Completely Randomized Factorial Design

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Completely Randomized Factorial Design With Two
Factors Example
A police department in a big city want to assess
their human relations course for new officers.
The independent variables are the type of
neighborhood the officers get to be assigned to
during the period of the course, factor A, and
the amount of time they spend in the course,
Factor B. Factor A has three levels
a1upper-class, a2middle-class and
a3inner-city. Factor B also has three levels
b15 hours, b210 hours and b315 hours. The
dependent (response) variable, y, is attitude
towards minority groups following the course.
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Completely Randomized Factorial Design With Two
Factors
a1b1 a1b2 a1b3 a2b1 a2b2 a2b3 a3b1 a3b2 a3b3
24 44 38 30 35 26 21 41 42
33 36 29 21 40 27 18 39 52
37 25 28 39 27 36 10 50 53
29 27 47 26 31 46 31 36 49
42 43 48 34 22 45 20 34 64
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Completely Randomized Factorial Design With Two
Factors
What do we want to compare?
A\B b1 b2 b3 Grand Means
a1 m11 m12 m13 m1.
a2 m21 m22 m23 m2.
a3 m31 m32 m33 m3.
Grand means m.1 m.2 m.3 m
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Completely Randomized Factorial Design With Two
Factors
Hypotheses
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Interactions
Assuming we have these two factors together in
one experiment and that we know the following
true means. Is there and interaction effect?
B A b1 b2 b3 Grand Means
a1 20 20 35 (202035)/3 25
a2 22 22 37 (222237)/3 27
a3 27 27 42 (272742)/3 32
Grand means (202227)/3 23 (202227)/3 23 (353742)/3 38 28
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Interactions
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Interactions
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Interactions
Assuming we have these two factors together in
one experiment and that we know the following
true means. Is there and interaction effect?
B A b1 b2 b3 Grand Means
a1 20 27 35 (202035)/3 27.33
a2 22 22 37 (222237)/3 27
a3 27 20 42 (272742)/3 29.66
Grand means (202227)/3 23 (202227)/3 23 (353742)/3 38 28
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Interactions
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Interactions
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Completely Randomized Factorial Design With Two
Factors
Linear Model
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Completely Randomized Factorial Design With Two
Factors
What are we comparing?
A/B b1 b2 b3 Grand Means
a1 m11 m a1 b1 (ab)11 m12 m a1 b2 (ab)12 m12 m a1 b3 (ab)13 m1. m a1
a2 m21 m a2 b1 (ab)21 m22 m a2 b2 (ab)22 m23 m a2 b3 (ab)23 m2. m a2
a3 m31 m a3 b1 (ab)31 m32 m a3 b2 (ab)32 m33 m a3 b3 (ab)33 m3. m a3
Grand means m.1 m b1 m.2 m b2 m.3 m b3 m
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Completely Randomized Factorial Design With Two
Factors
Hypotheses
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Interactions
If there is no interaction and we have these two
factors together in one experiment we will have
the following results (effects model)
A\B b1 b2 b3 Grand Means
a1 28(25-28)(20-25-2328) 28(25-28)0 28(25-28) 30-3
a2 28(27-28) 30-1
a3 28(32-28) 304
Grand means 28(23-28) 28-5 28(23-28) 28-5 28(38-28) 2810 28
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Interactions
Estimated means from the data
A\B b1 b2 b3 Grand Means
a1 33 35 38 35.33
a2 30 31 36 32.33
a3 30 40 52 40.67
Grand means 31 35.33 42 36.11
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Interactions
Estimated means from the data
A\B b1 b2 b3 Grand Means
a1 36.11-0.78-5.11 2.78 33 36.11-0.78-0.78 0.45 35 36.11-0.785.89 -3.22 38 36.11-0.78
a2 36.11-3.78-5.11 2.75 30 36.11-3.78-0.78 --0.55 31 36.11-3.785.89 -2.22 36 36.11-3.78
a3 36.114.59-5.11 -5.59 30 36.114.56-0.78 0.11 40 36.114.565.89 5.44 52 36.114.56
Grand means 36.11-5.11 36.11-0.78 36.115.89 36.11
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Interactions
Simple means vs. A-levels.
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Interactions
Simple means vs. B-levels.