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Factorial Experiments

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Title: 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
7
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

12
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

13
Notation for Means
  • When averaging over one or several subscripts we
    put a bar above the letter and replace the
    subscripts by
  • Example
  • y241

14
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

15
  • 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

16
  • 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.

17
  • 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 .

18
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.

22
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
26
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
28
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

49
(No Transcript)
50
Example 2
  • Paint Luster Experiment

51
(No Transcript)
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
(No Transcript)
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

91
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.

99
  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

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The Data
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Anova Table Treating Factors (Plant, Machine) as
crossed
120
Anova Table Two factor experiment B(machine)
nested in A (plant)
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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.

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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..

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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

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  • 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
128
  • Example 3 factors A , B fixed, C random

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
129
  • Example 3 factors A , B and C fixed

Source EMS F
A
B
C
AB
AC
BC
ABC
Error
130
Crossed and Nested Factors
131
  • 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

136
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.

137
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.
138
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
143
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
144
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.

146
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.

156
  • 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.

157
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.

159
  • 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

161
  • 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.

167
  • 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.

168
  • Example
  • A two factor (A and B) experiment, response
    variable y.
  • The SPSS data file

169
  • Using ANOVA SPSS package
  • Select the type of SS using model

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  • ANOVA table type I S.S

171
  • ANOVA table type II S.S

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  • ANOVA table type III S.S

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