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

Jean-Pierre Gauchi

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## 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

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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 ;
  fraction = d | maximum ;
  nrunfacs = m | minimum ;

MODEL
  estimate = effets ;
  nonnegligible = effetsnonneg ;
  resolution = r ;

BLOCKS
  nblkfacs = s | maximum ;
  nblocks = b | maximum ;
  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 * 'text' ;

RUN ;

QUIT ;

TITLE * 'text'
Exemples d'utilisation

de la procédure FACTEX du logiciel SAS/QC
# 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 outs -1, +1

proc print data; run;

## Sortie de FACTEX01.SAS

-> le plan complet 2^4

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
# 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 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

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>0</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>0</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>0</td><td>0</td></tr><tr><td>6</td><td>-1</td><td>0</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>0</td><td>-1</td><td>-1</td></tr><tr><td>11</td><td>0</td><td>-1</td><td>0</td></tr><tr><td>12</td><td>0</td><td>-1</td><td>1</td></tr><tr><td>13</td><td>0</td><td>0</td><td>-1</td></tr><tr><td>14</td><td>0</td><td>0</td><td>0</td></tr><tr><td>15</td><td>0</td><td>0</td><td>1</td></tr><tr><td>16</td><td>0</td><td>1</td><td>-1</td></tr><tr><td>17</td><td>0</td><td>1</td><td>0</td></tr><tr><td>18</td><td>0</td><td>1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>0</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>22</td><td>1</td><td>0</td><td>-1</td></tr><tr><td>23</td><td>1</td><td>0</td><td>0</td></tr><tr><td>24</td><td>1</td><td>0</td><td>1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>0</td></tr><tr><td>27</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
# 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;

proc print data;
run;

## Sortie de FACTEX03.SAS

→ le plan complet 2³ en deux répétitions

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
# 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; d'alias jusqu'à l'ordre 2

proc print data;
run;

# Sortie de FACTEX04.SAS

→ chaque expérience du plan complet 2⁵ est attribuée à l'un des deux blocs

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>Block</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>2</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>2</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>2</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>2</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>2</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>2</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>2</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>2</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>2</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>2</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>2</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>2</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>2</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>2</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>2</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>2</td></tr></tbody></table>
Block Pseudo-factor Confounding Rules

$$ [B1] = X1*X2*X3*X4*X5 \rightarrow \text{le facteur bloc est confondu avec l'interaction quintuple} $$

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.

<table><thead><tr><th>OBS</th><th>BLOCK</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>8</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>10</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>12</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>17</td><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>18</td><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>19</td><td>2</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>20</td><td>2</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>21</td><td>2</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>2</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>23</td><td>2</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>24</td><td>2</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>25</td><td>2</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>26</td><td>2</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>27</td><td>2</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>28</td><td>2</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>29</td><td>2</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>30</td><td>2</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>31</td><td>2</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

→ dans le fichier resul, les blocs apparaissent plus clairement.
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=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

<table><thead><tr><th>OBS</th><th>REACTEUR</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>r1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>r1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>r1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>4</td><td>r1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>r1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>r1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>r1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>r1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>9</td><td>r2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>r2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>r2</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>12</td><td>r2</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>r2</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>14</td><td>r2</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>r2</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>r2</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>17</td><td>r3</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>18</td><td>r3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>19</td><td>r3</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>20</td><td>r3</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>21</td><td>r3</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>r3</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>23</td><td>r3</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>24</td><td>r3</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>25</td><td>r4</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>26</td><td>r4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>27</td><td>r4</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>28</td><td>r4</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>29</td><td>r4</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>30</td><td>r4</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>31</td><td>r4</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>r4</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>OPERAT</th><th>LOT</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>jean</td><td>bayer</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>jean</td><td>bayer</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>jean</td><td>bayer</td></tr><tr><td>4</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>jean</td><td>bayer</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>jean</td><td>bayer</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>jean</td><td>bayer</td></tr><tr><td>7</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>jean</td><td>bayer</td></tr><tr><td>8</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>jean</td><td>bayer</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>jean</td><td>rp</td></tr><tr><td>10</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>jean</td><td>rp</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>jean</td><td>rp</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>jean</td><td>rp</td></tr><tr><td>13</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>jean</td><td>rp</td></tr><tr><td>14</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>jean</td><td>rp</td></tr><tr><td>15</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>jean</td><td>rp</td></tr><tr><td>16</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>jean</td><td>rp</td></tr><tr><td>17</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>paul</td><td>bayer</td></tr><tr><td>18</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>paul</td><td>bayer</td></tr><tr><td>19</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>paul</td><td>bayer</td></tr><tr><td>20</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>paul</td><td>bayer</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>paul</td><td>bayer</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>paul</td><td>bayer</td></tr><tr><td>23</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>paul</td><td>bayer</td></tr><tr><td>24</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>paul</td><td>bayer</td></tr><tr><td>25</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>paul</td><td>rp</td></tr><tr><td>26</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>paul</td><td>rp</td></tr><tr><td>27</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>paul</td><td>rp</td></tr><tr><td>28</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>paul</td><td>rp</td></tr><tr><td>29</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>paul</td><td>rp</td></tr><tr><td>30</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>paul</td><td>rp</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>paul</td><td>rp</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>paul</td><td>rp</td></tr></tbody></table>
# Programme FACTEX06.SAS

```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

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
# 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;

proc print data resul2;

run;

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

## SORTIE de FACTEX07.SAS

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>Block</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>4</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>4</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>2</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>3</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>3</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>2</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>3</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>2</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>2</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>3</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>4</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>4</td></tr></tbody></table>
## Factor Confounding Rules

$$X5 = X1*X2*X3*X4$$

## Block Pseudo-factor Confounding Rules

$$[B1] = X2*X3*X4$$

$$[B2] = X1*X3*X4 \rightarrow \text{deux facteurs blocs donc 4 blocs : un plan à facteurs à q niveaux (ici 2) partitionné en q^s blocs (ici q^s =4) contient s facteurs blocs (ici s=2) notés ici [B1] et [B2].}$$

## Aliasing Structure

X1
X2
X3
X4
X5

$$[B] = \begin{aligned} X1*X2 &= X3*X4*X5 \\ X1*X3 &= X2*X4*X5 \\ X1*X4 &= X2*X3*X5 \end{aligned}$$

$$[B] = \begin{aligned} X1*X5 &= X2*X3*X4 \\ X2*X3 &= X1*X4*X5 \\ X2*X4 &= X1*X3*X5 \end{aligned}$$

$$[B] = \begin{aligned} X2*X5 &= X1*X3*X4 \\ X3*X4 &= X1*X2*X5 \\ X3*X5 &= X1*X2*X4 \end{aligned}$$

$$X4*X5 = X1*X2*X3 \rightarrow \text{les facteurs blocs sont confondus avec les interactions doubles et triples}$$

<table><thead><tr><th>OBS</th><th>BLOCK</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>5</td><td>2</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>2</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>2</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>2</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>9</td><td>3</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>3</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>3</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>12</td><td>3</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>13</td><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>14</td><td>4</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>15</td><td>4</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>4</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

## Question :

*Comment retrouver à quel bloc appartient une expérience donnée i ? :*
Réponse :

$$n^{\text{bloc}}(i) = B1 + qB2 + q^2B3 + \dots + q^{s-1}B_s$$

codé de 0 à $q^{s-1}$ 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 → 0 et +1 → 1.

Applications au cas présent :

$$ \begin{array}{ccccc} X2 & X3 & X4 & X1 & X3 \end{array} \qquad \begin{array}{c} \downarrow \\ \downarrow \end{array} $$

$$ N^{\text{bloc}}(1) = (0 + 0 + 0) \cdot 1 + (0 + 0 + 0) \cdot 2 = 0 \rightarrow \text{bloc noté 1.} $$

$$ N^{\text{bloc}}(5) = (1 + 0 + 0) \cdot 1 + (0 + 0 + 0) \cdot 2 = 1 \rightarrow \text{bloc noté 2.} $$

$$ N^{\text{bloc}}(16) = (1 + 1 + 1) \cdot 1 + (1 + 1 + 1) \cdot 2 = 3 \rightarrow \text{bloc noté 4.} $$

$= 1 \pmod{2}$

$= 1 \pmod{2}$
## 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.
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th><th>X9</th><th>X10</th><th>X11</th><th>X12</th><th>X13</th><th>X14</th><th>X15</th><th>X16</th><th>X17</th><th>X18</th><th>X19</th><th>X20</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

→ Une solution à 32 expériences est trouvée ; l'optimum serait une matrice d'Hadamard à 24 expériences, voir plus loin.
Aliasing Structure

$$
\begin{aligned}
X1 = & X6*X37 = X7*X8 = X9*X10 = X11*X12 = X13*X14 = X15*X16 = X17*X18 \\
& = X19*X20 = X21*X22 = X23*X24 = X25*X26 = X27*X28 = X29*X30 = X31*X32 \\
& = X33*X34 = X35*X36 = X38*X39
\end{aligned}
$$

$$
\begin{aligned}
X2 = & X6*X36 = X7*X9 = X8*X10 = X11*X13 = X12*X14 = X15*X17 = X16*X18 \\
& = X19*X21 = X20*X22 = X23*X25 = X24*X26 = X27*X29 = X28*X30 = X31*X33 \\
& = X32*X34 = X35*X37 = X38*X40
\end{aligned}
$$

$$
\begin{aligned}
X3 = & X6*X34 = X7*X11 = X8*X12 = X9*X13 = X10*X14 = X15*X19 = X16*X20 \\
& = X17*X21 = X18*X22 = X23*X27 = X24*X28 = X25*X29 = X26*X30 = X31*X35 \\
& = X32*X36 = X33*X37
\end{aligned}
$$

$$
\begin{aligned}
X4 = & X6*X30 = X7*X15 = X8*X16 = X9*X17 = X10*X18 = X11*X19 = X12*X20 \\
& = X13*X21 = X14*X22 = X23*X31 = X24*X32 = X25*X33 = X26*X34 = X27*X35 \\
& = X28*X36 = X29*X37
\end{aligned}
$$

$$
\begin{aligned}
X5 = & X6*X22 = X7*X23 = X8*X24 = X9*X25 = X10*X26 = X11*X27 = X12*X28 \\
& = X13*X29 = X14*X30 = X15*X31 = X16*X32 = X17*X33 = X18*X34 = X19*X35 \\
& = X20*X36 = X21*X37
\end{aligned}
$$

$$
X6 = X1*X37 = X2*X36 = X3*X34 = X4*X30 = X5*X22 = X7*X38 = X8*X39 = X9*X40
$$

$$
X7 = X1*X8 = X2*X9 = X3*X11 = X4*X15 = X5*X23 = X6*X38 = X36*X40 = X37*X39
$$

$$
\begin{aligned}
X8 = & X1*X7 = X2*X10 = X3*X12 = X4*X16 = X5*X24 = X6*X39 = X35*X40 \\
& = X37*X38
\end{aligned}
$$

$$
\begin{aligned}
X9 = & X1*X10 = X2*X7 = X3*X13 = X4*X17 = X5*X25 = X6*X40 = X35*X39 \\
& = X36*X38
\end{aligned}
$$

$$
\begin{aligned}
X10 = & X1*X9 = X2*X8 = X3*X14 = X4*X18 = X5*X26 = X35*X38 = X36*X39 \\
& = X37*X40
\end{aligned}
$$

$$
\begin{aligned}
X11 = & X1*X12 = X2*X13 = X3*X7 = X4*X19 = X5*X27 = X32*X40 = X33*X39 \\
& = X34*X38
\end{aligned}
$$

$$
\begin{aligned}
X12 = & X1*X11 = X2*X14 = X3*X8 = X4*X20 = X5*X28 = X31*X40 = X33*X38 \\
& = X34*X39
\end{aligned}
$$

$$
\begin{aligned}
X13 = & X1*X14 = X2*X11 = X3*X9 = X4*X21 = X5*X29 = X31*X39 = X32*X38 \\
& = X34*X40
\end{aligned}
$$

$$
\begin{aligned}
X14 = & X1*X13 = X2*X12 = X3*X10 = X4*X22 = X5*X30 = X31*X38 = X32*X39 \\
& = X33*X40
\end{aligned}
$$

$$
\begin{aligned}
X15 = & X1*X16 = X2*X17 = X3*X19 = X4*X7 = X5*X31 = X28*X40 = X29*X39 \\
& = X30*X38
\end{aligned}
$$

$$
\begin{aligned}
X16 = & X1*X15 = X2*X18 = X3*X20 = X4*X8 = X5*X32 = X27*X40 = X29*X38 \\
& = X30*X39
\end{aligned}
$$

$$
\begin{aligned}
X17 = & X1*X18 = X2*X15 = X3*X21 = X4*X9 = X5*X33 = X27*X39 = X28*X38 \\
& = X30*X40
\end{aligned}
$$

$$
\begin{aligned}
X18 = & X1*X17 = X2*X16 = X3*X22 = X4*X10 = X5*X34 = X27*X38 = X28*X39 \\
& = X29*X40
\end{aligned}
$$

$$
\begin{aligned}
X19 = & X1*X20 = X2*X21 = X3*X15 = X4*X11 = X5*X35 = X24*X40 = X25*X39 \\
& = X26*X38
\end{aligned}
$$

$$
\begin{aligned}
X20 = & X1*X19 = X2*X22 = X3*X16 = X4*X12 = X5*X36 = X23*X40 = X25*X38 \\
& = X26*X39
\end{aligned}
$$

$$
\begin{aligned}
X21 = & X1*X22 = X2*X19 = X3*X17 = X4*X13 = X5*X37 = X23*X39 = X24*X38 \\
& = X26*X40
\end{aligned}
$$

$$
\begin{aligned}
X22 = & X1*X21 = X2*X20 = X3*X18 = X4*X14 = X5*X6 = X23*X38 = X24*X39 \\
& = X25*X40
\end{aligned}
$$

$$
\begin{aligned}
X23 = & X1*X24 = X2*X25 = X3*X27 = X4*X31 = X5*X7 = X20*X40 = X21*X39 \\
& = X22*X38
\end{aligned}
$$

$$
\begin{aligned}
X24 = & X1*X23 = X2*X26 = X3*X28 = X4*X32 = X5*X8 = X19*X40 = X21*X38 \\
& = X22*X39
\end{aligned}
$$

$$
\begin{aligned}
X25 = & X1*X26 = X2*X23 = X3*X29 = X4*X33 = X5*X9 = X19*X39 = X20*X38 \\
& = X22*X40
\end{aligned}
$$

$$
\begin{aligned}
X26 = & X1*X25 = X2*X24 = X3*X30 = X4*X34 = X5*X10 = X19*X38 = X20*X39 \\
& = X21*X40
\end{aligned}
$$
$$X27 = X1*X28 = X2*X29 = X3*X23 = X4*X35 = X5*X11 = X16*X40 = X17*X39$$
$$= X18*X38$$

$$X28 = X1*X27 = X2*X30 = X3*X24 = X4*X36 = X5*X12 = X15*X40 = X17*X38$$
$$= X18*X39$$

$$X29 = X1*X30 = X2*X27 = X3*X25 = X4*X37 = X5*X13 = X15*X39 = X16*X38$$
$$= X18*X40$$

$$X30 = X1*X29 = X2*X28 = X3*X26 = X4*X6 = X5*X14 = X15*X38 = X16*X39$$
$$= X17*X40$$

$$X31 = X1*X32 = X2*X33 = X3*X35 = X4*X23 = X5*X15 = X12*X40 = X13*X39$$
$$= X14*X38$$

$$X32 = X1*X31 = X2*X34 = X3*X36 = X4*X24 = X5*X16 = X11*X40 = X13*X38$$
$$= X14*X39$$

$$X33 = X1*X34 = X2*X31 = X3*X37 = X4*X25 = X5*X17 = X11*X39 = X12*X38$$
$$= X14*X40$$

$$X34 = X1*X33 = X2*X32 = X3*X6 = X4*X26 = X5*X18 = X11*X38 = X12*X39$$
$$= X13*X40$$

$$X35 = X1*X36 = X2*X37 = X3*X31 = X4*X27 = X5*X19 = X8*X40 = X9*X39$$
$$= X10*X38$$

$$X36 = X1*X35 = X2*X6 = X3*X32 = X4*X28 = X5*X20 = X7*X40 = X9*X38$$
$$= X10*X39$$

$$X37 = X1*X6 = X2*X35 = X3*X33 = X4*X29 = X5*X21 = X7*X39 = X8*X38$$
$$= X10*X40$$

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

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

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

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

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

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

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

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

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

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

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

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

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

$$X3*X38 = X6*X11 = X7*X34 = X8*X33 = X9*X32 = X10*X31 = X12*X37 = X13*X36$$
$$= X14*X35 = X15*X26 = X16*X25 = X17*X24 = X18*X23 = X19*X30$$
$$= X20*X29 = X21*X28 = X22*X27$$
$$
\begin{align*}
X3*X39 &= X6*X12 = X7*X33 = X8*X34 = X9*X31 = X10*X32 = X11*X37 = X13*X35 \\
&= X14*X36 = X15*X25 = X16*X26 = X17*X23 = X18*X24 = X19*X29 \\
&= X20*X30 = X21*X27 = X22*X28
\end{align*}
$$

$$
\begin{align*}
X3*X40 &= X6*X13 = X7*X32 = X8*X31 = X9*X34 = X10*X33 = X11*X36 = X12*X35 \\
&= X14*X37 = X15*X24 = X16*X23 = X17*X26 = X18*X25 = X19*X28 \\
&= X20*X27 = X21*X30 = X22*X29
\end{align*}
$$

$$
\begin{align*}
X4*X5 &= X6*X14 = X7*X31 = X8*X32 = X9*X33 = X10*X34 = X11*X35 = X12*X36 \\
&= X13*X37 = X15*X23 = X16*X24 = X17*X25 = X18*X26 = X19*X27 \\
&= X20*X28 = X21*X29 = X22*X30
\end{align*}
$$

$$
\begin{align*}
X4*X38 &= X6*X15 = X7*X30 = X8*X29 = X9*X28 = X10*X27 = X11*X26 = X12*X25 \\
&= X13*X24 = X14*X23 = X16*X37 = X17*X36 = X18*X35 = X19*X34 \\
&= X20*X33 = X21*X32 = X22*X31
\end{align*}
$$

$$
\begin{align*}
X4*X39 &= X6*X16 = X7*X29 = X8*X30 = X9*X27 = X10*X28 = X11*X25 = X12*X26 \\
&= X13*X23 = X14*X24 = X15*X37 = X17*X35 = X18*X36 = X19*X33 \\
&= X20*X34 = X21*X31 = X22*X32
\end{align*}
$$

$$
\begin{align*}
X4*X40 &= X6*X17 = X7*X28 = X8*X27 = X9*X30 = X10*X29 = X11*X24 = X12*X23 \\
&= X13*X26 = X14*X25 = X15*X36 = X16*X35 = X18*X37 = X19*X32 \\
&= X20*X31 = X21*X34 = X22*X33
\end{align*}
$$

$$
\begin{align*}
X5*X38 &= X6*X23 = X7*X22 = X8*X21 = X9*X20 = X10*X19 = X11*X18 = X12*X17 \\
&= X13*X16 = X14*X15 = X24*X37 = X25*X36 = X26*X35 = X27*X34 \\
&= X28*X33 = X29*X32 = X30*X31
\end{align*}
$$

$$
\begin{align*}
X5*X39 &= X6*X24 = X7*X21 = X8*X22 = X9*X19 = X10*X20 = X11*X17 = X12*X18 \\
&= X13*X15 = X14*X16 = X23*X37 = X25*X35 = X26*X36 = X27*X33 \\
&= X28*X34 = X29*X31 = X30*X32
\end{align*}
$$

$$
\begin{align*}
X5*X40 &= X6*X25 = X7*X20 = X8*X19 = X9*X22 = X10*X21 = X11*X16 = X12*X15 \\
&= X13*X18 = X14*X17 = X23*X36 = X24*X35 = X26*X37 = X27*X32 \\
&= X28*X31 = X29*X34 = X30*X33
\end{align*}
$$

$$
\begin{align*}
X6*X19 &= X7*X26 = X8*X25 = X9*X24 = X10*X23 = X11*X30 = X12*X29 = X13*X28 \\
&= X14*X27 = X15*X34 = X16*X33 = X17*X32 = X18*X31 = X20*X37 \\
&= X21*X36 = X22*X35
\end{align*}
$$

$$
\begin{align*}
X6*X27 &= X7*X18 = X8*X17 = X9*X16 = X10*X15 = X11*X22 = X12*X21 = X13*X20 \\
&= X14*X19 = X23*X34 = X24*X33 = X25*X32 = X26*X31 = X28*X37 \\
&= X29*X36 = X30*X35
\end{align*}
$$

$$
\begin{align*}
X6*X31 &= X7*X14 = X8*X13 = X9*X12 = X10*X11 = X15*X22 = X16*X21 = X17*X20 \\
&= X18*X19 = X23*X30 = X24*X29 = X25*X28 = X26*X27 = X32*X37 \\
&= X33*X36 = X34*X35
\end{align*}
$$
## 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
<table><thead><tr><th>OBS</th><th>BLOCK</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>3</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>5</td><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>6</td><td>2</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>7</td><td>2</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>8</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
## 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;

run;
## 2.2

```sas
/*** 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

```sas
/*** 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

```sas
/*** 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;
```
/**** 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;

run;
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;

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

size design=64;
examine aliasing(2) confounding design;
output out resul7 ;

run;
# SORTIE de FACTEX11.SAS

## SOLUTIONS

### 1

#### Design Points

##### Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

#### Factor Confounding Rules

$$X5 = X2 \times X3 \times X4$$

$$X6 = X1 \times X3 \times X4$$

$$X7 = X1 \times X2 \times X4$$

$$X8 = X1 \times X2 \times X3$$

#### Aliasing Structure

$$X1 = X2 \times X3 \times X8 = X2 \times X4 \times X7 = X2 \times X5 \times X6 = X3 \times X4 \times X6 = X3 \times X5 \times X7 = X4 \times X5 \times X8$$

$$X2 = X1 \times X3 \times X8 = X1 \times X4 \times X7 = X1 \times X5 \times X6 = X3 \times X4 \times X5 = X3 \times X6 \times X7 = X4 \times X6 \times X8$$

$$X3 = X1 \times X2 \times X8 = X1 \times X4 \times X6 = X1 \times X5 \times X7 = X2 \times X4 \times X5 = X2 \times X6 \times X7 = X4 \times X7 \times X8$$

$$X4 = X1 \times X2 \times X7 = X1 \times X3 \times X6 = X1 \times X5 \times X8 = X2 \times X3 \times X5 = X2 \times X6 \times X8 = X3 \times X7 \times X8$$

$$X5 = X1 \times X2 \times X6 = X1 \times X3 \times X7 = X1 \times X4 \times X8 = X2 \times X3 \times X4 = X2 \times X7 \times X8 = X3 \times X6 \times X8$$

$$X6 = X1 \times X2 \times X5 = X1 \times X3 \times X4 = X1 \times X7 \times X8 = X2 \times X3 \times X7 = X2 \times X4 \times X8 = X3 \times X5 \times X8$$

$$X7 = X1 \times X2 \times X4 = X1 \times X3 \times X5 = X1 \times X6 \times X8 = X2 \times X3 \times X6 = X2 \times X5 \times X8 = X3 \times X4 \times X8$$

$$X8 = X1 \times X2 \times X3 = X1 \times X4 \times X5 = X1 \times X6 \times X7 = X2 \times X4 \times X6 = X2 \times X5 \times X7 = X3 \times X4 \times X7$$

$$X1 \times X2 = X3 \times X8 = X4 \times X7 = X5 \times X6$$

$$X1 \times X3 = X2 \times X8 = X4 \times X6 = X5 \times X7$$

$$X1 \times X4 = X2 \times X7 = X3 \times X6 = X5 \times X8$$

$$X1 \times X5 = X2 \times X6 = X3 \times X7 = X4 \times X8$$

$$X1 \times X6 = X2 \times X5 = X3 \times X4 = X7 \times X8$$

$$X1 \times X7 = X2 \times X4 = X3 \times X5 = X6 \times X8$$

$$X1 \times X8 = X2 \times X3 = X4 \times X5 = X6 \times X7$$
Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>7</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>8</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

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 → **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
Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

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

$X5 = X6 = X7 = X8 \rightarrow \text{solution 2.2 impossible}$

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

$X3*X5 = X3*X6 = X3*X7 = X3*X8$

$X4*X5 = X4*X6 = X4*X7 = X4*X8$
Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

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 → solution 2.3 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
$$X3*X6 = X3*X7 = X3*X8$$

$$X4*X5$$

$$X4*X6 = X4*X7 = X4*X8$$

$$X5*X6 = X5*X7 = X5*X8$$
Fold-over

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>3</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>15</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>16</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr></tbody></table>

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 → solution 2.4 sans intérêt.
Pearson Correlation Coefficients / N = 16

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th></tr></thead><tbody><tr><td>X1</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th></tr></thead><tbody><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X7</th><th>X8</th><th>X1X2</th><th>X1X3</th><th>X1X4</th><th>X1X5</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X7</th><th>X8</th><th>X1X2</th><th>X1X3</th><th>X1X4</th><th>X1X5</th></tr></thead><tbody><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X1X6</th><th>X1X7</th><th>X1X8</th><th>X2X3</th><th>X2X4</th><th>X2X5</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X1X6</th><th>X1X7</th><th>X1X8</th><th>X2X3</th><th>X2X4</th><th>X2X5</th></tr></thead><tbody><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X2X6</th><th>X2X7</th><th>X2X8</th><th>X3X4</th><th>X3X5</th><th>X3X6</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X2X6</th><th>X2X7</th><th>X2X8</th><th>X3X4</th><th>X3X5</th><th>X3X6</th></tr></thead><tbody><tr><th>X5X7</th><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X5X8</th><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><th>X6X7</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X6X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><th>X7X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X3X7</th><th>X3X8</th><th>X4X5</th><th>X4X6</th><th>X4X7</th><th>X4X8</th></tr></thead><tbody><tr><th>X1</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X3</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X4</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X5</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X6</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X7</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X1X2</th><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><th>X1X3</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X1X4</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X1X5</th><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><th>X1X6</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X1X7</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X1X8</th><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2X3</th><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2X4</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2X5</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2X6</th><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><th>X2X7</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X3X4</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X3X5</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X3X7</th><th>X3X8</th><th>X4X5</th><th>X4X6</th><th>X4X7</th><th>X4X8</th></tr></thead><tbody><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td></td><td>X5X6</td><td>X5X7</td><td>X5X8</td><td>X6X7</td><td>X6X8</td><td>X7X8</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td></td><td>X5X6</td><td>X5X7</td><td>X5X8</td><td>X6X7</td><td>X6X8</td><td>X7X8</td></tr></tbody></table>
<table><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></table>
Corrélation de la solution 2.1

Pearson Correlation Coefficients / N = 24

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><th>X1</th><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2</th><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X3</th><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><th>X4</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><th>X5</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.33333</td></tr><tr><th>X6</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.33333</td></tr><tr><th>X7</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.33333</td></tr><tr><th>X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th></th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr><tr><th>X1</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X2</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X3</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><th>X4</th><td>0.33333</td><td>0.33333</td><td>0.33333</td><td>0.00000</td></tr><tr><th>X5</th><td>1.00000</td><td>0.33333</td><td>0.33333</td><td>0.00000</td></tr><tr><th>X6</th><td>0.33333</td><td>1.00000</td><td>0.33333</td><td>0.00000</td></tr><tr><th>X7</th><td>0.33333</td><td>0.33333</td><td>1.00000</td><td>0.00000</td></tr><tr><th>X8</th><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>

Valeur de 0.33 rédhibitoire
Corrélation de la solution 2.2

Pearson Correlation Coefficients / N = 32

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>X1</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td></td><td>X5</td><td>X6</td><td>X7</td><td>X8</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>1.00000</td><td>0.50000</td><td>0.50000</td><td>0.50000</td></tr><tr><td>X6</td><td>0.50000</td><td>1.00000</td><td>0.50000</td><td>0.50000</td></tr><tr><td>X7</td><td>0.50000</td><td>0.50000</td><td>1.00000</td><td>0.50000</td></tr><tr><td>X8</td><td>0.50000</td><td>0.50000</td><td>0.50000</td><td>1.00000</td></tr></tbody></table>

Valeur de 0.50 rédhibitoire
Corrélation de la solution 2.3

Pearson Correlation Coefficients / N = 48

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>X1</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td></td><td>X5</td><td>X6</td><td>X7</td><td>X8</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>1.00000</td><td>0.66667</td><td>0.66667</td></tr><tr><td>X7</td><td>0.00000</td><td>0.66667</td><td>1.00000</td><td>0.66667</td></tr><tr><td>X8</td><td>0.00000</td><td>0.66667</td><td>0.66667</td><td>1.00000</td></tr></tbody></table>

Valeur de 0.67 rédhibitoire
Corrélation de la solution 2.4

Pearson Correlation Coefficients / N = 32

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th></tr></thead><tbody><tr><td>X1</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th></tr></thead><tbody><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X7</th><th>X8</th><th>X1X2</th><th>X1X3</th><th>X1X4</th><th>X1X5</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X7</th><th>X8</th><th>X1X2</th><th>X1X3</th><th>X1X4</th><th>X1X5</th></tr></thead><tbody><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X1X6</th><th>X1X7</th><th>X1X8</th><th>X2X3</th><th>X2X4</th><th>X2X5</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X1X6</th><th>X1X7</th><th>X1X8</th><th>X2X3</th><th>X2X4</th><th>X2X5</th></tr></thead><tbody><tr><td>X2X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>

<table><thead><tr><th></th><th>X2X6</th><th>X2X7</th><th>X2X8</th><th>X3X4</th><th>X3X5</th><th>X3X6</th></tr></thead><tbody><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X2X6</th><th>X2X7</th><th>X2X8</th><th>X3X4</th><th>X3X5</th><th>X3X6</th></tr></thead><tbody><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td></td><td>X3X7</td><td>X3X8</td><td>X4X5</td><td>X4X6</td><td>X4X7</td><td>X4X8</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X3X7</th><th>X3X8</th><th>X4X5</th><th>X4X6</th><th>X4X7</th><th>X4X8</th></tr></thead><tbody><tr><td>X1X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X5X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td></td><td>X5X7</td><td>X5X8</td><td>X6X7</td><td>X6X8</td><td>X7X8</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr></tbody></table>
<table><thead><tr><th></th><th>X5X6</th><th>X5X7</th><th>X5X8</th><th>X6X7</th><th>X6X8</th><th>X7X8</th></tr></thead><tbody><tr><td>X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X2</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X3</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X4</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X1X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X1X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X1X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X2X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X2X6</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X7</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2X8</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>X3X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X3X6</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3X8</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X5</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X6</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X7</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X6</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X7</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X5X8</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X7</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X6X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>X7X8</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>

Comme on peut le voir en examinant la matrice de corrélations précédente, l'ajout de la fraction foldover ne donne pas la solution.
4.1

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
## Factor Confounding Rules

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

$X7 = X2*X3*X4*X5$

$X8 = X3*X4*X5$

## Aliasing Structure

$X1 = X6*X7$

$X2 = X7*X8$

$X3$

$X4$

$X5$

$X6 = X1*X7$

$X7 = X1*X6 = X2*X8$

$X8 = X2*X7$

$X1*X2 = X6*X8$

$X1*X3$

$X1*X4$

$X1*X5$

$X1*X8 = X2*X6$

$X2*X3$

$X2*X4$

$X2*X5$

$X3*X4 = X5*X8$

$X3*X5 = X4*X8$

$X3*X6$

$X3*X7$

$X3*X8 = X4*X5$

$X4*X6$

$X4*X7$

$X5*X6$

$X5*X7$

Si les autres interactions peuvent être considérées comme négligeables les 4 interactions d'intérêt sont maintenant estimables indépendamment. Au total 32 expériences auront été réalisées.

## 4.2

SAS donne le message « ERROR: No such design exists. »

En effet, il n'y a pas de fraction de taille 32 et de résolution V pour 8 facteurs.
4.3

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
## Factor Confounding Rules

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

$$X7 = X3*X4*X5$$

$$X8 = X2*X4*X5$$

## Aliasing Structure

X1

X2

X3

X4

X5

X6

X7

X8

$$X1*X2 = X6*X7$$

$$X1*X3 = X6*X8$$

$$X1*X4$$

$$X1*X5$$

$$X1*X6 = X2*X7 = X3*X8$$

$$X1*X7 = X2*X6$$

$$X1*X8 = X3*X6$$

$$X2*X3 = X7*X8$$

$$X2*X4 = X5*X8$$

$$X2*X5 = X4*X8$$

$$X2*X8 = X3*X7 = X4*X5$$

$$X3*X4 = X5*X7$$

$$X3*X5 = X4*X7$$

$$X4*X6$$

$$X5*X6$$

Solution non acceptable, il reste deux interactions d'intérêt aliasées.
4.4

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>12</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>15</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>16</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>17</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>18</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>19</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>20</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>21</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>23</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>24</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>25</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>26</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>27</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>28</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>29</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>30</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>31</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>32</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>33</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>34</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>35</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>36</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>37</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>38</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>39</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>40</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>41</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>42</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>43</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>44</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>45</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>46</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>47</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>48</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>49</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>50</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>51</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>52</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>53</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>54</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>55</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>56</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>57</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>58</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>59</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>60</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr></tbody></table>
<table><tr><td>61</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>62</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>63</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>64</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></table>

## Factor Confounding Rules

$$X7 = X2 \times X3 \times X4 \times X5 \times X6$$

$$X8 = X1 \times X4 \times X5 \times X6$$

## Aliasing Structure

X1
X2
X3
X4
X5
X6
X7
X8
X1 \times X2
X1 \times X3
X1 \times X4
X1 \times X5
X1 \times X6
X1 \times X7
X1 \times X8
X2 \times X3
X2 \times X4
X2 \times X5
X2 \times X6
X2 \times X7
X2 \times X8
X3 \times X4
X3 \times X5
X3 \times X6
X3 \times X7
X3 \times X8
X4 \times X5
X4 \times X6
X4 \times X7
X4 \times X8
X5 \times X6
X5 \times X7
X5 \times X8
X6 \times X7
X6 \times X8
X7 \times X8

Solution possible Mais peu économique ! !
La bonne solution

Si on examine dans la fraction originelle les colonnes des interactions qui nous intéressent on observe bien entendu qu'elles prennent toujours le même signe pour chaque expérience.

<table><thead><tr><th>OBS</th><th>X1X5</th><th>X2X6</th><th>X3X7</th><th>X4X8</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>14</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

Une solution est d'alors ajouter 4 expériences pour lesquelles les effets X1X5, X2X6, X3X7 et X4X8 varient orthogonalement c'est-à-dire la matrice suivante :

<table><thead><tr><th>OBS</th><th>X1X5</th><th>X2X6</th><th>X3X7</th><th>X4X8</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>3</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

La fraction complémentaire sera donc telle que les signes pris par les facteurs x1 à x8 conduisent pour les interactions d'intérêt aux signes de la matrice précédente :

<table><caption>Experiment</caption><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>3</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
Cependant, l'ajout de cette matrice à 4 expériences à la première matrice de 16 expériences ne constituera pas une matrice orthogonale. L'analyse devra se faire sur les résultats obtenus avec cette dernière, en prenant en compte un éventuel facteur bloc.
## Factor Confounding Rules

$$X3 = A1*A2*X1*X2$$

## Aliasing Structure

A1

A2

X1

X2

X3

A1*A2 = X1*X2*X3

A1*X1 = A2*X2*X3

A1*X2 = A2*X1*X3

A1*X3 = A2*X1*X2

A2*X1 = A1*X2*X3

A2*X2 = A1*X1*X3

A2*X3 = A1*X1*X2

X1*X2 = A1*A2*X3

X1*X3 = A1*A2*X2

X2*X3 = A1*A2*X1

<table><thead><tr><th>OBS</th><th>A1</th><th>A2</th><th>X1</th><th>X2</th><th>X3</th><th>X4 → le facteur à 4 niveaux</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-3</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-3</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-3</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-3</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>9</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>3</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>3</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>3</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td></tr></tbody></table>

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-3</td></tr><tr><td>2</td><td>-1</td><td>1</td><td>-1</td><td>-3</td></tr><tr><td>3</td><td>1</td><td>-1</td><td>-1</td><td>-3</td></tr><tr><td>4</td><td>1</td><td>1</td><td>1</td><td>-3</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>13</td><td>-1</td><td>-1</td><td>1</td><td>3</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>-1</td><td>3</td></tr><tr><td>15</td><td>1</td><td>-1</td><td>-1</td><td>3</td></tr><tr><td>16</td><td>1</td><td>1</td><td>1</td><td>3</td></tr></tbody></table>
→ Le codage {-3, -1, +1, +3} permet de mettre en évidence l'orthogonalité de la matrice du modèle suivante :

Correlation Analysis (Pearson Correlation Coefficients)

<table><thead><tr><th></th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>C12</th></tr></thead><tbody><tr><td>X1</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>C12</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr><tr><td>C13</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C14</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C23</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C24</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C34</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td></td><td>C13</td><td>C14</td><td>C23</td><td>C24</td><td>C34</td></tr><tr><td>X1</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X2</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>X4</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C12</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C13</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C14</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C23</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>C24</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>C34</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>
Programme FACTEX13.SAS

*/***************************FACTEX13.SAS ***************************/
/*** Construction d'une matrice factorielle fractionnaire mixte ***/
/*** par la methode OUTPUT : 2 facteurs a 4 niveaux et 1 facteur ***/
/*** a 2 niveaux ***/
;
options ps=65 ls=75 nodate nocenter;
title ' ' ;
;
proc factex;
factors a1 a2 b1 b2 x3;
model estimate=(b1|b2 x3 a1|a2)
  nonneg=(b1|b2|x3 a1|a2|x3);
size design=32;
output out=mixed [a1 a2]=a nvals=(-3 -1 1 3)
  [b1 b2]=b nvals=(-3 -1 1 3);
examine aliasing(3) confounding design;
proc print data=mixed;
data final; set mixed(drop=a1 a2 b1 b2);
proc print data=final;
run;

data final2; set final;
ab=a*b;ax3=a*x3;bx3=b*x3;
proc corr data=final2;
run;
SORTIE de FACTEX13.SAS

Design Points

Experiment

<table><thead><tr><th>Number</th><th>A1</th><th>A2</th><th>B1</th><th>B2</th><th>X3</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
<table><thead><tr><th>OBS</th><th>A1</th><th>A2</th><th>B1</th><th>B2</th><th>X3</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-3</td><td>-3</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-3</td><td>-3</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-3</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-3</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-3</td><td>-1</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-3</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-3</td><td>3</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-3</td><td>3</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-3</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-3</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>14</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>3</td></tr><tr><td>16</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-3</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-3</td></tr><tr><td>19</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>21</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>23</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>3</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>3</td></tr><tr><td>25</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>3</td><td>-3</td></tr><tr><td>26</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>3</td><td>-3</td></tr><tr><td>27</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>-1</td><td>3</td><td>1</td></tr><tr><td>28</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>1</td><td>3</td><td>1</td></tr><tr><td>29</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>-1</td><td>3</td><td>-1</td></tr><tr><td>30</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>1</td><td>3</td><td>-1</td></tr><tr><td>31</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-1</td><td>3</td><td>3</td></tr><tr><td>32</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td><td>3</td></tr></tbody></table>
<table><thead><tr><th>OBS</th><th>X3</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-3</td><td>-3</td></tr><tr><td>2</td><td>1</td><td>-3</td><td>-3</td></tr><tr><td>3</td><td>-1</td><td>-3</td><td>1</td></tr><tr><td>4</td><td>1</td><td>-3</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>-3</td><td>-1</td></tr><tr><td>6</td><td>1</td><td>-3</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-3</td><td>3</td></tr><tr><td>8</td><td>1</td><td>-3</td><td>3</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-3</td></tr><tr><td>10</td><td>1</td><td>1</td><td>-3</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>12</td><td>1</td><td>1</td><td>1</td></tr><tr><td>13</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>15</td><td>-1</td><td>1</td><td>3</td></tr><tr><td>16</td><td>1</td><td>1</td><td>3</td></tr><tr><td>17</td><td>-1</td><td>-1</td><td>-3</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>-3</td></tr><tr><td>19</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>20</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>21</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>22</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>23</td><td>-1</td><td>-1</td><td>3</td></tr><tr><td>24</td><td>1</td><td>-1</td><td>3</td></tr><tr><td>25</td><td>-1</td><td>3</td><td>-3</td></tr><tr><td>26</td><td>1</td><td>3</td><td>-3</td></tr><tr><td>27</td><td>-1</td><td>3</td><td>1</td></tr><tr><td>28</td><td>1</td><td>3</td><td>1</td></tr><tr><td>29</td><td>-1</td><td>3</td><td>-1</td></tr><tr><td>30</td><td>1</td><td>3</td><td>-1</td></tr><tr><td>31</td><td>-1</td><td>3</td><td>3</td></tr><tr><td>32</td><td>1</td><td>3</td><td>3</td></tr></tbody></table>

Correlation Analysis (Pearson Correlation Coefficients)

<table><thead><tr><th></th><th>X3</th><th>A</th><th>B</th><th>AB</th><th>AX3</th><th>BX3</th></tr></thead><tbody><tr><td>X3</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>A</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>B</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>AB</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>AX3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td><td>0.00000</td></tr><tr><td>BX3</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>1.00000</td></tr></tbody></table>
Programme FACTEX14.SAS

*/*******************************************************************FACTEX14.SAS*******************************************************************/
/* Construction d'une matrice factorielle fractionnaire asymétrique */
/*** par la methode du croisement des niveaux : ***/
*/
/*** 1 facteur a 3 niveaux et 4 facteurs a 2 niveaux ***/
;
options ps=65 ls=75 nodate nocenter;
title ' ' ;
;
proc factex;
factors x1-x4;
model resolution=4;
size design=8;
examine aliasing(2) confounding design;
output out=first;
run;
factors x5/nlev=3;
output out=final designrep=first; → chaque expérience de « first » est répétée pour la matrice formée par « factors x5/nlev=3 » (produit direct)
proc print data=final;
run;
SORTIE de FACTEX14.SAS

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>5</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>6</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>8</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

Factor Confounding Rules

$$X4 = X1*X2*X3$$

Aliasing Structure

X1

X2

X3

X4

$$X1*X2 = X3*X4$$

$$X1*X3 = X2*X4$$

$$X1*X4 = X2*X3$$

Design Points

Experiment

<table><thead><tr><th>Number</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>-1</td></tr><tr><td>2</td><td>0</td></tr><tr><td>3</td><td>1</td></tr></tbody></table>

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>0</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>4</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>5</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>0</td></tr><tr><td>6</td><td>-1</td><td>-1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>7</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>8</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>9</td><td>-1</td><td>1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>11</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>0</td></tr><tr><td>12</td><td>-1</td><td>1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>13</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>14</td><td>1</td><td>-1</td><td>-1</td><td>1</td><td>0</td></tr></tbody></table>
15 1 -1 -1 1 1
16 1 -1 1 -1 -1
17 1 -1 1 -1 0
18 1 -1 1 -1 1
19 1 1 -1 -1 -1
20 1 1 -1 -1 0
21 1 1 -1 -1 1
22 1 1 1 1 -1
23 1 1 1 1 0
24 1 1 1 1 1
Programme FACTEX15.SAS

*/*******************************************************************FACTEX15.SAS*******************************************************************/
/*********************************************************************/
;
/* Construction d'une matrice factorielle fractionnaire asymétrique */
/*** par la methode du croisement des niveaux, avec facteurs de ***/
*/
/*** bruit : 3 facteurs a 2 niveaux et 4 facteurs a 3 niveaux. ***/
;
options ps=65 ls=75 nodate nocenter;
title ' ' ;

proc factex; → pas de rubrique « model » : le plan complet est généré

factors time temperat humidity;
output out=first time nvals=(24 120)
 temperat nvals=(72 150)
 humidity nvals=(.25 .75);
run;
factors interfer connwall idepth glue / nlev=3;

size design=9; → plan gréco-latin

model resolution=3;
output out=final pointrep=first → autre façon de former le produit
direct

interfer cvals=('low' 'medium' 'high' )
connwall cvals=('thin' 'medium' 'thick' )
idepth cvals=('shallow' 'deep' 'medium' )
glue cvals=('low' 'high' 'medium' );

run;
proc print data=final;
run;

*/*******************************************************************/
SORTIE de FACTEX15.SAS

<table><thead><tr><th>OBS</th><th>INTERFER</th><th>CONNWALL</th><th>IDEPTH</th><th>GLUE</th><th>TIME</th><th>TEMPERAT</th><th>HUMIDITY</th></tr></thead><tbody><tr><td>1</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>2</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>3</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>4</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>5</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>6</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>7</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>8</td><td>low</td><td>thin</td><td>shallow</td><td>low</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>9</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>10</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>11</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>12</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>13</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>14</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>15</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>16</td><td>low</td><td>medium</td><td>medium</td><td>medium</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>17</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>18</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>19</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>20</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>21</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>22</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>23</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>24</td><td>low</td><td>thick</td><td>deep</td><td>high</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>25</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>26</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>27</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>28</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>29</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>30</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>31</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>32</td><td>medium</td><td>thin</td><td>medium</td><td>high</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>33</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>34</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>35</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>36</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>37</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>38</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>39</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>40</td><td>medium</td><td>medium</td><td>deep</td><td>low</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>41</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>42</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>43</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>44</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>45</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>46</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>47</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>48</td><td>medium</td><td>thick</td><td>shallow</td><td>medium</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>49</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>50</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>51</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>52</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>53</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>54</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>55</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>56</td><td>high</td><td>thin</td><td>deep</td><td>medium</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>57</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>24</td><td>72</td><td>0.25</td></tr></tbody></table>
<table><tr><td>58</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>59</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>60</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>61</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>120</td><td>72</td><td>0.25</td></tr></table>

<table><thead><tr><th>OBS</th><th>INTERFER</th><th>CONNWALL</th><th>IDEPTH</th><th>GLUE</th><th>TIME</th><th>TEMPERAT</th><th>HUMIDITY</th></tr></thead><tbody><tr><td>62</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>63</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>64</td><td>high</td><td>medium</td><td>shallow</td><td>high</td><td>120</td><td>150</td><td>0.75</td></tr><tr><td>65</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>24</td><td>72</td><td>0.25</td></tr><tr><td>66</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>24</td><td>72</td><td>0.75</td></tr><tr><td>67</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>24</td><td>150</td><td>0.25</td></tr><tr><td>68</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>24</td><td>150</td><td>0.75</td></tr><tr><td>69</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>120</td><td>72</td><td>0.25</td></tr><tr><td>70</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>120</td><td>72</td><td>0.75</td></tr><tr><td>71</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>120</td><td>150</td><td>0.25</td></tr><tr><td>72</td><td>high</td><td>thick</td><td>medium</td><td>low</td><td>120</td><td>150</td><td>0.75</td></tr></tbody></table>
$$ (2*X1) + X3 $$ $$ X1 + (2*X3) $$

$$ (2*X1) + (2*X3) = X2+X4 $$

$$ X1 + X4 = (2*X2) + (2*X3) $$

$$ (2*X1) + X4 $$ $$ X1 + (2*X4) $$

$$ (2*X1) + (2*X4) = X2+X3 $$

$$ (2*X2) + X3 $$ $$ X2 + (2*X3) $$

$$ (2*X2) + X4 $$ $$ X2 + (2*X4) $$

$$ (2*X3) + X4 $$ $$ X3 + (2*X4) $$
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>-1</td><td>0</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>4</td><td>-1</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>5</td><td>-1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>6</td><td>-1</td><td>1</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>-1</td><td>-1</td><td>-1</td><td>0</td></tr><tr><td>8</td><td>-1</td><td>-1</td><td>0</td><td>-1</td></tr><tr><td>9</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>10</td><td>1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>11</td><td>1</td><td>-1</td><td>0</td><td>0</td></tr><tr><td>12</td><td>1</td><td>-1</td><td>1</td><td>-1</td></tr><tr><td>13</td><td>1</td><td>1</td><td>-1</td><td>0</td></tr><tr><td>14</td><td>1</td><td>1</td><td>0</td><td>-1</td></tr><tr><td>15</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>16</td><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>17</td><td>1</td><td>-1</td><td>0</td><td>1</td></tr><tr><td>18</td><td>1</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>19</td><td>-1</td><td>-1</td><td>-1</td><td>0</td></tr><tr><td>20</td><td>-1</td><td>-1</td><td>0</td><td>-1</td></tr><tr><td>21</td><td>-1</td><td>-1</td><td>1</td><td>1</td></tr><tr><td>22</td><td>-1</td><td>1</td><td>-1</td><td>-1</td></tr><tr><td>23</td><td>-1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>24</td><td>-1</td><td>1</td><td>1</td><td>0</td></tr><tr><td>25</td><td>-1</td><td>-1</td><td>-1</td><td>1</td></tr><tr><td>26</td><td>-1</td><td>-1</td><td>0</td><td>0</td></tr><tr><td>27</td><td>-1</td><td>-1</td><td>1</td><td>-1</td></tr></tbody></table>

→ le résultat final est bien une matrice 3/4 puisque la matrice complète comporte $2^2 \times 3^2 = 36$ expériences
Programme FACTEX17.SAS

*/***************************FACTEX17.SAS ***************************/
/*** Construction des carres latins et derives ***/
;
;
options ps=65 ls=75 nodate nocenter;
title ' ';

proc factex;
factors x1-x3 / nlev=3;
size design=9; → carré latin 3x3
model res=3;
output out resul;
examine aliasing(2) confounding;
run;
proc print data resul;
run;

proc factex;
factors x1-x4 / nlev=3;
size design=9; → carré gréco-latin 3x3
model res=3;
output out resul;
examine aliasing(2) confounding;
run;
proc print data resul;
run;

proc factex;
factors x1-x3 / nlev=9;
size design=81; → carré latin 9x9
model res=3;
output out resul;
examine aliasing(2) confounding;
run;
proc print data resul;
run;

proc factex;
factors x1-x10 / nlev=9;
size design=81; → carré hyper-gréco-latin 9x9
model res=3;
output out resul;

run;
proc print data resul;
run;

→ en fait, FACTEX autorise k niveaux avec k = puissance d'un nombre premier :

2¹, 2², ..., 3¹, 3², ... 5¹, 5², ..., 7¹, ..., 11¹, ...
# SORTIE de FACTEX17.SAS

## Factor Confounding Rules

$$X3 = (2*X1) + (2*X2)$$

## Aliasing Structure

$$ 
\begin{aligned}
X1 &= (2*X2) + (2*X3) \\
(2*X1) &= X2 + X3
\end{aligned}
 $$

$$ 
\begin{aligned}
X2 &= (2*X1) + (2*X3) \\
(2*X2) &= X1 + X3
\end{aligned}
 $$

$$ 
\begin{aligned}
X3 &= (2*X1) + (2*X2) \\
(2*X3) &= X1 + X2
\end{aligned}
 $$

$$ 
\begin{aligned}
(2*X1) + X2 &= X1 + (2*X3) = (2*X2) + X3 \\
X1 + (2*X2) &= (2*X1) + X3 = X2 + (2*X3)
\end{aligned}
 $$

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>0</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>4</td><td>0</td><td>-1</td><td>1</td></tr><tr><td>5</td><td>0</td><td>0</td><td>0</td></tr><tr><td>6</td><td>0</td><td>1</td><td>-1</td></tr><tr><td>7</td><td>1</td><td>-1</td><td>0</td></tr><tr><td>8</td><td>1</td><td>0</td><td>-1</td></tr><tr><td>9</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

## Factor Confounding Rules

$$ 
\begin{aligned}
X3 &= (2*X1) + (2*X2) \\
X4 &= X1 + (2*X2)
\end{aligned}
 $$

## Aliasing Structure

$$ 
\begin{aligned}
X1 &= (2*X2) + (2*X3) = X2 + X4 = X3 + (2*X4) \\
(2*X1) &= X2 + X3 = (2*X2) + (2*X4) = (2*X3) + X4
\end{aligned}
 $$

$$ 
\begin{aligned}
X2 &= (2*X1) + (2*X3) = X1 + (2*X4) = X3 + X4 \\
(2*X2) &= X1 + X3 = (2*X1) + X4 = (2*X3) + (2*X4)
\end{aligned}
 $$

$$ 
\begin{aligned}
X3 &= (2*X1) + (2*X2) = X1 + X4 = X2 + (2*X4) \\
(2*X3) &= X1 + X2 = (2*X1) + (2*X4) = (2*X2) + X4
\end{aligned}
 $$

$$ 
\begin{aligned}
X4 &= X1 + (2*X2) = (2*X1) + X3 = X2 + (2*X3) \\
(2*X4) &= (2*X1) + X2 = X1 + (2*X3) = (2*X2) + X3
\end{aligned}
 $$
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>-1</td><td>-1</td></tr><tr><td>2</td><td>-1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>3</td><td>-1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>4</td><td>0</td><td>-1</td><td>1</td><td>0</td></tr><tr><td>5</td><td>0</td><td>0</td><td>0</td><td>-1</td></tr><tr><td>6</td><td>0</td><td>1</td><td>-1</td><td>1</td></tr><tr><td>7</td><td>1</td><td>-1</td><td>0</td><td>1</td></tr><tr><td>8</td><td>1</td><td>0</td><td>-1</td><td>0</td></tr><tr><td>9</td><td>1</td><td>1</td><td>1</td><td>-1</td></tr></tbody></table>

## Factor Confounding Rules

$$X3 = (8*X1) + (8*X2)$$

## Aliasing Structure

$$ 
\begin{aligned}
X1 &= (2*X2) + (6*X3) \\
(2*X1) &= X2 + (3*X3) \\
(3*X1) &= (6*X2) + (5*X3) \\
(4*X1) &= (8*X2) + (2*X3) \\
(5*X1) &= (7*X2) + (8*X3) \\
(6*X1) &= (3*X2) + (7*X3) \\
(7*X1) &= (5*X2) + (4*X3) \\
(8*X1) &= (4*X2) + X3
\end{aligned}
$$

$$ 
\begin{aligned}
X2 &= (2*X1) + (6*X3) \\
(2*X2) &= X1 + (3*X3) \\
(3*X2) &= (6*X1) + (5*X3) \\
(4*X2) &= (8*X1) + (2*X3) \\
(5*X2) &= (7*X1) + (8*X3) \\
(6*X2) &= (3*X1) + (7*X3) \\
(7*X2) &= (5*X1) + (4*X3) \\
(8*X2) &= (4*X1) + X3
\end{aligned}
$$

$$ 
\begin{aligned}
X3 &= (8*X1) + (8*X2) \\
(2*X3) &= (4*X1) + (4*X2) \\
(3*X3) &= (2*X1) + (2*X2) \\
(4*X3) &= (7*X1) + (7*X2) \\
(5*X3) &= (3*X1) + (3*X2) \\
(6*X3) &= X1 + X2 \\
(7*X3) &= (6*X1) + (6*X2) \\
(8*X3) &= (5*X1) + (5*X2)
\end{aligned}
$$

$$ 
\begin{aligned}
(2*X1) + X2 &= X1 + (6*X3) = (2*X2) + (3*X3) \\
(3*X1) + X2 &= (5*X1) + (6*X3) = (7*X2) + (5*X3) \\
(4*X1) + X2 &= (3*X1) + (6*X3) = (6*X2) + (2*X3) \\
(5*X1) + X2 &= (4*X1) + (6*X3) = (8*X2) + (8*X3) \\
(6*X1) + X2 &= (8*X1) + (6*X3) = (4*X2) + (7*X3) \\
(7*X1) + X2 &= (6*X1) + (6*X3) = (3*X2) + (4*X3) \\
(8*X1) + X2 &= (7*X1) + (6*X3) = (5*X2) + X3 \\
X1 + (2*X2) &= (2*X1) + (3*X3) = X2 + (6*X3) \\
(3*X1) + (2*X2) &= (4*X1) + (3*X3) = (8*X2) + (5*X3) \\
(4*X1) + (2*X2) &= (5*X1) + (3*X3) = (7*X2) + (2*X3) \\
(5*X1) + (2*X2) &= (3*X1) + (3*X3) = (6*X2) + (8*X3) \\
(6*X1) + (2*X2) &= (7*X1) + (3*X3) = (5*X2) + (7*X3) \\
(7*X1) + (2*X2) &= (8*X1) + (3*X3) = (4*X2) + (4*X3) \\
(8*X1) + (2*X2) &= (6*X1) + (3*X3) = (3*X2) + X3 \\
X1 + (3*X2) &= (7*X1) + (5*X3) = (5*X2) + (6*X3) \\
(2*X1) + (3*X2) &= (8*X1) + (5*X3) = (4*X2) + (3*X3)
\end{aligned}
$$
$$ (4*X1) + (3*X2) = X1 + (5*X3) = (2*X2) + (2*X3) $$

$$ (5*X1) + (3*X2) = (2*X1) + (5*X3) = X2 + (8*X3) $$

$$ (6*X1) + (3*X2) = (3*X1) + (5*X3) = (6*X2) + (7*X3) $$

$$ (7*X1) + (3*X2) = (4*X1) + (5*X3) = (8*X2) + (4*X3) $$

$$ (8*X1) + (3*X2) = (5*X1) + (5*X3) = (7*X2) + X3 $$

$$ X1 + (4*X2) = (6*X1) + (2*X3) = (3*X2) + (6*X3) $$

$$ (2*X1) + (4*X2) = (7*X1) + (2*X3) = (5*X2) + (3*X3) $$

$$ (3*X1) + (4*X2) = (2*X1) + (2*X3) = X2 + (5*X3) $$

$$ (5*X1) + (4*X2) = X1 + (2*X3) = (2*X2) + (8*X3) $$

$$ (6*X1) + (4*X2) = (5*X1) + (2*X3) = (7*X2) + (7*X3) $$

$$ (7*X1) + (4*X2) = (3*X1) + (2*X3) = (6*X2) + (4*X3) $$

$$ (8*X1) + (4*X2) = (4*X1) + (2*X3) = (8*X2) + X3 $$

$$ X1 + (5*X2) = (8*X1) + (8*X3) = (4*X2) + (6*X3) $$

$$ (2*X1) + (5*X2) = (6*X1) + (8*X3) = (3*X2) + (3*X3) $$

$$ (3*X1) + (5*X2) = X1 + (8*X3) = (2*X2) + (5*X3) $$

$$ (4*X1) + (5*X2) = (2*X1) + (8*X3) = X2 + (2*X3) $$

$$ (6*X1) + (5*X2) = (4*X1) + (8*X3) = (8*X2) + (7*X3) $$

$$ (7*X1) + (5*X2) = (5*X1) + (8*X3) = (7*X2) + (4*X3) $$

$$ (8*X1) + (5*X2) = (3*X1) + (8*X3) = (6*X2) + X3 $$

$$ X1 + (6*X2) = (4*X1) + (7*X3) = (8*X2) + (6*X3) $$

$$ (2*X1) + (6*X2) = (5*X1) + (7*X3) = (7*X2) + (3*X3) $$

$$ (3*X1) + (6*X2) = (6*X1) + (7*X3) = (3*X2) + (5*X3) $$

$$ (4*X1) + (6*X2) = (7*X1) + (7*X3) = (5*X2) + (2*X3) $$

$$ (5*X1) + (6*X2) = (8*X1) + (7*X3) = (4*X2) + (8*X3) $$

$$ (7*X1) + (6*X2) = X1 + (7*X3) = (2*X2) + (4*X3) $$

$$ (8*X1) + (6*X2) = (2*X1) + (7*X3) = X2 + X3 $$

$$ X1 + (7*X2) = (3*X1) + (4*X3) = (6*X2) + (6*X3) $$

$$ (2*X1) + (7*X2) = (4*X1) + (4*X3) = (8*X2) + (3*X3) $$

$$ (3*X1) + (7*X2) = (8*X1) + (4*X3) = (4*X2) + (5*X3) $$

$$ (4*X1) + (7*X2) = (6*X1) + (4*X3) = (3*X2) + (2*X3) $$

$$ (5*X1) + (7*X2) = (7*X1) + (4*X3) = (5*X2) + (8*X3) $$

$$ (6*X1) + (7*X2) = (2*X1) + (4*X3) = X2 + (7*X3) $$

$$ (8*X1) + (7*X2) = X1 + (4*X3) = (2*X2) + X3 $$

$$ X1 + (8*X2) = (5*X1) + X3 = (7*X2) + (6*X3) $$

$$ (2*X1) + (8*X2) = (3*X1) + X3 = (6*X2) + (3*X3) $$

$$ (3*X1) + (8*X2) = (7*X1) + X3 = (5*X2) + (5*X3) $$

$$ (4*X1) + (8*X2) = (8*X1) + X3 = (4*X2) + (2*X3) $$

$$ (5*X1) + (8*X2) = (6*X1) + X3 = (3*X2) + (8*X3) $$

$$ (6*X1) + (8*X2) = X1 + X3 = (2*X2) + (7*X3) $$

$$ (7*X1) + (8*X2) = (2*X1) + X3 = X2 + (4*X3) $$
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2</td><td>0</td><td>1</td><td>8</td></tr><tr><td>3</td><td>0</td><td>2</td><td>4</td></tr><tr><td>4</td><td>0</td><td>3</td><td>2</td></tr><tr><td>5</td><td>0</td><td>4</td><td>7</td></tr><tr><td>6</td><td>0</td><td>5</td><td>3</td></tr><tr><td>7</td><td>0</td><td>6</td><td>1</td></tr><tr><td>8</td><td>0</td><td>7</td><td>6</td></tr><tr><td>9</td><td>0</td><td>8</td><td>5</td></tr><tr><td>10</td><td>1</td><td>0</td><td>8</td></tr><tr><td>11</td><td>1</td><td>1</td><td>4</td></tr><tr><td>12</td><td>1</td><td>2</td><td>0</td></tr><tr><td>13</td><td>1</td><td>3</td><td>7</td></tr><tr><td>14</td><td>1</td><td>4</td><td>3</td></tr><tr><td>15</td><td>1</td><td>5</td><td>2</td></tr><tr><td>16</td><td>1</td><td>6</td><td>6</td></tr><tr><td>17</td><td>1</td><td>7</td><td>5</td></tr><tr><td>18</td><td>1</td><td>8</td><td>1</td></tr><tr><td>19</td><td>2</td><td>0</td><td>4</td></tr><tr><td>20</td><td>2</td><td>1</td><td>0</td></tr><tr><td>21</td><td>2</td><td>2</td><td>8</td></tr><tr><td>22</td><td>2</td><td>3</td><td>3</td></tr><tr><td>23</td><td>2</td><td>4</td><td>2</td></tr><tr><td>24</td><td>2</td><td>5</td><td>7</td></tr><tr><td>25</td><td>2</td><td>6</td><td>5</td></tr><tr><td>26</td><td>2</td><td>7</td><td>1</td></tr><tr><td>27</td><td>2</td><td>8</td><td>6</td></tr><tr><td>28</td><td>3</td><td>0</td><td>2</td></tr><tr><td>29</td><td>3</td><td>1</td><td>7</td></tr><tr><td>30</td><td>3</td><td>2</td><td>3</td></tr><tr><td>31</td><td>3</td><td>3</td><td>1</td></tr><tr><td>32</td><td>3</td><td>4</td><td>6</td></tr><tr><td>33</td><td>3</td><td>5</td><td>5</td></tr><tr><td>34</td><td>3</td><td>6</td><td>0</td></tr><tr><td>35</td><td>3</td><td>7</td><td>8</td></tr><tr><td>36</td><td>3</td><td>8</td><td>4</td></tr><tr><td>37</td><td>4</td><td>0</td><td>7</td></tr><tr><td>38</td><td>4</td><td>1</td><td>3</td></tr><tr><td>39</td><td>4</td><td>2</td><td>2</td></tr><tr><td>40</td><td>4</td><td>3</td><td>6</td></tr><tr><td>41</td><td>4</td><td>4</td><td>5</td></tr><tr><td>42</td><td>4</td><td>5</td><td>1</td></tr><tr><td>43</td><td>4</td><td>6</td><td>8</td></tr><tr><td>44</td><td>4</td><td>7</td><td>4</td></tr><tr><td>45</td><td>4</td><td>8</td><td>0</td></tr><tr><td>46</td><td>5</td><td>0</td><td>3</td></tr><tr><td>47</td><td>5</td><td>1</td><td>2</td></tr><tr><td>48</td><td>5</td><td>2</td><td>7</td></tr><tr><td>49</td><td>5</td><td>3</td><td>5</td></tr><tr><td>50</td><td>5</td><td>4</td><td>1</td></tr><tr><td>51</td><td>5</td><td>5</td><td>6</td></tr><tr><td>52</td><td>5</td><td>6</td><td>4</td></tr><tr><td>53</td><td>5</td><td>7</td><td>0</td></tr><tr><td>54</td><td>5</td><td>8</td><td>8</td></tr><tr><td>55</td><td>6</td><td>0</td><td>1</td></tr><tr><td>56</td><td>6</td><td>1</td><td>6</td></tr><tr><td>57</td><td>6</td><td>2</td><td>5</td></tr><tr><td>58</td><td>6</td><td>3</td><td>0</td></tr><tr><td>59</td><td>6</td><td>4</td><td>8</td></tr><tr><td>60</td><td>6</td><td>5</td><td>4</td></tr><tr><td>61</td><td>6</td><td>6</td><td>2</td></tr></tbody></table>
<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th></tr></thead><tbody><tr><td>62</td><td>6</td><td>7</td><td>7</td></tr><tr><td>63</td><td>6</td><td>8</td><td>3</td></tr><tr><td>64</td><td>7</td><td>0</td><td>6</td></tr><tr><td>65</td><td>7</td><td>1</td><td>5</td></tr><tr><td>66</td><td>7</td><td>2</td><td>1</td></tr><tr><td>67</td><td>7</td><td>3</td><td>8</td></tr><tr><td>68</td><td>7</td><td>4</td><td>4</td></tr><tr><td>69</td><td>7</td><td>5</td><td>0</td></tr><tr><td>70</td><td>7</td><td>6</td><td>7</td></tr><tr><td>71</td><td>7</td><td>7</td><td>3</td></tr><tr><td>72</td><td>7</td><td>8</td><td>2</td></tr><tr><td>73</td><td>8</td><td>0</td><td>5</td></tr><tr><td>74</td><td>8</td><td>1</td><td>1</td></tr><tr><td>75</td><td>8</td><td>2</td><td>6</td></tr><tr><td>76</td><td>8</td><td>3</td><td>4</td></tr><tr><td>77</td><td>8</td><td>4</td><td>0</td></tr><tr><td>78</td><td>8</td><td>5</td><td>8</td></tr><tr><td>79</td><td>8</td><td>6</td><td>3</td></tr><tr><td>80</td><td>8</td><td>7</td><td>2</td></tr><tr><td>81</td><td>8</td><td>8</td><td>7</td></tr></tbody></table>

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th><th>X9</th><th>X10</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2</td><td>0</td><td>1</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>3</td><td>0</td><td>2</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr><tr><td>4</td><td>0</td><td>3</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>5</td><td>0</td><td>4</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td></tr><tr><td>6</td><td>0</td><td>5</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>7</td><td>0</td><td>6</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>8</td><td>0</td><td>7</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>9</td><td>0</td><td>8</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>10</td><td>1</td><td>0</td><td>8</td><td>7</td><td>6</td><td>5</td><td>4</td><td>3</td><td>2</td><td>1</td></tr><tr><td>11</td><td>1</td><td>1</td><td>4</td><td>3</td><td>5</td><td>1</td><td>0</td><td>2</td><td>7</td><td>6</td></tr><tr><td>12</td><td>1</td><td>2</td><td>0</td><td>2</td><td>1</td><td>6</td><td>8</td><td>7</td><td>3</td><td>5</td></tr><tr><td>13</td><td>1</td><td>3</td><td>7</td><td>6</td><td>8</td><td>4</td><td>3</td><td>5</td><td>1</td><td>0</td></tr><tr><td>14</td><td>1</td><td>4</td><td>3</td><td>5</td><td>4</td><td>0</td><td>2</td><td>1</td><td>6</td><td>8</td></tr><tr><td>15</td><td>1</td><td>5</td><td>2</td><td>1</td><td>0</td><td>8</td><td>7</td><td>6</td><td>5</td><td>4</td></tr><tr><td>16</td><td>1</td><td>6</td><td>6</td><td>8</td><td>7</td><td>3</td><td>5</td><td>4</td><td>0</td><td>2</td></tr><tr><td>17</td><td>1</td><td>7</td><td>5</td><td>4</td><td>3</td><td>2</td><td>1</td><td>0</td><td>8</td><td>7</td></tr><tr><td>18</td><td>1</td><td>8</td><td>1</td><td>0</td><td>2</td><td>7</td><td>6</td><td>8</td><td>4</td><td>3</td></tr><tr><td>19</td><td>2</td><td>0</td><td>4</td><td>5</td><td>3</td><td>7</td><td>8</td><td>6</td><td>1</td><td>2</td></tr><tr><td>20</td><td>2</td><td>1</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td></tr><tr><td>21</td><td>2</td><td>2</td><td>8</td><td>6</td><td>7</td><td>2</td><td>0</td><td>1</td><td>5</td><td>3</td></tr><tr><td>22</td><td>2</td><td>3</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>0</td><td>1</td></tr><tr><td>23</td><td>2</td><td>4</td><td>2</td><td>0</td><td>1</td><td>5</td><td>3</td><td>4</td><td>8</td><td>6</td></tr><tr><td>24</td><td>2</td><td>5</td><td>7</td><td>8</td><td>6</td><td>1</td><td>2</td><td>0</td><td>4</td><td>5</td></tr><tr><td>25</td><td>2</td><td>6</td><td>5</td><td>3</td><td>4</td><td>8</td><td>6</td><td>7</td><td>2</td><td>0</td></tr><tr><td>26</td><td>2</td><td>7</td><td>1</td><td>2</td><td>0</td><td>4</td><td>5</td><td>3</td><td>7</td><td>8</td></tr><tr><td>27</td><td>2</td><td>8</td><td>6</td><td>7</td><td>8</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>28</td><td>3</td><td>0</td><td>2</td><td>8</td><td>5</td><td>4</td><td>1</td><td>7</td><td>6</td><td>3</td></tr><tr><td>29</td><td>3</td><td>1</td><td>7</td><td>4</td><td>1</td><td>0</td><td>6</td><td>3</td><td>5</td><td>2</td></tr><tr><td>30</td><td>3</td><td>2</td><td>3</td><td>0</td><td>6</td><td>8</td><td>5</td><td>2</td><td>1</td><td>7</td></tr><tr><td>31</td><td>3</td><td>3</td><td>1</td><td>7</td><td>4</td><td>3</td><td>0</td><td>6</td><td>8</td><td>5</td></tr><tr><td>32</td><td>3</td><td>4</td><td>6</td><td>3</td><td>0</td><td>2</td><td>8</td><td>5</td><td>4</td><td>1</td></tr><tr><td>33</td><td>3</td><td>5</td><td>5</td><td>2</td><td>8</td><td>7</td><td>4</td><td>1</td><td>0</td><td>6</td></tr><tr><td>34</td><td>3</td><td>6</td><td>0</td><td>6</td><td>3</td><td>5</td><td>2</td><td>8</td><td>7</td><td>4</td></tr><tr><td>35</td><td>3</td><td>7</td><td>8</td><td>5</td><td>2</td><td>1</td><td>7</td><td>4</td><td>3</td><td>0</td></tr><tr><td>36</td><td>3</td><td>8</td><td>4</td><td>1</td><td>7</td><td>6</td><td>3</td><td>0</td><td>2</td><td>8</td></tr></tbody></table>
<table><tr><td>37</td><td>4</td><td>0</td><td>7</td><td>3</td><td>2</td><td>6</td><td>5</td><td>1</td><td>8</td><td>4</td></tr><tr><td>38</td><td>4</td><td>1</td><td>3</td><td>2</td><td>7</td><td>5</td><td>1</td><td>6</td><td>4</td><td>0</td></tr><tr><td>39</td><td>4</td><td>2</td><td>2</td><td>7</td><td>3</td><td>1</td><td>6</td><td>5</td><td>0</td><td>8</td></tr><tr><td>40</td><td>4</td><td>3</td><td>6</td><td>5</td><td>1</td><td>8</td><td>4</td><td>0</td><td>7</td><td>3</td></tr><tr><td>41</td><td>4</td><td>4</td><td>5</td><td>1</td><td>6</td><td>4</td><td>0</td><td>8</td><td>3</td><td>2</td></tr><tr><td>42</td><td>4</td><td>5</td><td>1</td><td>6</td><td>5</td><td>0</td><td>8</td><td>4</td><td>2</td><td>7</td></tr><tr><td>43</td><td>4</td><td>6</td><td>8</td><td>4</td><td>0</td><td>7</td><td>3</td><td>2</td><td>6</td><td>5</td></tr><tr><td>44</td><td>4</td><td>7</td><td>4</td><td>0</td><td>8</td><td>3</td><td>2</td><td>7</td><td>5</td><td>1</td></tr><tr><td>45</td><td>4</td><td>8</td><td>0</td><td>8</td><td>4</td><td>2</td><td>7</td><td>3</td><td>1</td><td>6</td></tr><tr><td>46</td><td>5</td><td>0</td><td>3</td><td>1</td><td>8</td><td>2</td><td>6</td><td>4</td><td>7</td><td>5</td></tr><tr><td>47</td><td>5</td><td>1</td><td>2</td><td>6</td><td>4</td><td>7</td><td>5</td><td>0</td><td>3</td><td>1</td></tr><tr><td>48</td><td>5</td><td>2</td><td>7</td><td>5</td><td>0</td><td>3</td><td>1</td><td>8</td><td>2</td><td>6</td></tr><tr><td>49</td><td>5</td><td>3</td><td>5</td><td>0</td><td>7</td><td>1</td><td>8</td><td>3</td><td>6</td><td>4</td></tr><tr><td>50</td><td>5</td><td>4</td><td>1</td><td>8</td><td>3</td><td>6</td><td>4</td><td>2</td><td>5</td><td>0</td></tr><tr><td>51</td><td>5</td><td>5</td><td>6</td><td>4</td><td>2</td><td>5</td><td>0</td><td>7</td><td>1</td><td>8</td></tr><tr><td>52</td><td>5</td><td>6</td><td>4</td><td>2</td><td>6</td><td>0</td><td>7</td><td>5</td><td>8</td><td>3</td></tr><tr><td>53</td><td>5</td><td>7</td><td>0</td><td>7</td><td>5</td><td>8</td><td>3</td><td>1</td><td>4</td><td>2</td></tr><tr><td>54</td><td>5</td><td>8</td><td>8</td><td>3</td><td>1</td><td>4</td><td>2</td><td>6</td><td>0</td><td>7</td></tr><tr><td>55</td><td>6</td><td>0</td><td>1</td><td>4</td><td>7</td><td>8</td><td>2</td><td>5</td><td>3</td><td>6</td></tr><tr><td>56</td><td>6</td><td>1</td><td>6</td><td>0</td><td>3</td><td>4</td><td>7</td><td>1</td><td>2</td><td>5</td></tr><tr><td>57</td><td>6</td><td>2</td><td>5</td><td>8</td><td>2</td><td>0</td><td>3</td><td>6</td><td>7</td><td>1</td></tr><tr><td>58</td><td>6</td><td>3</td><td>0</td><td>3</td><td>6</td><td>7</td><td>1</td><td>4</td><td>5</td><td>8</td></tr><tr><td>59</td><td>6</td><td>4</td><td>8</td><td>2</td><td>5</td><td>3</td><td>6</td><td>0</td><td>1</td><td>4</td></tr><tr><td>60</td><td>6</td><td>5</td><td>4</td><td>7</td><td>1</td><td>2</td><td>5</td><td>8</td><td>6</td><td>0</td></tr><tr><td>61</td><td>6</td><td>6</td><td>2</td><td>5</td><td>8</td><td>6</td><td>0</td><td>3</td><td>4</td><td>7</td></tr></table>

<table><thead><tr><th>OBS</th><th>X1</th><th>X2</th><th>X3</th><th>X4</th><th>X5</th><th>X6</th><th>X7</th><th>X8</th><th>X9</th><th>X10</th></tr></thead><tbody><tr><td>62</td><td>6</td><td>7</td><td>7</td><td>1</td><td>4</td><td>5</td><td>8</td><td>2</td><td>0</td><td>3</td></tr><tr><td>63</td><td>6</td><td>8</td><td>3</td><td>6</td><td>0</td><td>1</td><td>4</td><td>7</td><td>8</td><td>2</td></tr><tr><td>64</td><td>7</td><td>0</td><td>6</td><td>2</td><td>4</td><td>1</td><td>3</td><td>8</td><td>5</td><td>7</td></tr><tr><td>65</td><td>7</td><td>1</td><td>5</td><td>7</td><td>0</td><td>6</td><td>2</td><td>4</td><td>1</td><td>3</td></tr><tr><td>66</td><td>7</td><td>2</td><td>1</td><td>3</td><td>8</td><td>5</td><td>7</td><td>0</td><td>6</td><td>2</td></tr><tr><td>67</td><td>7</td><td>3</td><td>8</td><td>1</td><td>3</td><td>0</td><td>5</td><td>7</td><td>4</td><td>6</td></tr><tr><td>68</td><td>7</td><td>4</td><td>4</td><td>6</td><td>2</td><td>8</td><td>1</td><td>3</td><td>0</td><td>5</td></tr><tr><td>69</td><td>7</td><td>5</td><td>0</td><td>5</td><td>7</td><td>4</td><td>6</td><td>2</td><td>8</td><td>1</td></tr><tr><td>70</td><td>7</td><td>6</td><td>7</td><td>0</td><td>5</td><td>2</td><td>4</td><td>6</td><td>3</td><td>8</td></tr><tr><td>71</td><td>7</td><td>7</td><td>3</td><td>8</td><td>1</td><td>7</td><td>0</td><td>5</td><td>2</td><td>4</td></tr><tr><td>72</td><td>7</td><td>8</td><td>2</td><td>4</td><td>6</td><td>3</td><td>8</td><td>1</td><td>7</td><td>0</td></tr><tr><td>73</td><td>8</td><td>0</td><td>5</td><td>6</td><td>1</td><td>3</td><td>7</td><td>2</td><td>4</td><td>8</td></tr><tr><td>74</td><td>8</td><td>1</td><td>1</td><td>5</td><td>6</td><td>2</td><td>3</td><td>7</td><td>0</td><td>4</td></tr><tr><td>75</td><td>8</td><td>2</td><td>6</td><td>1</td><td>5</td><td>7</td><td>2</td><td>3</td><td>8</td><td>0</td></tr><tr><td>76</td><td>8</td><td>3</td><td>4</td><td>8</td><td>0</td><td>5</td><td>6</td><td>1</td><td>3</td><td>7</td></tr><tr><td>77</td><td>8</td><td>4</td><td>0</td><td>4</td><td>8</td><td>1</td><td>5</td><td>6</td><td>2</td><td>3</td></tr><tr><td>78</td><td>8</td><td>5</td><td>8</td><td>0</td><td>4</td><td>6</td><td>1</td><td>5</td><td>7</td><td>2</td></tr><tr><td>79</td><td>8</td><td>6</td><td>3</td><td>7</td><td>2</td><td>4</td><td>8</td><td>0</td><td>5</td><td>6</td></tr><tr><td>80</td><td>8</td><td>7</td><td>2</td><td>3</td><td>7</td><td>0</td><td>4</td><td>8</td><td>1</td><td>5</td></tr><tr><td>81</td><td>8</td><td>8</td><td>7</td><td>2</td><td>3</td><td>8</td><td>0</td><td>4</td><td>6</td><td>1</td></tr></tbody></table>
Programme FACTEX18.SAS

*/***************************FACTEX18.SAS ***************************/
/*** Plan en blocs complètement randomisés ***/
;
;
options ps=65 ls=75 nodate nocenter;
title ' ';

proc factex;
factors bloc / nlev=3;
output out=genbloc bloc nvals=(1 2 3) randomize; → on randomise les 3 valeurs 1,2,3
run;

factors trait / nlev=10;
output out=pbcr
trait cvals=('a' 'b' 'c' 'd' 'e' 'f' 'g' 'h' 'j' 'k')
designrep=genbloc randomize; → les 10 lettres seront randomisées dans chaque bloc

run;

proc print data=pbcr;
run;
SORTIE de FACTEX18.SAS

<table><thead><tr><td>OBS</td><td>BLOC</td><td>TRAIT</td></tr></thead><tbody><tr><td>1</td><td>2</td><td>d</td></tr><tr><td>2</td><td>2</td><td>g</td></tr><tr><td>3</td><td>2</td><td>h</td></tr><tr><td>4</td><td>2</td><td>f</td></tr><tr><td>5</td><td>2</td><td>j</td></tr><tr><td>6</td><td>2</td><td>b</td></tr><tr><td>7</td><td>2</td><td>c</td></tr><tr><td>8</td><td>2</td><td>e</td></tr><tr><td>9</td><td>2</td><td>k</td></tr><tr><td>10</td><td>2</td><td>a</td></tr><tr><td>11</td><td>1</td><td>f</td></tr><tr><td>12</td><td>1</td><td>b</td></tr><tr><td>13</td><td>1</td><td>a</td></tr><tr><td>14</td><td>1</td><td>d</td></tr><tr><td>15</td><td>1</td><td>k</td></tr><tr><td>16</td><td>1</td><td>e</td></tr><tr><td>17</td><td>1</td><td>j</td></tr><tr><td>18</td><td>1</td><td>c</td></tr><tr><td>19</td><td>1</td><td>h</td></tr><tr><td>20</td><td>1</td><td>g</td></tr><tr><td>21</td><td>3</td><td>k</td></tr><tr><td>22</td><td>3</td><td>d</td></tr><tr><td>23</td><td>3</td><td>b</td></tr><tr><td>24</td><td>3</td><td>h</td></tr><tr><td>25</td><td>3</td><td>a</td></tr><tr><td>26</td><td>3</td><td>g</td></tr><tr><td>27</td><td>3</td><td>f</td></tr><tr><td>28</td><td>3</td><td>c</td></tr><tr><td>29</td><td>3</td><td>e</td></tr><tr><td>30</td><td>3</td><td>j</td></tr></tbody></table>