**********************************************************;
* A `Saturated' 8-run design extended by MIRRORING or
*   FOLDING IN ALL COLUMNS
*
* The basic 2_{III}^{7-4} fractional factorial design has 8 runs
*   with High/Low assignments for 7 factors A B C D E F G given by
*
* Dummy:   a   b   c   ab   ac   bc   abc
* Factor:  A   B   C    D    E    F    G
*   (1)   -1  -1  -1    1    1    1   -1
*   (2)    1  -1  -1   -1   -1    1    1
*   (3)   -1   1  -1   -1    1   -1    1
*   (4)    1   1  -1    1   -1   -1   -1
*   (5)   -1  -1   1   -1   -1   -1    1
*   (6)    1  -1   1    1    1   -1   -1
*   (7)   -1   1   1    1   -1    1   -1
*   (8)    1   1   1   -1    1    1    1
*
* (See FracFac74.sas.) This has the basic defining relations
*
*      D=AB  E=AC  F=BC
*
* with a complete set of defining relations
*
*   I = ABD = ACE = BCF = CDG = BEG = AFG = DEF
*    = ABCG = ABEF = ACDF = ADEG = BCDE = BDFG = CEFG = ABCDEFG
*
* We can augment this design by MIRRORING or FOLDING IN ALL COLUMNS
*   by adding 8 more runs with assignments
*
* Dummy:   -a  -b  -c  -ab  -ac  -bc  -abc
* Factor:   A   B   C    D    E    F    G
*    (9)    1   1   1   -1   -1   -1    1
*   (10)   -1   1   1    1    1   -1   -1
*   (11)    1  -1   1    1   -1    1   -1
*   (12)   -1  -1   1   -1    1    1    1
*   (13)    1   1  -1    1    1    1   -1
*   (14)   -1   1  -1   -1   -1    1    1
*   (15)    1  -1  -1   -1    1   -1    1
*   (16)   -1  -1  -1    1   -1   -1   -1
*
* Each 16x1 column in the augmented design is of the form
*   (A(over) -A)' where A is an 8x1 column. If (for example)
*
*      (A1)(A2)...(Ab)=I
*
*   is a relation in the 8-run design, then
*
*     (A1(over -A1)(A2(over) -A2)... =  (I(over) (-1)^b I)
*
*   is the corresponding identity in the 16-run design. This
*   means that the defining relations for the 16-run design
*   are exactly the relations of EVEN LENGTH in the 8-run
*   design, or
*
*      I = ABCG = ABEF = ACDF = ADEG = BCDE = BDFG
*
* In particular, the 16-run fully mirrored 2^{7-3} design has
*   resolution IV, or is a 2_{IV}^{7-3} design.
*
* This means that estimates of main effects are confounded at worst
*   with 3-way interactions, and 2-way interactions are confounded
*   at worst only with 2-way interactions. Ignoring 3-way and
*   higher interactions, we obtain unaliased estimates of
*
*     (Main effects) A  B  C  D  E  F  G
*     (Two-way interactions)
*     BD = CE = FG
*     AD = CF = EG
*     AE = BF = DG
*     AB = EF = CG
*     AC = DF = BG
*     BC = DE = AG
*     CD = BE = AF
*
* We apply this to an experiments at a water filtration plant
*   with 7 factors
*
*      A=Water_Supply (Low=Reservoir  H=Well)
*      B=Raw_Material (on site or off site)
*      C=Temperature
*      D=Recycle
*      E=Caustic_Soda  (Fast or slow)
*      F=Filter_Cloth
*      G=Holdup_Time
*
* The response variable was filtration time, so that low values
*   were desirable. Observation of the data for the first 8 runs
*   shows that the two lowest times were only with
*
*      A=C=E=High
*
* This suggested that the factors A C E were active. However, due to
*   the confounding patterns in the basic 2_{III}^{7-3} design, it
*   was possible that only two of  A C E  were active and estimates
*   of the third main effect were confounded with a significant
*   interaction of the two other effects. This suggested four
*   distinct possibilities for major effects that might have
*   led to the observed output:
*
*      (i) A C E    (ii) A C AC    (iii) A E AE    (iv)  C E CE
*
* The 8-run 2_{III}^{7-3} was augmented by 8 more runs to form the
*   fully mirrored Resolution IV 2_{IV}^{7-3} design. This allowed
*   estimates of main effects and their interactions to be
*   distinguished, and suggested case (iii) above.
*
* See Tables 6.10-6.13 (pages 252-257) in text
*
**********************************************************;

options ls=75 ps=60 nocenter pageno=1;

title1 'A 2_{III}^{7-4} design with 8 observations - YOUR NAME';
title2 'Analysis of the basic design suggests 4 distinct possibilities';
title3 '   (i) A C E    (ii) A C AC    (iii) A E AE    (iv)  C E CE';
title4 'for major effects. We do 8 more runs to obtain a 16-run';
title5 '  2_{IV}^{7-3} design that strongly suggests (iii).';
title6 'See text Tables 6.10 p253 and Table 6.12 p254';

data filtration;
     * yy1 are observations for first set of 8 runs;
     * yy2 are observations for the second set of 8 runs ;
     input Row  A B C  yy1 yy2;
     * Define High/Low relations for seven factors;
     D=A*B;  E=A*C;  F=B*C;  G=A*B*C;
     * Column yy1 is the first group of observations;
     Yield=yy1;  Rep=1;  output;
     * Column yy2 is the second group of 8 observations ;
     *  resulting from foldover in all columns ;
     A=-A;  B=-B;  C=-C;  D=-D;  E=-E;  F=-F;  G=-G;
     Yield=yy2;  Rep=2;  Row=Row+8;  output;
     * Descriptive names for the 7 factors;
     label A=Water_Supply  B=Raw_Material  C=Temperature  D=Recycle
       E=Caustic_Soda  F=Filter_Cloth  G=Holdup_Time;
datalines;
   1   -1  -1  -1    68.4   66.7
   2    1  -1  -1    77.7   65.0
   3   -1   1  -1    66.4   86.4
   4    1   1  -1    81.0   61.9
   5   -1  -1   1    78.6   47.8
   6    1  -1   1    41.2   59.0
   7   -1   1   1    68.7   42.6
   8    1   1   1    38.7   67.6
    run;


**********************************************************;
* Sort to keep the rows and replications in the proper order
**********************************************************;

proc sort;
     by Rep Row;  run;

proc print;
     title6 'The data as SAS sees it:';
     title7 '  D E F G  are functions of A B C';
     title8 'Levels are reflected in all columns in the second replication';
     run;

title2 'In the first replication, A, C, and E all seem to be large.';
title3 'The saturated 2_{iii}^{7-3} design has confounding relations';
title4 '  A=A+BD+CE+FG   B=B+AD+CF+EG   C=C+AE+BF+DG,';
title5 '  D=D+AB+CG+EF   E=E+AC+BG+DF   F=F+AG+BC+DE  G=G+AF+BE+CD';
title6 'Thus E could be an artifact of AC,  C of AE,  A of CE,  or';
title7 '  possibly only the main effects are important. Which of the four?';
title8 'Foldover in all columns allows us to separate the odd-order (main)';
title9 '  effects from the even-order (2-way) effects';

**********************************************************;
* Estimate parameters with and without foldover
* `by Rep' tells SAS to estimate regression parameters twice,
*   once within each replication
*
* The command `outest=parmrows in `proc reg' tells SAS to write
*   the effect parameter estimates for the two replications
*   to `data=parmrows as two rows.
*
* See FracFac74.sas for more comments.
**********************************************************;

proc reg data=filtration outest=parmrows;
     by Rep;
     model Yield = A B C D E F G;
     run;

data parmrows;
     set parmrows;
     if Rep=1 then Name="Parms1";
     else if Rep=2 then Name="Parms2";
     run;

proc transpose data=parmrows out=coldata name=Effect;
     id Name;
     run;

data coldata;
     set coldata;
     Book1=2*Parms1;  Book2=2*Parms2;
     if Effect ne "Intercept" and Effect ne "_RMSE_"
        and Effect ne "Yield" and Effect ne "Rep" then output;
     run;

proc print;
     title3 "PARAMETER AND BOOK PARAMETER VALUES IN TWO COLUMNS";
     title4 '(See as Table 6.11 p253)';
     title5 'In the second replication, C seems to be large';
     title6 '  but in the opposite direction';
     run;

**********************************************************;
* Compute the average and semi-difference to separate
*   the main effects from the second-order interactions
**********************************************************;

data coldata;
     set coldata;
     * Averages and Semi-Differences for foldover;
     Average=(Book1+Book2)/2;  SemiDf=(Book1-Book2)/2;
     run;

proc print;
     title3 'The average gives the main effects: C is now small';
     title4 'Semi-Df is the sum of three 2nd-order interactions';
     title5 'The large effects now seem to be  A  E  and  AE';
     title6 'AE is confounded with 2 other 2-way interations,';
     title7 '  but the most parsimonious explanation is that all';
     title8 '  factors other than A and E are inert.';
     run;


**********************************************************;
* Let's check our conclusions by doing a 2^3 design with
*   two observations per cell on A C E:
**********************************************************;

data filtration;
     set filtration;
     AC=A*C;  AE=A*E;  CE=C*E;  ACE=A*C*E;
     run;

proc reg data=filtration;
     title2 'ANALYSIS OF 2^3 DESIGN (2 OBSERVATIONS/CELL) FOR A C E';
     title3 'Only  E  AE  are significant, with only A borderline';
     model Yield = A C E AC AE CE ACE;
     run;

title2 'INTERACTION PLOT of A and E';

proc means nway noprint data=filtration;
     class A E;
     var Yield;
     output out=cmeans mean=;
     run;

data cmeans;
     set cmeans;
     if A<0 then APLOT="L";
     else APLOT="H";
     if E<0 then EPLOT="L";
     else EPLOT="H";
     label APLOT=Water_Supply  EPLOT=Caustic_Soda;
     run;

proc plot data=cmeans;
     plot Yield*A = Eplot / vpos=30;
     run;


title2 'AN ALTERNATIVE DERIVATION OF PARAMETER ESTIMATES:';
title3 "For the 16 runs together, `Average' estimates' are";
title4 "  `main effect' estimates and `Semi-Df' estimates";
title5 '  are interactions with Rep (the Replication number)';
title6 'This gives the 16 parameter estimates directly.';
title7 'The main effect of C is not significant';
title8 'RepC, which is a standin for AE, is strongly significant';

data filtall;
     set filtration;
     * Rep has to be +-1 for this to work;
     if Rep=1 then Rep=-1;
     else if Rep=2 then Rep=1;
     * Define 8 replication interaction effects;
     RepA=Rep*A;  RepB=Rep*B;  RepC=Rep*C;  RepD=Rep*D;
     RepE=Rep*E;  RepF=Rep*F;  RepG=Rep*G;    
           ;

proc reg data=filtall outest=filrows;
     * Carry out estimates for both replications together;
     model Yield = A B C D E F G
        Rep RepA RepB RepC RepD RepE RepF RepG;
     run;

proc transpose data=filrows out=filcols name=Effect;
     id _TYPE_;
     run;

data filcols;
     set filcols;
     BookParm=2*Parms;  AbsParms=abs(Parms);
     if Effect ne "Intercept" and Effect ne "_RMSE_"
        and Effect ne "Yield" then output;
     run;

proc sort data=filcols;
     by descending AbsParms;  run;

proc print data=filcols;
     title2 'SORTED PARAMETER ESTIMATES for 2_{IV}^{7-3} design';
     run;

title2 'IN THE SECOND REPLICATION, runs 10,11,13,16 have a complete sets';
title3 '  of levels for factors A C E. This means that we can resolve';
title4 '  C from AE with just 12 runs.';
title5 'Since a 2^3 design has 8 parameters, we can use these 12 runs';
title6 '  to get P-values for A C E and all of their interactions';
title7 '  with error-df=4';

**********************************************************;
* Keep all of the runs in the first replication,
*   but only runs 10,11,13,16 from the second replication
**********************************************************;

data filt12;
     set filtration;
     if Rep=1 then output;
     else if Row=10 or Row=11 or Row=13 or Row=16 then output;
     run;

proc print data=filt12;
     title8 'THE 12-RUN DATASET AS SAS SEES IT:';
     var Rep Row A B C D E F G yy1 yy2 Yield;
     run;

proc reg data=filt12;
     title5 'With 12 observations, only E AE are significant';
     title6 '  and C is not significant, with A CE bordlerline';
     title7 'Thus the additional half-replication was enough';
     title8 '  to resolve the C=AE ambiguity';
     model Yield = A C E AC AE CE ACE;
     run;


proc glm data=filt12;
     title6 'PROC GLM FOR THE 2^3 DESIGN WITH 12 OBSERVATIONS';
     title7 'This shows that the design is unbalanced,';
     title8 "  but the Type-III table conclusions are the same.";
     classes A C E;
     model Yield = A | C | E;
     run;