Factorial Experiments

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 = nt=nabc..

The treatment combinations can thought to be arranged in a k-dimensional rectangular block

A

1

2

a

B1 2 b

A

B

C

Another way of representing the treatment combinations in a factorial experiment

A

B

...

D

C

...

Example

In this example we are examining the effect of

We have n = 10 test animals randomly assigned to k = 6 diets

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.

The k = 6 diets are the 6 = 3×2 Level-Source combinations

1. High - Beef

2. High - Cereal

3. High - Pork

4. Low - Beef

5. Low - Cereal

6. Low - Pork

TableGains 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)

Levelof Protein High Protein Low protein

Sourceof Protein Beef Cereal Pork Beef Cereal Pork

Diet 1 2 3 4 5 6

73 98 94 90 107 49102 74 79 76 95 82118 56 96 90 97 73104 111 98 64 80 86

81 95 102 86 98 81107 88 102 51 74 97100 82 108 72 74 106

87 77 91 90 67 70117 86 120 95 89 61111 92 105 78 58 82

Mean 100.0 85.9 99.5 79.2 83.9 78.7Std. 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:

Two observations of film luster (Y) are taken for each treatment combination

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)

The data is tabulated below:Regular Dry Special DryMinutes 92 C 100 C 92C 100 C

1-mil Thickness20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.430 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.340 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.860 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9

2-mil Thickness20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.530 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.840 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

NotationLet 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 .

0

10

20

30

40

50

60

70

0 20 40 60

Factor A has no effect

A

B

0

10

20

30

40

50

60

70

0 20 40 60

Additive Factors

A

B

0

10

20

30

40

50

60

70

0 20 40 60

Interacting Factors

A

B

The testing in factorial experiments 1. Test first the higher order interactions.2. 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

3. The testing continues with for lower order interactions and main effects for factors which have not yet been determined to affect the response.

Level of Protein Beef Cereal Pork Overall

Low 79.20 83.90 78.70 80.60

Source of Protein

High 100.00 85.90 99.50 95.13

Overall 89.60 84.90 89.10 87.87

Example: Diet Example

Summary Table of Cell means

70

80

90

100

110

Beef Cereal Pork

Wei

ght

Gai

n

High Protein

Low Protein

Overall

Profiles of Weight Gain for Source and Level of Protein

70

80

90

100

110

High Protein Low Protein

Wei

ght

Gai

nBeef

Cereal

Pork

Overall

Profiles of Weight Gain for Source and Level of Protein

Models for factorial Experiments

Single Factor: A – a levels

yij = + i + ij i = 1,2, ... ,a; j = 1,2, ... ,n

01

a

ii

Random error – Normal, mean 0, std-dev.

i

iAyi when ofmean thei

Overall mean Effect on y of factor A when A = i

y11

y12

y13

y1n

y21

y22

y23

y2n

y31

y32

y33

y3n

ya1

ya2

ya3

yan

Levels of A1 2 3 a

observationsNormal dist’n

Mean of observations

1 2 3 a

+ 1

+ 2

+ 3

+ a

Definitions

a

iia 1

1mean overall

a

iiiii a

iA1

1 )en (Effect wh

Two Factor: A (a levels), B (b levels

yijk = + i + j+ ()ij + ijk

i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,n

0,0,0,01111

b

jij

a

iij

b

jj

a

ii

ij

ijji

ij jBiAy

and when ofmean the

Overall mean

Main effect of A Main effect of B

Interaction effect of A and 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 = + i + j+ ij + k + ()ik + ()jk+ ijk + ijkl

= + i + j+ k + ij + (ik + (jk

+ ijk + ijkl

i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n

0,,0,0,0,011111

c

kijk

a

iij

c

kk

b

jj

a

ii

Main effects Two factor Interactions

Three factor Interaction Random error

ijk = the mean of y when A = i, B = j, C = k

= + i + j+ k + ij + (ik + (jk

+ ijk

i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n

0,,0,0,0,011111

c

kijk

a

iij

c

kk

b

jj

a

ii


Three factor Interaction

Overall mean

Levels of C

Levels of B

Levels of A

Levels of B

Levels of A

No interaction

Levels of C

Levels of B

Levels of A Levels of A

A, B interact, No interaction with C

Levels of B

Levels of C

Levels of B

Levels of A Levels of A

A, B, C interact

Levels of B

Four Factor:

yijklm = + + j+ ()ij + k + ()ik + ()jk+ ()ijk + l+ ()il + ()jl+ ()ijl + ()kl + ()ikl + ()jkl+ ()ijkl + ijklm

=

+i + j+ k + l

+ ()ij + ()ik + ()jk + ()il + ()jl+ ()kl

+()ijk+ ()ijl + ()ikl + ()jkl

+ ()ijkl + ijklm

i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,d; m = 1,2, ... ,n

where 0 = i = j= ()ij k = ()ik = ()jk= ()ijk = l= ()il = ()jl = ()ijl = ()kl = ()ikl = ()jkl =

()ijkl

and denotes the summation over any of the subscripts.


Three factor Interactions

Overall mean

Four factor Interaction Random error

Estimation of Main Effects and Interactions • Estimator of Main effect of a Factor

• 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.

= Mean at level i of the factor - Overall Mean

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:

yyˆjj

yyyy kjjkjk

• 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:

yyyyyyyy lkjklljjkjkljkl

jklikiijjklklilijijkijklijkl yyyyyyyyy

yyyyyyy lkjikllj

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.

a

iiA nbSS

1

2

b

jjB naSS

1

2

a

i

b

jijAB nSS

1 1

2

a

i

b

j

n

kijijkError yySS

1 1 1

2

Analysis of Variance (ANOVA) Table Entries (Two factors – A and B)

The ANOVA Table

a

iiA nbcSS

1

2

b

jjB nacSS

1

2

a

i

b

jijAB ncSS

1 1

2

a

i

c

kikAC nbSS

1 1

2

b

j

c

kjkBC naSS

1 1

2

a

i

b

j

c

kijkABC nSS

1 1 1

2

a

i

b

j

c

k

n

lijkijklError yySS

1 1 1 1

2

Analysis of Variance (ANOVA) Table Entries (Three factors – A, B and C)

c

kkC nabSS

1

2

The ANOVA Table

Source SS df

A SSA a-1

B SSB b-1

C SSC c-1

AB SSAB (a-1)(b-1)

AC SSAC (a-1)(c-1)

BC SSBC (b-1)(c-1)

ABC SSABC (a-1)(b-1)(c-1)

Error SSError abc(n-1)

• 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

y

Main Effects for Factor A (Source of Protein)

Beef Cereal Pork

1.733 -2.967 1.233

yyˆ ii

Main Effects for Factor B (Level of Protein)

High Low

7.267 -7.267

yyˆjj

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

yy-y-y jiijij

Example 2

Paint Luster Experiment

Table: Means and Cell Frequencies

Means and Frequencies for the AB Interaction (Temp - Drying)

0

5

10

15

20

25

92 100

Temperature

Lus

ter

Regular Dry

Special Dry

Overall

Profiles showing Temp-Dry Interaction

Means and Frequencies for the AD Interaction (Temp- Thickness)

0

5

10

15

20

25

30

92 100

Temperature

Lus

ter

1-mil

2-mil

Overall

Profiles showing Temp-Thickness Interaction

The Main Effect of C (Length)

7060504030201012

13

14

15

16

Profile of Effect of Length on Luster

Length

Lu

ster



Experimental Design




Objectives

•Determine which factors have some effect on the response

•Which groups of factors interact





0

10

20

30

40

50

60

70

0 20 40 60

Factor A has no effect

A

B

0

10

20

30

40

50

60

70

0 20 40 60

Additive Factors

A

B

0

10

20

30

40

50

60

70

0 20 40 60

Interacting Factors

A

B



3. 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

a

ii

a

iiA yynbcnbcSS

1

2

1

2

Similar expressions for SSB , and SSC.

a

i

b

jjiij

a

iijAB yyyyncncSS

1 1

2

1

2

Similar expressions for SSBC , and SSAC.

Sum of squares entries

Finally

a

iikjABC nSS

1

2

a

i

b

j

c

kijkkiijijk yyyyyn

1 1 1 2 ikj yyy

a

i

b

j

c

k

n

lijkijklError yySS

1 1 1 1

2

The statistical model for the 3 factor Experiment

effectsmain effectmean kjiijk/y

error randomninteractiofactor 3nsinteractiofactor 2

ijk/ijkjkikij

Anova table for the 3 factor Experiment

Source SS df MS F p -value


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



3. 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

We have n = 10 test animals randomly assigned to k = 6 diets

• the level of protein A (High or Low) and • the source of protein B (Beef, Cereal, or

Pork) on weight gains (grams) in rats.

The k = 6 diets are the 6 = 3×2 Level-Source combinations

1. High - Beef

2. High - Cereal

3. High - Pork

4. Low - Beef

5. Low - Cereal

6. Low - Pork

TableGains 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)

Levelof Protein High Protein Low protein

Sourceof Protein Beef Cereal Pork Beef Cereal Pork

Diet 1 2 3 4 5 6

73 98 94 90 107 49102 74 79 76 95 82118 56 96 90 97 73104 111 98 64 80 86

81 95 102 86 98 81107 88 102 51 74 97100 82 108 72 74 106

87 77 91 90 67 70117 86 120 95 89 61111 92 105 78 58 82

Mean 100.0 85.9 99.5 79.2 83.9 78.7Std. 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->General Linear Model-> Univariate

The following dialog box appears

Select the dependent variable and the fixed factors

Press OK to perform the Analysis

The Output

Tests of Between-Subjects Effects

Dependent Variable: WTGN

4612.933a 5 922.587 4.300 .002

463233.1 1 463233.1 2159.036 .000

266.533 2 133.267 .621 .541

3168.267 1 3168.267 14.767 .000

1178.133 2 589.067 2.746 .073

11586.000 54 214.556

479432.0 60

16198.933 59

SourceCorrected Model

Intercept

SOURCE

LEVEL

SOURCE * LEVEL

Error

Total

Corrected Total

Type IIISum of

Squares dfMean

Square F Sig.

R Squared = .285 (Adjusted R Squared = .219)a.

Example – Four factor experiment

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

Two observations of film luster (Y) are taken for each treatment combination

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)

The data is tabulated below:Regular Dry Special DryMinutes 92 C 100 C 92C 100 C

1-mil Thickness20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.430 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.340 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.860 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9

2-mil Thickness20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.530 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.840 5.5 5.6 30.5 30.2 5.5 5.8 32.6 27.460 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


Dependent Variable: LUSTRE

6548.020a 31 211.226 76.814 .000

12586.035 1 12586.035 4577.000 .000

5039.225 1 5039.225 1832.550 .000

5.700 1 5.700 2.073 .160

70.285 3 23.428 8.520 .000

844.629 1 844.629 307.155 .000

15.504 1 15.504 5.638 .024

3.155 3 1.052 .383 .766

9.890 3 3.297 1.199 .326

6.422 3 2.141 .778 .515

511.325 1 511.325 185.947 .000

1.410 1 1.410 .513 .479

.150 1 .150 .055 .817

15.642 3 5.214 1.896 .150

11.520 3 3.840 1.396 .262

7.320 3 2.440 .887 .458

5.840 3 1.947 .708 .554

87.995 32 2.750

19222.050 64

6636.015 63


Intercept

TEMP

COND

LENGTH

THICK

TEMP * COND

TEMP * LENGTH

COND * LENGTH

TEMP * COND * LENGTH

TEMP * THICK

COND * THICK

TEMP * COND * THICK

LENGTH * THICK

TEMP * LENGTH * THICK

COND * LENGTH *THICK

TEMP * COND * LENGTH* THICK

Error

Total

Corrected Total

Type IIISum of

Squares dfMean

Square F Sig.


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 GainDependent

– 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 , 1, 2, 3, 1, 2, ()11 , ()21 , ()31 , ()12 , ()22 , ()32 , are fixed unknown constants

And ijk is random, normally distributed with mean 0 and variance 2.

Note:

ijkijjiijky

01111

b

jij

a

iij

n

jj

a

ii

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

where , 1, 2, 3, are fixed unknown constants

And ijk is random, normally distributed with mean 0 and variance 2.

j is normal with mean 0 and varianceand

()ij is normal with mean 0 and varianceNote:

ijkijjiijky

01

a

ii

2B

2AB

This model is called a variance components model

The Anova table for the two factor model

ijkijjiijky

Source SS df MS

A SSAa -1 SSA/(a – 1)

B SSAb - 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)

ijkijjiijky

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) MSError2

a

iia

nb

1

22

1

b

jjb

na

1

22

1

a

i

b

jijba

n

1 1

22

11

EMS = Expected Mean Square

The Anova table for the two factor model (A – fixed, B - random)

ijkijjiijky


A SSA a -1 MSA MSA/MSAB




a

iiAB a

nbn

1

222

1

22Bna

22ABn

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

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

Both fixed and random effects

Formulated by Schultz[1]

1. The EMS for Error is 2.2. The EMS for each ANOVA term contains

two or more terms the first of which is 2.3. 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 2 in the last term of each EMS is the same as the treatment designation.

5. The subscripts of all 2 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.

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

7. 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.

8. Replace 2 whose subscripts are composed entirely of fixed effects by the appropriate sum.

2

2 1 by 1

a

ii

A a

2

2 1 by 1 1

a

iji

AB a b

Example: 3 factors A, B, C – all are random effects

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2 2 2 2ABC AB AC An nc nb nbc

2 2 2 2 2ABC AB BC Bn nc na nac

2 2 2 2 2ABC BC AC Cn na nb nab

2 2 2ABC ABn nc

2 2 2ABC ACn nb

2 2 2ABC BCn na

2 2ABCn

2

AB ABCMS MS

AC ABCMS MS

BC ABCMS MS

ABC ErrorMS MS

Example: 3 factors A fixed, B, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2 2 2 2

1

1a

ABC AB AC ii

n nc nb nbc a

2 2 2

BC Bna nac

2 2 2BC Cna nab

2 2 2ABC ABn nc

2 2 2ABC ACn nb

2 2BCna

2 2ABCn

2

AB ABCMS MS

AC ABCMS MS

BC ErrorMS MS

ABC ErrorMS MS

C BCMS MS

B BCMS MS

Example: 3 factors A , B fixed, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2 2

1

1a

AC ii

nb nbc a

2 2Cnab

2 2ACnb

2 2BCna

2 2ABCn

2

AB ABCMS MS

AC ErrorMS MS

BC ErrorMS MS

ABC ErrorMS MS

C ErrorMS MS

B BCMS MS 2 2 2

1

1a

BC ji

na nac b

22 2

1 1

1 1a b

ABC iji j

n nc a b

A ACMS MS

Example: 3 factors A , B and C fixed

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2

1

1a

ii

nbc a

2

AB ErrorMS MS

AC ErrorMS MS

BC ErrorMS MS

ABC ErrorMS MS

C ErrorMS MS

B ErrorMS MS 2 2

1

1a

ji

nac b

22

1 1

1 1a b

iji j

nc a b

A ErrorMS MS

2 2

1

1c

kk

nbc c

22

1 1

1 1a c

iji k

nb a c

22

1 1

1 1b c

ijj k

na b c

22

1 1 1

1 1 1a b c

ijki j k

n a b c

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 DataDriver Tire Mileage Driver Tire Mileage

1 A 39.6 3 A 33.91 A 38.6 3 A 43.21 A 41.9 3 A 41.31 B 18.1 3 B 17.81 B 20.4 3 B 21.31 B 19 3 B 22.31 C 31.1 3 C 31.31 C 29.8 3 C 28.71 C 26.6 3 C 29.72 A 38.1 4 A 36.92 A 35.4 4 A 30.32 A 38.8 4 A 352 B 18.2 4 B 17.82 B 14 4 B 21.22 B 15.6 4 B 24.32 C 30.2 4 C 27.42 C 27.9 4 C 26.62 C 27.2 4 C 21

Asking SPSS to perform Univariate ANOVA

Select the dependent variable, fixed factors, random factors

The Output


Dependent Variable: MILEAGE

28928.340 1 28928.340 1270.836 .000

68.290 3 22.763a

2072.931 2 1036.465 71.374 .000

87.129 6 14.522b

68.290 3 22.763 1.568 .292

87.129 6 14.522b

87.129 6 14.522 2.039 .099

170.940 24 7.123c

SourceHypothesis

Error

Intercept

Hypothesis

Error

TIRE

Hypothesis

Error

DRIVER

Hypothesis

Error

TIRE * DRIVER

Type IIISum ofSquares df

MeanSquare F Sig.

MS(DRIVER)a.

MS(TIRE * DRIVER)b.

MS(Error)c.

The divisor for both the fixed and the random main effect is MSAB

This is contrary to the advice of some texts


ijkijjiijky






a

iiAB a

nbn

1

222

1

22Bna

22ABn

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


ijkijjiijky



B SSA b - 1 MSB MSB/MSAB



a

iiAB a

nbn

1

222

1

222BAB nan

22ABn

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 B

Levels of A

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.

error random within ofeffect factor ofeffect mean overall

ijkAB

ijA

iijky

The ANOVA table

Source SS df MS F p - value


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

Plant 1 2 machine 1 2 3 4 5 6 7 8 9 10 11 12

98.7 59.2 84.1 72.3 83.5 60.6 33.6 44.8 58.9 63.9 63.7 48.1 93.1 87.8 86.3 110.3 89.3 84.8 48.2 57.3 51.6 62.3 54.6 50.6

100.0 84.1 83.4 81.6 86.1 83.6 68.9 66.5 45.2 61.1 55.3 39.9 Plant 3 4 machine 13 14 15 16 17 18 19 20 21 22 23 24

83.6 76.1 64.2 69.2 77.4 61.0 64.2 35.5 46.9 37.0 43.8 30.0 84.6 55.4 58.4 86.7 63.3 81.3 50.3 30.8 43.1 47.8 62.4 43.0

90.6 92.3 75.4 60.8 76.6 73.8 32.1 36.3 40.8 41.0 60.8 56.9

Anova Table Treating Factors (Plant, Machine) as crossed


Dependent Variable: STRENGTH

21031.065a 23 914.394 7.972 .000

298531.4 1 298531.4 2602.776 .000

18174.761 3 6058.254 52.820 .000

1238.379 5 247.676 2.159 .074

1617.925 15 107.862 .940 .528

5505.469 48 114.697

325067.9 72

26536.534 71


Intercept

PLANT

MACHINE

PLANT * MACHINE

Error

Total

Corrected Total

Type IIISum of

Squares dfMean

Square F Sig.


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

Source Sum of Squares df Mean Square F p - valuePlant 18174.76119 3 6058.253731 52.819506 0.00000 Machine(Plant) 2856.303672 20 142.8151836 1.2451488 0.26171 Error 5505.469467 48 114.6972806










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

2 2 2 2 2ABC AB AC An nc nb nbc

2 2 2 2 2ABC AB BC Bn nc na nac

2 2 2 2 2ABC BC AC Cn na nb nab

2 2 2ABC ABn nc

2 2 2ABC ACn nb

2 2 2ABC BCn na

2 2ABCn

2

AB ABCMS MS

AC ABCMS MS

BC ABCMS MS

ABC ErrorMS MS

Example: 3 factors A fixed, B, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2 2 2 2

1

1a

ABC AB AC ii

n nc nb nbc a

2 2 2

BC Bna nac

2 2 2BC Cna nab

2 2 2ABC ABn nc

2 2 2ABC ACn nb

2 2BCna

2 2ABCn

2

AB ABCMS MS

AC ABCMS MS

BC ErrorMS MS

ABC ErrorMS MS

C BCMS MS

B BCMS MS

Example: 3 factors A , B fixed, C random

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2 2

1

1a

AC ii

nb nbc a

2 2Cnab

2 2ACnb

2 2BCna

2 2ABCn

2

AB ABCMS MS

AC ErrorMS MS

BC ErrorMS MS

ABC ErrorMS MS

C ErrorMS MS

B BCMS MS 2 2 2

1

1a

BC ji

na nac b

22 2

1 1

1 1a b

ABC iji j

n nc a b

A ACMS MS

Example: 3 factors A , B and C fixed

Source EMS F

A

B

C

AB

AC

BC

ABC

Error

2 2

1

1a

ii

nbc a

2

AB ErrorMS MS

AC ErrorMS MS

BC ErrorMS MS

ABC ErrorMS MS

C ErrorMS MS

B ErrorMS MS 2 2

1

1a

ji

nac b

22

1 1

1 1a b

iji j

nc a b

A ErrorMS MS

2 2

1

1c

kk

nbc c

22

1 1

1 1a c

iji k

nb a c

22

1 1

1 1b c

ijj k

na b c

22

1 1 1

1 1 1a b c

ijki j k

n a b c

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 B

Levels of A

The Analysis of Covariance

ANACOVA

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)



2. Categorical independent variables (Factors) A, B, C,…

The categorical independent variables A, B, C,… are set at specific values or levels.

Analysis of Covariance


2. Categorical independent variables (Factors) A, B, C,…

3. Continuous independent variables (covariates) X1, X2, …, Xp

Example

1. Dependent variable Y – weight gain

2. Categorical independent variables (Factors) i. A = level of protein in the diet (High, Low)

ii. B = source of protein (Beef, Cereal, Pork)

3. Continuous independent variables (covariates)

i. X1= initial wt. of animal.

Statistical Technique

Independent variables

continuous categorical

Multiple Regression ×

ANOVA ×

ANACOVA × ×

Dependent variable is continuous

It is possible to treat categorical independent variables in Multiple Regression using Dummy variables.

The Multiple Regression Model

0 1 1 p pY X X

The ANOVA Model

Main Effects Interactions

i j ijY

The ANACOVA Model

Main Effects Interactions

i j ijY

1 1 1 1Covariate Effects

X X

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 ModelSource S.S. d.f.

Main Effects

A SSA a - 1

B SSB b - 1

Interactions

AB SSAB (a – 1)(b – 1)

⁞


Total SSTotal n - 1

The ANACOVA ModelSource S.S. d.f.

Covariates SSCovaraites p

Main Effects

A SSA a - 1

B SSB b - 1

Interactions

AB SSAB (a – 1)(b – 1)

⁞


Total SSTotal n - 1

Example

1. Dependent variable Y – weight gain

2. Categorical independent variables (Factors) i. A = level of protein in the diet (High, Low)

ii. B = source of protein (Beef, Cereal, Pork)

3. Continuous independent variables (covariates)

X = initial wt. of animal.

The data

wtgn initial wt Level Source wtgn initial wt Level Source

112 1031 High Beef 56 1044 Low Beef126 1087 High Beef 86 1025 Low Beef88 890 High Beef 78 878 Low Beef97 1089 High Beef 69 1193 Low Beef91 894 High Beef 76 1024 Low Beef78 917 High Beef 65 1078 Low Beef86 972 High Beef 60 965 Low Beef83 899 High Beef 80 958 Low Beef

108 821 High Beef 78 1135 Low Beef104 846 High Beef 41 847 Low Beef42 1041 High Cereal 68 986 Low Cereal93 1108 High Cereal 67 1003 Low Cereal

102 1132 High Cereal 71 968 Low Cereal77 1023 High Cereal 76 1035 Low Cereal85 1090 High Cereal 85 1018 Low Cereal88 921 High Cereal 37 882 Low Cereal82 909 High Cereal 119 1053 Low Cereal41 1091 High Cereal 91 978 Low Cereal63 838 High Cereal 51 1057 Low Cereal88 935 High Cereal 57 1035 Low Cereal

104 1098 High Pork 96 965 Low Pork114 888 High Pork 67 1025 Low Pork78 1000 High Pork 85 970 Low Pork

111 993 High Pork 17 836 Low Pork109 1043 High Pork 67 961 Low Pork115 992 High Pork 54 931 Low Pork47 834 High Pork 105 1017 Low Pork

124 1005 High Pork 64 845 Low Pork80 905 High Pork 92 1092 Low Pork97 1059 High Pork 62 932 Low Pork

The ANOVA Table

Source Sum of Squares df Mean Square F Sig.

Initial (Covariate) 3357.8165 1 3357.82 9.075 0.00397LEVEL 6523.4815 1 6523.48 17.631 0.0001SOURCE 2013.6469 2 1006.82 2.721 0.07499LEVEL * SOURCE 2528.0163 2 1264.01 3.416 0.04022Error 19609.4835 53 369.99

Total 31966.8500 59

Using SPSS to perform ANACOVA

The data file

Select Analyze->General Linear Model -> Univariate

Choose the Dependent Variable, the Fixed Factor(s) and the Covaraites

The following ANOVA table appears


Dependent Variable: WTGN

12357.366a 6 2059.561 5.567 .000

24.883 1 24.883 .067 .796

3357.816 1 3357.816 9.075 .004

6523.482 1 6523.482 17.631 .000

2013.647 2 1006.823 2.721 .075

2528.016 2 1264.008 3.416 .040

19609.484 53 369.990

421265.0 60

31966.850 59


Intercept

INITIAL

LEVEL

SOURCE

LEVEL * SOURCE

Error

Total

Corrected Total

Type IIISum ofSquares df

MeanSquare F Sig.


40

60

80

100

120

140

700 800 900 1000 1100 1200 1300 1400

Covariate

Dep

end

ent

vari

able

The Process of Analysis of Covariance

Covariate

Ad

just

ed D

epen

den

t va

riab

leThe Process of Analysis of Covariance

40

60

80

100

120

140

700 800 900 1000 1100 1200 1300 1400

• 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.

Reduced Completemodel modelEffectSS RSS RSS

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 = + A + B + C + AB + AC + BC + ABC

A sequence of increasingly simpler models

1. Y = + A + B + C + AB + AC + BC + ABC

2. Y = + A+ B + C + AB + AC + BC

3. Y = + A + B+ C + AB + AC

4. Y = + A + B + C+ AB

5. Y = + A + B + C

6. Y = + A + B

7. Y = + A

8. Y =

Type I S.S.

2 1model modelABCSS RSS RSS I

3 2model modelBCSS RSS RSS I

4 3model modelACSS RSS RSS I

5 4model modelABSS RSS RSS I

6 5model modelCSS RSS RSS I

7 6model modelBSS RSS RSS I

8 7model modelASS RSS RSS I

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

1. Y = + A + B + C + AB + AC + BC + ABC (the three factor model)

2. Y = + A+ B + C + AB + AC + BC (the all two factor model)

3. Y = + 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

2 1model modelABCSS RSS RSS II

2modelABSS RSS Y A B C AC BC RSS II

3modelASS RSS Y B C RSS II

2modelACSS RSS Y A B C AB BC RSS II

2modelBCSS RSS Y A B C AB AC RSS II

3modelBSS RSS Y A C RSS II

3modelCSS RSS Y A B RSS II

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


Dependent Variable: Y

11545.858a 8 1443.232 45.554 .000

61603.201 1 61603.201 1944.440 .000

3666.552 2 1833.276 57.865 .000

809.019 2 404.509 12.768 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32


Intercept

A

B

A * B

Error

Total

Corrected Total

Ty pe I Sumof Squares df

MeanSquare F Sig.


ANOVA table – type II S.S



11545.858a 8 1443.232 45.554 .000

61603.201 1 61603.201 1944.440 .000

3358.643 2 1679.321 53.006 .000

809.019 2 404.509 12.768 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32


Intercept

A

B

A * B

Error

Total

Corrected Total

Ty pe IISum ofSquares df

MeanSquare F Sig.


ANOVA table – type III S.S



11545.858a 8 1443.232 45.554 .000

52327.002 1 52327.002 1651.647 .000

2812.027 2 1406.013 44.379 .000

1010.809 2 505.405 15.953 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32

SourceCorrec ted Model

Intercept

A

B

A * B

Error

Total

Correc ted Total

Ty pe IIISum ofSquares df

MeanSquare F Sig.


Next Topic Other Experimental Designs

Documents

Factorial Experiments