1. Factorial Experiments
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2. 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
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3. 2 Factor Model
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4. Model
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5. model
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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)
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13. ANOVA`
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14. HSTS312

15. 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
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16. 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
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17. 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.
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18. 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
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19. 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).
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20. 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
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21. 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.
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22. 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.
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23. 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
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