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Advanced ANOVA

Advanced ANOVA. 2-Way ANOVA Complex Factorial Designs The Factorial Design Partitioning The Variance For Multiple Effects Independent Main Effects of Factor A and Factor B Interactions. Total Variability. Effect Variability (MS Between). Error Variability (MS Within). The Source Table.

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Advanced ANOVA

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  1. Advanced ANOVA 2-Way ANOVA Complex Factorial Designs The Factorial Design Partitioning The Variance For Multiple Effects Independent Main Effects of Factor A and Factor B Interactions Anthony Greene

  2. Total Variability Effect Variability(MS Between) ErrorVariability(MS Within) The Source Table • Keeps track of all data in complex ANOVA designs • Source of SS, df, and Variance (MS) • Partitioning the SS, df and MS • All variability is attributable toeffect differences or error (all unexplained differences) Anthony Greene

  3. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Factor A Variability Factor B Variability Interaction Variability Stage 2 Anthony Greene

  4. Source Table for 1-Way ANOVA Effect Variability Error Variability

  5. 2-Way ANOVA • Used when two variables (any number of levels) are crossed in a factorial design • Factorial design allows the simultaneous manipulation of variables

  6. 2-Way ANOVA For Example: Consider two treatments for mood disorders • This design allows us to consider multiple variables • Importantly, it allows us to understand Interactions among variables

  7. 2-Way ANOVA Hypothetical Data: • You can see that the effects of the drug depend upon the disorder • This is referred to as an Interaction

  8. Example of a 2-way ANOVA: Main Effect A

  9. Example of a 2-way ANOVA: Main Effect B

  10. Example of a 2-way ANOVA: Main Effect A & B

  11. Example of a 2-way ANOVA: Interaction

  12. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Factor A Variability Factor B Variability Interaction Variability Stage 2 Anthony Greene

  13. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Denominator for all F-ratios Factor A Variability Factor B Variability Interaction Variability Stage 2 Numerator for Omnibus F-ratio Anthony Greene

  14. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Denominator for F-ratio Factor A Variability Factor B Variability Interaction Variability Stage 2 Numerator for Factor A F-ratio Anthony Greene

  15. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Denominator for F-ratio Factor A Variability Factor B Variability Interaction Variability Stage 2 Numerator for Factor B F-ratio

  16. Partitioning of Variability for Two-Way ANOVA Total Variability { Stage 1 Effect Variability(MS Between) Error Variability (MS Within) { Denominator for F-ratio Factor A Variability Factor B Variability Interaction Variability Stage 2 Numerator for Interaction F-ratio

  17. 2 Main Types of Interactions

  18. Simple Effects of An Interaction Anthony Greene

  19. Simple Effects of An Interaction Anthony Greene

  20. Simple Effects of An Interaction Anthony Greene

  21. Simple Effects of An Interaction Anthony Greene

  22. Simple Effects of An Interaction Anthony Greene

  23. Simple Effects of An Interaction Anthony Greene

  24. Simple Effects of An Interaction Anthony Greene

  25. Simple Effects of An Interaction Anthony Greene

  26. Simple Effects of An Interaction Anthony Greene

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  31. How To Make the Computations Anthony Greene

  32. Anthony Greene

  33. Higher Level ANOVA N-Way ANOVA: Any number of factorial variables may be crossed; for example, if you wanted to assess the effects of sleep deprivation: • Hours of sleep per night: 4, 5, 6, 7, 8 • Age: 20-30, 30-40, 40-50, 50-60, 60-70 • Gender: M, F You would need fifty samples Anthony Greene

  34. Higher Level ANOVA Mixed ANOVA: Any number of between subjects and repeated measures variables may be crossed For example, if you wanted to assess the effects of sleep deprivation using sleep per night as the repeated measure: • Hours of sleep per night: 4, 5, 6, 7, 8 • Age: 20-30, 30-40, 40-50, 50-60, 60-70 • Gender: M, F You would need 10 samples Anthony Greene

  35. How to Do a Mixed Factorial Design Total Variability { Stage 1 Effect Variability(MS Between) MS Within { Factor AVariability Factor BVariability InteractionVariability Individual Variability ErrorVariability Stage 2 Anthony Greene

  36. Two-Way ANOVA An experimenter wants to assess the simultaneous effects of having breakfast and enough sleep on academic performance. Factor A is a breakfast vs. no breakfast condition. Factor B is three sleep conditions: 4 hours, 6 hours & 8 hours of sleep. Each condition has 5 subjects. Anthony Greene

  37. Two-Way ANOVA Factor A has 2 levels, Factor B has 3 levels, and n = 5 (i.e., six conditions are required and each has five subjects). Fill in the missing values. Anthony Greene

  38. Two-Way ANOVA First the obvious: The degrees freedom for A and B are the number of levels minus 1 . The degrees freedom Between is the number of conditions (6 = 2x3) minus 1. Anthony Greene

  39. Two-Way ANOVA The interaction (AxB) is then computed: d.f.Between = d.f.A + d.f.B + d.f.AxB. OR d.f.AxB = d.f.A d.f.B Anthony Greene

  40. Two-Way ANOVA d.f.Within= Σd.f. each cell d.f.Total = N-1 = 29. d.f.Total= d.f.Between+ d.f.Within Anthony Greene

  41. Two-Way ANOVA Now you can compute MSBetween by dividing SS by d.f. Anthony Greene

  42. Two-Way ANOVA You can compute MSA by remembering that FA= MSA MSWithin, so 5 = ?/2. SSA is then found by remembering that MS = SS df,so 10 = ?/1 Anthony Greene

  43. Two-Way ANOVA Now SSB is computed by SSA + SSB + SSAxB = SSBetween MSB = SSB/dfB and MSAxB = SSAxB/dfAxB Anthony Greene

  44. Two-Way ANOVA MSWithin=SSWithin/dfWithin, solve for SS. Anthony Greene

  45. Two-Way ANOVA Now Solve for the missing F’s (Between, B, AxB). F=MS/MSWithin Anthony Greene

  46. Two-Way ANOVA An experimenter is interested in the effects of efficacy on self-esteem. She theorizes that lack of efficacy will result in lower self-esteem. She also wants to find out if there is a different effect for females than for males. She conducts an experiment on a sample of college students, half male and half female. She then puts them through one of three experimental conditions: no efficacy, moderate efficacy, and high efficacy. Then she measures level of self-esteem. Her results are below. Conduct a two-way ANOVA. Report all significant findings with α= 0.05. Anthony Greene

  47. Data No Moderate High Efficacy Efficacy Efficacy 1 4 7 Males 3 8 8 0 7 10 Females 2 10 16 5 7 13 4 8 15 Anthony Greene

  48. No Moderate High Efficacy Efficacy Efficacy 1 T=4 4 T=19 7 T=25 Males 3 SS=4.6 8 SS=8.6 8 SS=4.7 0 7 10 Tm= 48 Females 2 T=11 10 T=25 16 T=44 5 SS=4.6 7 SS=4.7 13 SS=4.7 Tf= 80 4 8 15 Tne=15 Tme=44 The=69 n=3 k=6 N=18 G=128 ∑x2=1260 Anthony Greene

  49. SSbetween SSbtw = ∑T2/n – G2/N SSbtw = (42 + 192 + 252 + 112 + 252 + 442)/3 –282/18 SSbtw = 1228-910.2=317.8 Anthony Greene

  50. SSsex, SSefficacy, SSinteraction SSsex = ∑T2sex/nsex – G2/N SSsex = (482 + 802)/9 – 910.2 SSsex = 56.9 SSefficacy= ∑T2e/ne– G2/N SSefficacy = (152 + 442 + 692)/6 – 910.2 SSefficacy = 243.47 SSinteraction = SSbetween – SSsex – SSefficacy SSinteraction= 317.8-56.9-243.47 SSinteraction= 17.43 Anthony Greene

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