# Optimal Blocking of Orthogonal Arrays in Designed Experiments

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Optimal Blocking of Orthogonal Arrays in Designed Experiments
Peter Goos

Optimal Blocking of Orthogonal Arrays in Designed Experiments
Peter Goos In collaboration with Eric Schoen and Bagus Sartono

Starting point Factorial experiments
Treatments are described by combinations of factor levels Interest is in main effects and two-factor interaction effects Experimental tests or runs need to be partitioned in blocks (due to different days, batches of raw material, …) Block effects are treated as fixed Experimenting is expensive, so we have small data !

Experiments in blocks Many processes have sources of variability that are uncontrollable. Examples are day-to-day variation, batch-to-batch variation, etc. When experimenting, this leads to groups of observations. The groups are called blocks. The grouping variable (day, batch) is called a blocking factor. Responses within each group are more homogeneous or similar than responses from different groups.

Outline PART 1: the number of observations exceeds the number of main effects and two-factor interactions Vitaming stability experiment 32 observations (8 blocks of size 4) 64 observations (16 blocks of size 4) PART 2: the number of observations is too small to estimate all two-factor interaction effects

Model of interest (in case all factors have 2 levels)

PART 1 The number of runs is big enough to estimate all main effects and two-factor interactions. Focus on 2-level factors n ≥ 1 + k + k(k‒1)/2

Context I have always advocated optimal design of experiments
Flexible in terms of numbers of runs Different types of factors Constraints on the factor levels Implicitly assuming that `traditional designs’ do a good job when the number of observations is a power of 2 or a multiple of 4 Today, I start talking about situations where n is a power of 2, as well as the number of blocks

Vitamin stability experiment
Vitamins degrade when exposed to light Can be stabilized when embedded in a special molecule, called a fatty molecule Five different fatty molecules Binding with sugar might help as well to stabilize the vitamins Experiment involving 6 two-level factors

Six factors Fatty molecules 1. Boundedness with sugar. 2. Oil Red O.
3. Oxybenzone. 4. Beta Carotene. 5. Sulisobenzone 6. Deoxybenzone Fatty molecules

Vitamin stability experiment
Vitamins degrade when exposed to light Can be stabilized when embedded in a special molecule, called a fatty molecule Five different fatty molecules Binding with sugar might help as well to stabilize the vitamins Experiment involving 6 two-level factors

Vitamin stability experiment
Vitamins degrade when exposed to light Can be stabilized when embedded in a special molecule, called a fatty molecule Five different fatty molecules Binding with sugar might help as well to stabilize the vitamins Experiment involving 6 two-level factors There is day-to-day variation in the process

Vitamin stability experiment
Vitamins degrade when exposed to light Can be stabilized when embedded in a special molecule, called a fatty molecule Five different fatty molecules Binding with sugar might help as well to stabilize the vitamins Experiment involving 6 two-level factors There is day-to-day variation in the process 4 runs per day are possible 8 days are available

Vitamin stability experiment
6 two-level factors 8 days of 4 runs or observations 32 runs in total model 6 main effects 15 two-factor interaction effects 1 intercept 7 contrasts for the 8-level blocking factor 29 parameters in total

Table 4B.3 in Wu & Hamada (26-1 design)

Traditional approach Design generator 6=12345 to choose 32 treatments or factor level combinations Block generators to arrange 32 runs in 8 blocks of 4 runs B1 = 135 B2 = 235 B3 = 145

32-run orthogonal design

Traditional approach Perfect design for main effects
Can be estimated independently, with maximum precision Estimates not affected by day-to-day variation No variance inflation No multicollinearity Not so for interaction effects 12 of the 15 interactions can be estimated independently, with maximum precision 3 interaction effects (12, 34, 56) cannot be estimated Perfect collinearity with the blocks

Double the number of runs ! 64 instead of 32 runs Full factorial design instead of half fraction 16 blocks of size 4 Table 3A in Wu & Hamada (2000)

Block generators to arrange 64 runs in 16 blocks of 4 runs B1 = 136 B2 = 1234 B3 = 3456 B4 = We can estimate all two-factor interaction effects except 12, 34 and 56

Conclusion The 64-run design is a waste of resources.
The traditional approach doesn’t work.

64 observations in 16 blocks of size 4 Do not start from full factorial design ! Instead, cleverly combine two half fractions of 32 observations arranged in 8 blocks of size 4

First half fraction Design generator 6=12345 to choose 32 treatments or factor level combinations Block generators to arrange 32 observations in 8 blocks of 4 runs B1 = 135 B2 = 235 B3 = 145 We can estimate all two-factor interaction effects except 12, 34 and 56 This was the original idea

Second half fraction Design generator 6=‒12345 to choose 32 treatments or factor level combinations Block generators to arrange 32 observations in 8 blocks of 4 runs B1 = 135  124 B2 = 235  134 B3 = 145  125 We can estimate all two-factor interaction effects except 23, 45 and 16

Result is a full factorial design From the first half of the experiment, we cannot estimate 12, 34 and 56 But we can estimate these effects from the second half From the second half of the experiment, we cannot estimate 23, 45 and 16 But we can estimate them from the first half

Some similar scenarios
5 two-level factors, 32 runs, 8 blocks of size 4: better to use two (cleverly selected) half fractions than it is to use a full factorial design 6 two-level factors, 64 runs, 16 blocks of size 4: better to use two 32-run half fractions than to use a full factorial but you can also combine a 32-run half fraction with a 16-run quarter fraction !

Do not trust tables in DOE textbooks ! Do not trust options for screening designs in your favorite software !

Advice to DOE textbook writers
Make clear that certain designs in the tables should not be used ! And describe the better alternatives.

Advice to DOE software developers
Make clear that certain screening design options should not be used ! And provide the better alternatives.

Do not trust tables in DOE textbooks ! Do not trust options for screening designs in your favorite software !

Throw away the DOE textbooks ! Do not trust options for screening designs in your favorite software !

Throw away the DOE textbooks ! Use optimal design of experiments !

D-optimal design I Calculate a 64-run D-optimal design with 16 blocks of size 4 Main effects + two-factor interactions Really easy with SAS, JMP, Design Expert, … D-optimal design is 3% better than the design produced by the semi-traditional approach Design is not orthogonally blocked

D-optimal design I

Optimal design D-optimality criterion: seeks designs that maximize determinant of information matrix Algorithms by Atkinson & Donev (1989) and Cook and Nachtsheim (1989) I used JMP’s coordinate-exchange algorithm

… but it does not solve the original problem …
This is interesting … … but it does not solve the original problem … … which was to find a 32-run two-level design in 8 blocks of size 4 for estimating main effects and two-factor interaction effects

D-optimal design II Calculate a 32-run D-optimal design with 8 blocks of size 4 Main effects + two-factor interactions Really easy with SAS, JMP, Design Expert, … All 2fis are estimable Design is not orthogonally blocked VIFs range from 1 to 2.6 only

D-optimal design II

Conclusion Part 1 64 runs 64-run textbook design was beaten by manually constructed design manually constructed design was beaten by optimal design 32-run textbook design was beaten by optimal design So, optimal designs do a better job than classical designs even in scenarios that are ideal for classical designs 42 42

PART 2 The number of runs is not big enough to estimate all main effects and two-factor interactions. Optimal design approach not feasible since information matrix is singular in that case. Factors with 2, 3 and 4 levels

Orthogonal arrays There exist many orthogonal arrays (OAs) that can be used as an experimental design 2-level arrays Regular full and fractional factorial designs Plackett-Burman designs Other nonregular arrays 3-level arrays: regular full and fractional factorial designs, nonregular arrays Mixed-level arrays: not all factors have the same number of levels (e.g. Taguchi’s L18) 44 44

Strength-2 (or resolution-III) arrays
Main effects can be estimated independently from each other But they are aliased with two-factor interactions Using complete catalogs of OAs, we sought optimal blocking patterns based on the concept of “generalized word-length pattern” Orthogonal blocking for main effects As little aliasing and confounding for two-factor interactions as possible We listed optimally blocked designs with 12, 16, 20, 24 and 27 runs 45 45

20 runs, eight 2-level factors, five blocks
Blocks X X X X X X X X 46 46

27 runs, nine 3-level factors, nine blocks
X X X X X X X X X 47 47

Strength-3 (or resolution IV) arrays
Main effects can be estimated independently From each other From two-factor interaction effects Two-factor interactions are aliased with each other Enumerating all possible blocking patterns for all OAs in catalogs was infeasible We used mixed integer linear programming instead to find blocking arrangements of good orthogonal arrays: Orthogonally blocked for main effects As little confounding between two-factor interactions and blocks as possible 48 48

Mixed integer linear programming
Input: A good OA which allows estimation of many two-factor interactions Number of blocks required Output: Optimal blocking pattern (orthogonal for the main effects) Tells you when it is infeasible to find such a pattern Implementations SAS/OR Matlab + CPLEX 49 49

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40 runs, one 5-level factor, six 2-level factors, four blocks 50 50

Conclusion Part 2 Catalogs of orthogonal arrays offer a starting point for designing blocked experiments For small numbers of observations, it is possible to completely enumerate all possible designs and select the best For larger numbers of observations, a mixed integer linear programming approach can be used to arrange an appropriate orthogonal array in blocks 51 51

Based on … Sartono, B., Goos, P., Schoen, E.D. (2014) Blocking Orthogonal Designs with Mixed Integer Linear Programming, Technometrics 56, to appear. Schoen E.D., Sartono B., Goos, P. (2013) Optimum blocking for general resolution-3 designs, Journal of Quality Technology 45, Goos, P., Jones, B. (2011) Optimal Design of Experiments: A Case-Study Approach, Wiley.

Optimal Blocking of Orthogonal Arrays in Designed Experiments
Peter Goos In collaboration with Eric Schoen, Bagus Sartono and Nha Vo-Thanh

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