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Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 13

Fractional Factorials

Confounding, Aliases, Design Resolution,

Pilot Plant Experiment :Aliasing/Confounding with Operators

Complete Factorial :

1/2 Replicate for Each of 2

Operators

45

80

C2

Catalyst

52

83

Operator 1

Operator 2

54

68

C1

40

Concentration

60

72

20

160

180

Temperature

Aliases : Main Effect for Temperatures and Main Effect for Operators

Main Effect for Operator

Aliased with

Main Effect for Temperature

Aliasing/Confounding of Effects :Pilot Plant Experimenty = Constant + Main Effects + Interaction Effects + Operator Effect + Error

M(Temp) = {180 Temp + Operator 2} - {160 Temp + Operator 1}

= 75.75 - 52.75

= 23.0

Does 23.0 Measure the Effect of

Temperatures, Operators, or Both ?

Aliasing/Confounding of Effects :Pilot Plant Experiment

y = Constant + Main Effects + Interaction Effects + Operator Effect + Error

M(Temp) = {180 Temp + Operator 2} - {160 Temp + Operator 1}

= 75.75 - 52.75

= 23.0

M(Cat) = {Cat C2 + (Operator 1 + Operator 2)/2}

- {Cat C1 + (Operator 1 + Operator 2)/2}

= {Cat C2 – Cat C1}

= 65.0 - 63.5 = 1.5

Operator Effect Not Aliased with the

Main Effect for Catalyst

c’cD = 0

Not Aliased

Main Effect for Temperature

cT’cD = 2

Aliased

Catalyst Effect

cC’cD = 0

Not Aliased

Pilot Plant Experiment :Aliased EffectsOperator Effect

Aliasing / Confounding of Factor Effects

Factor effects are Aliased or Confounded when differences in

average responses cannot uniquely be attributed to a single effect

- Unplanned confounding can result in loss of ability to evaluate important main effects and interactions
- Planned aliasing of unimportant interactions can enable the size of the experiment to be reduced while still enabling the estimation of important effects

Factor effects are Aliased or Confounded when they are estimated

by the same linear combination of response values

Factor effects are Partially Aliased or Partially Confounded when they are

estimated by nonorthogonal linear combinations of response values

General Confounding Principle for 2k Balanced Factoral Experiments

Effects Representations

Effect 1 = c1’y

Effect 2 = c2’y

Two Effects are Confounded or Aliased if

Aliases : c1 = const x c2

Partial Aliases :

Effects Representation for a Complete 23 Factorial

Lower Level = -1

Upper level = +1

Effect = c’y / Divisor

y = Vector of Responses or Average Responses

Aliasing with Operator

Same Alias if All Signs Reversed

Aliasing with Operators

Better design for operator aliasing?

Aliasing with Operators

Note: Operator effect is unconfounded with all effects except ABC;

Good choice of contrast for aliasing with operators

Summary

- Some designs have one or more factors aliased with one another
- Sums of squares measure the same effect or partially measure the same effect
- The sums of squares are not statistically independent
- Determining Aliases
- If two-level factors, multiply effect contrasts
- If nonzero, the effects are partially aliased
- If one is a multiple of another, the effects are aliased

Summary (con’t)

- Accommodation
- Eliminate one of the aliased effects
- Leave all In but properly interpret analysis of variance results (to be discussed in subsequent classes)

Two Types of Aliasing

Fractional Factorials in

Completely Randomized Designs:

Can’t Run All Combinations

Distinguish

Randomized Incomplete Block Designs :

Insufficient Homogeneous Experimental Units

or Homogeneous Test Conditions

in Each Block – Must Include Combinations

in Two or More Blocks

Fractional Factorials

- Pilot Plant Chemical Yield Study
- Temperature: 160, 180 oC
- Concentration: 20, 40 %
- Catalysts: 1, 2
- Too costly to run all 8 combinations
- Must run fewer combinations

Good Choice for a Fractional Factorial

Notation

Defining Equation (Contrast) The effect(s) aliased with the mean

I = ABC

Convention

Designate the mean by I (Identity)

Main Effects (s = 1) are unconfounded with

other main effects (R - s = 2)

Example : Half-Fraction of 23 (23-1)

Design ResolutionResolution R

Effects involving s factors are unconfounded

with effects involving fewer than R-s factors

Design Resolution

Resolution R

Effects involving s factors are unconfounded

with effects involving fewer than R-s factors

Resolution IV (R = 4)

Main Effects (s = 1) are unconfounded with

other main effects & two-factor interactions(R - s = 3)

Two-factor interactions (s = 2) are unconfounded with

main effects (R - s = 2); confounded with other

two-factor interactions

Confounding Pattern

Resolution III

Main Effects (s = 1) unaliased with other main effects (R - s = 2)

Importance of Design Resolution

- Quickly identifies the overall structure of the confounding pattern
- A design of resolution R is a complete factorial in any R-1 or fewer factors

A

C

C

B

B

A

C

A

Figure 7.3 Projections of a half fraction of a three-factor complete factorial

experiment (I=ABC).

Pilot Plant Experiment :Half Fraction

45

80

C2

Catalyst

52

83

54

68

C1

I = ABC

40

Concentration

60

72

20

160

180

Temperature

Pilot Plant Experiment : RIII is a Complete Factorial in any R-1 = 2 Factors

80

52

80

52

54

80

54

54

72

52

72

72

Catalyst

Concentration

Temperature

Importance of Fractional Factorial Experiments

Design Efficiency

Reduce the size of the experiment

through

intentional aliasing of relatively unimportant

effects

Effects Representation for a Complete 23 Factorial

Lower Level = -1

Upper level = +1

Effect = c’y / Divisor

y = Vector of responses or average responses for the run numbers

Designing a 1/2 Fraction of a 2k Complete Factorial

Resolution = k

- Write the effects representation for the main effects and the highest-order interaction for a complete factorial in k factors
- Randomly choose the +1 or -1 level for the highest-order interaction (defining contrast, defining equation)
- Eliminate all rows except those of the chosen level (+1 or -1) in the highest-order interaction
- Add randomly chosen repeat tests, if possible
- Randomize the test order or assignment to experimental units

Aliasing Pattern

- Write the defining equation (contrast) (I = Highest-order interaction)
- Symbolically multiply both sides of the defining equation by each of the other effects
- Reduce the right side of the equations: X x I = X X x X = X2 = I (powers mod(2) )

Resolution = III

(# factors in the

defining contrast)

Defining Equation: I = ABC

Aliases : A = AABC = BC

B = ABBC = AC

C = ABCC = AB

Designing Higher-Order Fractions

- Total number of factor-level combinations = 2k
- Experiment size desired = 2k/2p = 2k-p
- Choose p defining contrasts (equations)
- For each defining contrast randomly decide which level will be included in the design
- Select those combinations which simultaneously satisfy all the selected levels
- Add randomly selected repeat test runs
- Randomize

Acid Plant Corrosion Rate Study: Half Fraction

(I = - ABCDEF = +ABC = -DEF)

Implicit Contrast

-ABCDEF x ABC = -AABBCCDEF = -DEF

Design Resolution for Fractional Factorials

- Determine the p defining equations
- Determine the 2p - p - 1 implicit defining equations: symbolically multiply all of the defining equationsResolution = Smallest ‘Word’ length in the defining & implicit equations
- Each effect has 2p aliases

26-2 Fractional Factorials :Confounding Pattern

Build From 1/4 Fraction

RIII

I = ABCDEF = ABC = DEF

A = BCDEF = BC = ADEF

B = ACDEF = AC = BDEF

. . .

(I + ABCDEF)(I + ABC) = I + ABCDEF + ABC + DEF

Defining Contrasts

Implicit Contrast

RIV

I = ABCD = CDEF = ABEF

A = BCD = ACDEF = BEF

B = ACD = BCDEF = AEF

. . .

26-2 Fractional Factorials :Confounding PatternBuild From 1/2 Fraction

RIII

I = ABCDEF = ABC = DEF

A = BCDEF = BC = ADEF

B = ACDEF = AC = BDEF

. . .

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