1. DOCTORAL SEMINAR, SPRING SEMESTER 2007 Experimental Design & Analysis Two-Factor Experiments February 20, 2007

2. Two-Factor Experiments Two advantages  Economy  Detection of interaction effects

3. Economy a1a2a3 n=30n=30n=30 b1b2b3 n=30n=30n=30 b1b2b3 a1n=10n=10 n=10 a2n=10 n=10 n=10 a3n=10 n=10 n=10 Compare N for 2 one-factor experiments with 1 two-factor experiment N=180 N=90

4. Detection of Interactive Effects Factors may have multiplicative effect, rather than an additive one Interactions suggest important boundary conditions for hypothesized relationships, giving clues to nature of explanation

5. Two-Factor Analysis Sources of variance when A and B are independent variables  A  B  AxB  S/AxB The model is Y ij = μ + α i + β j + (αβ) ij +ε ij Overall grand mean Average effect of α Average effect of β Interaction effect of α, β (effect left in data after subtracting off lower-order effects) Error term, also known as S/AxB, or randomness

6. Two-Factor Analysis Y ijk = μ + α i + β j + (αβ) ij +ε ijk  We want to test 3 main hypotheses Main effect of A  H 0 : α 1 = α 2 = …= α a = 0 vs. H 1 : at least one α ≠ 0 Main effect of B  H 0 : β 1 = β 2 = …= β b = 0 vs. H 1 : at least one β ≠ 0 Interaction effect of AB  H 0 : αβ ij = 0 for all ij vs. H 1 : at least one αβ ≠ 0

7. Two-Factor Analysis Sources of variance in two-factor design  Total sum of squares: Difference between each score and grand mean is squared and then summed  The deviation of a score from the grand mean can be divided into 4 independent components 1st component - deviation of row mean from grand mean 2nd component - deviation of column mean from grand mean 3rd component - deviation of an individual's score from its corresponding cell mean (only affected by random variation) If we take these 3 components and subtract them from SS T we can find a remaining 4th source of variation, which is interaction effect

8. Two-Factor Analysis Sum of Squares Total = (X ijk – X … ) 2 Sum of Squares B = an(X.j – X … ) 2 Sum of Squares A = bn(X i. – X … ) 2 Sum of Squares S/AxB = n(X ijk – X ij ) 2.

9. Two-Factor Analysis Computations in two-way ANOVA involves 4 steps 1. Examining the model for sources of variance when A and B are independent variables A (with a levels) B (with b levels) AxB (interaction effect of A, B) S/AxB (subjects nested within factors A, B) 2. Determine degrees of freedom A: a-1 B: b-1 AxB: (a-1)(b-1) = ab - a - b +1 S/AxB: ab(n-1) = abn - ab Total: abn - 1

10. Two-Factor Analysis 3. Construct formulas for sums of squares using bracket terms [A], [B], [AB], [Y], [T]  Sums and means [A] = ΣA j 2 /bn[A] = bnΣY Aj 2 [B] = ΣB k 2 /an[B] = anΣY Bk 2 [AB] = ΣAB jk 2 /n[AB] = nΣY ijk 2 [Y] = ΣY ijk 2 [Y] = ΣY ijk 2 [T] = T 2 /abn [T] = abnY T 2  Bracket terms SS A = [A] – [T] SS B = [B] – [T] SS AxB = [AB] – [A] – [B] + [T] SS S/AB = [Y] – [T] See Keppel and Wickens, p. 217-218, for summary table of computational formulas

11. Two-Factor Analysis See Keppel and Wickens, p. 217-218, for summary table of computational formulas SourceSS computationdfMSf A[A]-[T]a-1SS A /df A MS A /MS S/AB B[B]-[T]b-1SS B /df B MS B /MS S/AB AxB[AB]-[A]-[B]+[T](a-1)(b-1) = ab-a-b+1 SS AxB df AxB MS AxB /MS S/AB S/AB[Y]-[AB]ab(n-1) = abn-ab SS S/AB df S/AB Total[Y]-[T]abn-1 4. Specify mean squares and F ratios for analysis

12. Numerical Example See Keppel and Wickens, p. 221 ControlDrug XDrug YControlDrug XDrug Y a1b1a1b1 a2b1a2b1 a3b1a3b1 a1b2a1b2 a2b2a2b2 a2b2a2b2 113915614 45166187 07 1096 71513 1513 1-hour deprivation24-hour deprivation AB jk 124056444840 ΣY 2 66468830530666450 Mean31014111210 Std dev3.164.763.923.925.484.08 Std error1.582.381.961.962.742.04 of mean

13. Numerical Example What is the total sum? What are the marginal sums? 1hour24hourSum Control124456 Drug X404888 Drug Y564096 Sum108132240

14. Two-Factor Analysis [T] = T 2 /abn = 240 2 /(3)(2)(4) = 2,400 [A] = ΣA j 2 /bn = 56 2 + 88 2 + 96 2 /(2)(4) = 2,512 [B] = ΣB k 2 /an = 108 2 + 132 2 /(3)(4) = 2,424 [AB] = ΣAB jk 2 /n = 12 2 + 40 2 + … + 48 2 + 40 2 /4 = 2,680 [Y] = ΣY ijk 2 = 66 + 468 + 830 + 530 + 666 + 450 = 3,010

15. Numerical Example

16. Main Effects and Interactions a1a1 a2a2 a2a2 a2a2 a1a1 a1a1 b1b1 b2b2 b1b1 b2b2 b1b1 b2b2 a2a2 a2a2 a2a2 a1a1 a1a1 a1a1 b1b1 b2b2 b1b1 b2b2 b1b1 b2b2

17. What’s the Story? Excitement ad Nutrition ad Children Adults Cereal rating

18. What’s the Story? “Not easy to use” “Not difficult to use” 10 seconds 45 seconds Product evaluation

19. What’s the Story? No advertising Advertising Milk Soft drink Gross margins

20. What’s the Story? Exceeded expectations Did not meet expectations Low expectations High expectations Satisfaction Met expectations

21. What’s the Story? Think of 2 reasons Think of 10 reasons Novices Experts BMW evaluation

22. Ceiling Effect

23. Ordinal Interactions