1. Factorial Experiments
Analysis of Variance
Experimental Design

2. k Categorical independent variables A, B, C, … (the Factors) Let
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 = nt=nabc..

4. The treatment combinations can thought to be arranged in a k-dimensional rectangular block1 2 b 1 2 A a

5. C B A

6. Another way of representing the treatment combinations in a factorial experiment
... 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 = 3×2 Level-Source combinations
High - Beef
High - Cereal
High - Pork
Low - Beef
Low - Cereal
Low - Pork

9. Gains in weight (grams) for rats under six diets
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 Pork
Diet
Mean
Std. Dev

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
2-mil Thickness

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
Low
Overall

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 observations Definitions Normal dist’n m1 m2 m3 ma
y11
y12
y13
y1n
y21
y22
y23
y2n
y31
y32
y33
y3n
ya1
ya2
ya3
yan
observations
Normal dist’n 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

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

35. + (ab)ij + (ag)ik + (bg)jk + (ad)il + (bd)jl+ (gd)kl
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 (Two factors – A and B)

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 =

46. Main Effects for Factor A (Source of Protein)
Beef Cereal Pork

47. Main Effects for Factor B (Level of Protein)
High Low

48. AB Interaction Effects
Source of Protein
Beef Cereal Pork
Level High
of Protein Low

50. Paint Luster Experiment
Example 2
Paint Luster Experiment

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)

59. 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.

60. 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

61. Sum of squares entries
Similar expressions for SSB , and SSC.
Similar expressions for SSBC , and SSAC.

62. Sum of squares entries
Finally

63. The statistical model for the 3 factor Experiment

64. 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

65. 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.

66. Examples
Using SPSS

67. 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

68. The k = 6 diets are the 6 = 3×2 Level-Source combinations
High - Beef
High - Cereal
High - Pork
Low - Beef
Low - Cereal
Low - Pork

69. Gains in weight (grams) for rats under six diets
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 Pork
Diet
Mean
Std. Dev

70. The data as it appears in SPSS

71. To perform ANOVA select Analyze->General Linear Model-> Univariate

72. The following dialog box appears

73. Select the dependent variable and the fixed factors
Press OK to perform the Analysis

74. The Output

75. 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

76. The data is tabulated below:
Regular Dry Special Dry
Minutes 92 C 100 C 92C 100 C
1-mil Thickness
2-mil Thickness

77. The Data as it appears in SPSS

78. The dialog box for performing ANOVA

79. The output

80. Random Effects and Fixed Effects Factors

81. 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

82. 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

83. 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

84. 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:

85. 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

86. The Anova table for the two factor model
Source SS df MS A
SSA
a -1
SSA/(a – 1) B
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)

87. 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
b - 1
MSB
MSB/MSError AB
SSAB
(a -1)(b -1)
MSAB
MSAB/MSError
Error
SSError
ab(n – 1)
MSError
EMS = Expected Mean Square

88. 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
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.

89. Rules for determining Expected Mean Squares (EMS) in an Anova Table
Both fixed and random effects
Formulated by Schultz[1]
Schultz E. F., Jr. “Rules of Thumb for Determining Expectations of Mean Squares in Analysis of Variance,”Biometrics, Vol 11, 1955,

90. The EMS for Error is s2.
The EMS for each ANOVA term contains two or more terms the first of which is s2.
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)
The subscript of s2 in the last term of each EMS is the same as the treatment designation.

91. 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.
When a capital letter is omitted from a subscript , the corresponding small letter appears in the coefficient.
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.

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

93. Example: 3 factors A, B, C – all are random effects
Source
EMS F A B C AB AC BC
ABC
Error

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

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

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

97. 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

98. The Data

99. Asking SPSS to perform Univariate ANOVA

100. Select the dependent variable, fixed factors, random factors

101. The Output
The divisor for both the fixed and the random main effect is MSAB
This is contrary to the advice of some texts

102. 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
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

103. 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
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

104. Crossed and Nested Factors

105. 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

106. 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

107. Example: A company has a = 4 plants for producing paper
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

108. Machines (B) are nested within plants (A)
The model for a two factor experiment with B nested within A.

109. 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)

110. Example: A company has a = 4 plants for producing paper
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

111. The Data

112. Anova Table Treating Factors (Plant, Machine) as crossed

113. Anova Table: Two factor experiment B(machine) nested in A (plant)