1. ANALYSIS OF VARIANCE (ANOVA)
In spite of its name it serves for comparing means,
not for comparing variances.
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2. Remember: Two-sample t test
Two independent samples
Assuming the equality of variance for the two populations:
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3. One-way ANOVA
Several independent samples of size
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4. The sample means are different, even if there is no difference between groups
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5. The sample means are more different. Is the difference significant?
group
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6. order of experiments is important
Example 51
(Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 165) Blood coagulation times (seconds) with four different diets
order of experiments is important
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7. ANOVA1

8. ANOVA1

9. Open Data Table: blood.xls Analyze>Fit Y by X Y: CTIME X: DIET
click on the red triangle, choose Display options
error bar
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10. The variance within the i-th group
(deviations from the group mean)
The pooled within-groups variance (if constant across groups):
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11. Variance of the group mean, if there is no real difference (H0)
but sampling fluctuation
repetitions
as it is estimated from the group means
(if p=const)
more generally (if pconst)
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12. if there exists real difference (H1)
(between)
(within)
if there exists real difference (H1)
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13. The ANOVA Table
Effect of factor A is significant (H0 hypothesis is rejected) if
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14. Balanced design: p1=p2=...=pr=p
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15. Click on the red triangle, choose Means/Anova
Example (cont.)
Click on the red triangle, choose Means/Anova F0
between p
within
decision?
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16. Minitab>Stat>ANOVA>One-Way
between
within
decision?
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17. pi different
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18. Statistics>Advanced Linear/Nonlinear Models>
>General Linear Models>One-way ANOVA
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19. Summary tab: Descriptive cell statistics
Summary tab: Test all effects
between
within
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20. Assumptions expected value of the ij experimental errors is zero
the ij experimental errors are independent both within groups (across j) and between groups (across i)
the variance of experimental errors is constant
the ij experimental errors follow normal distribution
to be checked!
In order to avoid misunderstanding for error variance
will be used
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21. experimental error (not just measurement error)
i-th level of the factor (i-th diet)
j-th repetition in the i-th group
Model
experimental error (not just measurement error)
measured value
true value
expected value
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22. i effect of the i-th level (i-th diet)
means model
i effect of the i-th level (i-th diet)
 is a common value; r+1 parameters
i=1,…,r
sum to zero
set to zero
effects model
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23. effect, only r-1 independent
Estimates
grand mean
effect, only r-1 independent
group mean
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24. Estimation =0
grand mean
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25. effect, only r-1 of them are independentfüggetlen
mean of the i-th group
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26. Fisher-Cochran-theorem
all
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27. Summary tab: Coefficients
sum to zero
sigma-restricted
set to zero
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28. Summary tab: Coefficients sigma-restricted
set to zero
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29. Confidence interval for the expected value of group means
Point estimator:
Interval estimator :
degrees of freedom:
Confidence interval for thee expected value of the i-th group:
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30. ANOVA1

31. All of them are different?
rejected
All of them are different?
Comparisons: planned, post hoc
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32. df would be only n2+n3-2=6+6-2=10 and pooling
df for is
LSD test
(Least Significant Difference)
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33. cik contrast coefficients
Generalisation:
(kth null hypothesis)
cik contrast coefficients
contrast
c11=0, c21=1, c31=-1, c41=0
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34. orthogonal contrasts if kl independent comparisons
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35. ?
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36. for a comparison α* (e.g. 0.05) (individual error rate)
comparisons (1-2, 1-3, 1-4, 2-3, 2-4, 3-4)
for a comparison α* (e.g. 0.05)
(individual error rate)
not committing type I error: 1- α*
not committing type I error at any of r independent comparisons:
committing type I error at some comparison:
(family error rate)
e.g.
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37. In case of non-independent comparisons
Bonferroni inequality
e.g. for 6 non-independent comparisons 60.05=0.3
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38. Post hoc comparisons
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39. B-A
C-A
D-A
C-B
D-B
D-C
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40. B-A
C-A
D-A
C-B
D-B
D-C
Minitab12

41. Post-hoc tab: Bonferroni
Post hoc comparisons
Post-hoc tab: LSD
Post-hoc tab: Bonferroni
·6=
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42. Planned comparisons Planned comps tab: Specify contrasts
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43. ANOVA1

44. Calculate the effect estimates:
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45. ANOVA1

46. is rejected
None of them are equal?
further questions:
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47. Click on the red triangle next to Oneway analysis,
choose Compare Means, Each Pair, Student’s t
threshold in t
(LSD)
significant: C-A, C-D,...
C and B are connected by A
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48. Click on the red triangle next to Oneway analysis,
choose Compare Means, All Pairs, Tukey HSD
more conservative
(less easily states significance)
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49. Power: probability of detecting an existing difference
Click on the red triangle next to Oneway analysis,
choose Power
If =2.5, with 20 experiments
(5 for each diet) we will be able to detect Delta as large as 3 with 98.6% probability
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50. E.g. if 1=-3, 2=3, 3= 4=0
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51. The size of detectable difference
Statistics>Power Analysis>Several Means, ANOVA 1-Way
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52. E.g. if 1=-3, 2=3, 3= 4=0
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53. The size of detectable difference
Minitab>Power and Sample Size>One-Way ANOVA
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54. Required sample size for detecting difference of 5 units
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55. Checking the assumption on homogeneity of variances
click on the red triangle,
choose Unequal Variances p
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56. Contrast analysis
click
LSMeans Contrast
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57. A+, B+, C-, D-
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58. Homogeneity of (within-group) variance
Minitab>Stat>ANOVA>Homogeneity of variance
sensitive to the normaality assumption
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59. Homogeneity of (within-group) variance
Minitab>Stat>ANOVA>Homogeneity of variance
sensitive to the normality assumption
ANOVA
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60. Checking residuals
(Graphs)
ANOVA
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61. ANOVA
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62. ANOVA
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63. ANOVA
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64. Homogeneity of variance?
More results>Assumptions tab: Homogeneity of variances ...
Bartlett test
sensitive to normality
Levene test
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65. Checking assumptions by examining the residuals
Residuals 1 tab
Normality
Pred & resids Predicted results
(histogram)
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66. Residuals 2 tab
X: Order Y: Resids
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67. Checking residuals
Graphs>Four in One
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68. Two-way ANOVA
All levels of the first factor are combined with all levels of the second factor equal number of repetitions in each cell (balanced design).
The structure of the design ensures the orthogonality.
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69. Survival time of animals
Example 52
(Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 228)
Survival time of animals
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70. ANOVA1

71. ANOVA1

72. poison (i) treatment (j)
i=1,…,r; j=1,…,q, k=1,…,p
Model
repetition (k)
poison (i) treatment (j)
means model
(all are equal)
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73. effect of the i-th poison j-th treatment interaction
Model
effect of the i-th poison j-th treatment interaction
effects model
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74. The ANOVA Table
34(4-1)=36
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75. Open Data Table: poison.xls, (poison, treatment Nominal)
Analyze>Fit Model
Y: survival
Add: poison, treatment (Macros, Full factorial)
Emphasis: Minimal Report N!
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76. Click on the red triangle next to Response SURVIVAL,
choose Factor Profiling, Profiler and Interaction Plots
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77. 34(4-1)=36
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78. (r-1)(q-1)=(3-1)(4-1)=6 independent
r-1=3-1=2 independent
q-1=4-1=3 independent
(r-1)(q-1)=(3-1)(4-1)=6 independent
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79. Minitab>Stat>ANOVA>Two-Way
poison.mtw
Minitab>Stat>ANOVA>Two-Way
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80. Minitab>Stat>ANOVA>Main Effects Plot

81. Minitab>Stat>ANOVA>Interactions Plot

82. Advanced Linear/Nonlinear Models>
Statistics>
Advanced Linear/Nonlinear Models>
>General Linear Models>Factorial ANOVA>
Means tab: Observed, unweighted, Plot
Summary tab: All effects
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83. Checking the assumptions by plotting residuals
Click on the red triangle next to Response SURVIVAL,
choose Row Diagnostics, Plot Residual by Predicted
is not justified
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84. Box-Cox transformation
Click on the red triangle next to Response SURVIVAL,
choose Factor Profiling, Box-Cox Y transformation
variance stabilising transformation
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85. Homogeneity of variance?
More results>Assumptions tab: Homogeneity of variances
sensitive to normality
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86. Checking the assumptions by plotting residuals
Residuals1 tab: Pred. & resid.
is not justified
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87. Checking the assumptions by plotting residuals
satisfied?
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88. ANOVA1

89. Box-Cox transformation
File>Open: (Program Files>StatSoft>Statistica8>Examples>Macros>
>Analysis Examples>BoxCox)
variance stabilising transformation
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90. ANOVA1

91. if
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92. straight line is fitted
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93. ANOVA1

94. New column: recsurv=1/survival Repeat the analysis not transformed
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95. Click on the red triangle next to Response SURVIVAL,
choose Save Columns, Residuals
Analyze: Distributions, Normal Quantile Plot
random fluctuation around the line: Normal
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96. Minitab>Stat>Basic Statistics>Store Descriptive Statistics
Mean, Standard Deviation
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97. ANOVA1

98. Minitab>Stat>Regression
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99. Box-Cox transformation
Minitab>Control Charts>Box-Cox transformation
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100. ANOVA1

101. ANOVA1

102. ANOVA1

103. The effects are more convincing (F values are larger), p for interaction is 0.112 → 0.387
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104. The residuals
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105. Minitab>Stat>ANOVA>Interactions Plot
y to 1/y
Minitab>Calc
Minitab>Stat>ANOVA>Interactions Plot
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106. ANOVA1

107. Minitab>Stat>ANOVA>Two-Way

108. Checking residuals
Graphs>Four in One
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109. Comparisons Do Poisons 1 and 2 differ? Planned comparisons fülön
Compute
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110. estimated effect
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