Factorial Experiments

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Factorial Experiments. Analysis of Variance Experimental Design. Dependent variable Y k Categorical independent variables A, B, C, â€¦ (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.

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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 = nt=nabc..

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

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 gainsY (grams) in rats.

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

The k = 6 diets are the 6 = 3×2 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 Pork

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 92 105 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 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.
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

Models for factorial Experiments

Single Factor: A – a levels

yij = m + ai + eiji = 1,2, ... ,a; j = 1,2, ... ,n

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

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 dist’n

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

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

Overall mean

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

where0 = S ai = S bj= S (ab)ij= Sgk = S (ag)ik = S(bg)jk= S (abg)ijk = Sdl= 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.

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.
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.)
Main Effects for Factor A (Source of Protein)

Beef Cereal Pork

1.733 -2.967 1.233

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

Example 2

Paint Luster Experiment

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

Treats

1 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

i = 1,2,…, t

j = 1,2,…, b

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

m = overall mean

ti = the effect of the ith treatment

bj = the effect of the jth Block

eij = random error

• 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

Treats

1 2 3 … t

Experimental units randomly assigned to treatments

Randomized Block Design

Blocks

All treats appear once in each block

Pairs

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

The Model for a randomized Block Experiment

i = 1,2,…, t

j = 1,2,…, b

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

m = overall mean

ti = the effect of the ith treatment

bj = the effect of the jth Block

eij = random error

Incomplete Block Designs

Randomized Block Design
• We want to compare t treatments
• Group the N = bt experimentalunits 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.
• 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.
• 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.

1

2

2

3

1

3

4

5

5

6

4

6

• 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.
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.

1

2

2

3

1

3

4

5

1

4

• This incomplete block design is connected.
• All treatment differences are estimable.
• Some treatment differences are estimated with a higher precision than others.
Definition

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

• if all treatments appear in exactly r blocks.
• This ensures that each treatment is estimated with the same precision
• if all treatment pairs i and i* appear together in exactly l blocks.
• This ensures that each treatment difference is estimated with the same precision.
• The value of l is the same for each treatment pair.
Some Identities

Let b = the number of blocks.

t = the number of treatments

k = the block size

r = the number of times a treatment appears in the experiment.

l = the number of times a pair of treatment appears together in the same block

• bk = rt
• Both sides of this equation are found by counting the total number of experimental units in the experiment.
• r(k-1) = l (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 Design

A Balanced Incomplete Block Design

(b = 15, k = 4, t = 6, r = 10, l = 6)

An Example
• 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.

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

Analysis of Block Experiments

Analysis of Block Experiments

The 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

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

Let

if treat n is applied to mthunit in jthblock

otherwise

Note

Now

if treat n is applied to mthunit in jthblock

otherwise

Thus

Also

and

Thus

and

Summary: The Least Squares Estimates

The Residual Sum of Squares

Summary: The Least Squares Estimates

The Residual Sum of Squares