ANOVA II (Part 2) Class 18

Interactions are Non-Additive Relationships Between Factors   1. Additive: When presence of one factor changes the expression of another factor consistently, across all levels. 2. Non-Additive: When the presence of one factor changes the expression of another factor differently, at different levels.

Ordinal and Disordinal Interactions

Eyeballing Interactions and Main Effects * Dem GOP * * X X X North South X Dem GOP * * * X X North South

Birth Order Main Effect: Gender Main Effect: Interaction: NO NO NO

Birth Order Main Effect: Gender Main Effect: Interaction: YES NO NO

Birth Order Main Effect: Gender Main Effect: Interaction: NO YES NO

Birth Order Main Effect: Gender Main Effect: Interaction: YES YES NO

Birth Order Main Effect: Gender Main Effect: Interaction: NO NO YES

Birth Order Main Effect: Gender Main Effect: Interaction: YES NO YES

Birth Order Main Effect: Gender Main Effect: Interaction: NO YES YES

Birth Order Main Effect: Gender Main Effect: Interaction: YES YES YES

To What Degree Does a Person Who Discloses Personal Problems Appear "Active"? (3) (2) (2) (3) Birth Order Means Main Effects are? Interaction is? Simple effects are? Diff. betwen males & females, youngest/oldest How birth order effect is moderated by gender Youngest females v. Oldest females, for example Note: Condition ns in parentheses

ANOVA: A MACHINE FOR SEPARATING TREATMENT EFFECTS (T) FROM ERROR (E)

ABS Matrix (Treatment Combinations) Design and Notation for Two-Factor Design Experimental Design Factor A Factor A: Birth Order a1 = oldest a2 = youngest Factor B: Gender b1 = male b2 = female Factor B a1 a2 n b1 s = 3 s = 2 b2 s = 2 s = 3 5 5 n 5 5 Total n = 10 ABS Matrix (Treatment Combinations) ab11 ab12 ab21 ab22 ABS111 ABS121 ABS211 ABS221 ABS112 ABS122 ABS212 ABS222 ABS113 ABS223 AB Matrix Levels of Factor A a1 a2 Marginal Sum b1 AB11 AB21 B1 b2 AB12 AB22 B2 A1 A2 T Levels of Factor B Marginal sum

Conceptual Approach to Two Way ANOVA    SS total = SS between groups + SS within groups One-way ANOVA SS between groups = Factor A and its levels, e.g., birth order: level 1 = older level 2 = younger Two-way ANOVA Factor A and its levels (e.g., birth order; older/younger) Factor B and its levels (e.g., gender; male / female) The interaction between Factors A and B (e.g., how ratings of help seeker are jointly affected by birth order and gender)

Distributions of All Four Conditions Total Mean (4.32)

Gender Effect (collapsing across birth order) Total Mean (4.32)

Birth Order Effect (collapsing across gender) Total Mean (4.32)

Interaction: Gender * Birth Order Total Mean (4.32)

Understanding Effects of Individual Treatment Groups   How much can the variance of any particular treatment group be explained by: Factor A Factor B The interaction of Factors A and B Quantification of AB Interaction AB - T = (A effect) + (B effect) + (A x B Interaction) AB - T = (A - T) + (B - T) + (AB - A - B + T) (AB - A - B + T) = Interaction AKA "residual" (AB - T) - (A - T) - (B - T) = Interaction Error Term in Two-Way ANOVA Error = (ABS - AB)

Deviation of an Individual Score in Two Way ANOVA ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bij + T) + (ABSijk – ABij) Total Mean Ind. score Factor A Effect Factor B Effect Interaction AXB Effect Error (w’n Effect) (Birth Order) (Gender) (Birth * Gender)

Variance for All Factors Degrees of Freedom in 2-Way ANOVA   Between Groups Factor A (Birth Order) df A = a - 1 2 – 1 = 1 Factor B (Gender) df B = b – 1 2 – 1 = 1 Interaction Effect Factor A X Factor B (Birth X Gender) dfA X B = (a –1) (b – 1) (2-1) x (2-1) = 1 Error Effect Subject Variance   df s/AB = n - ab 10 – (2 x 2) = 6 Total Effect Variance for All Factors   df Total = n – 1 10 – 1 = 9  

Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Birth Order Gender Youngest Oldest Sum 9.00 Males 4.50 4.50 11.00 Females 5.50 5.50 Sum 10.00 10.00 20.00 df A = a - 1 df B = b – 1 dfA X B = (a –1) (b – 1) NOTE: “Fictional sums” for demonstration.

Conceptualizing Degrees of Freedom(df) For Conditions and Factors in Factorial ANOVA Factor A Factor B a1 a2 a3 Sum b1 # # X # b2 # # X # b3 X X X X Sum # # X T # = free to vary; T has been computed X = determined by #s Once # are established, Xs are known df Formulas: Factor A = (Σa – 1) Factor B = (Σb – 1) A X B = (Σa – 1) * (Σb – 1)

Analysis of Variance Summary Table: Two Factor (Two Way) ANOVA Source of Variation Sum of Squares df Mean Square F Ratio (SS) (MS)   A SSA a - 1 dfA MSA MSS/AB B SSB b - 1 SSb dfb MSB A X B SSA X B (a - 1)(b - 1) SSAB dfA X B MSA X B Within (S/AB) SSS/A ab (s- 1) SSS/AB dfS/AB Total SST abs - 1 MSA = MSB = MSAB = MSS/AB =

F Ratios for 2-Way ANOVA

Effect of Multi-Factorial Design on Significance Levels: Gender Main Effects   Mean Men Women Sum of Sqrs. Betw'n dt MS Within df MS Within F p One Way 4.78 3.58 3.42 1 22.45 8 2.81 1.22 .30 Two Way 5.09 6 .85 4.03 .09

Source Sum of Squares df Mean Square F Sig. Source Sum of Squares df ONEWAY ANOVA AND GENDER MAIN EFFECT Source Sum of Squares df Mean Square F Sig. Gender 3.42 1 1.22 .34 Error 22.45 8 2.81   TWO-WAY ANOVA AND GENDER MAIN EFFECT Source Sum of Squares df Mean Square F Sig. Gender 3.42 1 4.03 .09 Birth Order 16.02 18.87 .005 Interaction 3.75 4.42 .08 Error 5.09 6 0.85   Total 9 Oneway F: 3.42 = 1.22 Twoway F: 3.42 = 4.42 2.81 .85