ANOVA II (Part 2) Class 18

FINAL EXAM Rutgers Date / Time Tuesday Dec. 15, 3:00 – 6:00 Prefer: Dec. 15, 1:00 – 4:00 What works for you???

Suppress Group: n = 10, σ = 0.69 Confide Group: n = 10, σ = 1.51 Appreciate Friends if Suppress or Express Scary Movie Thoughts and Feelings Does variability of suppress group differ from variability of confide group? Friend Appreciation Score How do conditions differ? Suppress Cond Express Cond

Levene's Test of Homogeneity of Variance Levene's Test output

Relation of Z scores to Raw Scores Question: Does converting raw scores to Z scores (std. scores) change the shape of score distribution? Answer: Although scores change, rel. btwn scores the same 2 SDs

Birth Order Means Multiple Factor Tests

Anatomy of Factorial Design "FX" = "Effects"

Limitations of Main Effects Show “what” but not “why” Fail to account for the “what ifs” –Cannot show moderation –Cannot account for underlying causes

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 Ordinal Interaction Disordinal Interaction

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

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

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

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

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

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

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

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

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

Birth Order Means To What Degree Does a Person Who Discloses Personal Problems Appear "Active"? 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 (3) (2) (3) Note: Condition ns in parentheses

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

Partitioning the Sum of Squares

Development of Two-Way ANOVA Analytic Components 1. Individual scores  Condition (cell) sums 2. Condition sums  Condition means 3. Cond. means – ind. scores  Deviations  Deviations 2 4. Deviations 2  Sums of squares (SS between, SS within ) 5. Sum Sqrs / df  Mean squares (Between and Within) 6. MS Between  F Ratio MS Within F (X, Y df)  Probability of null ( p ) p  Accept null, or accept alt.

Birth Order and Ratings of “Activity” Deviation Scores AS Total Between Within (AS – T) = (A – T) +(AS –A) 1.33 (-2.97)= (-1.17) +(-1.80) 2.00(-2.30)=(-1.17) +(-1.13) 3.33(-0.97)=(-1.17) + ( 0.20) 4.33(0.03)=(-1.17) +( 1.20) 4.67(0.37)=(-1.17) + ( 1.54) Level a 1: Oldest Child: Mean (A 1 )= 3.13 Level a 2: Youngest Child : Mean (A 2 )= (0.03)= (1.17) +(-1.14) 5.00(0.07)= (1.17) +(-0.47) 5.33(1.03)= (1.17) + (-0.14) 5.67(1.37)= (1.17) +( 0.20) 7.00(2.70)= (1.17) + ( 1.53) Sum: (0) = (0) + (0) Mean of Total Scores ( T) = 4.30

Sum of Squared Deviations Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations SS Total = SS Between + SS Within

Computing Sums of Squares from Deviation Scores Birth Order and Activity Ratings (continued) SS = Sum of squared diffs, AKA “sum of squares” SS T =Sum of squares., total (all subjects) SS A = Sum of squares, between groups (treatment) SS s/A =Sum of squares, within groups (error) SS T = (2.97) 2 + (2.30) 2 + … + (1.37) 2 + (2.70) 2 = SS A = (-1.17) 2 + (-1.17) 2 + … + (1.17) 2 + (1.17) 2 = SS s/A = (-1.80) 2 + (-1.13) 2 + … + (0.20) 2 + (1.53) 2 = Total (SS A + SS s/A ) = 25.88

Variance CodeCalculationMeaning Mean Square Between Groups MS A SS A df A Between groups variance Mean Square Within Groups MS S/A SS S/A df S/A Within groups variance Variance CodeCalculationDataResult Mean Square Between Groups MS A SS A df A Mean Square Within Groups MS S/A SS S/A df S/A Mean Squares Calculations

F Ratio Computation F = = 8.78 F = MS A = Between Group Variance MS S/A Within Group Variance

Design and Notation for Two-Factor Design Experimental Design Factor A Factor B a1 a2 n b1 s = 3 s = 2 b2 s = 2 s = 3 ABS Matrix (Treatment Combinations) ab 11 ab 12 ab 21 ab 22 ABS 111 ABS 121 ABS 211 ABS 221 ABS 112 ABS 122 ABS 212 ABS 222 ABS 113 ABS 223 AB Matrix Levels of Factor A Levels of Factor B a1 a2 Marginal Sum b1 AB 11 AB 21 B 1 b2 AB 12 AB 22 B 2 A 1 A 2 T Marginal sum Factor A: Birth Order a1 = oldest a2 = youngest Factor B: Gender b1 = male b2 = female n Total n = 10

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 SS between groups = 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)

Total Mean (4.32) 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

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) Ind. score Total Mean Factor A Effect Factor B Effect Interaction AXB Effect Error (w’n Effect) (Birth Order) (Gender) (Birth * Gender)

Degrees of Freedom in 2-Way ANOVA Between Groups Factor A (Birth Order) df A = a - 12 – 1 = 1 Factor B (Gender) df B = b – 12 – 1 = 1 Interaction Effect Factor A X Factor B (Birth X Gender) df A X B = (a –1) (b – 1) (2-1) x (2-1) = 1 Error Effect Subject Variance df s/AB = ab(s – 1) df s/AB = n - ab 10 – (2 x 2) = 6 Total Effect Variance for All Factors df Total = abs – 1 df Total = n – 1 10 – 1 = 9

Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Birth Order GenderYoungest Oldest Sum Males Sum Females NOTE: “Fictional sums” for demonstration.

Conceptualizing Degrees of Freedom(df) For Conditions and Factors in Factorial ANOVA Factor A Factor Ba 1 a 2 a 3 Sum b 1 # # X # b 2 # # X # b 3 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 ASS A a - 1SS A df A MS A MS S/AB BSS B b - 1SS b df b MS B MS S/AB A X BSS A X B (a - 1)(b - 1)SS AB df A X B MS A X B MS S/AB Within (S/AB) SS S/A ab (s- 1)SS S/AB df S/AB TotalSS T abs - 1 Source of Variation Sum of Squares df Mean Square F Ratio (SS)(MS)

F Ratios for 2-Way ANOVA

Effect of Multi-Factorial Design on Significance Levels: Gender Main Effects Mean Men Mean Women Sum of Sqrs. Betw'n dt Betw'n MS Betw'n Sum of Sqrs. Within df Withi n MS Withi n Fp One Way Two Way

ONEWAY ANOVA AND GENDER MAIN EFFECT SourceSum of Squares dfMean Square FSig. Gender Error SourceSum of Squares dfMean Square FSig. Gender Birth Order Interaction Error Total 9 TWO-WAY ANOVA AND GENDER MAIN EFFECT Oneway F: 3.42 =1.22Twoway F : 3.42 =