4.4 Blocking Unreplicated 2K Factorial Designs • If Data for Every

Home , Blocking (statistics)

4.4 Blocking Unreplicated 2k Factorial Designs • If data for every combination of factor levels cannot be collected under identical experimental conditions for an unreplicated 2k design, then blocks containing only a fraction of the 2k experimental runs should be formed.

• For example, due to time constraints, the design is often fractionated into blocks such that each block of experimental runs will correspond to diﬀerent time units (e.g, days or work shifts).

• The main problem is the assignment of the factor level combinations to blocks so that the main eﬀects and the interaction eﬀects of interest are not confounded with blocks.

• When a block effect is completely confounded with a main effect or an interaction effect, calculation of the two effects, as well as the sum of squares, are identical. This means we cannot separate the effect estimates in the statistical analysis.

• The common blocking method for 2k designs is to confound blocks with certain high order interactions. We will show how to form 2, 4, 8,... blocks from an unreplicated 2k design.

• Textbooks describe two equivalent ways to construct the blocks: using deﬁning contrasts and using principal blocks. I will not be covering these topics because there is a simpler way.

4.4.1 The Unreplicated 2k Design in Two Blocks of Size 2k−1 • When fractionating an unreplicated 2k design into 2 blocks of size 2k−1, it is recommended to confound the 2 block eﬀects with the k−factor interaction (the highest order interaction).

– For a 23 design, we would create 2 blocks of size 4 such that the block eﬀect is confounded with the (or ) interaction. – For a 24 design, we would create 2 blocks of size 4 such that the block eﬀect is confounded with the (or ) interaction.

• After the two blocks are formed, we randomly assign block to the experimental blocking variable. If blocks are two days, then we randomly assign Block 1 to either day 1 or day 2.

• Then, once blocks are assigned, you randomize the order of the 2k−1 rows within each block.

• Important: When you include blocks in the model, you cannot also include the highest-order interaction because of complete confounding. Therefore, you cannot separate any blocking eﬀect from the highest-order interaction eﬀect.

Confounding the hightest-order interaction in a 2k design when forming 2 blocks:

1. Generate a table containing the 2k possible combinations of + and − signs for the k factors. Let the columns be labeled A, B, C, . . ..

2. Create a column for the highest-order interaction. Multiply the entries in the A, B, C, . . . columns for each row. This will yield either a + or − sign.

3. Put all rows containing a + sign into the ﬁrst block, and all rows containing a − sign into the second block.

58 Example: The following table summarizes the eight treatment combinations and the signs for calculating eﬀects in the 23 design along with a column for the highest order interaction (ABC).

ABC ABC ABC ABC Block (1) − − − − a + − − 1 a + − − + b − + − 1 b − + − + c − − + 1 ab + + − − abc + + + 1 c − − + + (1) − − − 2 ac + − + − ab + + − 2 bc − + + − ac + − + 2 abc + + + + bc − + + 2

Example: The following table summarizes the 16 treatment combinations and the signs for calculating eﬀects in the 24 design along with a column for the highest order interaction (ABCD).

ABCD ABCD ABCD ABCD Block (1) − − − − + (1) − − − − 1 a + − − − − ab + + − − 1 b − + − − − ac + − + − 1 ab + + − − + bc − + + − 1 c − − + − − ad + − − + 1 ac + − + − + bd − + − + 1 bc − + + − + cd − − + + 1 abc + + + − − abcd + + + + 1 d − − − + − a + − − − 2 ad + − − + + b − + − − 2 bd − + − + + c − − + − 2 abd + + − + − abc + + + − 2 cd − − + + + d − − − + 2 acd + − + + − abd + + − + 2 bcd − + + + − acd + − + + 2 abcd + + + + + bcd − + + + 2

4.4.2 The Unreplicated 2k Design in Four Blocks of Size 2k−2 Confounding two interactions in a 2k design when forming 4 blocks:

1. Select two interactions as block generators. You can use the table of suggested generators in the text and notes.

2. Generate a table containing the 2k possible combinations of + and − signs for the k factors. Let the columns be labeled A, B, C, . . ..

3. Create two columns for the two generating interactions. Multiply the entries in the A, B, C, . . . columns for each row. This will yield either a + or − sign in each of the two generator columns.

4. There are four possible + and − sign combinations: (++), (+−), (−+), and (−−). Put all rows containing a (++) into the Block 1, (+−) into the Block 2, (−+) into the Block 3, and (−−) into the Block 4.

59 Example: The following table summarizes the 16 treatment combinations and the signs for calculating eﬀects in the 24 design along with two column for the two interaction generators (ABC and BCD).

ABCD ABC BCD ABCD ABC BCD Block (1) − − − − − − b − + − − 1 + a + − − − + − c − − + − 1 + b − + − − + + ad + − − + 1 + ab + + − − − + abcd + + + + 1 + c − − + − + + a + − − − 2 − ac + − + − − + abc + + + − 2 − bc − + + − − − bd − + − + 2 − abc + + + − + − cd − − + + 2 − d − − − + − + ab + + − − 3 − ad + − − + + + ac + − + − 3 − bd − + − + + − d − − − + 3 − abd + + − + − − bcd − + + + 3 − cd − − + + + − (1) − − − − 4 + acd + − + + − − bc − + + − 4 + bcd − + + + − + abd + + − + 4 + abcd + + + + + + acd + − + + 4 +

• After the four blocks are formed, we randomly assign blocks to the experimental blocking variable. If blocks are four days, then we randomly assign Blocks 1 to 4 to Days 1 to 4.

• Then, once blocks are assigned, you randomize the order of the 2k−2 rows within each block.

• Note that we formed 4 blocks (3 d.f.) formed from two interaction effects (2 d.f.). Thus, there is one additional effect confounded with blocks. This effect is the generalized interaction of the two generating interactions.

• The generalized interaction (GI) is the product of the two generating interactions.

– In this example, the GI is ABC × BCD = = . Note that if we have an even power (e.g. B2 or C2) we get a column of all ones, and for an odd power we get the original column back (e.g. B = B3 = B5). – In the column we get all + or − within a block. – If you select any two of the three interactions (ABC, BCD, ), then you would form the same blocks.

• Important: When you include 4 blocks in the model, you cannot also include any of the three interactions (2 generators and the GI) because of complete confounding. Therefore, you cannot separate any blocking eﬀect from the these particular interaction eﬀects.

– In this example, we cannot include ABC, BCD, or in the model if blocks are also in the model. Or, in the regression model, we cannot include

4.4.3 An Example of Good vs Poor Blocking • In the following example, the goal was to run a 24 design in four blocks of size 4. It is impossible to ﬁnd a blocked design that allows estimation of all two-factor interactions. However, it is possible to include 5 of the 6 two-factor interactions in the blocking model.

60 • In the “Good Blocking Example”, blocks are confounded with the ABC, ACD, and BD effects. It is “good” because the experimenter did not expect/believe a B ∗ D interaction would be significant. Thus, the design allowed it be the one two-factor interaction confounded with blocks. • In the “Poor Blocking Example”, the blocks are confounded with the BCD, ABD, and AC effects. It is “bad” because the experimenter did not take the time to plan which two-factor interaction should be confounded with blocks. In this case, the A ∗ C effect may be important but cannot be included in the model with blocks. GOOD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4 ASSIGNMENTS TO BLOCKS -- SORTED BY BLOCK Obs A B C D ABC ACD BD BLOCK 1 -1 -1 -1 -1 -1 -1 1 1 2 1 -1 1 -1 -1 -1 1 1 3 1 1 -1 1 -1 -1 1 1 4 -1 1 1 1 -1 -1 1 1 5 -1 1 -1 -1 1 -1 -1 2 6 1 1 1 -1 1 -1 -1 2 7 1 -1 -1 1 1 -1 -1 2 8 -1 -1 1 1 1 -1 -1 2 9 1 1 -1 -1 -1 1 -1 3 10 -1 1 1 -1 -1 1 -1 3 11 -1 -1 -1 1 -1 1 -1 3 12 1 -1 1 1 -1 1 -1 3 13 1 -1 -1 -1 1 1 1 4 14 -1 -1 1 -1 1 1 1 4 15 -1 1 -1 1 1 1 1 4 16 1 1 1 1 1 1 1 4 ANALYSIS WITH BLOCKS The GLM Procedure Dependent Variable: YIELD Sum of Source DF Squares Mean Square F Value Pr > F Model 12 283.2500000 23.6041667 8.33 0.0534 Error 3 8.5000000 2.8333333 Corrected Total 15 291.7500000 R-Square Coeff Var Root MSE YIELD Mean 0.970865 9.687775 1.683251 17.37500 Source DF Type III SS Mean Square F Value Pr > F BLOCK 3 4.250 <-- 1.4167 0.50 0.7082 <- SS = 4.250 A 1 81.000 81.0000 28.59 0.0128 B 1 1.000 1.0000 0.35 0.5943 C 1 16.000 16.0000 5.65 0.0979 D 1 42.250 42.2500 14.91 0.0307 A*B 1 2.250 2.2500 0.79 0.4385 < No B*D A*C 1 72.250 72.2500 25.50 0.0150 < interaction A*D 1 64.000 64.0000 22.59 0.0177 < in the B*C 1 0.250 0.2500 0.09 0.7858 < model. C*D 1 0.000 0.0000 0.00 1.0000 < Level of ------YIELD------BLOCK N Mean Std Dev 1 4 17.000 5.09901951 < Relatively small 2 4 18.000 5.29150262 < block-to-block 3 4 16.750 4.99165971 < variability. 4 4 17.750 4.11298756 <

61 GOOD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4 ANALYSIS WITH BLOCKS AND WITH ABC ACD BD The GLM Procedure Dependent Variable: YIELD Sum of Source DF Squares Mean Square F Value Pr > F Model 12 283.2500000 23.6041667 8.33 0.0534 Error 3 8.5000000 2.8333333 Corrected Total 15 291.7500000 R-Square Coeff Var Root MSE YIELD Mean 0.970865 9.687775 1.683251 17.37500 Source DF Type I SS Mean Square F Value Pr > F A 1 81.000 81.0000 28.59 0.0128 B 1 1.000 1.0000 0.35 0.5943 C 1 16.000 16.0000 5.65 0.0979 D 1 42.250 42.2500 14.91 0.0307 A*B 1 2.250 2.2500 0.79 0.4385 A*C 1 72.250 72.2500 25.50 0.0150 A*D 1 64.000 64.0000 22.59 0.0177 B*C 1 0.250 0.2500 0.09 0.7858 C*D 1 0.000 0.0000 0.00 1.0000 B*D 1 0.000 <- 0.0000 0.00 1.0000 < Combined SS A*B*C 1 4.000 <- 4.0000 1.41 0.3203 < = 4.250 A*C*D 1 0.250 <- 0.2500 0.09 0.7858 < = Block SS in BLOCK 0 0.000 . . . < previous model.

SAS Code for Good Blocking Example

*****************************************************************; *** A 2**4 DESIGN: GOOD BLOCKING EXAMPLE (4 BLOCKS OF SIZE 4) ***; *****************************************************************; DATA IN; DO D = -1 TO 1 BY 2; DO C = -1 TO 1 BY 2; DO B = -1 TO 1 BY 2; DO A = -1 TO 1 BY 2; INPUT YIELD @@; ABC = A * B * C; * GENERATOR 1; ACD = A * C * D; * GENERATOR 2; BD = B * D; * GENERALIZED INTERACTION; IF ABC=-1 AND ACD=-1 THEN BLOCK=1; IF ABC= 1 AND ACD=-1 THEN BLOCK=2; IF ABC=-1 AND ACD= 1 THEN BLOCK=3; IF ABC= 1 AND ACD= 1 THEN BLOCK=4; OUTPUT; END; END; END; END; LINES; 12 18 13 16 17 15 20 15 10 25 13 24 19 21 17 23 ; TITLE ’GOOD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4’; TITLE2 ’EFFECTS CONFOUNDED WITH BLOCKS: ABC, ACD, BD’; PROC SORT DATA=IN; BY BLOCK; PROC PRINT DATA=IN; VAR A B C D ABC ACD BD BLOCK; TITLE2 "ASSIGNMENTS TO BLOCKS -- SORTED BY BLOCK"; ***********************************************************; *** FOUR BLOCK ANALYSIS: POOL HIGHER ORDER INTERACTIONS ***; *** CONFOUNDED WITH BLOCKS: ABC ACD BD ***; ***********************************************************; PROC GLM DATA=IN; CLASS A B C D BLOCK; MODEL YIELD = BLOCK A B C D A*B A*C A*D B*C C*D / SS3; MEANS BLOCK; TITLE2 "ANALYSIS WITH BLOCKS"; PROC GLM DATA=IN; CLASS A B C D BLOCK; MODEL YIELD = A B C D A*B A*C A*D B*C C*D B*D A*B*C A*C*D BLOCK / SS1; TITLE2 "ANALYSIS WITH BLOCKS AND WITH ABC ACD BD"; RUN;

62 Bad Blocking Example (BCD, ABD, and AC confounded with Blocks) BAD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4 ASSIGNMENTS TO BLOCKS -- SORTED BY BLOCK Obs A B C D BCD ABD AC BLOCK 1 -1 -1 -1 -1 -1 -1 1 1 2 1 1 1 -1 -1 -1 1 1 3 -1 1 -1 1 -1 -1 1 1 4 1 -1 1 1 -1 -1 1 1 5 1 1 -1 -1 1 -1 -1 2 6 -1 -1 1 -1 1 -1 -1 2 7 1 -1 -1 1 1 -1 -1 2 8 -1 1 1 1 1 -1 -1 2 9 1 -1 -1 -1 -1 1 -1 3 10 -1 1 1 -1 -1 1 -1 3 11 1 1 -1 1 -1 1 -1 3 12 -1 -1 1 1 -1 1 -1 3 13 -1 1 -1 -1 1 1 1 4 14 1 -1 1 -1 1 1 1 4 15 -1 -1 -1 1 1 1 1 4 16 1 1 1 1 1 1 1 4 ANALYSIS WITH BLOCKS The GLM Procedure Dependent Variable: YIELD Sum of Source DF Squares Mean Square F Value Pr > F Model 12 283.5000000 23.6250000 8.59 0.0512 Error 3 8.2500000 2.7500000 Corrected Total 15 291.7500000 R-Square Coeff Var Root MSE YIELD Mean 0.971722 9.544244 1.658312 17.37500 Source DF Type III SS Mean Square F Value Pr > F BLOCK 3 76.750 <-- 25.5833 9.30 0.0498 <-- Is this due A 1 81.000 81.0000 29.45 0.0123 to block B 1 1.000 1.0000 0.36 0.5890 effects or to C 1 16.000 16.0000 5.82 0.0948 confounded D 1 42.250 42.2500 15.36 0.0295 effects? A*B 1 2.250 2.2500 0.82 0.4324 A*D 1 64.000 64.0000 23.27 0.0170 B*C 1 0.250 0.2500 0.09 0.7827 B*D 1 0.000 0.0000 0.00 1.0000 C*D 1 0.000 0.0000 0.00 1.0000 Level of ------YIELD------BLOCK N Mean Std Dev 1 4 15.250 4.03112887 < Much larger 2 4 18.750 4.19324854 < block-to-block 3 4 20.250 2.62995564 < variability than 4 4 15.250 5.56027577 < previous example. BAD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4 ANALYSIS WITH BLOCKS AND WITH BCD ABD AD The GLM Procedure Dependent Variable: YIELD Sum of Source DF Squares Mean Square F Value Pr > F Model 12 283.5000000 23.6250000 8.59 0.0512 Error 3 8.2500000 2.7500000 Corrected Total 15 291.7500000 R-Square Coeff Var Root MSE YIELD Mean 0.971722 9.544244 1.658312 17.37500

63 Source DF Type I SS Mean Square F Value Pr > F A 1 81.000 81.0000 29.45 0.0123 B 1 1.000 1.0000 0.36 0.5890 C 1 16.000 16.0000 5.82 0.0948 D 1 42.250 42.2500 15.36 0.0295 A*B 1 2.250 2.2500 0.82 0.4324 A*D 1 64.000 64.0000 23.27 0.0170 B*C 1 0.250 0.2500 0.09 0.7827 B*D 1 0.000 0.0000 0.00 1.0000 C*D 1 0.000 0.0000 0.00 1.0000 A*C 1 72.250 72.2500 26.27 0.0144 < Combined SS B*C*D 1 2.250 2.2500 0.82 0.4324 < = 76.750 A*B*D 1 2.250 2.2500 0.82 0.4324 < = Block SS in BLOCK 0 0.000 . . . < previous model.

SAS Code for Bad Blocking Example:

DM ’LOG;CLEAR;OUT;CLEAR;’; OPTIONS NODATE NONUMBER PS=60 LS=78; ODS LISTING; *****************************************************************; *** A 2**4 DESIGN: BAD BLOCKING EXAMPLE (4 BLOCKS OF SIZE 4) ***; *****************************************************************; DATA IN; DO D = -1 TO 1 BY 2; DO C = -1 TO 1 BY 2; DO B = -1 TO 1 BY 2; DO A = -1 TO 1 BY 2; INPUT YIELD @@; BCD = B * C * D; * GENERATOR 1; ABD = A * B * D; * GENERATOR 2; AC = A * C; * GENERALIZED INTERACTION; IF BCD=-1 AND ABD=-1 THEN BLOCK=1; IF BCD= 1 AND ABD=-1 THEN BLOCK=2; IF BCD=-1 AND ABD= 1 THEN BLOCK=3; IF BCD= 1 AND ABD= 1 THEN BLOCK=4; OUTPUT; END; END; END; END; LINES; 12 18 13 16 17 15 20 15 10 25 13 24 19 21 17 23 ; TITLE ’BAD BLOCKING EXAMPLE OF A 2^4 DESIGN IN 4 BLOCKS OF SIZE 4’; TITLE2 ’EFFECTS CONFOUNDED WITH BLOCKS: BCD, ABD, AC’; PROC SORT DATA=IN; BY BLOCK; PROC PRINT DATA=IN; VAR A B C D BCD ABD AC BLOCK; TITLE2 "ASSIGNMENTS TO BLOCKS -- SORTED BY BLOCK"; **********************************************************; *** FOUR BLOCK ANALYIS: POOL HIGHER ORDER INTERACTIONS ***; *** CONFOUNDED WITH BLOCKS: BCD ABD AC ***; **********************************************************; PROC GLM DATA=IN; CLASS A B C D BLOCK; MODEL YIELD = BLOCK A B C D A*B A*D B*C B*D C*D / SS3; MEANS BLOCK; TITLE2 "ANALYSIS WITH BLOCKS"; PROC GLM DATA=IN; CLASS A B C D BLOCK; MODEL YIELD = A B C D A*B A*D B*C B*D C*D A*C B*C*D A*B*D BLOCK / SS1; TITLE2 "ANALYSIS WITH BLOCKS AND WITH BCD ABD AD"; RUN;

64 65 4.4.4 The Unreplicated 2k Design in 2p Blocks of Size 2k−p Goal: Generalize the results for the generation of 2 blocks of size 2k−1 or 4 blocks of size 2k−2 to the general case of generating a total 2p blocks of size 2k−p. Confounding p interactions in a 2k design when forming 2p blocks: 1. Select p interactions as block generators. You can use the table of suggested generators in the text and notes. 2. Generate a table containing the 2k possible combinations of + and − signs for the k factors. Let the columns be labeled A, B, C, . . .. 3. Create p columns for the p generating interactions. Multiply the entries in the A, B, C, . . . columns for each row. This will yield either a + or − sign in each of the p generator columns. 4. There are 2p possible + and − sign combinations: (±1, ±1,..., ±1). Put all rows containing the same sequence of ±1 values into the same block (which will form 2p blocks of size 2k−p). Example: The following table summarizes the 32 treatment combinations and the signs for calculating eﬀects in the 25 design along with three columns for the three interaction generators (ABE, BCE, and CDE). ABCDE ABE BCE CDE (1) − − − − − − − − a + − − − − + − − b − + − − − + + − ab + + − − − − + − c − − + − − − + + ac + − + − − + + + bc − + + − − + − + abc + + + − − − − + d − − − + − − − + ad + − − + − + − + bd − + − + − + + + abd + + − + − − + + cd − − + + − − + − acd + − + + − + + − bcd − + + + − + − − abcd + + + + − − − − e − − − − + + + + ae + − − − + − + + be − + − − + − − + abe + + − − + + − + ce − − + − + + − − ace + − + − + − − − bce − + + − + − + − abce + + + − + + + − de − − − + + + + − ade + − − + + − + − bde − + − + + − − − abde + + − + + + − − cde − − + + + + − + acde + − + + + − − + bcde − + + + + − + + abcde + + + + + + + +

66 The blocking structure (plus the generalized interactions):

ABCDE ABE BCE CDE Block e − − − − + + + 1 + + + + ac + − + − − + + 1 + + + + bd − + − + − + + 1 + + + + abcde + + + + + + + 1 + + + + b − + − − − + + 2 + − − − acd + − + + − + + 2 + − − − abce + + + − + + + 2 + − − − de − − − + + + + 2 + − − − bc − + + − − + − 3 − − + − ad + − − + − + − 3 − − + − abe + + − − + + − 3 − − + − cde − − + + + + − 3 − − + − a + − − − − + − 4 − + − + bcd − + + + − + − 4 − + − + ce − − + − + + − 4 − + − + abde + + − + + + − 4 − + − + c − − + − − − + 5 − + − − abd + + − + − − + 5 − + − − ae + − − − + − + 5 − + − − bcde − + + + + − + 5 − + − − ab + + − − − − + 6 − − + + cd − − + + − − + 6 − − + + bce − + + − + − + 6 − − + + ade + − − + + − + 6 − − + + abc + + + − − − − 7 + − − + d − − − + − − − 7 + − − + be − + − − + − − 7 + − − + acde + − + + + − − 7 + − − + (1) − − − − − − − 8 + + + − abcd + + + + − − − 8 + + + − ace + − + − + − − 8 + + + − bde − + − + + − − 8 + + + −

• After the 2p blocks are formed, we randomly assign blocks to the experimental blocking variable. If blocks are 2p days, then we randomly assign Blocks 1 to 2p to Days 1 to 2p.

• Then, once blocks are assigned, you randomize the order of the 2k−p rows within each block.

• Note that we formed 8 blocks (7 d.f.) formed from three interaction effects (3 d.f.). Thus, there are 4 additional effects confounded with blocks. These effects are the generalized interaction of the three generating interactions.

• The generalized interactions (GIs) are the products of each combination of two or more generating interactions.

67 For this example of a 25 design in 8 blocks of size 4:

• There are 7 df for blocks, but only 3 generating interactions (ABE, BCE, and CDE). This means there are 4 generalized interactions to account for the 4 remaining df.

• These generalized interactions are found by ﬁnding the eﬀect resulting from multiplication of each combination of 2 or 3 generators.

(ABE)(BCE) = AB2CE2 =

(ABE)(CDE) = ABCDE2 =

(BCE)(CDE) = BC2DE2 =

(ABE)(BCE)(CDE) = AB2C2DE3 = =

• In each of the GI columns we get all + or − within a block. If you select any three of the seven interactions (ABE, BCE, CDE ), then you would form the same 8 blocks (but just labelled 1 to 8 in a diﬀerent order).

• The 7 df for blocks corresponds to the 7 df associated with the ABE, BCE, CDE interaction eﬀects. Therefore, none of these 7 interactions can be included in a model with blocks.

• This means the ”biggest” model we can fit having the 8 defined blocks contains all main effects (A, B, C, D, E) and all two-way and higher interactions except for those 7 just listed.

• The 7 df for blocks corresponds 7 df associated with these 7 eﬀects. Therefore, the terms cannot be included in a regression model with blocks.

68 4.4.5 An Example of Using SAS to Block a 2k Design ********************************************; *** GENERATE A 2**5 DESIGN WITH 4 BLOCKS ***; ********************************************; PROC FACTEX; FACTORS X1 X2 X3 X4 X5 / NLEV=2; <-- List the factors SIZE DESIGN=32; <-- Design size BLOCKS NBLOCKS=4; <-- Number of blocks EXAMINE CONFOUNDING DESIGN; MODEL EST=(X1|X2|X3|X4|X5@2); <-- Specify model OUTPUT OUT = BLOCK54 X1 CVALS=(’LOW’ ’HIGH’) +--> X2 CVALS=(’TOP’ ’BOT’) | Specify levels X3 NVALS=(0 100) | CVAL if categorical X4 NVALS=(-2 2) | NVAL if numerical X5 CVALS=(’NEW’ ’OLD’); +--> TITLE ’GENERATION OF A 2**5 DESIGN IN 4 BLOCKS’;

PROC PRINT DATA=BLOCK54; RUN;

======GENERATION OF A 2**5 DESIGN IN 4 BLOCKS ======Design Points Experiment Number X1 X2 X3 X4 X5 Block ------1 -1 -1 -1 -1 -1 2 2 -1 -1 -1 -1 1 3 3 -1 -1 -1 1 -1 3 4 -1 -1 -1 1 1 2 5 -1 -1 1 -1 -1 1 6 -1 -1 1 -1 1 4 7 -1 -1 1 1 -1 4 8 -1 -1 1 1 1 1 9 -1 1 -1 -1 -1 1 10 -1 1 -1 -1 1 4 11 -1 1 -1 1 -1 4 12 -1 1 -1 1 1 1 13 -1 1 1 -1 -1 2 14 -1 1 1 -1 1 3 15 -1 1 1 1 -1 3 16 -1 1 1 1 1 2 17 1 -1 -1 -1 -1 4 18 1 -1 -1 -1 1 1 19 1 -1 -1 1 -1 1 20 1 -1 -1 1 1 4 21 1 -1 1 -1 -1 3 22 1 -1 1 -1 1 2 23 1 -1 1 1 -1 2 24 1 -1 1 1 1 3 25 1 1 -1 -1 -1 3 26 1 1 -1 -1 1 2 27 1 1 -1 1 -1 2 28 1 1 -1 1 1 3 29 1 1 1 -1 -1 4 30 1 1 1 -1 1 1 31 1 1 1 1 -1 1 32 1 1 1 1 1 4

69 GENERATION OF A 2**5 DESIGN IN 4 BLOCKS Block Pseudo-factor Confounding Rules [B1] = X2*X3*X4*X5 <--- Generators: BCDE , ADE [B2] = X1*X4*X5 ABC = GI

OBS BLOCK X1 X2 X3 X4 X5 1 1 LOW TOP 100 -2 NEW 2 1 LOW TOP 100 2 OLD 3 1 LOW BOT 0 -2 NEW 4 1 LOW BOT 0 2 OLD 5 1 HIGH TOP 0 -2 OLD 6 1 HIGH TOP 0 2 NEW 7 1 HIGH BOT 100 -2 OLD 8 1 HIGH BOT 100 2 NEW 9 2 LOW TOP 0 -2 NEW 10 2 LOW TOP 0 2 OLD 11 2 LOW BOT 100 -2 NEW 12 2 LOW BOT 100 2 OLD 13 2 HIGH TOP 100 -2 OLD 14 2 HIGH TOP 100 2 NEW 15 2 HIGH BOT 0 -2 OLD 16 2 HIGH BOT 0 2 NEW 17 3 LOW TOP 0 -2 OLD 18 3 LOW TOP 0 2 NEW 19 3 LOW BOT 100 -2 OLD 20 3 LOW BOT 100 2 NEW 21 3 HIGH TOP 100 -2 NEW 22 3 HIGH TOP 100 2 OLD 23 3 HIGH BOT 0 -2 NEW 24 3 HIGH BOT 0 2 OLD 25 4 LOW TOP 100 -2 OLD 26 4 LOW TOP 100 2 NEW 27 4 LOW BOT 0 -2 OLD 28 4 LOW BOT 0 2 NEW 29 4 HIGH TOP 0 -2 NEW 30 4 HIGH TOP 0 2 OLD 31 4 HIGH BOT 100 -2 NEW 32 4 HIGH BOT 100 2 OLD