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ANOVA II (Part 2)

ANOVA II (Part 2). Class 16. Implications of Interaction 1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts.

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ANOVA II (Part 2)

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  1. ANOVA II (Part 2) Class 16

  2. Implications of Interaction 1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts. 3. Analysis of findings, when an interaction is present, will focus on individual treatment means rather than on overall factor (IV) means. 4. Interaction indicates moderation.

  3. 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.

  4. Ordinal and Disordinal Interactions XXXX Interaction YYY Interaction

  5. Ordinal and Disordinal Interactions Ordinal Interaction Disordinal Interaction

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

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

  8. Birth Order Main Effect: NO Gender Main Effect: YES Interaction: N0

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

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

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

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

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

  14. Birth Order Means

  15. Development of ANOVA Analytic Components 1. Individual scores  Condition (cell) sums 2. Condition sums  Condition means 3. Cond. means – ind. scores  Deviations  Deviations2 4. Deviations2  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.

  16. Birth Order and Ratings of “Activity” Deviation Scores AS Total Between Within (AS – T) = (A – T) + (AS –A) Level a1: Oldest Child 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 a2: Youngest Child 4.33 (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 scores: Oldest = 3.13 Youngest = 5.47 Total = 4.30

  17. Sum of Squared Deviations Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations SSTotal = SSBetween + SSWithin

  18. Computing Sums of Squares from Deviation Scores Birth Order and Activity Ratings (continued) SS = Sum of squared diffs, AKA “sum of squares” SST = Sum of squares., total (all subjects) SSA = Sum of squares, between groups (treatment) SSs/A = Sum of squares, within groups (error) SST = (2.97)2 + (2.30)2 + … + (1.37)2 + (2.70)2 = 25.88 SSA = (-1.17)2 + (-1.17)2 + … + (1.17)2 + (1.17)2 = 13.61 SSs/A = (-1.80)2 + (-1.13)2 + … + (0.20)2 + (1.53)2 = 12.27 Total (SSA + SSs/A) = 25.88

  19. Variance Variance Code Code Calculation Calculation Data Result Meaning Mean Square Between Groups MSA SSA dfA Between groups variance Mean Square Between Groups MSA SSA dfA 13.61 1 13.61 Mean Square Within Groups MSS/A SSS/A dfS/A Within groups variance Mean Square Within Groups MSS/A SSS/A dfS/A 12.27 8 1.53 Mean Squares Calculations

  20. F = 13.61 1.51 = 8.78 F Ratio Computation F = MSA = Between Group Variance MSS/A Within Group Variance

  21. Conceptual Approach to Two Way ANOVA SS total = SS between groups + SS within groups Oneway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) XXXX YYYY

  22. Conceptual Approach to Two Way ANOVA SS total = SS between groups + SS within groups Oneway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) Twoway 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)

  23. Distributions of All Four Conditions Total Mean (4.32)

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

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

  26. 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 Effects AB - T = (A effect) + (B effect) + ????? AB - T = (A - T) + (B - T) + (AB - A - B + T) (AB - A - B + T) = ??? AKA "????" (AB - T) - (? - T) - (? - T) = Interaction Error Term in Two-Way ANOVA Error = (ABS - AB)

  27. 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 Effects 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)

  28. Deviation of an Individual Score in Two Way ANOVA ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij) Total Mean Ind. score ??? Effect ??? Effect ??? (w’n Effect) ??? Effect

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

  30. Variance for All Factors Subject Variance Factor A X Factor B dfA X B = (a –1) (b – 1) dfs/AB = ab(s – 1) dfTotal = abs – 1 (2-1) x (2-1) = 1 dfs/AB = n - ab dfTotal = n – 1 20 – 1 = 19 20 – (2 x 2) = 16 Degrees of Freedom in 2-Way ANOVA Between Groups Factor A df A = a - 1 2 – 1 = 1 Factor B df B = b – 1 2 – 1 = 1 Interaction Effect Error Effect Total Effect

  31. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Factor A Factor B a1 a2 Sum b1 # X B1 b2X X X Sum A1X T Factor A = Birth Order Factor B = Gender # = Known quantity

  32. 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 NOTE: “Fictional sums” for demonstration.

  33. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Factor A Factor B a1 a2 a3 Sum b1 # # X B1 b2 # # XB2 b3X X X X Sum A1 A2X T A, B, T, # = free to vary; X = determined by #s Once # are established, Xs are known

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

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

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

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