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# Analysis of Variance

Outline of Section. One-Way Anova (repetition)Blonds have more funK-Way AnovaMultifactorial ExperimentsInteractionsAnalysis of Covariance (AnCoVa)Adjustment for comtinuous confoundersRepeated Measures AnovaClustered observationsRandom Effects Models"Repeated Measures AnCoVa". . . . One Télécharger la présentation ## Analysis of Variance

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1. Analysis of Variance

2. Outline of Section One-Way Anova (repetition) Blonds have more fun K-Way Anova Multifactorial Experiments Interactions Analysis of Covariance (AnCoVa) Adjustment for comtinuous confounders Repeated Measures Anova Clustered observations Random Effects Models �Repeated Measures AnCoVa�

3. One-Way ANOVA (repetition) Data Example (blonds have more fun) Assumptions/Model Analysis in SPSS Post-hoc Tests Output Model Checking

4. Pain Thresholds of Blonds and Brunettes

5. The data

6. Statistical model Model Hypothesis Same threshold

7. F-test Compare between-variation to within-variation Type III Sums of squares Partial SS

8. ANOVA in SPSS

9. A first glance at the data

10. Main analysis

11. Analysis of differences Which hair colors are different? Trend? Specific other hypothesis Post hoc analysis

12. Contrasts

13. Specific contrasts

14. Post hoc analysis

15. Blonds Output

16. More Blonds Output

17. Blonds over (and out)

18. Take Home Messages Assumptions Independent observations Normality Variance homogeneity Output Anova-table Regression coefficients Post hoc Model check Residual plots

19. Multi factorial Experiments Some Motivating Examples �Clean� K-way Example Potthoff & Roy Growth Data Heart Beat Variability and n3-PUFA Blood Pressure and Weight

20. Simulated Anova-data The effect of some treatment is assesed in 100 individuals Placebo (N = 50), treatment (N = 50) The effect is the difference between measurement after intervention and baseline Maybe difference in effect between gender The treatment may be inefficient on diabetics

21. Anthropometry and Diabetes 232 persons with impared insulin sensitivity Males, N = 106 Females, N = 126 Antrhropometric measures Height, weight, waist and hip Insuline sensitivity index

22. The Pothoff and Roy data Data from 27 children (11 girls, 16 boys) Data obtained at ages 8, 10, 12 and 14 Distance from the centre of the pituitary to the pterygo-maxillary fissure. Hypoteses Differences between gender Differences over time

23. Potthoff and Roy data

24. Blood Pressure vs BMI (simulated data)

25. K-way ANOVA Example Statistical notation Build model SPSS Interpretation of output Assumptions (controlling the model)

26. Simulated Anova-data The effect of some treatment is assesed in 100 individuals Placebo (N = 50), treatment (N = 50) Maybe difference in effect between gender The treatment may be inefficient on diabetics

27. Hypotheses The treatment has an effect Differences in effect between gender Differences in effect between healthy and diabetics More�.

28. A simple 2-way set-up

29. A general 2-way Anova

30. Testing hypotheses No interaction Testing: g = 0 No effect of treatment Testing: a = 0 No difference in effect between gender Testing: b = 0 No effect at all

31. Anova in SPSS

32. Output Design and Descriptives

33. Output Anova-table

34. Output Regression Coefficients

35. Output

36. Full analysis (3-way) All main effects Treatment, Gender and Diabetes All pairwise interactions Treat*Gender, Treat*Diab and Gender*Diab The 3-way interaction Treat*Gender*Diab

37. Output Anova-table

38. Output Regression coefficients

39. Marginal means Main effects

40. Marginal means Interactions

41. Marginal means interaction

42. Conclusion

43. Take Home Messages Assumptions Independent observations Normality Variance homogeneity Output Anova-table Regression coefficients Interactions!!!! Post hoc Model check Residual plots

44. AnCoVa Analysis of Covariance The variation in the response is explained by categorical as well as continuous covariates Adjustment for confounding Categorical confounder (gender, smoking etc) gives (k+1)-way ANOVA Continuous confounder gives ANCOVA

45. Anthropometry and Diabetes 232 persons with impared insulin sensitivity Males, N = 106 Females, N = 126 Antrhropometric measures Height, weight, waist and hip Insuline sensitivity index

46. (Too) Simple Research Question Do female and males have different expected weights? Are females more fat? Do felames and males have different expected waist measure? Do females have the same shape as males?

47. Weight vs. gender

48. Weight vs. height

49. SPSS

50. Conclusion (weight)

51. Waist vs. Gender

52. Waist vs. Hip

53. Confounding vs. Effect Modification

55. Take Home Messages Assumptions Independent observations Normality Variance homogeneity Output Anova-table Regression coefficients Post hoc Model check Residual plots

56. Repeated Measures Anova Data Example (Potthoff & Roy) Repetition of unpaired/paired comparison Chosing the �right� SD Anova Approach (wrong analysis) Model Checking Repeated measures anova assumptions Spherificity & normal distribution SPSS Long vs Wide Format Output

57. The Pothoff and Roy data Data from 27 children (11 girls, 16 boys) Data obtained at ages 8, 10, 12 and 14 Distance from the centre of the pituitary to the pterygo-maxillary fissure. Hypoteses Differences between gender Differences over time

58. Potthoff and Roy data

59. A paired comparison Are the growth from 12th to 14th week significant?

60. Growth from 12th to 14th week

61. Paired T-test

62. The difference of interest We are evaluating is the average in group 1 and the average in group 2 SD is the standard error of the difference For paired data with variance homogeneity

63. Model assumptions for the paired analysis Normal distribution of the differences Not neccessarily of X and Y Variance homogeneity Independence (between subjects) Not neccessarily between X and Y

64. How are the assumptions checked? Normal distribution of differences QQ-plot Variance homogeneity Diff-sum-plot Independence per design

65. Why not make an unpaired T-test?

66. Paired versus unpaired

67. ANOVA analysis on Potthoff and Roy data NB!! This is a wrong analysis

68. Output from Anova

69. Assumptions Normality Variance homogeneity Independence

70. Independence

71. Model diagnostics Normality: ? Variance homogeneity: ? Residuals versus id ? Observations are clustered Wrong variance (similar to paired/unpaired)

72. Model

73. Repeated Measeures Model Each individual has its own level Yij = mi + a j (gender) + eij m i is normal (Between Subject Variation) eij is normal (Within Subject Variation) Variance structure Compound symetry Intra class Sphericity

74. Hypotheses Full model Different growth patterns between gender Parallel growth pattern between gender Difference in size between gender Same growth pattern No difference in size between gender No growth, different sizes between gender No growth

75. A two-way Anova approach Factors are: Time and Individual Advantages Easy to model We can test differences in growthpattern between gender We can test growth vs no-growth Disadvantages Impossible to test effects between gender, i.e. parallel growth patterns between gender

76. Assumptions of Repeated Measures Normal distribution Variance homogeneity between factors Sphericity (within) Homogeneity of variance of all differences (over time) If violated the F-statistic is adjusted

77. Correction for non-sphericity Calculation of epsilons e = 1 if spericity condition is met, otherwise e < 1 F-statistic F ~ F(e df1; e df2) NB! Log-transformation may do the job

78. Data structure and reshape

79. SPSS

80. SPSS

81. Output

82. Output Anova-table

83. Output Analysis of time-trend

84. Output Between-subject Effects

85. Output

86. Output

87. Conclusion No interaction, i.e. Same growth pattern in both gender Linear growth Growth rate ? 0

88. How to proceed without the interaction term Use General Linear Model -> Univariate Id as Random factor Or use General Linear Model -> Variance Component

89. Within and Between terms

90. Take Home Messages Assumptions Normality Variance homogeneity Sperificity Output Anova-table Regression coefficients Between subject and within subject comparisons Model check Residual plots

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