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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
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Foreword
Thanks to the Isaac Newton Institute and DAE organisers
Credit to Andre Kobilinsky
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Plan
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Plan
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Solution : (1) 23 complete factorial design on A, B, C
Unit A B C
1 + + +2 + + −3 + − +4 + − −5 − + +6 − + −7 − − +8 − − −
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Solution : (2) defining relation D = ABC
Unit A B C D = ABC
1 + + + +2 + + − −3 + − + −4 + − − +5 − + + −6 − + − +7 − − + +8 − − − −
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : aliasing I
> alias(ex1.key, model = ~(A + B + C + D)^2)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
A ; B ; C ; D
ALIASED TREATMENT EFFECTS
A B = C D
A C = B D
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
First visit to column 5
First visit to column 6
First visit to column 7
The search is closed: max.sol = 1 solution(s) found
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : design key matrix
> print(ex2.key)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : aliasing I
> alias(ex2.key, model = ~(A + B + C + D + E + F + G)^2)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
nil
ALIASED TREATMENT EFFECTS
A = B D = C E = F G
B = A D = C F = E G
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : aliasing II
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
TREATMENT EFFECTS CONFOUNDED WITH BLOCK EFFECTS
nil
UNALIASED BLOCK EFFECTS
nil
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : aliasing III
--- Synthesis on the aliased treatment effects for prime 2 ---
unaliased trt.aliased blc.aliased
[1,] 0 28 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Result : defining relationships I
> summary(ex2.key)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Plan
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixture of 2-level and 4-level factors
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 5 predefined columns
First visit to column 6
First visit to column 7
First visit to column 8
The search is closed: max.sol = 1 solution(s) found
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixture of 2-level and 4-level factors
Result : design key matrix
> print(mxd1.key)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixture of 2-level and 4-level factors
Result : aliasing I> alias(mxd1.key, model = ~(A + B + C + D + E + F)^2)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
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
ALIASED TREATMENT EFFECTS
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixture of 2-level and 4-level factors
Result : aliasing II
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
TREATMENT EFFECTS CONFOUNDED WITH BLOCK EFFECTS
nil
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixture of 2-level and 4-level factors
Result : aliasing III
UNALIASED BLOCK EFFECTS
nil
--- Synthesis on the aliased treatment effects for prime 2 ---
unaliased trt.aliased blc.aliased
[1,] 16 33 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Search process information
Determination of ineligible factorial terms
Determination of ineligible pseudofactorial terms
Independent searches for prime(s) : 2 3
Key-matrix search for prime p = 2
There are 2 predefined columns
First visit to column 3
The search is closed: max.sol = 1 solution(s) found
Key-matrix search for prime p = 3
There are 2 predefined columns
First visit to column 3
The search is closed: max.sol = 1 solution(s) found
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Result : design key matrix part 2
> print(ls.key[[2]])
Number of solutions: 1 for prime 3
ROW_2 COL_2 B
ROW_2 1 0 1
COL_2 0 1 1
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Result : aliasing I
> alias(ls.key, model = ~(ROW + COL + A + B)^2)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
nil
ALIASED TREATMENT EFFECTS
ROW_1 = COL_1 A
COL_1 = ROW_1 A
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Result : aliasing II
A = ROW_1 COL_1
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,] 0 6 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Result : aliasing III
********** Prime 3 design **********
--- Solution 1 for prime 3 ---
UNALIASED TREATMENT EFFECTS
nil
ALIASED TREATMENT EFFECTS
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Mixtures involving different primes
Result : aliasing IV
TREATMENT EFFECTS CONFOUNDED WITH BLOCK EFFECTS
nil
UNALIASED BLOCK EFFECTS
nil
--- Synthesis on the aliased treatment effects for prime 3 ---
unaliased trt.aliased blc.aliased
[1,] 0 9 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Plan
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Symmetric split-plot design
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 4 predefined columns
First visit to column 5
First visit to column 6
First visit to column 7
The search is closed: max.sol = 1 solution(s) found
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Symmetric split-plot design
Result : design key matrix
> print(split.key)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Symmetric split-plot design
Result : aliasing I
> alias(split.key, model = ~REP * SUB + A * B)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
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
ALIASED TREATMENT EFFECTS
SUB_1 = A_1
SUB_2 = A_2
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Symmetric split-plot design
Result : aliasing II
SUB_1 SUB_2 = A_1 A_2
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,] 16 6 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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)
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Search process information
Determination of ineligible factorial terms
Determination of ineligible pseudofactorial terms
Independent searches for prime(s) : 2 3
Key-matrix search for prime p = 2
There are 2 predefined columns
First visit to column 3
The search is closed: max.sol = 1 solution(s) found
Key-matrix search for prime p = 3
There is 1 predefined column
First visit to column 2
The search is closed: max.sol = 1 solution(s) found
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : design key matrix part 1
> print(split.key[[1]])
Number of solutions: 1 for prime 2
REP_1 REP_2 B
REP_1 1 0 0
REP_2 0 1 0
*U* 0 0 1
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : design key matrix part 2
> print(split.key[[2]])
Number of solutions: 1 for prime 3
SUB A
SUB 1 1
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : aliasing I
> alias(split.key, model = ~REP * SUB + A * B)
********** Prime 2 design **********
--- Solution 1 for prime 2 ---
UNALIASED TREATMENT EFFECTS
REP_1 ; REP_2 ; REP_1 REP_2 ; B
ALIASED TREATMENT EFFECTS
nil
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : aliasing II
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 0 0
********** Prime 3 design **********
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : aliasing III
--- Solution 1 for prime 3 ---
UNALIASED TREATMENT EFFECTS
nil
ALIASED TREATMENT EFFECTS
SUB = A
TREATMENT EFFECTS CONFOUNDED WITH BLOCK EFFECTS
nil
UNALIASED BLOCK EFFECTS
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Asymmetric split-plot design
Result : aliasing IV
nil
--- Synthesis on the aliased treatment effects for prime 3 ---
unaliased trt.aliased blc.aliased
[1,] 0 2 0
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
Plan
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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")
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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
Herve Monod INRA, MIA-Jouy Unit, France
Construction of flexible fractional factorial designs and its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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 its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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 its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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 its implementation in R
Introduction Fractions with 2-level factors Asymmetric fractions Multi-stratum designs From design key to design
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 its implementation in R
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