Factorial Experiments

1
Factorial Experiments

• Analysis of Variance
• Experimental Design

2

• Dependent variable Y
• k Categorical independent variables A, B, C,
  (the Factors)
• Let
• a the number of categories of A
• b the number of categories of B
• c the number of categories of C
• etc.

3
The Completely Randomized Design

• We form the set of all treatment combinations
  the set of all combinations of the k factors
• Total number of treatment combinations
• t abc.
• In the completely randomized design n
  experimental units (test animals , test plots,
  etc. are randomly assigned to each treatment
  combination.
• Total number of experimental units N ntnabc..

4
The treatment combinations can thought to be
arranged in a k-dimensional rectangular block
B
1
2
b
1
2
A
a
5
C
B
A
6
Another way of representing the treatment
combinations in a factorial experiment
C
B
...
A
...
D
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Example

• In this example we are examining the effect of

The level of protein A (High or Low) and The
source of protein B (Beef, Cereal, or Pork) on
weight gains Y (grams) in rats.
We have n 10 test animals randomly assigned to
k 6 diets
8
The k 6 diets are the 6 32 Level-Source
combinations

• High - Beef

• High - Cereal

• High - Pork

• Low - Beef

• Low - Cereal

• Low - Pork

9
Table Gains in weight (grams) for rats under six
diets differing in level of protein (High or
Low) and s ource of protein (Beef, Cereal, or
Pork)
Level
of Protein High Protein Low protein
Source of Protein Beef Cereal Pork Beef Cereal P
ork
Diet 1 2 3 4 5 6
73 98 94 90 107 49 102 74 79 76 95 82 118 56
96 90 97 73 104 111 98 64 80 86 81 95 102 86
98 81 107 88 102 51 74 97 100 82 108 72 74 106
87 77 91 90 67 70 117 86 120 95 89 61 111 9
2 105 78 58 82
Mean 100.0 85.9 99.5 79.2 83.9 78.7 Std.
Dev. 15.14 15.02 10.92 13.89 15.71 16.55
10
Example Four factor experiment

• Four factors are studied for their effect on Y
  (luster of paint film). The four factors are

1) Film Thickness - (1 or 2 mils)
2) Drying conditions (Regular or Special)
3) Length of wash (10,30,40 or 60 Minutes), and
4) Temperature of wash (92 C or 100 C)
Two observations of film luster (Y) are taken for
each treatment combination
11

• The data is tabulated below
• Regular Dry Special Dry
• Minutes 92 ?C 100 ?C 92?C 100 ?C
• 1-mil Thickness
• 20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.4
• 30 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.3
• 40 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.8
• 60 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9
• 2-mil Thickness
• 20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.5
• 30 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.8
• 40 5.5 5.6 30.5 30.2 5.5 5.8 32.6 27.4
• 60 7.2 6.0 31.4 29.6 8.0 9.9 33.5 29.5

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Notation

• Let the single observations be denoted by a
  single letter and a number of subscripts
• yijk..l
• The number of subscripts is equal to
• (the number of factors) 1
• 1st subscript level of first factor
• 2nd subscript level of 2nd factor
• Last subsrcript denotes different observations on
  the same treatment combination

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Notation for Means

• When averaging over one or several subscripts we
  put a bar above the letter and replace the
  subscripts by
• Example
• y241

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Profile of a Factor

• Plot of observations means vs. levels of the
  factor.
• The levels of the other factors may be held
  constant or we may average over the other levels

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• Definition
• A factor is said to not affect the response if
  the profile of the factor is horizontal for all
  combinations of levels of the other factors
• No change in the response when you change the
  levels of the factor (true for all combinations
  of levels of the other factors)
• Otherwise the factor is said to affect the
  response

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• Definition
• Two (or more) factors are said to interact if
  changes in the response when you change the level
  of one factor depend on the level(s) of the other
  factor(s).
• Profiles of the factor for different levels of
  the other factor(s) are not parallel
• Otherwise the factors are said to be additive .
• Profiles of the factor for different levels of
  the other factor(s) are parallel.

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• If two (or more) factors interact each factor
  effects the response.
• If two (or more) factors are additive it still
  remains to be determined if the factors affect
  the response
• In factorial experiments we are interested in
  determining
• which factors effect the response and
• which groups of factors interact .

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Factor A has no effect
B
A
19
Additive Factors
B
A
20
Interacting Factors
B
A
21

• The testing in factorial experiments
• Test first the higher order interactions.
• If an interaction is present there is no need to
  test lower order interactions or main effects
  involving those factors. All factors in the
  interaction affect the response and they interact
• The testing continues with for lower order
  interactions and main effects for factors which
  have not yet been determined to affect the
  response.

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Example Diet Example Summary Table of Cell means
Source of Protein
Level of Protein Beef Cereal Pork Overall
High 100.00 85.90 99.50 95.13
Low 79.20 83.90 78.70 80.60
Overall 89.60 84.90 89.10 87.87
23
Profiles of Weight Gain for Source and Level of
Protein
24
Profiles of Weight Gain for Source and Level of
Protein
25
Models for factorial Experiments

• Single Factor A a levels
• yij m ai eij i 1,2, ... ,a j 1,2,
  ... ,n

Random error Normal, mean 0, std-dev. s
Overall mean
Effect on y of factor A when A i
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Levels of A
1
2
3
a
y11 y12 y13 y1n
y21 y22 y23 y2n
y31 y32 y33 y3n
ya1 ya2 ya3 yan
observations Normal distn
m1
m2
m3
ma
Mean of observations
m a1
m a2
m a3
m aa
Definitions
27

• Two Factor A (a levels), B (b levels
• yijk m ai bj (ab)ij eijk
•   i 1,2, ... ,a j 1,2, ... ,b k 1,2,
  ... ,n

Overall mean
Interaction effect of A and B
Main effect of A
Main effect of B
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Table of Means
29
Table of Effects Overall mean, Main effects,
Interaction Effects
30

• Three Factor A (a levels), B (b levels), C (c
  levels)
• yijkl m ai bj (ab)ij gk (ag)ik
  (bg)jk (abg)ijk eijkl
• m ai bj gk (ab)ij (ag)ik (bg)jk
• (abg)ijk eijkl
•  
• i 1,2, ... ,a j 1,2, ... ,b k 1,2, ...
  ,c l 1,2, ... ,n

Main effects
Two factor Interactions
Three factor Interaction
Random error
31

• mijk the mean of y when A i, B j, C k
• m ai bj gk (ab)ij (ag)ik (bg)jk
• (abg)ijk
•  
• i 1,2, ... ,a j 1,2, ... ,b k 1,2,
  ... ,c l 1,2, ... ,n

Two factor Interactions
Overall mean
Main effects
Three factor Interaction
32
No interaction
Levels of C
Levels of B
Levels of B
Levels of A
Levels of A
33
A, B interact, No interaction with C
Levels of C
Levels of B
Levels of B
Levels of A
Levels of A
34
A, B, C interact
Levels of C
Levels of B
Levels of B
Levels of A
Levels of A
35

• Four Factor
• yijklm m ai bj (ab)ij gk (ag)ik
  (bg)jk (abg)ijk dl (ad)il (bd)jl (abd)ijl
  (gd)kl (agd)ikl (bgd)jkl (abgd)ijkl
  eijklm
• m
• ai bj gk dl
• (ab)ij (ag)ik (bg)jk (ad)il (bd)jl
  (gd)kl
• (abg)ijk (abd)ijl (agd)ikl (bgd)jkl
• (abgd)ijkl eijklm
• i 1,2, ... ,a j 1,2, ... ,b k 1,2, ...
  ,c l 1,2, ... ,d m 1,2, ... ,n
• where 0 S ai S bj S (ab)ij S gk S (ag)ik
  S(bg)jk S (abg)ijk S dl S (ad)il S (bd)jl
  S (abd)ijl S (gd)kl S (agd)ikl S (bgd)jkl
  
• S (abgd)ijkl
• and S denotes the summation over any of the
  subscripts.

Overall mean
Two factor Interactions
Main effects
Three factor Interactions
Four factor Interaction
Random error
36
Estimation of Main Effects and Interactions

• Estimator of Main effect of a Factor

Mean at level i of the factor - Overall Mean

• Estimator of k-factor interaction effect at a
  combination of levels of the k factors

Mean at the combination of levels of the k
factors - sum of all means at k-1 combinations
of levels of the k factors sum of all means at
k-2 combinations of levels of the k factors - etc.
37
Example

• The main effect of factor B at level j in a four
  factor (A,B,C and D) experiment is estimated by

• The two-factor interaction effect between factors
  B and C when B is at level j and C is at level k
  is estimated by

38

• The three-factor interaction effect between
  factors B, C and D when B is at level j, C is at
  level k and D is at level l is estimated by

• Finally the four-factor interaction effect
  between factors A,B, C and when A is at level i,
  B is at level j, C is at level k and D is at
  level l is estimated by

39
Anova Table entries

• Sum of squares interaction (or main) effects
  being tested (product of sample size and levels
  of factors not included in the interaction)
  (Sum of squares of effects being tested)
• Degrees of freedom df product of (number of
  levels - 1) of factors included in the
  interaction.

40
Analysis of Variance (ANOVA) Table Entries (Two
factors A and B)
41
The ANOVA Table
42
Analysis of Variance (ANOVA) Table Entries
(Three factors A, B and C)
43
The ANOVA Table
44

• The Completely Randomized Design is called
  balanced
• If the number of observations per treatment
  combination is unequal the design is called
  unbalanced. (resulting mathematically more
  complex analysis and computations)
• If for some of the treatment combinations there
  are no observations the design is called
  incomplete. (some of the parameters - main
  effects and interactions - cannot be estimated.)

45

• Example Diet example
• Mean
• 87.867
•  

46

• Main Effects for Factor A (Source of Protein)
• Beef Cereal Pork
• 1.733 -2.967 1.233

47

• Main Effects for Factor B (Level of Protein)
• High Low
• 7.267 -7.267
•  

48

• AB Interaction Effects
• Source of Protein
• Beef Cereal Pork
• Level High 3.133 -6.267 3.133
• of Protein Low -3.133 6.267 -3.133

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Example 2

• Paint Luster Experiment

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52
Table Means and Cell Frequencies
53
Means and Frequencies for the AB Interaction
(Temp - Drying)
54
Profiles showing Temp-Dry Interaction
55
Means and Frequencies for the AD Interaction
(Temp- Thickness)
56
Profiles showing Temp-Thickness Interaction
57
The Main Effect of C (Length)
58
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59
Factorial Experiments

• Analysis of Variance
• Experimental Design

60

• Dependent variable Y
• k Categorical independent variables A, B, C,
  (the Factors)
• Let
• a the number of categories of A
• b the number of categories of B
• c the number of categories of C
• etc.

61

• Objectives
• Determine which factors have some effect on the
  response
• Which groups of factors interact

62
The Completely Randomized Design

• We form the set of all treatment combinations
  the set of all combinations of the k factors
• Total number of treatment combinations
• t abc.
• In the completely randomized design n
  experimental units (test animals , test plots,
  etc. are randomly assigned to each treatment
  combination.
• Total number of experimental units N ntnabc..

63
Factor A has no effect
B
A
64
Additive Factors
B
A
65
Interacting Factors
B
A
66

• The testing in factorial experiments
• Test first the higher order interactions.
• If an interaction is present there is no need to
  test lower order interactions or main effects
  involving those factors. All factors in the
  interaction affect the response and they interact
• The testing continues with for lower order
  interactions and main effects for factors which
  have not yet been determined to affect the
  response.

67
Anova table for the 3 factor Experiment
Source SS df MS F p -value
A SSA a - 1 MSA MSA/MSError
B SSB b - 1 MSB MSB/MSError
C SSC c - 1 MSC MSC/MSError
AB SSAB (a - 1)(b - 1) MSAB MSAB/MSError
AC SSAC (a - 1)(c - 1) MSAC MSAC/MSError
BC SSBC (b - 1)(c - 1) MSBC MSBC/MSError
ABC SSABC (a - 1)(b - 1)(c - 1) MSABC MSABC/MSError
Error SSError abc(n - 1) MSError
68
Sum of squares entries
Similar expressions for SSB , and SSC.
Similar expressions for SSBC , and SSAC.
69
Sum of squares entries
Finally
70
The statistical model for the 3 factor Experiment
71
Anova table for the 3 factor Experiment
Source SS df MS F p -value
A SSA a - 1 MSA MSA/MSError
B SSB b - 1 MSB MSB/MSError
C SSC c - 1 MSC MSC/MSError
AB SSAB (a - 1)(b - 1) MSAB MSAB/MSError
AC SSAC (a - 1)(c - 1) MSAC MSAC/MSError
BC SSBC (b - 1)(c - 1) MSBC MSBC/MSError
ABC SSABC (a - 1)(b - 1)(c - 1) MSABC MSABC/MSError
Error SSError abc(n - 1) MSError
72

• The testing in factorial experiments
• Test first the higher order interactions.
• If an interaction is present there is no need to
  test lower order interactions or main effects
  involving those factors. All factors in the
  interaction affect the response and they interact
• The testing continues with lower order
  interactions and main effects for factors which
  have not yet been determined to affect the
  response.

73
Examples

• Using SPSS

74
Example

• In this example we are examining the effect of

• the level of protein A (High or Low) and
• the source of protein B (Beef, Cereal, or Pork)
  on weight gains (grams) in rats.

We have n 10 test animals randomly assigned to
k 6 diets
75
The k 6 diets are the 6 32 Level-Source
combinations

• High - Beef

• High - Cereal

• High - Pork

• Low - Beef

• Low - Cereal

• Low - Pork

76
Table Gains in weight (grams) for rats under six
diets differing in level of protein (High or
Low) and s ource of protein (Beef, Cereal, or
Pork)
Level
of Protein High Protein Low protein
Source of Protein Beef Cereal Pork Beef Cereal P
ork
Diet 1 2 3 4 5 6
73 98 94 90 107 49 102 74 79 76 95 82 118 56
96 90 97 73 104 111 98 64 80 86 81 95 102 86
98 81 107 88 102 51 74 97 100 82 108 72 74 106
87 77 91 90 67 70 117 86 120 95 89 61 111 9
2 105 78 58 82
Mean 100.0 85.9 99.5 79.2 83.9 78.7 Std.
Dev. 15.14 15.02 10.92 13.89 15.71 16.55
77
The data as it appears in SPSS
78
To perform ANOVA select Analyze-gtGeneral Linear
Model-gt Univariate
79
The following dialog box appears
80
Select the dependent variable and the fixed
factors
Press OK to perform the Analysis
81
The Output
82
Example Four factor experiment

• Four factors are studied for their effect on Y
  (luster of paint film). The four factors are

1) Film Thickness - (1 or 2 mils)
2) Drying conditions (Regular or Special)
3) Length of wash (10,30,40 or 60 Minutes), and
4) Temperature of wash (92 C or 100 C)
Two observations of film luster (Y) are taken for
each treatment combination
83

• The data is tabulated below
• Regular Dry Special Dry
• Minutes 92 ?C 100 ?C 92?C 100 ?C
• 1-mil Thickness
• 20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.4
• 30 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.3
• 40 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.8
• 60 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9
• 2-mil Thickness
• 20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.5
• 30 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.8
• 40 5.5 5.6 30.5 30.2 5.5 5.8 32.6 27.4
• 60 7.2 6.0 31.4 29.6 8.0 9.9 33.5 29.5

84
The Data as it appears in SPSS
85
The dialog box for performing ANOVA
86
The output
87
Random Effects and Fixed Effects Factors
88

• So far the factors that we have considered are
  fixed effects factors
• This is the case if the levels of the factor are
  a fixed set of levels and the conclusions of any
  analysis is in relationship to these levels.
• If the levels have been selected at random from a
  population of levels the factor is called a
  random effects factor
• The conclusions of the analysis will be directed
  at the population of levels and not only the
  levels selected for the experiment

89
Example - Fixed Effects

• Source of Protein, Level of Protein, Weight Gain
• Dependent
• Weight Gain
• Independent
• Source of Protein,
• Beef
• Cereal
• Pork
• Level of Protein,
• High
• Low

90
Example - Random Effects

• In this Example a Taxi company is interested in
  comparing the effects of three brands of tires
  (A, B and C) on mileage (mpg). Mileage will also
  be effected by driver. The company selects b 4
  drivers at random from its collection of drivers.
  Each driver has n 3 opportunities to use each
  brand of tire in which mileage is measured.
• Dependent
• Mileage
• Independent
• Tire brand (A, B, C),
• Fixed Effect Factor
• Driver (1, 2, 3, 4),
• Random Effects factor

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The Model for the fixed effects experiment

• where m, a1, a2, a3, b1, b2, (ab)11 , (ab)21 ,
  (ab)31 , (ab)12 , (ab)22 , (ab)32 , are fixed
  unknown constants
• And eijk is random, normally distributed with
  mean 0 and variance s2.
• Note

92
The Model for the case when factor B is a random
effects factor

• where m, a1, a2, a3, are fixed unknown constants
• And eijk is random, normally distributed with
  mean 0 and variance s2.
• bj is normal with mean 0 and variance
• and
• (ab)ij is normal with mean 0 and variance
• Note

This model is called a variance components model
93
The Anova table for the two factor model
Source SS df MS
A SSA a -1 SSA/(a 1)
B SSA b - 1 SSB/(a 1)
AB SSAB (a -1)(b -1) SSAB/(a 1) (a 1)
Error SSError ab(n 1) SSError/ab(n 1)
94
The Anova table for the two factor model (A, B
fixed)
Source SS df MS EMS F
A SSA a -1 MSA MSA/MSError
B SSA b - 1 MSB MSB/MSError
AB SSAB (a -1)(b -1) MSAB MSAB/MSError
Error SSError ab(n 1) MSError
EMS Expected Mean Square
95
The Anova table for the two factor model (A
fixed, B - random)
Source SS df MS EMS F
A SSA a -1 MSA MSA/MSAB
B SSA b - 1 MSB MSB/MSError
AB SSAB (a -1)(b -1) MSAB MSAB/MSError
Error SSError ab(n 1) MSError
Note The divisor for testing the main effects of
A is no longer MSError but MSAB.
96
Rules for determining Expected Mean Squares (EMS)
in an Anova Table
Both fixed and random effects Formulated by
Schultz1

1. Schultz E. F., Jr. Rules of Thumb for
   Determining Expectations of Mean Squares in
   Analysis of Variance,Biometrics, Vol 11, 1955,
   123-48.

97

1. The EMS for Error is s2.
2. The EMS for each ANOVA term contains two or more
   terms the first of which is s2.
3. All other terms in each EMS contain both
   coefficients and subscripts (the total number of
   letters being one more than the number of
   factors) (if number of factors is k 3, then the
   number of letters is 4)
4. The subscript of s2 in the last term of each EMS
   is the same as the treatment designation.

98

1. The subscripts of all s2 other than the first
   contain the treatment designation. These are
   written with the combination involving the most
   letters written first and ending with the
   treatment designation.
2. When a capital letter is omitted from a subscript
   , the corresponding small letter appears in the
   coefficient.
3. For each EMS in the table ignore the letter or
   letters that designate the effect. If any of the
   remaining letters designate a fixed effect,
   delete that term from the EMS.

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1. Replace s2 whose subscripts are composed entirely
   of fixed effects by the appropriate sum.

100

• Example 3 factors A, B, C all are random
  effects

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
101

• Example 3 factors A fixed, B, C random

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
102

• Example 3 factors A , B fixed, C random

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
103

• Example 3 factors A , B and C fixed

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
104
Example - Random Effects

• In this Example a Taxi company is interested in
  comparing the effects of three brands of tires
  (A, B and C) on mileage (mpg). Mileage will also
  be effected by driver. The company selects at
  random b 4 drivers at random from its
  collection of drivers. Each driver has n 3
  opportunities to use each brand of tire in which
  mileage is measured.
• Dependent
• Mileage
• Independent
• Tire brand (A, B, C),
• Fixed Effect Factor
• Driver (1, 2, 3, 4),
• Random Effects factor

105
The Data
106
Asking SPSS to perform Univariate ANOVA
107
Select the dependent variable, fixed factors,
random factors
108
The Output
The divisor for both the fixed and the random
main effect is MSAB
This is contrary to the advice of some texts
109
The Anova table for the two factor model (A
fixed, B - random)
Source SS df MS EMS F
A SSA a -1 MSA MSA/MSAB
B SSA b - 1 MSB MSB/MSError
AB SSAB (a -1)(b -1) MSAB MSAB/MSError
Error SSError ab(n 1) MSError
Note The divisor for testing the main effects of
A is no longer MSError but MSAB.
References Guenther, W. C. Analysis of Variance
Prentice Hall, 1964
110
The Anova table for the two factor model (A
fixed, B - random)
Source SS df MS EMS F
A SSA a -1 MSA MSA/MSAB
B SSA b - 1 MSB MSB/MSAB
AB SSAB (a -1)(b -1) MSAB MSAB/MSError
Error SSError ab(n 1) MSError
Note In this case the divisor for testing the
main effects of A is MSAB . This is the approach
used by SPSS.
References Searle Linear Models John Wiley, 1964
111
Crossed and Nested Factors
112

• The factors A, B are called crossed if every
  level of A appears with every level of B in the
  treatment combinations.

Levels of B





Levels of A
113

• Factor B is said to be nested within factor A if
  the levels of B differ for each level of A.

Levels of A
Levels of B
114

• Example A company has a 4 plants for producing
  paper. Each plant has 6 machines for producing
  the paper. The company is interested in how
  paper strength (Y) differs from plant to plant
  and from machine to machine within plant

Plants
Machines
115

• Machines (B) are nested within plants (A)

The model for a two factor experiment with B
nested within A.
116
The ANOVA table
Source SS df MS F p - value
A SSA a - 1 MSA MSA/MSError
B(A) SSB(A) a(b 1) MSB(A) MSB(A) /MSError
Error SSError ab(n 1) MSError
Note SSB(A ) SSB SSAB and a(b 1) (b 1)
(a - 1)(b 1)
117

• Example A company has a 4 plants for producing
  paper. Each plant has 6 machines for producing
  the paper. The company is interested in how
  paper strength (Y) differs from plant to plant
  and from machine to machine within plant.
• Also we have n 5 measurements of paper
  strength for each of the 24 machines

118
The Data
119
Anova Table Treating Factors (Plant, Machine) as
crossed
120
Anova Table Two factor experiment B(machine)
nested in A (plant)
121
Analysis of Variance

• Factorial Experiments

122

• Dependent variable Y
• k Categorical independent variables A, B, C,
  (the Factors)
• Let
• a the number of categories of A
• b the number of categories of B
• c the number of categories of C
• etc.

123
The Completely Randomized Design

• We form the set of all treatment combinations
  the set of all combinations of the k factors
• Total number of treatment combinations
• t abc.
• In the completely randomized design n
  experimental units (test animals , test plots,
  etc. are randomly assigned to each treatment
  combination.
• Total number of experimental units N ntnabc..

124
Random Effects and Fixed Effects Factors
125

• fixed effects factors
• he levels of the factor are a fixed set of levels
  and the conclusions of any analysis is in
  relationship to these levels.
• random effects factor
• If the levels have been selected at random from a
  population of levels.
• The conclusions of the analysis will be directed
  at the population of levels and not only the
  levels selected for the experiment

126

• Example 3 factors A, B, C all are random
  effects

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
127

• Example 3 factors A fixed, B, C random

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
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• Example 3 factors A , B fixed, C random

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
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• Example 3 factors A , B and C fixed

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
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Crossed and Nested Factors
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• Factor B is said to be nested within factor A if
  the levels of B differ for each level of A.

Levels of A
Levels of B
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The Analysis of Covariance

• ANACOVA

133
Multiple Regression

1. Dependent variable Y (continuous)
2. Continuous independent variables X1, X2, , Xp

The continuous independent variables X1, X2, ,
Xp are quite often measured and observed (not set
at specific values or levels)
134
Analysis of Variance

1. Dependent variable Y (continuous)
2. Categorical independent variables (Factors) A, B,
   C,

The categorical independent variables A, B, C,
are set at specific values or levels.
135
Analysis of Covariance

1. Dependent variable Y (continuous)
2. Categorical independent variables (Factors) A, B,
   C,
3. Continuous independent variables (covariates) X1,
   X2, , Xp

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Example

• Dependent variable Y weight gain
• Categorical independent variables (Factors)
• A level of protein in the diet (High, Low)
• B source of protein (Beef, Cereal, Pork)
• Continuous independent variables (covariates)
• X1 initial wt. of animal.

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Dependent variable is continuous
Statistical Technique Independent variables Independent variables
Statistical Technique continuous categorical
Multiple Regression
ANOVA
ANACOVA
It is possible to treat categorical independent
variables in Multiple Regression using Dummy
variables.
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The Multiple Regression Model
139
The ANOVA Model
140
The ANACOVA Model
141
ANOVA Tables
142
The Multiple Regression Model
Source S.S. d.f.
Regression SSReg p
Error SSError n p - 1
Total SSTotal n - 1
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The ANOVA Model
Source S.S. d.f.
Main Effects Main Effects Main Effects
A SSA a - 1
B SSB b - 1
Interactions Interactions Interactions
AB SSAB (a 1)(b 1)
? ? ?
Error SSError n p - 1
Total SSTotal n - 1
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The ANACOVA Model
Source S.S. d.f.
Covariates SSCovaraites p
Main Effects Main Effects Main Effects
A SSA a - 1
B SSB b - 1
Interactions Interactions Interactions
AB SSAB (a 1)(b 1)
? ? ?
Error SSError n p - 1
Total SSTotal n - 1
145
Example

• Dependent variable Y weight gain
• Categorical independent variables (Factors)
• A level of protein in the diet (High, Low)
• B source of protein (Beef, Cereal, Pork)
• Continuous independent variables (covariates)
• X initial wt. of animal.

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The data
147
The ANOVA Table
148
Using SPSS to perform ANACOVA
149
The data file
150
Select Analyze-gtGeneral Linear Model -gt Univariate
151
Choose the Dependent Variable, the Fixed
Factor(s) and the Covaraites
152
The following ANOVA table appears
153
The Process of Analysis of Covariance
Dependent variable
Covariate
154
The Process of Analysis of Covariance
Adjusted Dependent variable
Covariate
155

• The dependent variable (Y) is adjusted so that
  the covariate takes on its average value for each
  case
• The effect of the factors ( A, B, etc) are
  determined using the adjusted value of the
  dependent variable.

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• ANOVA and ANACOVA can be handled by Multiple
  Regression Package by the use of Dummy variables
  to handle the categorical independent variables.
• The results would be the same.

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Analysis of unbalanced Factorial Designs

• Type I, Type II, Type III
• Sum of Squares

158
Sum of squares for testing an effect

• modelComplete model with the effect in.
• modelReduced model with the effect out.

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• Type I SS
• Type I estimates of the sum of squares associated
  with an effect in a model are calculated when
  sums of squares for a model are calculated
  sequentially

• Example
• Consider the three factor factorial experiment
  with factors A, B and C.
• The Complete model
• Y m A B C AB AC BC ABC

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• A sequence of increasingly simpler models
• Y m A B C AB AC BC ABC
• Y m A B C AB AC BC
• Y m A B C AB AC
• Y m A B C AB
• Y m A B C
• Y m A B
• Y m A
• Y m

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• Type I S.S.

162

• Type II SS
• Type two sum of squares are calculated for an
  effect assuming that the Complete model contains
  every effect of equal or lesser order. The
  reduced model has the effect removed ,

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• The Complete models
• Y m A B C AB AC BC ABC (the
  three factor model)
• Y m A B C AB AC BC (the all two
  factor model)
• Y m A B C (the all main effects model)

The Reduced models For a k-factor effect the
reduced model is the all k-factor model with the
effect removed
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• Type III SS
• The type III sum of squares is calculated by
  comparing the full model, to the full model
  without the effect.

166

• Comments
• When using The type I sum of squares the effects
  are tested in a specified sequence resulting in a
  increasingly simpler model. The test is valid
  only the null Hypothesis (H0) has been accepted
  in the previous tests.
• When using The type II sum of squares the test
  for a k-factor effect is valid only the all
  k-factor model can be assumed.
• When using The type III sum of squares the tests
  require neither of these assumptions.

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• An additional Comment
• When the completely randomized design is balanced
  (equal number of observations per treatment
  combination) then type I sum of squares, type II
  sum of squares and type III sum of squares are
  equal.

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• Example
• A two factor (A and B) experiment, response
  variable y.
• The SPSS data file

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• Using ANOVA SPSS package
• Select the type of SS using model

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• ANOVA table type I S.S

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• ANOVA table type II S.S

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• ANOVA table type III S.S

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Next Topic Other Experimental Designs