Factorial Experiments Analysis of Variance Experimental Design

Factorial Experiments

Analysis of VarianceExperimental 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 - Beef2. High - Cereal3. 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 5 673 98 94 90 107 49

102 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 G

ain

High ProteinLow ProteinOverall

Profiles of Weight Gain for Source and Level of Protein

70

80

90

100

110

High Protein Low Protein

Wei

ght G

ain

Beef

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 the

i 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 levelsyijk = + 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

Main effects Two factor Interactions

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, ... ,nwhere 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.

Main effects Two factor Interactions

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 dfA SSA a-1B SSB b-1C 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 exampleMean

= 87.867 

y

Main Effects for Factor A (Source of Protein)

Beef Cereal Pork1.733 -2.967 1.233

yyˆ ii

Main Effects for Factor B (Level of Protein)

High Low7.267 -7.267

yyˆjj

AB Interaction Effects

Source of ProteinBeef Cereal Pork

Level High 3.133 -6.267 3.133of 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

Lust

er

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

Lust

er

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

Lus

ter

The Randomized Block Design

• Suppose a researcher is interested in how several treatments affect a continuous response variable (Y).

• The treatments may be the levels of a single factor or they may be the combinations of levels of several factors.

• Suppose we have available to us a total of N = nt experimental units to which we are going to apply the different treatments.

The Completely Randomized (CR) design randomly divides the experimental units into t groups of size n and randomly assigns a treatment to each group.

The Randomized Block Design • divides the group of experimental units into

n homogeneous groups of size t. • These homogeneous groups are called

blocks. • The treatments are then randomly assigned

to the experimental units in each block - one treatment to a unit in each block.

Experimental Designs• In many experiments were are interested in

comparing a number of treatments. (the treatments maybe combinations of levels of several factors.)

• The objective of Experimental design is to reduce the magnitude of random error resulting in more powerful tests to detect experimental effects

The Completely Randomizes Design

Treats1 2 3 … t

Experimental units randomly assigned to treatments

Randomized Block Design

Blocks

All treats appear once in each block

The Model for a randomized Block Experiment

ijjiijy

i = 1,2,…, t j = 1,2,…, b

yij = the observation in the jth block receiving the ith treatment

= overall meani = the effect of the ith treatment

j = the effect of the jth Blockij = random error

The Anova Table for a randomized Block Experiment

Source S.S. d.f. M.S. F p-valueTreat SST t-1 MST MST /MSE

Block SSB n-1 MSB MSB /MSE

Error SSE (t-1)(b-1) MSE

• A randomized block experiment is assumed to be a two-factor experiment.

• The factors are blocks and treatments.

• The is one observation per cell. It is assumed that there is no interaction between blocks and treatments.

• The degrees of freedom for the interaction is used to estimate error.

Experimental Designs• In many experiments were are interested in

comparing a number of treatments. (the treatments maybe combinations of levels of several factors.)

• The objective of Experimental design is to reduce the magnitude of random error resulting in more powerful tests to detect experimental effects

The Completely Randomized Design

Treats1 2 3 … t

Experimental units randomly assigned to treatments

Randomized Block Design

Blocks

All treats appear once in each block

The matched pair designan experimental design for comparing two

treatments

Pairs

The matched pair design is a randomized block design for t = 2 treatments

The Model for a randomized Block Experiment

ijjiijy

i = 1,2,…, t j = 1,2,…, b

yij = the observation in the jth block receiving the ith treatment

= overall meani = the effect of the ith treatment

j = the effect of the jth Blockij = random error

The Anova Table for a randomized Block Experiment

Source S.S. d.f. M.S. F p-valueTreat SST t-1 MST MST /MSE

Block SSB n-1 MSB MSB /MSE

Error SSE (t-1)(b-1) MSE

Incomplete Block Designs

Randomized Block Design

• We want to compare t treatments• Group the N = bt experimental units into b

homogeneous blocks of size t.• In each block we randomly assign the t treatments

to the t experimental units in each block.• The ability to detect treatment to treatment

differences is dependent on the within block variability.

Comments• The within block variability generally increases

with block size.• The larger the block size the larger the within

block variability.• For a larger number of treatments, t, it may not be

appropriate or feasible to require the block size, k, to be equal to the number of treatments.

• If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

Commentsregarding Incomplete block designs

• When two treatments appear together in the same block it is possible to estimate the difference in treatments effects.

• The treatment difference is estimable.

• If two treatments do not appear together in the same block it not be possible to estimate the difference in treatments effects.

• The treatment difference may not be estimable.

Example• Consider the block design with 6 treatments

and 6 blocks of size two.

• The treatments differences (1 vs 2, 1 vs 3, 2 vs 3, 4 vs 5, 4 vs 6, 5 vs 6) are estimable.

• If one of the treatments is in the group {1,2,3} and the other treatment is in the group {4,5,6}, the treatment difference is not estimable.

1

2

2

3

1

3

4

5

5

6

4

6

Definitions

• Two treatments i and i* are said to be connected if there is a sequence of treatments i0 = i, i1, i2, … iM = i* such that each successive pair of treatments (ij and ij+1) appear in the same block

• In this case the treatment difference is estimable.

• An incomplete design is said to be connected if all treatment pairs i and i* are connected.

• In this case all treatment differences are estimable.

Example• Consider the block design with 5 treatments

and 5 blocks of size two.

• This incomplete block design is connected. • All treatment differences are estimable.• Some treatment differences are estimated

with a higher precision than others.

1

2

2

3

1

3

4

5

1

4

DefinitionAn incomplete design is said to be a Balanced Incomplete Block Design.

1. if all treatments appear in exactly r blocks.• This ensures that each treatment is estimated with

the same precision2. if all treatment pairs i and i* appear together in exactly

blocks.• This ensures that each treatment difference is

estimated with the same precision.• The value of is the same for each treatment pair.

Some IdentitiesLet b = the number of blocks.

t = the number of treatmentsk = the block sizer = the number of times a treatment appears in the experiment. = the number of times a pair of treatment appears together in the same block

1. bk = rt• Both sides of this equation are found by counting the

total number of experimental units in the experiment.2. r(k-1) = (t – 1)

• Both sides of this equation are found by counting the total number of experimental units that appear with a specific treatment in the experiment.

BIB DesignA Balanced Incomplete Block Design(b = 15, k = 4, t = 6, r = 10, = 6)

Block Block Block 1 1 2 3 4 6 3 4 5 6 11 1 3 5 6

2 1 4 5 6 7 1 2 3 6 12 2 3 4 6

3 2 3 4 6 8 1 3 4 5 13 1 2 5 6

4 1 2 3 5 9 2 4 5 6 14 1 3 4 6

5 1 2 4 6 10 1 2 4 5 15 2 3 4 5

An Example A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal.

For this purpose: • subjects will be asked to taste and compare these

cereals scoring them on a scale of 0 - 100. • For practical reasons it is decided that each subject

should be asked to taste and compare at most four of the six cereals.

• For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal.

The design and the data is tabulated below:

Subject Taste Scores (Brands)

1 51 (A) 55 (B) 69 (C) 83 (D) 2 48 (A) 87 (D) 56 (E) 22 (F) 3 65 (B) 91 (C) 67 (E) 35 (F) 4 42 (A) 48 (B) 65 (C) 43 (E) 5 36 (A) 58 (B) 69 (D) 7 (F) 6 79 (C) 85 (D) 56 (E) 25 (F) 7 54 (A) 60 (B) 90 (C) 21 (F) 8 62 (A) 92 (C) 94 (D) 63 (E) 9 39 (B) 71 (D) 47 (E) 11 (F) 10 51 (A) 59 (B) 84 (D) 51 (E) 11 39 (A) 74 (C) 61 (E) 25 (F) 12 69 (B) 78 (C) 78 (D) 22 (F) 13 63 (A) 74 (B) 59 (E) 32 (F) 14 55 (A) 74 (C) 78 (D) 34 (F) 15 73 (B) 83 (C) 92 (D) 68 (E)

Analysis of Block Experiments

Analysis of Block ExperimentsThe purpose of such experiments is to estimate the effects of treatments applied to some material (experimental units, subjects etc.) grouped into relatively homogeneous groups (blocks).

The variability within the groups (blocks) will be considerably less than if the subjects were left ungrouped.

This will lead to a more powerful analysis for comparing the treatments.

The basic model for block experiments

jiij block in treat toreponsemean theLet

Suppose we have t treatments and b blocks of size k.

The Assumption of Additivityblocks) ofnt (Independe ikkjij

written.becan Hence ij

0 with j

ji

ijiij

0 with j

jji

block in the nsobservatio of vector the1

th

kj jk

yLet

if treat n is applied to mth unit in jth block

0

1 where j

mnj

mntkj xxX

otherwise

τX1yβτ

jkjj

bt

E

then , if Thus 2

1

2

1

.0with

then

If

2

1

2

1

2

1

1

1β

βτ

100X

010X001X

y

yy

y

y

yy

y

bb

b

bk

E

EE

E

Not of full rank

.orthogonal is such that is

wherelet

0 condition side thehandle To

0

21

1

0

U

uuuUU

1U

1β

bb

bb

0

Hence 1

0

vβUβ1

βU

b

0

and 2

1

100

vu

vuvu

vUv

U1βUUβ

b

b

Thus i vu i

θXvτ

u1X

u1Xu1X

y

bb

E 22

11

and

rank full of is here X

121 ˆvar and ˆ Thus

XXθyXXXθ

Now

bb

b

u1X

u1Xu1X

1u1u1uXXX

XX

22

11

21

111

jjj

jjj

jjj

jjj

u11uX1u

u1XXX

and

b

b

y

yy

1u1u1uXXX

yX

2

1

21

111

BUT

y1u

yX

jjj

jjj

with

totals treatmentof vector 1

jjj

tT

TyXT

lsblock tota of vector 11

bbB

B

y1

y1B

and

Now define the incidence matrix

.block in appears treat. timesof no. where

jin

n

ij

ijbt

N

.replicated is treat timesof no. the where

00

0000

and also 1

1

2

1

inr

r

rr

r

rr

jiji

tt

Rr

Note and bkb k11Nr1N

if treat n is applied to mth unit in jth block

01

where jmn

jmnj xxX

otherwise

Now

tj

j

j

jj

n

nn

00

0000

2

1

XX

Thus

RXX

t

jjj

r

rr

00

0000

2

1

Nn1X of col 2

1

thj

tj

j

j

j j

n

nn

Also

N1X1X1X 21

b

and

Thus

and

UN

u

uu

1X1X1Xu1X

2

1

21

b

bj

jj

NUX1u j

jj

Finally

IUU

u

uu

uuu

uuu11u

kkk

k

b

b

jjj

jjj

2

1

21

.orthogonal is Since1

0

U1U

b

Therefore

jjj

jjj

jjj

jjj

u11uX1u

u1XXXXX

INUUNR

k

UΩNNUIΩNUUΩNΩ

XXkkk

k111

11 and

11 where NUUNRΩ k

Now orthogonal is since 0

100 U11UUUUI

b

11 and NUUNRΩ k

11IUU

b1or

111 N11INR

bk

111 N11NNNR

bkk

111 rrNNR

bkk

ˆˆˆ Thus 1 yXXXvτθ

BUT

UΩNNUIΩNUUΩNΩ

kkk

k111

1

B1rBNTΩ

B11INTΩ

BUUNTΩ

BUUΩNTΩτ

GbkG

k

bk

k

k

where

ˆor

1

11

1

1

1BNTΩ

rBNTΩτ

bkG

k

bkG

k

1

1

ˆ Hence

rrNNRΩ bkk111But

rrrrrr1Nr

1rr1NN1R

1rrNNR1Ω

Thus11

111

bkk

bkk

1rΩ

and

TΩNUBUUΩNNUIv kkk

111ˆ Also

TΩNUUBUUΩNNUIUvUβ

kkk111ˆˆ

and

TΩNUUBUUΩNNUUBUU

kkk111

2

TΩNBUUΩNNBUU

kk11

BUUNTΩNBUU

kk11

τNBτNB11I ˆˆ 111 kbk

0ˆˆˆ Since τrτN1B1τNB1 G

GGGGGG

G

G

k

k

bkG

k

ˆ

1

1

1

B1

BN1T1

BNT1

1rBNTΩrτr

Summary: The Least Squares Estimates

τNBβ ˆˆ 1 k

BNTQ

1QΩ1BNTΩτ

k

bkG

bkG

k

1

1

where

ˆ

The Residual Sum of SquaresθXyyy RSS

vτUBTyy ˆˆ

vUBτTyy ˆˆ βBτTyy ˆˆ

τNBBτTyy ˆˆ 1 k

Hence

0 since 1 GGk BN1T1Q1

BBτBNTyy

kk

11 ˆ τNBBτTyy ˆˆ 1 kRSS

BBτQyy k

1ˆ

BB1QΩQyy kbk

G 11QΩ

bkG

1QBBQΩQyy bk

Gk1

BBQΩQyy k

1

Summary: The Least Squares Estimates

τNBβ ˆˆ 1 k

BNTQ1QΩτ

kbkG 1 whereˆ

The Residual Sum of Squares

BBQΩQyy kRSS 1