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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. **OutputDesign and Descriptives

**33. **OutputAnova-table

**34. **OutputRegression 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. **OutputAnova-table

**38. **OutputRegression coefficients

**39. **Marginal meansMain effects

**40. **Marginal meansInteractions

**41. **Marginal meansinteraction

**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. **AnCoVaAnalysis 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. **OutputAnova-table

**83. **OutputAnalysis of time-trend

**84. **OutputBetween-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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