1. Multisample inference: Analysis of Variance
Chapter 12
Multisample inference:
Analysis of Variance
EPI809/Spring 2008

2. Learning Objectives 1. Describe Analysis of Variance (ANOVA)
2. Explain the Rationale of ANOVA
3. Compare Experimental Designs
4. Test the Equality of 2 or More Means
Completely Randomized Design
Randomized Block Design
Factorial Design
As a result of this class, you will be able to ...
EPI809/Spring 2008

3. Analysis of Variance
A analysis of variance is a technique that partitions the total sum of squares of deviations of the observations about their mean into portions associated with independent variables in the experiment and a portion associated with error
EPI809/Spring 2008

4. Analysis of Variance
The ANOVA table was previously discussed in the context of regression models with quantitative independent variables, in this chapter the focus will be on nominal independent variables (factors)
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5. Analysis of Variance
A factor refers to a categorical quantity under examination in an experiment as a possible cause of variation in the response variable.
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6. Analysis of Variance
Levels refer to the categories, measurements, or strata of a factor of interest in the experiment.
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7. Types of Experimental Designs
This teleology is based on the number of explanatory variables & nature of relationship between X & Y.
Completely Randomized
Randomized Block
Factorial
One-Way Anova
Two-Way Anova
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8. Completely Randomized Design
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9. Completely Randomized Design
1. Experimental Units (Subjects) Are Assigned Randomly to Treatments
Subjects are Assumed Homogeneous
2. One Factor or Independent Variable
2 or More Treatment Levels or groups
3. Analyzed by One-Way ANOVA
EPI809/Spring 2008

10. One-Way ANOVA F-Test
Tests the Equality of 2 or More (p) Population Means
Variables
One Nominal Independent Variable
One Continuous Dependent Variable
Note:
There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable.
Ask, what are nominal & interval scales?
EPI809/Spring 2008

11. One-Way ANOVA F-Test Assumptions
1. Randomness & Independence of Errors
2. Normality
Populations (for each condition) are Normally Distributed
3. Homogeneity of Variance
Populations (for each condition) have Equal Variances
EPI809/Spring 2008

12. One-Way ANOVA F-Test Hypotheses
H0: 1 = 2 = 3 = ... = p
All Population Means are Equal
No Treatment Effect
Ha: Not All j Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
NOT 1  2  ...  p
EPI809/Spring 2008

13. One-Way ANOVA F-Test Hypotheses
H0: 1 = 2 = 3 = ... = p
All Population Means are Equal
No Treatment Effect
Ha: Not All j Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
NOT 1 = 2 = ... = p
Or i ≠ j for some i, j.
f(X) X  =  =  1 2 3
f(X) X  =   1 2 3
EPI809/Spring 2008

14. One-Way ANOVA Basic Idea
1. Compares 2 Types of Variation to Test
Equality of Means
2. If Treatment Variation Is Significantly Greater Than Random Variation then Means Are Not Equal
3.Variation Measures Are Obtained by ‘Partitioning’ Total Variation
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15. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
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16. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
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17. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
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18. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling
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19. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling
Sum of Squares Among
Sum of Squares Between
Sum of Squares Treatment
Among Groups Variation
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20. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling
Sum of Squares Among
Sum of Squares Between
Sum of Squares Treatment (SST)
Among Groups Variation
Sum of Squares Within
Sum of Squares Error (SSE)
Within Groups Variation
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21. Total Variation Y Response, Y Group 1 Group 2 Group 3
EPI809/Spring 2008

22. Treatment Variation Y3 Y Y2 Y1 Response, Y Group 1 Group 2 Group 3
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23. Random (Error) Variation
Response, Y
Y3
Y2
Y1
Group 1
Group 2
Group 3
EPI809/Spring 2008

24. One-Way ANOVA F-Test Test Statistic
F = MST / MSE
MST Is Mean Square for Treatment
MSE Is Mean Square for Error
2. Degrees of Freedom
1 = p -1
2 = n - p
p = # Populations, Groups, or Levels
n = Total Sample Size
EPI809/Spring 2008

25. One-Way ANOVA Summary Table
Source of
Degrees
Sum of
Mean F
Variation of
Squares
Square
Freedom
(Variance)
n = sum of sample sizes of all populations.
c = number of factor levels
All values are positive. Why? (squared terms)
Degrees of Freedom & Sum of Squares are additive; Mean Square is NOT.
Treatment
p - 1
SST
MST =
MST
SST/(p - 1)
MSE
Error
n - p
SSE
MSE =
SSE/(n - p)
Total
n - 1
SS(Total) =
SST+SSE
EPI809/Spring 2008

26. One-Way ANOVA F-Test Critical Value
If means are equal, F = MST / MSE  1. Only reject large F!
Reject H 
Do Not
Reject H F F a ( p  1 , n  p )
Always One-Tail!
© T/Maker Co.
EPI809/Spring 2008

27. One-Way ANOVA F-Test Example
As a vet epidemiologist you want to see if 3 food supplements have different mean milk yields. You assign 15 cows, 5 per food supplement.
Question: At the .05 level, is there a difference in mean yields?
Food1 Food2 Food
EPI809/Spring 2008

28. One-Way ANOVA F-Test Solution
H0: 1 = 2 = 3
Ha: Not All Equal
 = .05
1 = 2 2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
MST 23 .
5820 F    25 . 6
MSE .
9211
Reject at  = .05
 = .05
There Is Evidence Pop. Means Are Different F
3.89
EPI809/Spring 2008

29. Summary Table Solution
Source of
Degrees of
Sum of
Mean F
Variation
Freedom
Squares
Square
(Variance)
Food
3 - 1 = 2
25.60
Error
= 12
.9211
Total
= 14
EPI809/Spring 2008

30. SAS CODES FOR ANOVA proc anova; /* or PROC GLM */ class group;
Data Anova;
input group$ milk
cards;
food food food
food food food
food food food
food food food
food food food ;
run;
proc anova; /* or PROC GLM */
class group;
model milk=group;
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31. SAS OUTPUT - ANOVA Sum of
Source DF Squares Mean Square F Value Pr > F
Model <.0001
Error
Corrected Total
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32. Pair-wise comparisons
Needed when the overall F test is rejected
Can be done without adjustment of type I error if other comparisons were planned in advance (least significant difference - LSD method)
Type I error needs to be adjusted if other comparisons were not planned in advance (Bonferroni’s and scheffe’s methods)
EPI809/Spring 2008

33. Fisher’s Least Significant Difference (LSD) Test
To compare level 1 and level 2
Compare this to t/2 = Upper-tailed value or - t/2 lower-tailed from Student’s t-distribution for /2 and (n - p) degrees of freedom
MSE = Mean square within from ANOVA table
n = Number of subjects
p = Number of levels
EPI809/Spring 2008

34. Bonferroni’s method To compare level 1 and level 2
Adjust the significance level α by taking the new significance level α*
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35. SAS CODES FOR multiple comparisons
proc anova;
class group;
model milk=group;
means group/ lsd bon;
run;
EPI809/Spring 2008

36. SAS OUTPUT - LSD t Tests (LSD) for milk
NOTE: This test controls the Type I comparisonwise error rate,
not the experimentwise error rate.
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of t = t.975,12
Least Significant Difference
Means with the same letter are not significantly different.
t Grouping Mean N group
A food1
B food2
C food3
EPI809/Spring 2008

37. SAS OUTPUT - Bonferroni
Bonferroni (Dunn) t Tests for milk
NOTE: This test controls the Type I experimentwise error rate
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of t =t1-0.05/3/2,12
Minimum Significant Difference
Means with the same letter are not significantly different.
Bon Grouping Mean N group
A food1
B food2
C food3
EPI809/Spring 2008

38. Randomized Block Design
EPI809/Spring 2008

39. Randomized Block Design
1. Experimental Units (Subjects) Are Assigned Randomly within Blocks
Blocks are Assumed Homogeneous
2. One Factor or Independent Variable of Interest
2 or More Treatment Levels or Classifications
3. One Blocking Factor
EPI809/Spring 2008

40. Randomized Block Design
Factor Levels:
(Treatments)
A, B, C, D
Experimental Units
 Treatments are randomly assigned within blocks
Block 1 A C D B
Block 2
Block 3 .
Block b
Are the mean training times the same for 3 different methods?
9 subjects
3 methods (factor levels)
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41. Randomized Block F-Test
1. Tests the Equality of 2 or More (p) Population Means
2. Variables
One Nominal Independent Variable
One Nominal Blocking Variable
One Continuous Dependent Variable
Note:
There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable.
Ask, what are nominal & interval scales?
EPI809/Spring 2008

42. Randomized Block F-Test Assumptions
1. Normality
Probability Distribution of each Block-Treatment combination is Normal
2. Homogeneity of Variance
Probability Distributions of all Block-Treatment combinations have Equal Variances
EPI809/Spring 2008

43. Randomized Block F-Test Hypotheses
H0: 1 = 2 = 3 = ... = p
All Population Means are Equal
No Treatment Effect
Ha: Not All j Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
1  2  ...  p Is wrong
EPI809/Spring 2008

44. Randomized Block F-Test Hypotheses
H0: 1 = 2 = ... = p
All Population Means are Equal
No Treatment Effect
Ha: Not All j Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
1  2  ...  p Is wrong
f(X) X  =  =  1 2 3
f(X) X  =   1 2 3
EPI809/Spring 2008

45. The F Ratio for Randomized Block Designs
SS=SSE+SSB+SST

46. Randomized Block F-Test Test Statistic
F = MST / MSE
MST Is Mean Square for Treatment
MSE Is Mean Square for Error
2. Degrees of Freedom
1 = p -1
2 = n – b – p +1
p = # Treatments, b = # Blocks, n = Total Sample Size
EPI809/Spring 2008

47. Randomized Block F-Test Critical Value
If means are equal, F = MST / MSE  1. Only reject large F!
Reject H 
Do Not
Reject H F F a ( p  1 , n  p )
Always One-Tail!
© T/Maker Co.
EPI809/Spring 2008

48. Randomized Block F-Test Example
You wish to determine which of four brands of tires has the longest tread life. You randomly assign one of each brand (A, B, C, and D) to a tire location on each of 5 cars. At the .05 level, is there a difference in mean tread life?
Tire Location
Block
Left Front
Right Front
Left Rear
Right Rear
Car 1
A: 42,000
C: 58,000
B: 38,000
D: 44,000
Car 2
B: 40,000
D: 48,000
A: 39,000
C: 50,000
Car 3
C: 48,000
D: 39,000
B: 36,000
Car 4
A: 41,000
D: 42,000
C: 43,000
Car 5
D: 51,000
A: 44,000
C: 52,000
B: 35,000
EPI809/Spring 2008

49. Randomized Block F-Test Solution
H0: 1 = 2 = 3= 4
Ha: Not All Equal
 = .05
1 = 3 2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
F =
Reject at  = .05
 = .05
There Is Evidence Pop. Means Are Different F
3.49
EPI809/Spring 2008

50. SAS CODES FOR ANOVA data block; input Block$ trt$ resp @@; cards;
Car1 A: Car1 C: Car1 B: Car1 D: 44000
Car2 B: Car2 D: Car2 A: Car2 C: 50000
Car3 C: Car3 D: Car3 B: Car3 A: 39000
Car4 A: Car4 B: Car4 D: Car4 C: 43000
Car5 D: Car5 A: Car5 C: Car5 B: 35000 ;
run;
proc anova;
class trt block;
model resp=trt block;
Means trt /lsd bon;
EPI809/Spring 2008

51. Dependent Variable: resp
SAS OUTPUT - ANOVA
Dependent Variable: resp
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE resp Mean
Source DF Anova SS Mean Square F Value Pr > F
trt
Block
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52. SAS OUTPUT - LSD
Means with the same letter are not significantly different.
t Grouping Mean N trt
A C:
B D: B
C B A: C
C B:
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53. SAS OUTPUT - Bonferroni
Means with the same letter are not significantly different.
Bon Grouping Mean N trt
A C: A
B A D: B
B C A: C
C B:
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54. Factorial Experiments
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55. Factorial Design
1. Experimental Units (Subjects) Are Assigned Randomly to Treatments
Subjects are Assumed Homogeneous
2. Two or More Factors or Independent Variables
Each Has 2 or More Treatments (Levels)
3. Analyzed by Two-Way ANOVA
EPI809/Spring 2008

56. Advantages of Factorial Designs
1. Saves Time & Effort
e.g., Could Use Separate Completely Randomized Designs for Each Variable
2. Controls Confounding Effects by Putting Other Variables into Model
3. Can Explore Interaction Between Variables
EPI809/Spring 2008

57. Two-Way ANOVA
Tests the Equality of 2 or More Population Means When Several Independent Variables Are Used
Same Results as Separate One-Way ANOVA on Each Variable
But Interaction Can Be Tested
EPI809/Spring 2008

58. Two-Way ANOVA Assumptions
1. Normality
Populations are Normally Distributed
2. Homogeneity of Variance
Populations have Equal Variances
3. Independence of Errors
Independent Random Samples are Drawn
EPI809/Spring 2008

59. Two-Way ANOVA Data Table
Factor
Factor B A 1 2
... b
Observation k 1 Y Y
... Y
111
121
1b1
Yijk Y Y
... Y
112
122
1b2 2 Y Y
... Y
211
221
2b1 Y Y
... YX
212
222
2b2
Level i Factor A
Level j Factor B : : : : : a Y Y
... Y
a11
a21
ab1 Y Y
... Y
a12
a22
ab2
EPI809/Spring 2008

60. Two-Way ANOVA Null Hypotheses
1. No Difference in Means Due to Factor A
H0: 1. = 2. =... = a.
2. No Difference in Means Due to Factor B
H0: .1 = .2 =... = .b
3. No Interaction of Factors A & B
H0: ABij = 0
EPI809/Spring 2008

61. Two-Way ANOVA Total Variation Partitioning
SS(Total)
Variation Due to Treatment A
Variation Due to Treatment B
SSB
SSA
Variation Due to Interaction
Variation Due to Random Sampling
SS(AB)
SSE
EPI809/Spring 2008

62. Two-Way ANOVA Summary Table
Source of
Degrees of
Sum of
Mean F
Variation
Freedom
Squares
Square A
a - 1
SS(A)
MS(A)
MS(A)
(Row)
MSE B
b - 1
SS(B)
MS(B)
MS(B)
(Column)
MSE AB
(a-1)(b-1)
SS(AB)
MS(AB)
MS(AB)
(Interaction)
MSE
Error
n - ab
SSE
MSE
Same as Other Designs
Total
n - 1
SS(Total)
EPI809/Spring 2008

63. Interaction
1. Occurs When Effects of One Factor Vary According to Levels of Other Factor
2. When Significant, Interpretation of Main Effects (A & B) Is Complicated
3. Can Be Detected
In Data Table, Pattern of Cell Means in One Row Differs From Another Row
In Graph of Cell Means, Lines Cross
EPI809/Spring 2008

64. Graphs of Interaction
Effects of Gender (male or female) & dietary group (sv, lv, nor) on systolic blood pressure
Interaction
No Interaction
Average
Average
Response
male
Response
male
female
female sv lv
nor sv lv
nor
EPI809/Spring 2008

65. Two-Way ANOVA F-Test Example
Effect of diet (sv-strict vegetarians, lv-lactovegetarians, nor-normal) and gender (female, male) on systolic blood pressure.
Question: Test for interaction and main effects at the .05 level.
EPI809/Spring 2008

66. SAS CODES FOR ANOVA data factorial; input dietary$ sex$ sbp; cards;
sv male
sv male
sv male
sv male
sv male
sv male
sv female 102.6
sv female 99
sv female 83 .6
sv female 99.6
sv female 112.6
lv male 116.5
lv male 118.5
lv male 119.5
lv male 110.5
lv male 115.5
lv male 105.2
nor male 128.3
nor male 129.3
nor male 126.3
nor male 127.3
nor male 125.3
nor female 119.1
nor female 119.2
nor female 115.6
nor female 119.9
nor female 119.8
nor female 119.7 ;
run;
EPI809/Spring 2008

67. SAS CODES FOR ANOVA proc glm; class dietary sex;
model sbp=dietary sex dietary*sex;
run;
model sbp=dietary sex;
EPI809/Spring 2008

68. Dependent Variable: sbp
SAS OUTPUT - ANOVA
Dependent Variable: sbp
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE sbp Mean
Source DF Type I SS Mean Square F Value Pr > F
dietary
sex
dietary*sex
Source DF Type III SS Mean Square F Value Pr > F
dietary
sex
dietary*sex
EPI809/Spring 2008

69. Linear Contrast
Linear Contrast is a linear combination of the means of populations
Purpose: to test relationship among different group means
with
Example: 4 populations on treatments T1, T2, T3 and T4.
Contrast T T T T4 relation to test
L μ1 - μ3 = 0
L / / μ1 – μ2/2 – μ3/2 = 0
EPI809/Spring 2008

70. T-test for Linear Contrast (LSD)
Construct a t statistic involving k group means. Degrees of freedom of t - test: df = n-k.
Construct
To test H0:
Compare with critical value t1-α/2,, n-k.
Reject H0 if |t| ≥ t1-α/2,, n-k.
SAS uses contrast statement and performs an F – test df (1, n-k);
Or estimate statement and perform a t-test df (n-k).
EPI809/Spring 2008

71. T-test for Linear Contrast (Scheffe)
Construct multiple contrasts involving k group means. Trying to search for significant contrast
Construct
To test H0:
Compare with critical value.
Reject H0 if |t| ≥ a
EPI809/Spring 2008

72. SAS Code for contrast testing
proc glm;
class trt block;
model resp=trt block;
Means trt /lsd bon scheffe;
contrast 'A - B = 0' trt ;
contrast 'A - B/2 - C/2 = 0' trt ;
contrast 'A - B/3 - C/3 -D/3 = 0' trt ;
contrast 'A + B - C - D = 0' trt ;
lsmeans trt/stderr pdiff;
lsmeans trt/stderr pdiff adjust=scheffe; /* Scheffe's test */
lsmeans trt/stderr pdiff adjust=bon; /* Boneferoni's test */
estimate ‘A - B' trt ;
run;
EPI809/Spring 2008

73. Regression representation of Anova
EPI809/Spring 2008

74. Regression representation of Anova
One-way anova:
Two-way anova:
SAS uses a different constraint
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75. Regression representation of Anova
One-way anova: Dummy variables of factor with p levels
This is the parameterization used by SAS
EPI809/Spring 2008

76. Conclusion: should be able to
1. Recognize the applications that uses ANOVA
2. Understand the logic of analysis of variance.
3. Be aware of several different analysis of variance designs and understand when to use each one.
4. Perform a single factor hypothesis test using analysis of variance manually and with the aid of SAS or any statistical software.
EPI809/Spring 2008

77. Conclusion: should be able to
5. Conduct and interpret post-analysis of variance pairwise comparisons procedures.
6. Recognize when randomized block analysis of variance is useful and be able to perform the randomized block analysis.
7. Perform two factor analysis of variance tests with replications using SAS and interpret the output.
EPI809/Spring 2008

78. Key Terms Between-Sample Variation Levels One-Way Analysis of Variance
Completely Randomized Design
Experiment-Wide Error Rate
Factor
Levels
One-Way Analysis of Variance
Total Variation
Treatment
Within-Sample Variation
EPI809/Spring 2008