ANOVA TABLE Factorial Experiment Completely Randomized Design

ANOVA TABLE

Factorial Experiment

Completely Randomized Design

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

2̂

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

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 lower order interactions and main effects for factors which have not yet been determined to affect the response.

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

Tests of Between-Subjects Effects

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

Tests of Between-Subjects Effects

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

SourceCorrected Model

Intercept

PLANT

MACHINE

PLANT * MACHINE

Error

Total

Corrected Total

Type IIISum of

Squares dfMean

Square F Sig.

R Squared = .793 (Adjusted R Squared = .693)a.

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

Graph

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7

Machine

Plant 1

Plant 2

Plant 3

Plant 5

Pap

er S

tren

gth

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ANOVA TABLE Factorial Experiment Completely Randomized Design