukey (HSD =.0).5.90.5.5.5 0.5.00.80 0.90.0.0 8//007 P7 Analysis of Variance
Scheffé Test Compute a value called C for each pair of means. C corrects for multiple pairwise comparisons. Need to compute C only once if the sample sizes are equal for all groups. To make a decision you compare the computed C obt to C crit. 8//007 P7 Analysis of Variance Computing C crit crit ( )( ) C = k F C crit = crit ( )(.7) C =.59 crit If the computed value of C exceeds the critical value, then the two means are significantly different based on the alpha level of F crit. 8//007 P7 Analysis of Variance 5 Computing C obt C obt = MS X W X + n n C obt.00.90 =.9 + 0 0.90 C obt =.8 C =.5 obt 8//007 P7 Analysis of Variance
Advantages of Scheffé Evaluates each pair of means at a time. Corrects for differing sample sizes. More conservative than Tukey. 8//007 P7 Analysis of Variance 7 Effect Size: Eta Squared (η ) η = Treatment.87 η = 5.7 η =.5 8//007 P7 Analysis of Variance 8 Effect Size: Omega Squared K MS ω = + MS ( ) B T W W ω.87 x ω =.9 5.7 +.9 ( ) 5.05 ω = 5.8.79 ω = 8//007 P7 Analysis of Variance 9
Assumptions. The observations within each sample are independent.. The population from which the samples are selected is normally distributed.. The population from which the samples are selected have equal variances (homogeneity of variance) 8//007 P7 Analysis of Variance 0 Another ANOVA Example Sal O. Gysm felt that the perceived difficulty of logic problems would influence performance on these problems. Sal developed a set of problems and gave them to three groups. One group was told that the problems was easy, another was told that they were moderately difficult, and the third was told that they were difficult. The dependent variable was the number of problems solved. 8//007 P7 Analysis of Variance ANOVA: Example Easy 9 8 7 M = 8.0 Moderate 0 M =.0 Difficult 5 M =.0 8//007 P7 Analysis of Variance
Logic Problem Results No of problems solved 8 7 5 0 Easy Moderate Difficult 8//007 P7 Analysis of Variance ANOVA: Summary Table Source df MS F Between Within. 8.00 7..7 7.00.5* * p <.05 8//007 P7 Analysis of Variance Post hoc Analysis: Tukey HSD 9 8 7 5 0 HSD =. 8 Easy Moderate Difficult 8//007 P7 Analysis of Variance 5
Effect Size : Eta : ( η ) η = between total. η = 7. η =.8 8//007 P7 Analysis of Variance Effect Size : Omega K MS ω = + MS ( ) B T W W ( ). 7.00 ω = 7.+ 7.00 0. ω = 57. ω =.55 8//007 P7 Analysis of Variance 7 Another Practice Problem Tess Tosterone is studying aggression among adolescent girls. She believes that there is a relationship between the level of interaction a girl has with her mother and the girl s level of aggression. She has identified fifteen girls who fall into one of three maternal interaction levels (low, medium, and high) and has measured their aggression scores. The scores are shown on the next slide. 8//007 P7 Analysis of Variance 8
Data Summary Table Low interaction 5 9 M =.00 Moderate Interaction 8 5 M = 5.00 High Interaction 0 0 0 M =.00 8//007 P7 Analysis of Variance 9 ANOVA Summary Table Sum of Squares df Mean Square F Between 70.00 5.00 9. Within.00.8.00 8//007 P7 Analysis of Variance 50 Tukey HSD HSD = q ( α, dfwithin, k).77 MS n within.8 5 HSD =.0 8//007 P7 Analysis of Variance 5
Which Means are Different? 7 5 0 5 Low Moderate High 8//007 P7 Analysis of Variance 5 Eta η = between total 70.00.00.0 8//007 P7 Analysis of Variance 5 Repeated Measures ANOVA 8//007 P7 Analysis of Variance 5
Sources of Within Group. Measurement error.. Individual differences among the subjects.. Random error. 8//007 P7 Analysis of Variance 55 Sources of Between Group. TREATMENT EFFECT.. Individual differences among the subjects.. Measurement Error.. Random error. 8//007 P7 Analysis of Variance 5 Partitioning the Variance Between Groups (Conditions) Within Groups Between Subjects Error 8//007 P7 Analysis of Variance 57
Partitioning the Variance df = Between Groups Between + Between Subjects = Between BetweenSs df + df + df BetweenSs + Error Error Error (N - ) (N - k) (n ) (N k) -(n -) 8//007 P7 Analysis of Variance 58 The F-test F (k-)(n-) = MS Treatment MS Error 8//007 P7 Analysis of Variance 59 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?. What is the statistical hypothesis? 8//007 P7 Analysis of Variance 0
Subject 5 7 8 9 Mean Baseline week 0 9 7 5 5 0 0 7 9 7 7 8..00 Treatment Week 5 8 0 5 5 7 8 8 7 5 5 8 5 8 9 9. 5.78.78 Subject 7 97 8 5 5 9 8. 8//007 P7 Analysis of Variance Mean Headache Strength by Week 5 0 5 0 5 0 5 8//007 P7 Analysis of Variance Summary Table Source df MS F Between Weeks Within Between Subjects Error 8 9.0 8.7 0.0.0 7.0 85.0 Error. 8//007 P7 Analysis of Variance
Post hoc Tests Tukey s HSD Replace MS within with Ms error. Replace df within with df error. Scheffé Replace MS within with Ms error. Replace df within with df error. 8//007 P7 Analysis of Variance Effect Size η = between total 9. η =. η =.77 8//007 P7 Analysis of Variance 5 Factorial ANOVA 8//007 P7 Analysis of Variance
Definition Experimental design in which there are two or more independent variables and one dependent variable. 8//007 P7 Analysis of Variance 7 Problem # Effects of Music on Mood Clarissa Thompson 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. 8//007 P7 Analysis of Variance 8 Music Study Descriptive Statistics Music Type Aggressive Calm Classical 5.00 8.7. Popular 5.9 9.7 0.5 5. 9.00. 8//007 P7 Analysis of Variance 9
Relationship between music type and mood M ood 0 50 0 0 0 0 0 Classical Popular Calm Aggressive 8//007 P7 Analysis of Variance 70 Main Effect The independent influence that one independent variable alone has on the dependent variable. 8//007 P7 Analysis of Variance 7 Factorial ANOVA: One Main Effect 0 Job Satisfaction 8 Workers Slackers 0 Nonsmokers Smokers 8//007 P7 Analysis of Variance 7
Factorial ANOVA: Main Effects Teacher Satisfaction 0 8 0 Easy Final Hard Final F students A students 8//007 P7 Analysis of Variance 7 Interaction The combined effects of two or more independent variables on the dependent variable. 8//007 P7 Analysis of Variance 7 Factorial Graphs: Interaction 0 Job Satisfaction 8 Smoking OK No Smoking 0 Nonsmokers Smokers 8//007 P7 Analysis of Variance 75
Partitioning Sources of Between Treatments Within Treatments Factor A Factor B Interaction 8//007 P7 Analysis of Variance 7 The 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. 8//007 P7 Analysis of Variance 77 Model Experience Under 0 years Chocolate Chips No Yes n=5 n=5 Over 0 years n=5 n=5 8//007 P7 Analysis of Variance 78
Under 0 Over 0 Mean No Chips 9 0 8 9 0 9 9 0 9. Chips M= 9. M=.8 9 0 9 0 M= 9.8 M= 0.0 5.9 Mean 5. 9.9 7.75 8//007 P7 Analysis of Variance 79 Chocolate Chip Study 0 Impractical Ideas 8 0.8 Under 0 Over 0 No CHIPS Chips 8//007 P7 Analysis of Variance 80 x Factorial ANOVA Chocolate study summary table Source Between Group Experience Chocolate Chips AXB Within Group * p <.0.95 9.5 8.5 7.05.80 5.75 df 9 MS 9.5 8.5 7.05.05 F 88.05* 5.9* 7.* 8//007 P7 Analysis of Variance 8
Experience Chips Experience x Chips Effect Size ω..5.7 η.8.50.78 8//007 P7 Analysis of Variance 8 Factorial ANOVA: Notation Number of independent variables x x factorial ANOVA Levels of each independent variable. 8//007 P7 Analysis of Variance 8 Factorial ANOVA Assumptions. The observations within each treatment condition are independent.. The population distribution is relatively normal.. The variances within each treatment condition are equal. 8//007 P7 Analysis of Variance 8
8//007 P7 Analysis of Variance 85