F formula F = Variance (differences) between sample means Variance (differences) expected from sampling error

Simple data table - Audience vs. No Audience No AudienceAudience X = 5X = 7

Simple data table - Low Esteem vs. High Esteem Self-Esteem LowHigh X = 9X = 3

Audience vs. No Audience & High Esteem vs. Low Esteem No AudienceAudience High Low

Audience vs. No Audience & High Esteem vs. Low Esteem No AudienceAudience HighX = 3 LowX = 7X = 11

Audience vs. No Audience & High Esteem vs. Low Esteem No AudienceAudience HighX = 3 LowX = 7X = 11 X = 5X = 7

Audience vs. No Audience & High Esteem vs. Low Esteem No AudienceAudience HighX = 3 LowX = 7X = 11X = 9 X = 5X = 7

Graph of Means HighLow Mean Number of Errors No Audience Audience

Single Factor ANOVA vs. 2 Factor ANOVA Single Independent Variable Dependent Variable Single Factor - One-way ANOVA Now: Two Independent Variables Dependent Variable Two Factor ANOVA

Program type vs. Class size Factor B (class size) 18-student class24-student class30-student class Factor A (program) Program I Scores for n = 15 subjects taught by program I in a class of 18 Scores for n = 15 subjects taught by program I in a class of 24 Scores for n = 15 subjects taught by program I in a class of 30 Program II Scores for n = 15 subjects taught by program II in a class of 18 Scores for n = 15 subjects taught by program II in a class of 24 Scores for n = 15 subjects taught by program II in a class of 30

Two-factor analysis of variance permits us to test: 1.Mean difference between the 2 teaching programs 2.Mean differences between the 3 class sizes 3.Combinations of teaching program and class size

Three separate hypothesis tests in one ANOVA Three F-ratios: F = Variance (differences) between sample means Variance (differences) expected from sampling error

Program type vs. Class size data 1 18-student class 24-student class 30-student class Program I X = 85X = 77X = 75X = 79 Program II X = 75X = 67X = 65X = 69 X = 80X = 72X = 70

Hypotheses for Teaching program Factor ATeaching Program (Teaching program has no effect on math scores) (Teaching program has an effect on math scores) F = Variance (differences) between treatment means for Factor A Variance (differences) expected from sampling error

Program type vs. Class size data 1 18-student class 24-student class 30-student class Program I X = 85X = 77X = 75X = 79 Program II X = 75X = 67X = 65X = 69 X = 80X = 72X = 70

Hypotheses for Class Size Factor BClass Size (Class size has no effect on math scores) (Class size has an effect on math scores) F = Variance (differences) between treatment means for Factor B Variance (differences) expected from sampling error At least one population mean is different

Program type vs. Class size data 1 18-student class 24-student class 30-student class Program I X = 85X = 77X = 75X = 79 Program II X = 75X = 67X = 65X = 69 X = 80X = 72X = 70

Program type vs. Class size data 2 18-student class 24-student class 30-student class Program I X = 80X = 77X = 80X = 79 Program II X = 80X = 67X = 60X = 69 X = 80X = 72X = 70

Graphs of Math Test Score Means 18 Students Mean math test scores 24 Students 30 Students Program I Program II 18 Students 24 Students 30 Students Mean math test scores Program I Program II

Thus two-factor AVOVA composed of 3 distinct hypothesis tests: 1.The main effect of A (called the A-effect) 2.The main effect of B (called the B-effect) 3.The interaction (called the AxB interaction)

(a) Data showing a main effect for factor A, but no B-effect and no interaction B1B1 B2B2 A1A1 20 A 1 mean = 20 A2A2 10 A 2 mean = 10 B 1 mean = 15 B 2 mean = 15 No difference 10-point difference

(b) Data showing a main effects for both factor A and factor B, but no interaction B1B1 B2B2 A1A1 1030A 1 mean = 20 A2A2 2040A 2 mean = 30 B 1 mean = 15 B 2 mean = point difference 10-point difference

(c) Data showing no main effect for either factor A or factor B, but an interaction B1B1 B2B2 A1A1 1020A 1 mean = 15 A2A2 2010A 2 mean = 15 B 1 mean = 15 B 2 mean = 15 No difference

Null and Alternate Hypotheses for an Interaction Interaction H 0 : There is no interaction between factors A and B. H 1 : There is an interaction between factors A and B. OR H 0 : The effect of factor A does not depend on the levels of factor B (and B does not depend on A). H 1 : The effect of one factor does depend on the levels of the other factor (and B does not depend on A). F = Variance (differences) not explained by main effects Variance (differences) expected from sampling error

Differences in Variabiltiy Between Treatment Variability 1.Treatment (factor A, B, or AxB interaction) 2.Individual differences (difference of SS in each treatment condition) 3.Experimental error Variability Within Treatments (Chance) 1.Individual differences 2.Experimental error F = Treatment effect + Individual Differences + Experimental Error Individual Differences + Experimental Error

Treatment (Cell) Combinations Factor B Level B 1 Level B 2 Level B 3 Factor A Level A 1 Treatment (cell) A 1 B 1 Treatment (cell) A 1 B 2 Treatment (cell) A 1 B 3 Level A 2 Treatment (cell) A 2 B 1 Treatment (cell) A 2 B 2 Treatment (cell) A 2 B 3

Breakdown of Variability Sources Total Variability Within-treatments Variability Between-treatments Variability Factor A Variability Factor B Variability Interaction Variability Stage 2 Stage 1

2 - Factor Data Table Factor B B1B1 B2B2 B3B3 Factor A A1A AB = 10 SS = AB = 35 SS = AB = 15 SS = 18 A 1 = 60 A2A AB = 20 SS = AB = 5 SS = AB = 5 SS = 8 A 2 = 30 B 1 = 30B 2 = 40B 3 = 20 N = 30 G = 90 ∑x 2 = 520

2 - Factor Data Table w/ Cell Means Factor B B1B1 B2B2 B3B3 Factor A A1A AB = 10 SS = AB = 35 SS = AB = 15 SS = 18 A 1 = 60 A2A AB = 20 SS = AB = 5 SS = AB = 5 SS = 8 A 2 = 30 B 1 = 30B 2 = 40B 3 = 20 N = 30 G = 90 ∑x 2 = 520

2 - Factor Data Table w/ Cell Means & Marginal Means Factor B B1B1 B2B2 B3B3 Factor A A1A AB = 10 SS = AB = 35 SS = AB = 15 SS = 18 A 1 = 60 A2A AB = 20 SS = AB = 5 SS = AB = 5 SS = 8 A 2 = 30 B 1 = 30B 2 = 40B 3 = 20 N = 30 G = 90 ∑x 2 = 520

Breakdown of Variability Sources and Formulas Interaction SS found by subtraction = 80 Stage 2 Stage 1 TotalBetween TreatmentsWithin TreatmentsFactor AFactor B

Breakdown of Degrees of Freedom and Formulas Interaction df = df A x df B = 2 Stage 2 Stage 1 Total df = N - 1 = 29 Between Treatments df = ab - 1 = 5 Within Treatments df = N - ab = 24 Factor A df = a - 1 = 1 Factor B df = b - 1 = 2

Breakdown of both Degrees of Freedom and SS Formulas Interaction SS is found by subtraction df = (a - 1)(b - 1) Stage 2 Stage 1 Total df = N - 1 Between Treatments df = ab - 1 Within Treatments df = N - ab Factor A df = a - 1 Factor B df = b - 1

2 F-ratio distributions Distribution of F-ratios df = 2.24 Distribution of F-ratios df = 1.24

2-Factor Source Table SourceSSdfMS Fp <.05 Between Treatments1305 Factor A (program)301 F(1,24)= 6.0 √ Factor B (class size)20210F(1,24)= 2.0 n.s. AxB Interaction80240F(1,24)= 8.0 √ Within Treatments Total25029

Plot of the mean scores of Factor A and B B1B1 B2B2 B3B3 Factor B Mean score A2A2 A1A1

Schacter (1968) Obesity and Eating Behavior Hypothesis: obese individuals do not respond to internal biological signals of hunger

Variables 2 Independent Variables or Factors –Weight (obese vs. normal) –Fullness (full stomach vs. empty stomach) Dependent Variable –Number of crackers eaten by each subject

Weight vs. Fullness Factor B (fullness) EmptyFull Factor A (weight) Normaln = 20 Obesen = 20

Weight vs. Fullness Factor B (fullness) Empty Stomach Full Stomach Factor A (weight) Normal n = 20 X = 22 AB = 440 SS = 1540 n = 20 X = 15 AB = 300 SS = 1270 A 1 = 740 Obese n = 20 X = 17 AB = 340 SS = 1320 n = 20 X = 18 AB = 360 SS = 1266 A 2 = 700 B 1 = 780B 2 = 660 G = 1440  x 2 = 31,836 N = 80

Source Table for Weight vs. Fullness SourceSSdfMS Fp <.05 Between Treatments5203 Factor A (weight)201 F(1,76)= 0.28 n.s. Factor B (fullness)1801 F(1,76)= 2.54 n.s. AxB Interaction3201 F(1,76)= 4.51 √ Within Treatments Total591679

Plot of mean number of crackers eaten for each group EmptyFull Normal2215 Obese1718 Mean Number of Crackers Eaten Mean Number of Crackers Eaten Empty Stomach Full Stomach Obese Normal

Graph of Schacter 1

Schacter Graph 2

Conclusion to Weight vs. Fullness Problem The means and standard deviations are presented in Table 1. The two-factor analysis of variance showed no significant main effect for the weight factor, F(1,76) = 0.28, p >.05; no significant main effect for the fullness factor, F(1,76) = 2.54, p >.05; but the interaction between weight and fullness was significant, F(1,76) = 4.51, p <.05. Mean number of crackers eaten in each treatment condition Fullness Empty Stomach Full Stomach Weight Normal M = 22.0 SD = 9.00 M = 15.0 SD = 8.18 Obese M = 17.0 SD = 8.34 M = 18.0 SD = 8.16 TABLE 1

Data table - Treatment 1 vs. 2 & Males vs. Females Treatment 1Treatment Males3 7Females T 1 = 40T 2 = 60 SS 1 = 48SS 2 = 48

Data table for Treatment 1 vs. 2 & M vs. F Factor A (Treatment) Treatment 1Treatment 2 Factor B (Sex) Males AB = 10 SS = AB = 20 SS = 4 B 1 = 30 Females AB = 30 SS = AB = 40 SS = 4 B 2 = 70 A 1 = 40A 2 = 60

Assumptions for the 2-factor ANOVA: (Independent Measures) 1.Observations within each sample are independent 2.Populations from which the samples are drawn are normal 3.Populations from which the samples are selected must have equal variances (homogeneity of variance)