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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

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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

    54. Adjusted analysis

    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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