construction of flexible fractional factorial designs and ... · introductionfractions with 2-level...

$: Construction of flexible fractional factorial designs and ... · IntroductionFractions with 2-level factors Asymmetric fractions Multi-stratum designsFrom design key to design Construction$
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design

Construction of flexible fractional factorialdesigns and its implementation in R

Herve MonodINRA, MIA-Jouy Unit, France

Isaac Newton Institute

11 octobre 2011

Herve Monod INRA, MIA-Jouy Unit, France

Construction of flexible fractional factorial designs and its implementation in R


Foreword

Thanks to the Isaac Newton Institute and DAE organisers

Credit to Andre Kobilinsky




Outline

1 Introduction

2 Fractions with 2-level factors

3 Asymmetric fractionsMixture of 2-level and 4-level factorsMixtures involving different primes

4 Multi-stratum designsSymmetric split-plot designAsymmetric split-plot design

5 From design key to design




Plan

1 Introduction








Introduction

Bibliography

1978 : Patterson and Bailey. Design keys for factorialexperiments. Appl.Statist.

1991 : Kobilinsky and Monod. Experimental design generatedby group morphism. An introduction. Scand. J. Statist.

2008 : Pistone and Rogantin. Indicator function and complexcoding for mixed fractional factorial designs. JSPI




Unpublished

1994 : Kobilinsky. Automatic generation of asymmetricalregular designs.

2005 : Kobilinsky. PLANOR : program for the automaticgeneration of regular experimental designs.

2011 : Kobilinsky, Bouvier, Monod. PLANOR : an R library forthe automatic generation of regular fractional factorial designs




Software implementations of regular fractional designs

SAS : proc factex, + Kuhfeld and Tobias (2005)Technometrics

GENSTAT : agfraction (stored keys)

R, package FrF2 (Ulrike Groemping)

But no tool coping with

mixed numbers of factor levels

flexible model assumptions

integration of hierarchy constraints




Plan

1 Introduction








Example 1

Standard example : 24−1 design of resolution 4

Treatments : 4 factors at 2 levels A, B, C , D

Units : N = 23 = 8

Model :

main effects + 2-factor interactionsestimate main effects




Solution : (1) 23 complete factorial design on A, B, C

Unit A B C

1 + + +2 + + −3 + − +4 + − −5 − + +6 − + −7 − − +8 − − −




Solution : (2) defining relation D = ABC

Unit A B C D = ABC

1 + + + +2 + + − −3 + − + −4 + − − +5 − + + −6 − + − +7 − − + +8 − − − −




Consequences : pairwise aliasing between factorial effects 1

1 = ABCDA = BCDB = ACDC = ABDD = ABC

AB = CDAC = BDAD = BC

interactions of order ≥ 3 assumed zero ⇒ main effects areestimable




In additive notation (integers modulo 2)

Unit A B C D =A+B +C

000 0 0 0 0001 0 0 1 1010 0 1 0 1011 0 1 1 0100 1 0 0 1101 1 0 1 0110 1 1 0 0111 1 1 1 1




Planor detailed syntax

> library("planor",lib="/home/hmonod/R/library")

> ex1.fac <- planor.factors(factors=c("A","B","C","D"),

+ nlevels=2)

> ex1.mod <- planor.model(

+ model= ~ A+B+C+D+ A:B + A:C + A:D + B:C + B:D + C:D,

+ estimate=~ A + B + C + D)

> ex1.key <- planor.designkey(factors=ex1.fac,

+ model=ex1.mod,

+ nunits=8,

+ base=~A+B+C)




Planor shorter syntax

> library("planor",lib="/home/hmonod/R/library")

> ex1.key <- planor.designkey(

+ factors=LETTERS[1:4], nlevels=2,

+ model= ~(A + B + C + D)^2,

+ estimate=~ A + B + C + D,

+ nunits=8,

+ base=~A+B+C)




Search process information

Determination of ineligible factorial terms

Determination of ineligible pseudofactorial terms

Independent searches for prime(s) : 2

Key-matrix search for prime p = 2

There are 3 predefined columns

First visit to column 4

The search is closed: max.sol = 1 solution(s) found




Result : design key matrix

> print(ex1.key)

********** Prime 2 design **********

--- Solution 1 for prime 2 ---

A B C D

A 1 0 0 1

B 0 1 0 1

C 0 0 1 1




Result : aliasing I

> alias(ex1.key, model = ~(A + B + C + D)^2)

********** Prime 2 design **********


UNALIASED TREATMENT EFFECTS

A ; B ; C ; D

ALIASED TREATMENT EFFECTS

A B = C D

A C = B D




Result : aliasing II

A D = B C

TREATMENT EFFECTS CONFOUNDED WITH BLOCK EFFECTS

nil

UNALIASED BLOCK EFFECTS

nil

--- Synthesis on the aliased treatment effects for prime 2 ---

unaliased trt.aliased blc.aliased

[1,] 4 6 0




Example 2 : 27−4 of resolution 3

7 factors at 2 levels, N = 8, model : main effects.

> ex2.key <- planor.designkey(

+ factors=LETTERS[1:7], nlevels=2,

+ model= ~A + B + C + D + E + F + G,

+ nunits=8,

+ base=~A+B+C)





> print(ex2.key)

********** Prime 2 design **********


A B C D E F G

A 1 0 0 1 1 0 1

B 0 1 0 1 0 1 1

C 0 0 1 0 1 1 1




Result : aliasing I

> alias(ex2.key, model = ~(A + B + C + D + E + F + G)^2)

********** Prime 2 design **********



nil


A = B D = C E = F G

B = A D = C F = E G





C = A E = B F = D G

D = A B = C G = E F

E = A C = B G = D F

F = A G = B C = D E

G = A F = B E = C D


nil


nil




Result : aliasing III



[1,] 0 28 0




Result : defining relationships I

> summary(ex2.key)

********** Prime 2 design **********


DESIGN KEY MATRIX

A B C D E F G

A 1 0 0 1 1 0 1

B 0 1 0 1 0 1 1

C 0 0 1 0 1 1 1




Result : defining relationships II

TREATMENT EFFECTS CONFOUNDED WITH THE MEAN

1 = A B D

1 = A C E

1 = B C F

1 = D E F

1 = C D G

1 = B E G

1 = A F G

1 = B C D E

1 = A C D F

1 = A B E F

1 = A B C G

1 = A D E G




Result : defining relationships III

1 = B D F G

1 = C E F G

1 = A B C D E F G

BLOCK-and-TREATMENT EFFECTS CONFOUNDED WITH THE MEAN

nil

WEIGHT PROFILES

Treatment effects confounded with the mean: 3^7 4^7 7^1

Treatment effects confounded with block effects: none

Treatment pseudo-effects confounded with the mean: 3^7 4^7 7^1

Treatment pseudo-effects confounded with block effects: none




Plan

1 Introduction








Mixture of 2-level and 4-level factors

Example 3 : 4224−3 of resolution 4

5 factors at 4 and 2 levels, N = 32Model : main effects + 2-factor interactionsEstimate main effects

> mxd1.key <- planor.designkey(

+ factors=LETTERS[1:6],

+ nlevels=c(4,4,rep(2,4)),

+ resolution=4,

+ nunits=32,

+ base=~A+B+C)






> print(mxd1.key)

********** Prime 2 design **********


A_1 A_2 B_1 B_2 C D E F

A_1 1 0 0 0 0 1 0 1

A_2 0 1 0 0 0 0 1 1

B_1 0 0 1 0 0 1 0 1

B_2 0 0 0 1 0 0 1 1

C 0 0 0 0 1 1 1 1





Result : aliasing I> alias(mxd1.key, model = ~(A + B + C + D + E + F)^2)

********** Prime 2 design **********



A_1 ; A_2 ; A_1 A_2 ; B_1 ; B_2 ; B_1 B_2 ; C ; D ; E ; F ; A_1 B_2 ; A_1 B_1 B_2 ; A_2 B_1 ; A_2 B_1 B_2 ; A_1 A_2 B_1 ; A_1 A_2 B_2


A_1 B_1 = C D = E F

A_2 B_2 = C E = D F

A_1 A_2 B_1 B_2 = C F = D E

A_1 C = B_1 D

A_2 C = B_2 E






A_1 A_2 C = B_1 B_2 F

A_1 D = B_1 C

A_2 D = B_2 F

A_1 A_2 D = B_1 B_2 E

A_1 E = B_1 F

A_2 E = B_2 C

A_1 A_2 E = B_1 B_2 D

A_1 F = B_1 E

A_2 F = B_2 D

A_1 A_2 F = B_1 B_2 C


nil







nil



[1,] 16 33 0




Mixtures involving different primes

Example 4 : Latin square of order 6

6×6 row-column design with A and B at 2 and 3 levels

> ls.key <- planor.designkey(

+ factors=c("ROW","COL","A","B"),

+ nlevels=c(6,6,2,3),

+ block=~ROW+COL,

+ model=~ROW + COL + A*B,

+ nunits=36, base=~ROW+COL)








Independent searches for prime(s) : 2 3













Result : design key matrix part 1

> print(ls.key[[1]])

Number of solutions: 1 for prime 2

ROW_1 COL_1 A

ROW_1 1 0 1

COL_1 0 1 1






> print(ls.key[[2]])


ROW_2 COL_2 B

ROW_2 1 0 1

COL_2 0 1 1





Result : aliasing I

> alias(ls.key, model = ~(ROW + COL + A + B)^2)

********** Prime 2 design **********



nil


ROW_1 = COL_1 A

COL_1 = ROW_1 A






A = ROW_1 COL_1


nil


nil



[1,] 0 6 0






********** Prime 3 design **********



nil


ROW_2 = COL_2^2 B

COL_2 = ROW_2^2 B

B = ROW_2 COL_2

ROW_2 COL_2^2 = ROW_2^2 B^2 = COL_2 B





Result : aliasing IV


nil


nil



[1,] 0 9 0




Plan

1 Introduction








Symmetric split-plot design

Example 5 : split-plot design, powers of 2 only

4 replicates of 4 subblocks of size 2Factor A at 4 levels, constant in subblocksFactor B at 2 levels, no constraint

> split.key <- planor.designkey(

+ factors=c("REP","SUB","A","B"), nlevels=c(4,4,4,2),

+ hiera=~A/(REP*SUB),

+ listofmodels=list( c(~ REP*SUB + A*B, ~ B + A:B),

+ c(~ REP + A*B, ~ A )),

+ nunits=32, base=~REP+SUB)






> print(split.key)

********** Prime 2 design **********


REP_1 REP_2 SUB_1 SUB_2 A_1 A_2 B

REP_1 1 0 0 0 0 0 0

REP_2 0 1 0 0 0 0 0

SUB_1 0 0 1 0 1 0 0

SUB_2 0 0 0 1 0 1 0

*U* 0 0 0 0 0 0 1





Result : aliasing I

> alias(split.key, model = ~REP * SUB + A * B)

********** Prime 2 design **********



REP_1 ; REP_2 ; REP_1 REP_2 ; B ; REP_1 SUB_1 ; REP_1 SUB_2 ; REP_1 SUB_1 SUB_2 ; REP_2 SUB_1 ; REP_2 SUB_2 ; REP_2 SUB_1 SUB_2 ; REP_1 REP_2 SUB_1 ; REP_1 REP_2 SUB_2 ; REP_1 REP_2 SUB_1 SUB_2 ; A_1 B ; A_2 B ; A_1 A_2 B


SUB_1 = A_1

SUB_2 = A_2






SUB_1 SUB_2 = A_1 A_2


nil


nil



[1,] 16 6 0




Asymmetric split-plot design

Example 6 : split-plot design

4 replicates of 3 subblocks of size 2Factor A at 3 levels, constant in subblocksFactor B at 2 levels, no constraint

> split.key <- planor.designkey(

+ factors=c("REP","SUB","A","B"), nlevels=c(4,3,3,2),

+ hiera=~A/(REP*SUB),

+ listofmodels=list( c(~ REP*SUB + A*B, ~ B + A:B),

+ c(~ REP + A*B, ~ A )),

+ nunits=24, base=~REP+SUB)








Independent searches for prime(s) : 2 3






There is 1 predefined column








> print(split.key[[1]])


REP_1 REP_2 B

REP_1 1 0 0

REP_2 0 1 0

*U* 0 0 1






> print(split.key[[2]])


SUB A

SUB 1 1





Result : aliasing I

> alias(split.key, model = ~REP * SUB + A * B)

********** Prime 2 design **********



REP_1 ; REP_2 ; REP_1 REP_2 ; B


nil







nil


nil



[1,] 4 0 0

********** Prime 3 design **********








nil


SUB = A


nil






Result : aliasing IV

nil



[1,] 0 2 0




Plan

1 Introduction








Possible post-treatments

compare several solutions

1 set option max.sol more than 12 function summary and/or aberration3 own function(s) to calculate own criteria

construct the design

randomize




Example 6 continued

Construct and randomize the split-plot design

> split.plan <- planor.design(split.key,

+ randomize=~REP/SUB/UNITS)

Extraction of a design key from an object of class listofkeyrings




Result : randomized design

> finalplan <- split.plan@design

> finalplan <- finalplan[

+ order(finalplan$REP,finalplan$SUB),

+ c("REP","SUB","A","B")]

> rownames(finalplan) <- NULL

> colnames(finalplan) <- c("R","S","A","B")




REP 1

R S A B

1 1 1 1 1

2 1 1 1 2

3 1 2 2 2

4 1 2 2 1

5 1 3 3 2

6 1 3 3 1

REP 2

R S A B

7 2 1 3 2

8 2 1 3 1

9 2 2 1 1

10 2 2 1 2

11 2 3 2 1

12 2 3 2 2

REP 3

R S A B

13 3 1 3 2

14 3 1 3 1

15 3 2 1 2

16 3 2 1 1

17 3 3 2 1

18 3 3 2 2

REP 4

R S A B

19 4 1 3 1

20 4 1 3 2

21 4 2 1 2

22 4 2 1 1

23 4 3 2 2

24 4 3 2 1




Discussion

Advantages

quite flexible

starts from the model and ends in a randomized design

R environment allows to go further

Limits

generates only orthogonal and regular designs

not computing-time efficient

Perspectives

increase flexibility

increase speed

finer selection criteria

Testers welcome !




Discussion

Advantages

quite flexible

starts from the model and ends in a randomized design

R environment allows to go further

Limits

generates only orthogonal and regular designs

not computing-time efficient

Perspectives

increase flexibility

increase speed

finer selection criteria

Testers welcome !Herve Monod INRA, MIA-Jouy Unit, France


construction of flexible fractional factorial designs and ... · introductionfractions with 2-level...

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