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Factorial Experiments

- Analysis of Variance
- Experimental Design

- 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.

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..

The treatment combinations can thought to be

arranged in a k-dimensional rectangular block

B

1

2

b

1

2

A

a

C

B

A

Another way of representing the treatment

combinations in a factorial experiment

C

B

...

A

...

D

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

The k 6 diets are the 6 32 Level-Source

combinations

- High - Beef

- High - Cereal

- High - Pork

- Low - Beef

- Low - Cereal

- Low - Pork

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

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

- 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

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

Notation for Means

- When averaging over one or several subscripts we

put a bar above the letter and replace the

subscripts by - Example
- y241

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

- 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

- 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.

- 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 .

Factor A has no effect

B

A

Additive Factors

B

A

Interacting Factors

B

A

- 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.

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

Profiles of Weight Gain for Source and Level of

Protein

Profiles of Weight Gain for Source and Level of

Protein

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

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

- 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

Table of Means

Table of Effects Overall mean, Main effects,

Interaction Effects

- 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

- 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

No interaction

Levels of C

Levels of B

Levels of B

Levels of A

Levels of A

A, B interact, No interaction with C

Levels of C

Levels of B

Levels of B

Levels of A

Levels of A

A, B, C interact

Levels of C

Levels of B

Levels of B

Levels of A

Levels of A

- 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

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.

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

- 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

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.

Analysis of Variance (ANOVA) Table Entries (Two

factors A and B)

The ANOVA Table

Analysis of Variance (ANOVA) Table Entries

(Three factors A, B and C)

The ANOVA Table

- 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.)

- Example Diet example
- Mean
- 87.867

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

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

- 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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Table Means and Cell Frequencies

Means and Frequencies for the AB Interaction

(Temp - Drying)

Profiles showing Temp-Dry Interaction

Means and Frequencies for the AD Interaction

(Temp- Thickness)

Profiles showing Temp-Thickness Interaction

The Main Effect of C (Length)

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Factorial Experiments

- Analysis of Variance
- Experimental Design

- 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.

- Objectives
- Determine which factors have some effect on the

response - Which groups of factors interact

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..

Factor A has no effect

B

A

Additive Factors

B

A

Interacting Factors

B

A

- 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.

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

Sum of squares entries

Similar expressions for SSB , and SSC.

Similar expressions for SSBC , and SSAC.

Sum of squares entries

Finally

The statistical model for the 3 factor Experiment

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

- 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.

Examples

- Using SPSS

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

The k 6 diets are the 6 32 Level-Source

combinations

- High - Beef

- High - Cereal

- High - Pork

- Low - Beef

- Low - Cereal

- Low - Pork

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

The data as it appears in SPSS

To perform ANOVA select Analyze-gtGeneral Linear

Model-gt Univariate

The following dialog box appears

Select the dependent variable and the fixed

factors

Press OK to perform the Analysis

The Output

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

- 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

The Data as it appears in SPSS

The dialog box for performing ANOVA

The output

Random Effects and Fixed Effects Factors

- 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

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

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

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

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

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)

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

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.

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.

- 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.

- 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.

- Replace s2 whose subscripts are composed entirely

of fixed effects by the appropriate sum.

- Example 3 factors A, B, C all are random

effects

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A fixed, B, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A , B fixed, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A , B and C fixed

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

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

The Data

Asking SPSS to perform Univariate ANOVA

Select the dependent variable, fixed factors,

random factors

The Output

The divisor for both the fixed and the random

main effect is MSAB

This is contrary to the advice of some texts

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

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

Crossed and Nested Factors

- 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

- 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

- 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

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

The model for a two factor experiment with B

nested within A.

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)

- 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

The Data

Anova Table Treating Factors (Plant, Machine) as

crossed

Anova Table Two factor experiment B(machine)

nested in A (plant)

Analysis of Variance

- Factorial Experiments

- 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.

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..

Random Effects and Fixed Effects Factors

- 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

- Example 3 factors A, B, C all are random

effects

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A fixed, B, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A , B fixed, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

- Example 3 factors A , B and C fixed

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

Crossed and Nested Factors

- 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

The Analysis of Covariance

- ANACOVA

Multiple Regression

- Dependent variable Y (continuous)
- 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)

Analysis of Variance

- Dependent variable Y (continuous)
- Categorical independent variables (Factors) A, B,

C,

The categorical independent variables A, B, C,

are set at specific values or levels.

Analysis of Covariance

- Dependent variable Y (continuous)
- Categorical independent variables (Factors) A, B,

C, - Continuous independent variables (covariates) X1,

X2, , Xp

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.

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.

The Multiple Regression Model

The ANOVA Model

The ANACOVA Model

ANOVA Tables

The Multiple Regression Model

Source S.S. d.f.

Regression SSReg p

Error SSError n p - 1

Total SSTotal n - 1

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

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

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.

The data

The ANOVA Table

Using SPSS to perform ANACOVA

The data file

Select Analyze-gtGeneral Linear Model -gt Univariate

Choose the Dependent Variable, the Fixed

Factor(s) and the Covaraites

The following ANOVA table appears

The Process of Analysis of Covariance

Dependent variable

Covariate

The Process of Analysis of Covariance

Adjusted Dependent variable

Covariate

- 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.

- 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.

Analysis of unbalanced Factorial Designs

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

Sum of squares for testing an effect

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

- 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

- 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

- Type I S.S.

- 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 ,

- 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.

- 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.

- 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.

- Example
- A two factor (A and B) experiment, response

variable y. - The SPSS data file

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

- ANOVA table type I S.S

- ANOVA table type II S.S

- ANOVA table type III S.S

Next Topic Other Experimental Designs