ANOVA TABLE

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ANOVA TABLE

• Factorial Experiment
• Completely Randomized Design

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Anova table for the 3 factor Experiment
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Sum of squares entries
Similar expressions for SSB , and SSC.
Similar expressions for SSBC , and SSAC.
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Sum of squares entries
Finally
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The statistical model for the 3 factor Experiment
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Anova table for the 3 factor Experiment
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• 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.

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Examples

• Using SPSS

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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 (grams) in rats.

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

• High - Beef

• High - Cereal

• High - Pork

• Low - Beef

• Low - Cereal

• Low - Pork

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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
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The data as it appears in SPSS
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To perform ANOVA select Analyze-gtGeneral Linear
Model-gt Univariate
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The following dialog box appears
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Select the dependent variable and the fixed
factors
Press OK to perform the Analysis
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The Output
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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
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• 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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The Data as it appears in SPSS
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The dialog box for performing ANOVA
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The output
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Random Effects and Fixed Effects Factors
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• 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

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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

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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

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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
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The Anova table for the two factor model
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The Anova table for the two factor model (A, B
fixed)
EMS Expected Mean Square
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The Anova table for the two factor model (A
fixed, B - random)
Note The divisor for testing the main effects of
A is no longer MSError but MSAB.
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Rules for determining Expected Mean Squares (EMS)
in an Anova Table
Both fixed and random effects Formulated by
Schultz1

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

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• 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.

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• 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.

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

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

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• Example 3 factors A fixed, B, C random

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• Example 3 factors A , B fixed, C random

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• Example 3 factors A , B and C fixed

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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

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The Data
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Asking SPSS to perform Univariate ANOVA
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Select the dependent variable, fixed factors,
random factors
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The Output
The divisor for both the fixed and the random
main effect is MSAB
This is contrary to the advice of some texts
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The Anova table for the two factor model (A
fixed, B - random)
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
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The Anova table for the two factor model (A
fixed, B - random)
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
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Crossed and Nested Factors
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• 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
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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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• 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
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• Machines (B) are nested within plants (A)

The model for a two factor experiment with B
nested within A.
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The ANOVA table
Note SSB(A ) SSB SSAB and a(b 1) (b 1)
(a - 1)(b 1)
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• 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
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Anova Table Two factor experiment B(machine)
nested in A (plant)