Exemples d'utilisation de la procédure FACTEX du logiciel SAS/QC (version 6.12)

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Exemples d’utilisation de la procédure FACTEX du

logiciel SAS/QC (version 6.12)

Jean-Pierre Gauchi

To cite this version:

Jean-Pierre Gauchi. Exemples d’utilisation de la procédure FACTEX du logiciel SAS/QC (version 6.12). [Rapport Technique] RT2000-1, auto-saisine. 2000. �hal-01604513�

Institut National de la Recherche Agronomique

Centre de Versailles

Unité de Biométrie

Exemples d’utilisation

de la procédure FACTEX

du logiciel SAS/QC (version 6.12)

Rapport technique n°2000-1

Jean-Pierre Gauchi

Février 2000

La procédure FACTEX

(version 6.12)

INTRODUCTION

La procédure FACTEX est une procédure interactive qui permet de construire :

„ des plans factoriels complets à nombre quelconque de niveaux, avec

partition en blocs ou non,

„ des plans factoriels fractionnaires avec partition en blocs ou non,

„ des plans factoriels complets ou fractionnaires asymétriques,

„ des carrés latins et dérivés,

„ des plans en blocs randomisés,

„ certains plans en blocs incomplets (avec l'étape DATA).

En plus de ces constructions, la procédure permet :

„ d'examiner la structure d'alias et les règles de concomitance,

„ de modifier la taille du plan,

„ de jouer sur la taille des blocs,

„ de choisir les effets à estimer,

„ de jouer sur la résolution du fractionnement,

„ de randomiser les expériences,

„ de répliquer les expériences,

„ de recoder les niveaux formels en niveaux appropriés,

„ d'exporter le plan vers une étape DATA.

STRUCTURE GENERALE D'INVOCATION DE LA PROCEDURE

PROC FACTEX

nocheck ;

FACTORS

names / nlev

=

q

;

SIZE

design

=

n | minimum

;

ou

fraction

=

d | maximum

;

ou

nrunfacs

=

m | minimum ;

MODEL

estimate

=

effets

nonnegligible

=

effetsnonneg

;

resolution

=

r ;

BLOCKS

nblkfacs

=

s | maximum

;

ou

nblocks

=

b | maximum

;

ou

size = k | minimum

;

EXAMINE

aliasing(u)

confounding

design

;

OUTPUT

OUT

=

name

effet nvals = , cvals =

blockname

=

name nvals = , cvals =

[name1 name2 ....]=name nvals = , cvals =

designrep

=

novalran =

pointrep

=

randomize

=

;

FOOTNOTE

n 'text' ;

RUN

;

QUIT

;

TITLE

n

'text'

Exemples d’utilisation

Programme FACTEX01.SAS

/***************************FACTEX01.SAS **************************/; /******** exemple elementaire d'utilisation ********/; /******** de la procedure FACTEX du logiciel SAS/QC ********/; ;

;

options ps=1000 ls=75 pageno=1 nodate nocenter; title ' ' ;

;

proc factex;

factors x1 x2 x3 x4;

→

par défaut chacun des 4 facteurs aura 2 niveaux

output out=resul; codés –1, +1

proc print data=resul; run;

Sortie de FACTEX01.SAS

→

le plan complet 24 OBS X1 X2 X3 X4 1 -1 -1 -1 -1 2 -1 -1 -1 1 3 -1 -1 1 -1 4 -1 -1 1 1 5 -1 1 -1 -1 6 -1 1 -1 1 7 -1 1 1 -1 8 -1 1 1 1 9 1 -1 -1 -1 10 1 -1 -1 1 11 1 -1 1 -1 12 1 -1 1 1 13 1 1 -1 -1 14 1 1 -1 1 15 1 1 1 -1 16 1 1 1 1

Programme FACTEX02.SAS

/***************************FACTEX02.SAS **************************/; /******** exemple elementaire d'utilisation ********/; /******** de la procedure FACTEX du logiciel SAS/QC ********/; ;

;

options ps=1000 ls=75 pageno=1 nodate nocenter; title ' ' ;

;

proc factex;

factors x1 x2 x3/nlev=3;

output out=resul x1 nvals=(-1 0 1)

→

pour 3 niveaux le codage par défaut

x2 nvals=(-1 0 1) est 0,1,2 , on le change ici en –1,0,+1 x3 nvals=(-1 0 1);

proc print data=resul; run;

Sortie de FACTEX02.SAS

→

le plan complet 33 OBS X1 X2 X3 1 -1 -1 -1 2 -1 -1 0 3 -1 -1 1 4 -1 0 -1 5 -1 0 0 6 -1 0 1 7 -1 1 -1 8 -1 1 0 9 -1 1 1 10 0 -1 -1 11 0 -1 0 12 0 -1 1 13 0 0 -1 14 0 0 0 15 0 0 1 16 0 1 -1 17 0 1 0 18 0 1 1 19 1 -1 -1 20 1 -1 0 21 1 -1 1 22 1 0 -1 23 1 0 0 24 1 0 1 25 1 1 -1 26 1 1 0 27 1 1 1

Programme FACTEX03.SAS

/***************************FACTEX03.SAS **************************/; /******** exemple elementaire d'utilisation ********/; /******** de la procedure FACTEX du logiciel SAS/QC ********/; ;

;

proc factex;

factors x1 x2 x3;

→

le plan factoriel complet 23 _{comporte 8 expériences,}

size design=16; ici chaque expérience sera doublée output out=resul;

proc print data=resul; run;

Sortie de FACTEX03.SAS

→

le plan complet 23 en deux répétitions

OBS X1 X2 X3 1 -1 -1 -1 2 -1 -1 1 3 -1 1 -1 4 -1 1 1 5 1 -1 -1 6 1 -1 1 7 1 1 -1 8 1 1 1 --- 9 -1 -1 -1 10 -1 -1 1 11 -1 1 -1 12 -1 1 1 13 1 -1 -1 14 1 -1 1 15 1 1 -1 16 1 1 1

Programme FACTEX04.SAS

/***************************FACTEX04.SAS **************************/; /******** exemple elementaire d'utilisation ********/; /******** de la procedure FACTEX du logiciel SAS/QC ********/; ;

proc factex;

factors x1 x2 x3 x4 x5;

blocks size=16;

→

le plan complet 25 _{sera réparti sur 2 blocs de 16}

expériences chacun

model est=(x1|x2|x3|x4|x5@2);

→

le modèle que l’on souhaite estimer est :

examine aliasing(2) confounding design;

→

on demande la structure

output out=resul; d’alias jusqu’à l’ordre 2 proc print data=resul;

run;

Sortie de FACTEX04.SAS

→

chaque expérience du plan complet 25 est attribuée à l’un des deux blocs Design Points Experiment Number X1 X2 X3 X4 X5 Block --- 1 -1 -1 -1 -1 -1 1 2 -1 -1 -1 -1 1 2 3 -1 -1 -1 1 -1 2 4 -1 -1 -1 1 1 1 5 -1 -1 1 -1 -1 2 6 -1 -1 1 -1 1 1 7 -1 -1 1 1 -1 1 8 -1 -1 1 1 1 2 9 -1 1 -1 -1 -1 2 10 -1 1 -1 -1 1 1 11 -1 1 -1 1 -1 1 12 -1 1 -1 1 1 2 13 -1 1 1 -1 -1 1 14 -1 1 1 -1 1 2 15 -1 1 1 1 -1 2 16 -1 1 1 1 1 1 17 1 -1 -1 -1 -1 2 18 1 -1 -1 -1 1 1 19 1 -1 -1 1 -1 1 20 1 -1 -1 1 1 2 21 1 -1 1 -1 -1 1 22 1 -1 1 -1 1 2 23 1 -1 1 1 -1 2 24 1 -1 1 1 1 1 25 1 1 -1 -1 -1 1 26 1 1 -1 -1 1 2 27 1 1 -1 1 -1 2

Block Pseudo-factor Confounding Rules

[B1] = X1*X2*X3*X4*X5

→

le facteur bloc est confondu avec l’interaction

quintuple Aliasing Structure X1 X2 X3 X4 X5 X1*X2 X1*X3 X1*X4 X1*X5 X2*X3 X2*X4 X2*X5 X3*X4 X3*X5 X4*X5

→

dans chaque bloc la résolution est de V : les effets principaux sont

estimables indépendamment entre eux et indépendamment des interactions doubles qui, elles-mêmes, seront estimables indépendamment entre elles. OBS BLOCK X1 X2 X3 X4 X5 1 1 -1 -1 -1 -1 -1 2 1 -1 -1 -1 1 1 3 1 -1 -1 1 -1 1 4 1 -1 -1 1 1 -1 5 1 -1 1 -1 -1 1 6 1 -1 1 -1 1 -1 7 1 -1 1 1 -1 -1 8 1 -1 1 1 1 1 9 1 1 -1 -1 -1 1 10 1 1 -1 -1 1 -1 11 1 1 -1 1 -1 -1 12 1 1 -1 1 1 1 13 1 1 1 -1 -1 -1 14 1 1 1 -1 1 1 15 1 1 1 1 -1 1 16 1 1 1 1 1 -1 17 2 -1 -1 -1 -1 1 18 2 -1 -1 -1 1 -1 19 2 -1 -1 1 -1 -1 20 2 -1 -1 1 1 1 21 2 -1 1 -1 -1 -1 22 2 -1 1 -1 1 1 23 2 -1 1 1 -1 1 24 2 -1 1 1 1 -1 25 2 1 -1 -1 -1 -1 26 2 1 -1 -1 1 1 27 2 1 -1 1 -1 1 28 2 1 -1 1 1 -1 29 2 1 1 -1 -1 1 30 2 1 1 -1 1 -1 31 2 1 1 1 -1 -1 32 2 1 1 1 1 1

Programme FACTEX05.SAS

/***************************FACTEX05.SAS **************************/; /* exemples de plans factoriels fractionnaires avec ou sans blocs */; ;

;

options ps=65 ls=75 nodate nocenter; title ' ' ;

;

/********************************/; /** facteur bloc à 4 modalites **/; proc factex;

factors x1 x2 x3 x4 x5;

blocks size=8 ;

→

comme les blocs sont de taille 8, le facteur bloc

aura 4 modalités dont les noms figurent ci-dessous model est =(x1|x2|x3|x4|x5 @2) ; ↓

examine aliasing(2) confounding;

output out=resul1 blockname=reacteur cvals=('r1' 'r2' 'r3' 'r4');

proc print data=resul1; run;

/********************************/;

/*******************************************/; /** 2 facteurs blocs à 2 modalites chacun **/; proc factex;

factors operat lot;

output out=resul2 operat cvals=('jean' 'paul') lot cvals=('bayer' 'rp')

pointrep=8;

→

chacune des 4 expériences sera répétée

8 fois data resul3; merge resul1(drop=reacteur) resul2; proc print data=resul3;

run;

/*******************************************/;

→

le résultat sera un plan factoriel complet en 4 blocs de 8 expériences,

chaque bloc étant caractérisé par un pattern donné des facteurs blocs ; les 4 patterns possibles sont (--,-+,+-,++).

SORTIE de FACTEX05.SAS

Block Pseudo-factor Confounding Rules

[B1] = X2*X3*X4*X5 [B2] = X1*X4*X5

→

le facteur bloc étant construit à partir de 2 facteurs blocs à 2 niveaux

chacun, la structure d’alias est facile à établir et à comprendre. Aliasing Structure X1 X2 X3 X4 X5 X1*X2 X1*X3 X1*X4 X1*X5 X2*X3 X2*X4 X2*X5 X3*X4 X3*X5 X4*X5 OBS REACTEUR X1 X2 X3 X4 X5 1 r1 -1 -1 1 -1 -1 2 r1 -1 -1 1 1 1 3 r1 -1 1 -1 -1 -1 4 r1 -1 1 -1 1 1 5 r1 1 -1 -1 -1 1 6 r1 1 -1 -1 1 -1 7 r1 1 1 1 -1 1 8 r1 1 1 1 1 -1 9 r2 -1 -1 -1 -1 -1 10 r2 -1 -1 -1 1 1 11 r2 -1 1 1 -1 -1 12 r2 -1 1 1 1 1 13 r2 1 -1 1 -1 1 14 r2 1 -1 1 1 -1 15 r2 1 1 -1 -1 1 16 r2 1 1 -1 1 -1 17 r3 -1 -1 -1 -1 1 18 r3 -1 -1 -1 1 -1 19 r3 -1 1 1 -1 1 20 r3 -1 1 1 1 -1 21 r3 1 -1 1 -1 -1 22 r3 1 -1 1 1 1 23 r3 1 1 -1 -1 -1 24 r3 1 1 -1 1 1 25 r4 -1 -1 1 -1 1 26 r4 -1 -1 1 1 -1 27 r4 -1 1 -1 -1 1 28 r4 -1 1 -1 1 -1 29 r4 1 -1 -1 -1 -1 30 r4 1 -1 -1 1 1 31 r4 1 1 1 -1 -1 32 r4 1 1 1 1 1

OBS X1 X2 X3 X4 X5 OPERAT LOT 1 -1 -1 1 -1 -1 jean bayer 2 -1 -1 1 1 1 jean bayer 3 -1 1 -1 -1 -1 jean bayer 4 -1 1 -1 1 1 jean bayer 5 1 -1 -1 -1 1 jean bayer 6 1 -1 -1 1 -1 jean bayer 7 1 1 1 -1 1 jean bayer 8 1 1 1 1 -1 jean bayer 9 -1 -1 -1 -1 -1 jean rp 10 -1 -1 -1 1 1 jean rp 11 -1 1 1 -1 -1 jean rp 12 -1 1 1 1 1 jean rp 13 1 -1 1 -1 1 jean rp 14 1 -1 1 1 -1 jean rp 15 1 1 -1 -1 1 jean rp 16 1 1 -1 1 -1 jean rp 17 -1 -1 -1 -1 1 paul bayer 18 -1 -1 -1 1 -1 paul bayer 19 -1 1 1 -1 1 paul bayer 20 -1 1 1 1 -1 paul bayer 21 1 -1 1 -1 -1 paul bayer 22 1 -1 1 1 1 paul bayer 23 1 1 -1 -1 -1 paul bayer 24 1 1 -1 1 1 paul bayer 25 -1 -1 1 -1 1 paul rp 26 -1 -1 1 1 -1 paul rp 27 -1 1 -1 -1 1 paul rp 28 -1 1 -1 1 -1 paul rp 29 1 -1 -1 -1 -1 paul rp 30 1 -1 -1 1 1 paul rp 31 1 1 1 -1 -1 paul rp 32 1 1 1 1 1 paul rp

Programme FACTEX06.SAS

/***************************FACTEX06.SAS **************************/; /* exemples de plans factoriels fractionnaires avec ou sans blocs */; ;

;

options ps=65 ls=75 nodate nocenter; title ' ' ; ; /********************************/; /** fraction de resolution V **/; proc factex; factors x1 x2 x3 x4 x5; model resolution=5; size design=16 ;

examine aliasing(3) confounding; output out=resul1;

proc print data=resul1; run;

/********************************/;

SORTIE de FACTEX06.SAS

Factor Confounding Rules

X5 = X1*X2*X3*X4 Aliasing Structure X1 X2 X3 X4 X5 X1*X2 = X3*X4*X5 X1*X3 = X2*X4*X5 X1*X4 = X2*X3*X5 X1*X5 = X2*X3*X4 X2*X3 = X1*X4*X5 X2*X4 = X1*X3*X5 X2*X5 = X1*X3*X4 X3*X4 = X1*X2*X5 X3*X5 = X1*X2*X4 X4*X5 = X1*X2*X3

→

La fraction de résolution V OBS X1 X2 X3 X4 X5 1 -1 -1 -1 -1 1 2 -1 -1 -1 1 -1 3 -1 -1 1 -1 -1 4 -1 -1 1 1 1 5 -1 1 -1 -1 -1 6 -1 1 -1 1 1 7 -1 1 1 -1 1 8 -1 1 1 1 -1 9 1 -1 -1 -1 -1 10 1 -1 -1 1 1 11 1 -1 1 -1 1 12 1 -1 1 1 -1 13 1 1 -1 -1 1 14 1 1 -1 1 -1 15 1 1 1 -1 -1 16 1 1 1 1 1

Programme FACTEX07.SAS

/***************************FACTEX07.SAS **************************/; /* exemples de plans factoriels fractionnaires avec ou sans blocs */; ; /**********************/; /** fraction et bloc **/; proc factex; factors x1 x2 x3 x4 x5; size fraction=2;

blocks size=minimum;

→

on demande les blocs les plus petits possibles.

model est=(x1 x2 x3 x4 x5)

nonneg=(x1|x2|x3|x4|x5@2);

→

on ne souhaite pas estimer les

interactions doubles indépendamment mais on souhaite qu’elles soient présentes dans le modèle.

examine aliasing(3) confounding design; output out=resul2;

proc print data=resul2; run; /**********************/;

SORTIE de FACTEX07.SAS

Design Points Experiment Number X1 X2 X3 X4 X5 Block --- 1 -1 -1 -1 -1 1 1 2 -1 -1 -1 1 -1 4 3 -1 -1 1 -1 -1 4 4 -1 -1 1 1 1 1 5 -1 1 -1 -1 -1 2 6 -1 1 -1 1 1 3 7 -1 1 1 -1 1 3 8 -1 1 1 1 -1 2 9 1 -1 -1 -1 -1 3 10 1 -1 -1 1 1 2 11 1 -1 1 -1 1 2 12 1 -1 1 1 -1 3 13 1 1 -1 -1 1 4 14 1 1 -1 1 -1 1 15 1 1 1 -1 -1 1 16 1 1 1 1 1 4

Factor Confounding Rules

X5 = X1*X2*X3*X4

Block Pseudo-factor Confounding Rules

[B1] = X2*X3*X4

[B2] = X1*X3*X4

→

deux facteurs blocs donc 4 blocs : un plan à facteurs à

q niveaux (ici 2) partitionné en qs _{blocs (ici q}s ₌₄₎

contient s facteurs blocs (ici s=2) notés ici [B1] et [B2]. Aliasing Structure X1 X2 X3 X4 X5 [B] = X1*X2 = X3*X4*X5 X1*X3 = X2*X4*X5 X1*X4 = X2*X3*X5 [B] = X1*X5 = X2*X3*X4 X2*X3 = X1*X4*X5 X2*X4 = X1*X3*X5 [B] = X2*X5 = X1*X3*X4 X3*X4 = X1*X2*X5 X3*X5 = X1*X2*X4

X4*X5 = X1*X2*X3

→

les facteurs blocs sont confondus avec les

interactions doubles et triples

OBS BLOCK X1 X2 X3 X4 X5 1 1 -1 -1 -1 -1 1 2 1 -1 -1 1 1 1 3 1 1 1 -1 1 -1 4 1 1 1 1 -1 -1 5 2 -1 1 -1 -1 -1 6 2 -1 1 1 1 -1 7 2 1 -1 -1 1 1 8 2 1 -1 1 -1 1 9 3 -1 1 -1 1 1 10 3 -1 1 1 -1 1 11 3 1 -1 -1 -1 -1 12 3 1 -1 1 1 -1 13 4 -1 -1 -1 1 -1 14 4 -1 -1 1 -1 -1 15 4 1 1 -1 -1 1 16 4 1 1 1 1 1

Réponse :

n°bloc(i) = B1 + qB2 + q²B3 + … +qs-1Bs ↑

codé de 0 à qs-1 dans FACTEX

B1, …, Bs prennent les valeurs prises par les facteurs blocs (modulo q), obtenues par les règles de concomitance (page précédente), les niveaux des facteurs eux-mêmes étant codés de 0 à q-1 dans FACTEX : -1 Æ 0 et +1 Æ 1.

Applications au cas présent :

X2 X3 X4 X1 X3 X4 ↓ ↓

N°bloc(1)=(0 + 0 + 0)*1 + (0 + 0 + 0)*2 = 0 Æ bloc noté 1. N°bloc(5)=(1 + 0 + 0)*1 + (0 + 0 + 0)*2 = 1 Æ bloc noté 2. N°bloc(16)=(1 + 1 + 1)*1 + (1 + 1 + 1)*2 = 3 Æ bloc noté 4.

↑ ↑

Programme FACTEX08.SAS

/***************************FACTEX08.SAS **************************/; /********** exemples de plans factoriels fractionnaires ***********/; ;

;

options ps=65 ls=75 nodate nocenter; title ' ' ;

;

/**********************************/; /** fraction de resolution III **/; proc factex;

factors x1-x20;

model est=(x1-x20);

→

aucune interaction n’est demandée, implicitement plan

de résolution III cherché size design=minimum ;

examine aliasing(2) confounding; output out=resul1;

proc print data=resul1; run;

/********************************/;

SORTIE de FACTEX08.SAS

Factor Confounding Rules

X6 = X1*X2*X3*X4*X5 X7 = X2*X3*X4*X5 X8 = X1*X3*X4*X5 X9 = X3*X4*X5 X10 = X1*X2*X4*X5 X11 = X2*X4*X5 X12 = X1*X4*X5 X13 = X4*X5 X14 = X1*X2*X3*X5 X15 = X2*X3*X5 X16 = X1*X3*X5 X17 = X3*X5 X18 = X1*X2*X5 X19 = X2*X5 X20 = X1*X5

→

Résolution III : les effets principaux sont estimables indépendamment

Aliasing Structure X1 = X5*X20 = X6*X7 = X8*X9 = X10*X11 = X12*X13 = X14*X15 = X16*X17 = X18*X19 X2 = X5*X19 = X6*X8 = X7*X9 = X10*X12 = X11*X13 = X14*X16 = X15*X17 = X18*X20 X3 = X5*X17 = X6*X10 = X7*X11 = X8*X12 = X9*X13 = X14*X18 = X15*X19 = X16*X20 X4 = X5*X13 = X6*X14 = X7*X15 = X8*X16 = X9*X17 = X10*X18 = X11*X19 = X12*X20 X5 = X1*X20 = X2*X19 = X3*X17 = X4*X13 X6 = X1*X7 = X2*X8 = X3*X10 = X4*X14 X7 = X1*X6 = X2*X9 = X3*X11 = X4*X15 X8 = X1*X9 = X2*X6 = X3*X12 = X4*X16 X9 = X1*X8 = X2*X7 = X3*X13 = X4*X17 X10 = X1*X11 = X2*X12 = X3*X6 = X4*X18 X11 = X1*X10 = X2*X13 = X3*X7 = X4*X19 X12 = X1*X13 = X2*X10 = X3*X8 = X4*X20 X13 = X1*X12 = X2*X11 = X3*X9 = X4*X5 X14 = X1*X15 = X2*X16 = X3*X18 = X4*X6 X15 = X1*X14 = X2*X17 = X3*X19 = X4*X7 X16 = X1*X17 = X2*X14 = X3*X20 = X4*X8 X17 = X1*X16 = X2*X15 = X3*X5 = X4*X9 X18 = X1*X19 = X2*X20 = X3*X14 = X4*X10 X19 = X1*X18 = X2*X5 = X3*X15 = X4*X11 X20 = X1*X5 = X2*X18 = X3*X16 = X4*X12 X1*X2 = X5*X18 = X6*X9 = X7*X8 = X10*X13 = X11*X12 = X14*X17 = X15*X16 = X19*X20 X1*X3 = X5*X16 = X6*X11 = X7*X10 = X8*X13 = X9*X12 = X14*X19 = X15*X18 = X17*X20 X1*X4 = X5*X12 = X6*X15 = X7*X14 = X8*X17 = X9*X16 = X10*X19 = X11*X18 = X13*X20 X2*X3 = X5*X15 = X6*X12 = X7*X13 = X8*X10 = X9*X11 = X14*X20 = X16*X18 = X17*X19 X2*X4 = X5*X11 = X6*X16 = X7*X17 = X8*X14 = X9*X15 = X10*X20 = X12*X18 = X13*X19 X3*X4 = X5*X9 = X6*X18 = X7*X19 = X8*X20 = X10*X14 = X11*X15 = X12*X16 = X13*X17 X5*X6 = X7*X20 = X8*X19 = X9*X18 = X10*X17 = X11*X16 = X12*X15 = X13*X14 X5*X7 = X6*X20 = X8*X18 = X9*X19 = X10*X16 = X11*X17 = X12*X14 = X13*X15 X5*X8 = X6*X19 = X7*X18 = X9*X20 = X10*X15 = X11*X14 = X12*X17 = X13*X16 X5*X10 = X6*X17 = X7*X16 = X8*X15 = X9*X14 = X11*X20 = X12*X19 = X13*X18 X5*X14 = X6*X13 = X7*X12 = X8*X11 = X9*X10 = X15*X20 = X16*X19 = X17*X18

→ La structure d’alias n’est pas symétrique : par exemple les 4 premiers effets principaux sont aliasés avec 8 interactions doubles tandis que les 16 autres le sont avec 4 interactions doubles seulement.

OBS X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 2 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 3 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 4 -1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 5 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 6 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 7 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 8 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 9 -1 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 10 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 11 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 12 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 13 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 14 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 15 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 16 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 17 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 18 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 19 1 -1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 20 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 21 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 22 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 23 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 24 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 25 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 26 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 27 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 28 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 29 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 30 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 31 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

→

Une solution à 32 expériences est trouvée ; l’optimum serait une matrice

Programme FACTEX09.SAS

/***************************FACTEX09.SAS **************************/; /********** exemples de plans factoriels fractionnaires ***********/; ;

/**********************************/; /** fraction de resolution III **/; proc factex;

factors x1-x40; model est=(x1-x40); size design=minimum ;

examine aliasing(2) confounding; output out=resul2;

proc print data=resul2; run;

/********************************/;

SORTIE de FACTEX09.SAS

Factor Confounding Rules

X7 = X1*X2*X3*X4*X5*X6 X8 = X2*X3*X4*X5*X6 X9 = X1*X3*X4*X5*X6 X10 = X3*X4*X5*X6 X11 = X1*X2*X4*X5*X6 X12 = X2*X4*X5*X6 X13 = X1*X4*X5*X6 X14 = X4*X5*X6 X15 = X1*X2*X3*X5*X6 X16 = X2*X3*X5*X6 X17 = X1*X3*X5*X6 X18 = X3*X5*X6 X19 = X1*X2*X5*X6 X20 = X2*X5*X6 X21 = X1*X5*X6 X22 = X5*X6 X23 = X1*X2*X3*X4*X6 X24 = X2*X3*X4*X6 X25 = X1*X3*X4*X6 X26 = X3*X4*X6 X27 = X1*X2*X4*X6 X28 = X2*X4*X6 X29 = X1*X4*X6 X30 = X4*X6 X31 = X1*X2*X3*X6 X32 = X2*X3*X6 X33 = X1*X3*X6 X34 = X3*X6 X35 = X1*X2*X6 X36 = X2*X6 X37 = X1*X6 X38 = X1*X2*X3*X4*X5 X39 = X2*X3*X4*X5 X40 = X1*X3*X4*X5

Aliasing Structure X1 = X6*X37 = X7*X8 = X9*X10 = X11*X12 = X13*X14 = X15*X16 = X17*X18 = X19*X20 = X21*X22 = X23*X24 = X25*X26 = X27*X28 = X29*X30 = X31*X32 = X33*X34 = X35*X36 = X38*X39 X2 = X6*X36 = X7*X9 = X8*X10 = X11*X13 = X12*X14 = X15*X17 = X16*X18 = X19*X21 = X20*X22 = X23*X25 = X24*X26 = X27*X29 = X28*X30 = X31*X33 = X32*X34 = X35*X37 = X38*X40 X3 = X6*X34 = X7*X11 = X8*X12 = X9*X13 = X10*X14 = X15*X19 = X16*X20 = X17*X21 = X18*X22 = X23*X27 = X24*X28 = X25*X29 = X26*X30 = X31*X35 = X32*X36 = X33*X37 X4 = X6*X30 = X7*X15 = X8*X16 = X9*X17 = X10*X18 = X11*X19 = X12*X20 = X13*X21 = X14*X22 = X23*X31 = X24*X32 = X25*X33 = X26*X34 = X27*X35 = X28*X36 = X29*X37 X5 = X6*X22 = X7*X23 = X8*X24 = X9*X25 = X10*X26 = X11*X27 = X12*X28 = X13*X29 = X14*X30 = X15*X31 = X16*X32 = X17*X33 = X18*X34 = X19*X35 = X20*X36 = X21*X37 X6 = X1*X37 = X2*X36 = X3*X34 = X4*X30 = X5*X22 = X7*X38 = X8*X39 = X9*X40 X7 = X1*X8 = X2*X9 = X3*X11 = X4*X15 = X5*X23 = X6*X38 = X36*X40 = X37*X39 X8 = X1*X7 = X2*X10 = X3*X12 = X4*X16 = X5*X24 = X6*X39 = X35*X40 = X37*X38 X9 = X1*X10 = X2*X7 = X3*X13 = X4*X17 = X5*X25 = X6*X40 = X35*X39 = X36*X38 X10 = X1*X9 = X2*X8 = X3*X14 = X4*X18 = X5*X26 = X35*X38 = X36*X39 = X37*X40 X11 = X1*X12 = X2*X13 = X3*X7 = X4*X19 = X5*X27 = X32*X40 = X33*X39 = X34*X38 X12 = X1*X11 = X2*X14 = X3*X8 = X4*X20 = X5*X28 = X31*X40 = X33*X38 = X34*X39 X13 = X1*X14 = X2*X11 = X3*X9 = X4*X21 = X5*X29 = X31*X39 = X32*X38 = X34*X40 X14 = X1*X13 = X2*X12 = X3*X10 = X4*X22 = X5*X30 = X31*X38 = X32*X39 = X33*X40 X15 = X1*X16 = X2*X17 = X3*X19 = X4*X7 = X5*X31 = X28*X40 = X29*X39 = X30*X38 X16 = X1*X15 = X2*X18 = X3*X20 = X4*X8 = X5*X32 = X27*X40 = X29*X38 = X30*X39 X17 = X1*X18 = X2*X15 = X3*X21 = X4*X9 = X5*X33 = X27*X39 = X28*X38 = X30*X40 X18 = X1*X17 = X2*X16 = X3*X22 = X4*X10 = X5*X34 = X27*X38 = X28*X39 = X29*X40 X19 = X1*X20 = X2*X21 = X3*X15 = X4*X11 = X5*X35 = X24*X40 = X25*X39 = X26*X38 X20 = X1*X19 = X2*X22 = X3*X16 = X4*X12 = X5*X36 = X23*X40 = X25*X38 = X26*X39 X21 = X1*X22 = X2*X19 = X3*X17 = X4*X13 = X5*X37 = X23*X39 = X24*X38 = X26*X40 X22 = X1*X21 = X2*X20 = X3*X18 = X4*X14 = X5*X6 = X23*X38 = X24*X39 = X25*X40 X23 = X1*X24 = X2*X25 = X3*X27 = X4*X31 = X5*X7 = X20*X40 = X21*X39 = X22*X38 X24 = X1*X23 = X2*X26 = X3*X28 = X4*X32 = X5*X8 = X19*X40 = X21*X38 = X22*X39 X25 = X1*X26 = X2*X23 = X3*X29 = X4*X33 = X5*X9 = X19*X39 = X20*X38 = X22*X40 X26 = X1*X25 = X2*X24 = X3*X30 = X4*X34 = X5*X10 = X19*X38 = X20*X39 = X21*X40

X27 = X1*X28 = X2*X29 = X3*X23 = X4*X35 = X5*X11 = X16*X40 = X17*X39 = X18*X38 X28 = X1*X27 = X2*X30 = X3*X24 = X4*X36 = X5*X12 = X15*X40 = X17*X38 = X18*X39 X29 = X1*X30 = X2*X27 = X3*X25 = X4*X37 = X5*X13 = X15*X39 = X16*X38 = X18*X40 X30 = X1*X29 = X2*X28 = X3*X26 = X4*X6 = X5*X14 = X15*X38 = X16*X39 = X17*X40 X31 = X1*X32 = X2*X33 = X3*X35 = X4*X23 = X5*X15 = X12*X40 = X13*X39 = X14*X38 X32 = X1*X31 = X2*X34 = X3*X36 = X4*X24 = X5*X16 = X11*X40 = X13*X38 = X14*X39 X33 = X1*X34 = X2*X31 = X3*X37 = X4*X25 = X5*X17 = X11*X39 = X12*X38 = X14*X40 X34 = X1*X33 = X2*X32 = X3*X6 = X4*X26 = X5*X18 = X11*X38 = X12*X39 = X13*X40 X35 = X1*X36 = X2*X37 = X3*X31 = X4*X27 = X5*X19 = X8*X40 = X9*X39 = X10*X38 X36 = X1*X35 = X2*X6 = X3*X32 = X4*X28 = X5*X20 = X7*X40 = X9*X38 = X10*X39 X37 = X1*X6 = X2*X35 = X3*X33 = X4*X29 = X5*X21 = X7*X39 = X8*X38 = X10*X40 X38 = X1*X39 = X2*X40 = X6*X7 = X8*X37 = X9*X36 = X10*X35 = X11*X34 = X12*X33 = X13*X32 = X14*X31 = X15*X30 = X16*X29 = X17*X28 = X18*X27 = X19*X26 = X20*X25 = X21*X24 = X22*X23 X39 = X1*X38 = X6*X8 = X7*X37 = X9*X35 = X10*X36 = X11*X33 = X12*X34 = X13*X31 = X14*X32 = X15*X29 = X16*X30 = X17*X27 = X18*X28 = X19*X25 = X20*X26 = X21*X23 = X22*X24 X40 = X2*X38 = X6*X9 = X7*X36 = X8*X35 = X10*X37 = X11*X32 = X12*X31 = X13*X34 = X14*X33 = X15*X28 = X16*X27 = X17*X30 = X18*X29 = X19*X24 = X20*X23 = X21*X26 = X22*X25 X1*X2 = X6*X35 = X7*X10 = X8*X9 = X11*X14 = X12*X13 = X15*X18 = X16*X17 = X19*X22 = X20*X21 = X23*X26 = X24*X25 = X27*X30 = X28*X29 = X31*X34 = X32*X33 = X36*X37 = X39*X40 X1*X3 = X6*X33 = X7*X12 = X8*X11 = X9*X14 = X10*X13 = X15*X20 = X16*X19 = X17*X22 = X18*X21 = X23*X28 = X24*X27 = X25*X30 = X26*X29 = X31*X36 = X32*X35 = X34*X37 X1*X4 = X6*X29 = X7*X16 = X8*X15 = X9*X18 = X10*X17 = X11*X20 = X12*X19 = X13*X22 = X14*X21 = X23*X32 = X24*X31 = X25*X34 = X26*X33 = X27*X36 = X28*X35 = X30*X37 X1*X5 = X6*X21 = X7*X24 = X8*X23 = X9*X26 = X10*X25 = X11*X28 = X12*X27 = X13*X30 = X14*X29 = X15*X32 = X16*X31 = X17*X34 = X18*X33 = X19*X36 = X20*X35 = X22*X37 X1*X40 = X2*X39 = X6*X10 = X7*X35 = X8*X36 = X9*X37 = X11*X31 = X12*X32 = X13*X33 = X14*X34 = X15*X27 = X16*X28 = X17*X29 = X18*X30 = X19*X23 = X20*X24 = X21*X25 = X22*X26 X2*X3 = X6*X32 = X7*X13 = X8*X14 = X9*X11 = X10*X12 = X15*X21 = X16*X22 = X17*X19 = X18*X20 = X23*X29 = X24*X30 = X25*X27 = X26*X28 = X31*X37 = X33*X35 = X34*X36 X2*X4 = X6*X28 = X7*X17 = X8*X18 = X9*X15 = X10*X16 = X11*X21 = X12*X22 = X13*X19 = X14*X20 = X23*X33 = X24*X34 = X25*X31 = X26*X32 = X27*X37 = X29*X35 = X30*X36 X2*X5 = X6*X20 = X7*X25 = X8*X26 = X9*X23 = X10*X24 = X11*X29 = X12*X30 = X13*X27 = X14*X28 = X15*X33 = X16*X34 = X17*X31 = X18*X32 = X19*X37 = X21*X35 = X22*X36 X3*X4 = X6*X26 = X7*X19 = X8*X20 = X9*X21 = X10*X22 = X11*X15 = X12*X16 = X13*X17 = X14*X18 = X23*X35 = X24*X36 = X25*X37 = X27*X31 = X28*X32 = X29*X33 = X30*X34 X3*X5 = X6*X18 = X7*X27 = X8*X28 = X9*X29 = X10*X30 = X11*X23 = X12*X24 = X13*X25 = X14*X26 = X15*X35 = X16*X36 = X17*X37 = X19*X31 = X20*X32 = X21*X33 = X22*X34 X3*X38 = X6*X11 = X7*X34 = X8*X33 = X9*X32 = X10*X31 = X12*X37 = X13*X36 = X14*X35 = X15*X26 = X16*X25 = X17*X24 = X18*X23 = X19*X30 = X20*X29 = X21*X28 = X22*X27

X3*X39 = X6*X12 = X7*X33 = X8*X34 = X9*X31 = X10*X32 = X11*X37 = X13*X35 = X14*X36 = X15*X25 = X16*X26 = X17*X23 = X18*X24 = X19*X29 = X20*X30 = X21*X27 = X22*X28 X3*X40 = X6*X13 = X7*X32 = X8*X31 = X9*X34 = X10*X33 = X11*X36 = X12*X35 = X14*X37 = X15*X24 = X16*X23 = X17*X26 = X18*X25 = X19*X28 = X20*X27 = X21*X30 = X22*X29 X4*X5 = X6*X14 = X7*X31 = X8*X32 = X9*X33 = X10*X34 = X11*X35 = X12*X36 = X13*X37 = X15*X23 = X16*X24 = X17*X25 = X18*X26 = X19*X27 = X20*X28 = X21*X29 = X22*X30 X4*X38 = X6*X15 = X7*X30 = X8*X29 = X9*X28 = X10*X27 = X11*X26 = X12*X25 = X13*X24 = X14*X23 = X16*X37 = X17*X36 = X18*X35 = X19*X34 = X20*X33 = X21*X32 = X22*X31 X4*X39 = X6*X16 = X7*X29 = X8*X30 = X9*X27 = X10*X28 = X11*X25 = X12*X26 = X13*X23 = X14*X24 = X15*X37 = X17*X35 = X18*X36 = X19*X33 = X20*X34 = X21*X31 = X22*X32 X4*X40 = X6*X17 = X7*X28 = X8*X27 = X9*X30 = X10*X29 = X11*X24 = X12*X23 = X13*X26 = X14*X25 = X15*X36 = X16*X35 = X18*X37 = X19*X32 = X20*X31 = X21*X34 = X22*X33 X5*X38 = X6*X23 = X7*X22 = X8*X21 = X9*X20 = X10*X19 = X11*X18 = X12*X17 = X13*X16 = X14*X15 = X24*X37 = X25*X36 = X26*X35 = X27*X34 = X28*X33 = X29*X32 = X30*X31 X5*X39 = X6*X24 = X7*X21 = X8*X22 = X9*X19 = X10*X20 = X11*X17 = X12*X18 = X13*X15 = X14*X16 = X23*X37 = X25*X35 = X26*X36 = X27*X33 = X28*X34 = X29*X31 = X30*X32 X5*X40 = X6*X25 = X7*X20 = X8*X19 = X9*X22 = X10*X21 = X11*X16 = X12*X15 = X13*X18 = X14*X17 = X23*X36 = X24*X35 = X26*X37 = X27*X32 = X28*X31 = X29*X34 = X30*X33 X6*X19 = X7*X26 = X8*X25 = X9*X24 = X10*X23 = X11*X30 = X12*X29 = X13*X28 = X14*X27 = X15*X34 = X16*X33 = X17*X32 = X18*X31 = X20*X37 = X21*X36 = X22*X35 X6*X27 = X7*X18 = X8*X17 = X9*X16 = X10*X15 = X11*X22 = X12*X21 = X13*X20 = X14*X19 = X23*X34 = X24*X33 = X25*X32 = X26*X31 = X28*X37 = X29*X36 = X30*X35 X6*X31 = X7*X14 = X8*X13 = X9*X12 = X10*X11 = X15*X22 = X16*X21 = X17*X20 = X18*X19 = X23*X30 = X24*X29 = X25*X28 = X26*X27 = X32*X37 = X33*X36 = X34*X35

O X X X X X X X X X X X B X X X X X X X X X 1 1 1 1 1 1 1 1 1 1 2 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 2 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 3 -1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 4 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 5 -1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 6 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 7 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 8 -1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 9 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 10 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 11 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 12 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 13 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 14 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 15 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 16 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 17 -1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 18 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 19 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 20 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 21 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 22 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 23 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 24 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 25 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 26 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 27 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 O X X X X X X X X X X X X X X X X X X X X B 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 2 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 3 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 4 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 5 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 6 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 7 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 8 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 9 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 10 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 11 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 12 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 13 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 14 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 15 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 16 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 17 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 18 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 19 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 20 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 21 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 22 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 23 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 24 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 25 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 26 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1

27 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 O X X X X X X X X X X X B X X X X X X X X X 1 1 1 1 1 1 1 1 1 1 2 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 28 -1 1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 29 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 30 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 31 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 32 -1 1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 33 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 34 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 35 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 36 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 37 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 38 1 -1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 39 1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 40 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 41 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 42 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 43 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 44 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 45 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 46 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 47 1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 48 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 49 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 50 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 51 1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 52 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 53 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 54 1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 O X X X X X X X X X X X X X X X X X X X X B 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 28 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 29 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 30 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 31 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 32 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 33 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 34 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 35 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 36 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 37 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 38 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 39 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 40 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 41 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 42 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 43 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 44 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 45 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 46 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1

53 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 54 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 O X X X X X X X X X X X B X X X X X X X X X 1 1 1 1 1 1 1 1 1 1 2 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 55 1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 56 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 57 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 58 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 59 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 60 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 61 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 62 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 63 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 64 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 O X X X X X X X X X X X X X X X X X X X X B 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 S 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 55 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 56 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 57 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 58 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 59 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 60 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 61 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 62 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 63 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 64 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

→

Une solution à 64 expériences est trouvée ; l’optimum serait une matrice

d’Hadamard à 44 expériences.

Pour avoir les matrices d’Hadamard, il suffit d’invoquer les

macros suivantes (présentes dans le module QC):

%adxgen;

%adxinit;

%adxpbd(plan,nf);

Æ le plan souhaité sera dans « plan » et nf et le nombre de facteurs

proc print data=plan;

run;

Programme FACTEX10.SAS

/***************************FACTEX10.SAS **************************/; /********** exemples de plans factoriels fractionnaires ***********/; ;

;

options ps=65 ls=75 nodate nocenter; title ' ' ;

;

/*******************************************************/; /** fraction de resolution III d'un plan 2**6 **/; /** avec blocs de taille 4 **/; ;

proc factex; factors x1-x6; model est=(x1-x6);

size design=minimum;

→

on ne peut pas imposer l’option « minimum »

simultanément aux instructions « size » et « blocks ».

blocks size=4;

examine aliasing(2) confounding; output out=resul1;

proc print data=resul1; run;

/*******************************************************/;

SORTIE de FACTEX10.SAS

Factor Confounding Rules

X4 = X1*X2*X3 X5 = X2*X3 X6 = X1*X3

Block Pseudo-factor Confounding Rules

[B1] = X1*X2 Aliasing Structure X1 = X3*X6 = X4*X5 X2 = X3*X5 = X4*X6 X3 = X1*X6 = X2*X5 X4 = X1*X5 = X2*X6 X5 = X1*X4 = X2*X3 X6 = X1*X3 = X2*X4 [B] = X1*X2 = X3*X4 = X5*X6

OBS BLOCK X1 X2 X3 X4 X5 X6 1 1 -1 1 -1 1 -1 1 2 1 -1 1 1 -1 1 -1 3 1 1 -1 -1 1 1 -1 4 1 1 -1 1 -1 -1 1 5 2 -1 -1 -1 -1 1 1 6 2 -1 -1 1 1 -1 -1 7 2 1 1 -1 -1 -1 -1 8 2 1 1 1 1 1 1

Programme FACTEX11.SAS

/**********************FACTEX11.SAS********************************/;

/*********************** etude sequentielle ***********************/; /*** Construction d'une matrice complementaire pour lever une ***/; /*** concomitance . ***/; ;

options ps=1000 ls=75 nodate nocenter; title ' ' ;

;

1

/*** En premier lieu une matrice fractionnaire pour 8 facteurs ***/; /*** en 16 experiences est construite. ***/; ; options linesize=75; proc factex; factors x1-x8; model res=4; size design=16;

examine aliasing(3) confounding design; output out=resul0;

run;

2

/*** Apres analyse des premiers resultats, il semble que les ***/ /*** interactions x1*x5 x2*x6 x3*x7 x4*x8 soient significati- ***/ /*** ves, toutes ou en partie. Il faut donc des experiences ***/ /*** supplementaires pour lever leurs concomitances. ***/ ;

2.1

/*** Tentative numero 1 ***/ ; model estimate=(x1*x5 x2*x6 x3*x7 x4*x8); size design=minimum;

examine aliasing(2) confounding design; output out=resul1;

2.2

/*** Tentative numero 2 ***/ ;

model estimate=(x1*x5 x2*x6 x3*x7 x4*x8); size design=16;

examine aliasing(2) confounding design; output out=resul2; run;

2.3

/*** Tentative numero 3 ***/ ; model estimate=(x1*x5 x2*x6 x3*x7 x4*x8); size design=32;

examine aliasing(2) confounding design; output out=resul3;

run;

2.4

/*** Tentative numero 4 ***/ ;

data fold1;set resul0;

x1=-x1;x2=-x2;x3=-x3;x4=-x4;x5=-x5;x6=-x6;x7=-x7;x8=-x8;

proc print data=fold1;

data fold2;set fold1; x1x2=x1*x2;x1x3=x1*x3;x1x4=x1*x4;x1x5=x1*x5;x1x6=x1*x6;x1x7=x1*x7;x1x8=x1*x8; x2x3=x2*x3;x2x4=x2*x4;x2x5=x2*x5;x2x6=x2*x6;x2x7=x2*x7;x2x8=x2*x8; x3x4=x3*x4;x3x5=x3*x5;x3x6=x3*x6;x3x7=x3*x7;x3x8=x3*x8; x4x5=x4*x5;x4x6=x4*x6;x4x7=x4*x7;x4x8=x4*x8; x5x6=x5*x6;x5x7=x5*x7;x5x8=x5*x8; x6x7=x6*x7;x6x8=x6*x8; x7x8=x7*x8;

proc corr data=fold2 noprob nosimple; run;

3

/*** Examen des concomitances sur le plan constitue de la reunion ***/ /*** des deux fractions pour les 4 tentatives precedentes ***/ ;

data tot1;set resul0 resul1;

proc corr data=tot1 noprob nosimple;

run;

data tot2;set resul0 resul2;

proc corr data=tot2 noprob nosimple;

run;

data tot3;set resul0 resul3;

proc corr data=tot3 noprob nosimple;

run;

data tot4;set resul0 fold1;

/* examen des concomitances entre interactions d ordre 2 */ /* pour la tentative 4 */ ; x1x2=x1*x2;x1x3=x1*x3;x1x4=x1*x4;x1x5=x1*x5;x1x6=x1*x6;x1x7=x1*x7;x1x8=x1*x8; x2x3=x2*x3;x2x4=x2*x4;x2x5=x2*x5;x2x6=x2*x6;x2x7=x2*x7;x2x8=x2*x8; x3x4=x3*x4;x3x5=x3*x5;x3x6=x3*x6;x3x7=x3*x7;x3x8=x3*x8; x4x5=x4*x5;x4x6=x4*x6;x4x7=x4*x7;x4x8=x4*x8; x5x6=x5*x6;x5x7=x5*x7;x5x8=x5*x8; x6x7=x6*x7;x6x8=x6*x8; x7x8=x7*x8;

proc corr data=tot4 noprob nosimple;

4

/**** approches ab initio ****/;

4.1

proc factex; factors x1-x8; model est=(x1-x8 x1*x5 x2*x6 x3*x7 x4*x8); size design=32;

examine aliasing(2) confounding design; output out=resul4; run;

4.2

proc factex; factors x1-x8; model est=(x1-x8 x1*x5 x2*x6 x3*x7 x4*x8) nonneg=(x1*x2 x1*x3 x1*x4 x1*x6 x1*x7 x1*x8 x2*x3 x2*x4 x2*x5 x2*x7 x2*x8 x3*x4 x3*x5 x3*x6 x3*x8 x4*x5 x4*x6 x4*x7 x5*x6 x5*x7 x5*x8 x6*x7 x6*x8 x7*x8); size design=32;

examine aliasing(2) confounding design; output out=resul5; run;

4.3

proc factex; factors x1-x8; model resolution=4; size design=32;

examine aliasing(2) confounding design; output out=resul6;

4.4

proc factex; factors x1-x8; model resolution=5;

size design=64;

examine aliasing(2) confounding design; output out=resul7 ;

SORTIE de FACTEX11.SAS

SOLUTIONS

1

Design Points Experiment Number X1 X2 X3 X4 X5 X6 X7 X8 --- 1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 1 1 1 1 -1 3 -1 -1 1 -1 1 1 -1 1 4 -1 -1 1 1 -1 -1 1 1 5 -1 1 -1 -1 1 -1 1 1 6 -1 1 -1 1 -1 1 -1 1 7 -1 1 1 -1 -1 1 1 -1 8 -1 1 1 1 1 -1 -1 -1 9 1 -1 -1 -1 -1 1 1 1 10 1 -1 -1 1 1 -1 -1 1 11 1 -1 1 -1 1 -1 1 -1 12 1 -1 1 1 -1 1 -1 -1 13 1 1 -1 -1 1 1 -1 -1 14 1 1 -1 1 -1 -1 1 -1 15 1 1 1 -1 -1 -1 -1 1 16 1 1 1 1 1 1 1 1 Factor Confounding Rules

X5 = X2*X3*X4 X6 = X1*X3*X4 X7 = X1*X2*X4 X8 = X1*X2*X3 Aliasing Structure X1 = X2*X3*X8 = X2*X4*X7 = X2*X5*X6 = X3*X4*X6 = X3*X5*X7 = X4*X5*X8 = X6*X7*X8 X2 = X1*X3*X8 = X1*X4*X7 = X1*X5*X6 = X3*X4*X5 = X3*X6*X7 = X4*X6*X8 = X5*X7*X8 X3 = X1*X2*X8 = X1*X4*X6 = X1*X5*X7 = X2*X4*X5 = X2*X6*X7 = X4*X7*X8 = X5*X6*X8 X4 = X1*X2*X7 = X1*X3*X6 = X1*X5*X8 = X2*X3*X5 = X2*X6*X8 = X3*X7*X8 = X5*X6*X7 X5 = X1*X2*X6 = X1*X3*X7 = X1*X4*X8 = X2*X3*X4 = X2*X7*X8 = X3*X6*X8 = X4*X6*X7 X6 = X1*X2*X5 = X1*X3*X4 = X1*X7*X8 = X2*X3*X7 = X2*X4*X8 = X3*X5*X8 = X4*X5*X7 X7 = X1*X2*X4 = X1*X3*X5 = X1*X6*X8 = X2*X3*X6 = X2*X5*X8 = X3*X4*X8 = X4*X5*X6 X8 = X1*X2*X3 = X1*X4*X5 = X1*X6*X7 = X2*X4*X6 = X2*X5*X7 = X3*X4*X7 = X3*X5*X6

X1*X2 = X3*X8 = X4*X7 = X5*X6

X1*X3 = X2*X8 = X4*X6 = X5*X7

X1*X4 = X2*X7 = X3*X6 = X5*X8

X1*X5 = X2*X6 = X3*X7 = X4*X8

X1*X6 = X2*X5 = X3*X4 = X7*X8

X1*X7 = X2*X4 = X3*X5 = X6*X8

X1*X8 = X2*X3 = X4*X5 = X6*X7

2.1

Design Points Experiment Number X1 X2 X3 X4 X5 X6 X7 X8 --- 1 -1 -1 -1 -1 -1 -1 -1 1 2 -1 -1 1 1 1 1 1 -1 3 -1 1 -1 1 1 1 1 -1 4 -1 1 1 -1 -1 -1 -1 1 5 1 -1 -1 1 1 1 1 1 6 1 -1 1 -1 -1 -1 -1 -1 7 1 1 -1 -1 -1 -1 -1 -1 8 1 1 1 1 1 1 1 1 Factor Confounding Rules

X4 = X1*X2*X3 X5 = X1*X2*X3 X6 = X1*X2*X3 X7 = X1*X2*X3 X8 = X2*X3 Aliasing Structure 0 = X4*X5 = X4*X6 = X4*X7 = X5*X6 = X5*X7 = X6*X7 X1 = X4*X8 = X5*X8 = X6*X8 = X7*X8 X2 = X3*X8 X3 = X2*X8 X4 = X5 = X6 = X7 = X1*X8

Æ

solution 2.1 impossible X8 = X1*X4 = X1*X5 = X1*X6 = X1*X7 = X2*X3 X1*X2 = X3*X4 = X3*X5 = X3*X6 = X3*X7 X1*X3 = X2*X4 = X2*X5 = X2*X6 = X2*X7

2.2

Design Points Experiment Number X1 X2 X3 X4 X5 X6 X7 X8 --- 1 -1 -1 -1 -1 1 1 1 1 2 -1 -1 -1 1 -1 -1 -1 -1 3 -1 -1 1 -1 -1 -1 -1 -1 4 -1 -1 1 1 1 1 1 1 5 -1 1 -1 -1 -1 -1 -1 -1 6 -1 1 -1 1 1 1 1 1 7 -1 1 1 -1 1 1 1 1 8 -1 1 1 1 -1 -1 -1 -1 9 1 -1 -1 -1 -1 -1 -1 -1 10 1 -1 -1 1 1 1 1 1 11 1 -1 1 -1 1 1 1 1 12 1 -1 1 1 -1 -1 -1 -1 13 1 1 -1 -1 1 1 1 1 14 1 1 -1 1 -1 -1 -1 -1 15 1 1 1 -1 -1 -1 -1 -1 16 1 1 1 1 1 1 1 1 Factor Confounding Rules

X5 = X1*X2*X3*X4 X6 = X1*X2*X3*X4 X7 = X1*X2*X3*X4 X8 = X1*X2*X3*X4 Aliasing Structure 0 = X5*X6 = X5*X7 = X5*X8 = X6*X7 = X6*X8 = X7*X8 X1 X2 X3 X4 X5 = X6 = X7 = X8

Æ

solution 2.2 impossible X1*X2 X1*X3 X1*X4 X1*X5 = X1*X6 = X1*X7 = X1*X8 X2*X3 X2*X4 X2*X5 = X2*X6 = X2*X7 = X2*X8 X3*X4 X3*X5 = X3*X6 = X3*X7 = X3*X8 X4*X5 = X4*X6 = X4*X7 = X4*X8

2.3

Design Points Experiment Number X1 X2 X3 X4 X5 X6 X7 X8 --- 1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 1 1 1 1 3 -1 -1 -1 1 -1 1 1 1 4 -1 -1 -1 1 1 -1 -1 -1 5 -1 -1 1 -1 -1 1 1 1 6 -1 -1 1 -1 1 -1 -1 -1 7 -1 -1 1 1 -1 -1 -1 -1 8 -1 -1 1 1 1 1 1 1 9 -1 1 -1 -1 -1 1 1 1 10 -1 1 -1 -1 1 -1 -1 -1 11 -1 1 -1 1 -1 -1 -1 -1 12 -1 1 -1 1 1 1 1 1 13 -1 1 1 -1 -1 -1 -1 -1 14 -1 1 1 -1 1 1 1 1 15 -1 1 1 1 -1 1 1 1 16 -1 1 1 1 1 -1 -1 -1 17 1 -1 -1 -1 -1 1 1 1 18 1 -1 -1 -1 1 -1 -1 -1 19 1 -1 -1 1 -1 -1 -1 -1 20 1 -1 -1 1 1 1 1 1 21 1 -1 1 -1 -1 -1 -1 -1 22 1 -1 1 -1 1 1 1 1 23 1 -1 1 1 -1 1 1 1 24 1 -1 1 1 1 -1 -1 -1 25 1 1 -1 -1 -1 -1 -1 -1 26 1 1 -1 -1 1 1 1 1 27 1 1 -1 1 -1 1 1 1 28 1 1 -1 1 1 -1 -1 -1 29 1 1 1 -1 -1 1 1 1 30 1 1 1 -1 1 -1 -1 -1 31 1 1 1 1 -1 -1 -1 -1 32 1 1 1 1 1 1 1 1

Factor Confounding Rules

X6 = X1*X2*X3*X4*X5 X7 = X1*X2*X3*X4*X5 X8 = X1*X2*X3*X4*X5 Aliasing Structure 0 = X6*X7 = X6*X8 = X7*X8 X1 X2 X3 X4 X5 X6 = X7 = X8

Æ

solution 2.3 impossible X1*X2 X1*X3 X1*X4 X1*X5 X1*X6 = X1*X7 = X1*X8

X3*X6 = X3*X7 = X3*X8 X4*X5

X4*X6 = X4*X7 = X4*X8 X5*X6 = X5*X7 = X5*X8

2.4

Fold-over OBS X1 X2 X3 X4 X5 X6 X7 X8 1 1 1 1 1 1 1 1 1 2 1 1 1 -1 -1 -1 -1 1 3 1 1 -1 1 -1 -1 1 -1 4 1 1 -1 -1 1 1 -1 -1 5 1 -1 1 1 -1 1 -1 -1 6 1 -1 1 -1 1 -1 1 -1 7 1 -1 -1 1 1 -1 -1 1 8 1 -1 -1 -1 -1 1 1 1 9 -1 1 1 1 1 -1 -1 -1 10 -1 1 1 -1 -1 1 1 -1 11 -1 1 -1 1 -1 1 -1 1 12 -1 1 -1 -1 1 -1 1 1 13 -1 -1 1 1 -1 -1 1 1 14 -1 -1 1 -1 1 1 -1 1 15 -1 -1 -1 1 1 1 1 -1 16 -1 -1 -1 -1 -1 -1 -1 -1

Par construction elle revient à poser :

X1 = -X3*X4*X6 X2 = -X3*X4*X5 X3 = -X1*X4*X6 X4 = -X1*X2*X7 X5 = -X2*X3*X4 X6 = -X1*X3*X4 X7 = -X1*X2*X4 X8 = -X1*X2*X3

D’où :

X1*X2 = -X3*X8 = -X4*X7 = +X5*X6

X1*X3 = -X2*X8 = -X4*X6 = +X5*X7

X1*X4 = -X2*X7 = -X3*X6 = +X5*X8

X1*X5 = +X2*X6 = +X3*X7 = +X4*X8

X1*X6 = +X2*X5 = -X3*X4 = -X7*X8

X1*X7 = -X2*X4 = +X3*X5 = -X6*X8

X1*X8 = -X2*X3 = +X4*X5 = -X6*X7

On retrouve cette structure d’alias dans la matrice de

corrélation suivante

Æ solution 2.4 sans intérêt.

Pearson Correlation Coefficients / N = 16 X1 X2 X3 X4 X5 X6 X1 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 X3 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 X4 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 X5 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 X6 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

X1 X2 X3 X4 X5 X6 X4X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X7X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X7 X8 X1X2 X1X3 X1X4 X1X5 X1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X7 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X8 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 X1X2 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 X1X3 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 X1X4 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 X1X5 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X1X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X6 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000

X7 X8 X1X2 X1X3 X1X4 X1X5 X2X7 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 X2X8 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 X3X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X6 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 X3X7 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X3X8 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 X4X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X6 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 X4X7 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 X4X8 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X5X6 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 X5X7 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 X5X8 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 X6X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X7X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X6 X1X7 X1X8 X2X3 X2X4 X2X5 X1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X3 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X4 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

X1X6 X1X7 X1X8 X2X3 X2X4 X2X5 X1X5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X1X6 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X1X7 0.00000 1.00000 0.00000 0.00000 1.00000 0.00000 X1X8 0.00000 0.00000 1.00000 1.00000 0.00000 0.00000 X2X3 0.00000 0.00000 1.00000 1.00000 0.00000 0.00000 X2X4 0.00000 1.00000 0.00000 0.00000 1.00000 0.00000 X2X5 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X2X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X4 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 X3X5 0.00000 1.00000 0.00000 0.00000 1.00000 0.00000 X3X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X3X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X5 0.00000 0.00000 1.00000 1.00000 0.00000 0.00000 X4X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X4X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X5X8 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X6X7 0.00000 0.00000 1.00000 1.00000 0.00000 0.00000 X6X8 0.00000 1.00000 0.00000 0.00000 1.00000 0.00000 X7X8 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000