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).
A
−
+
−
+
−
+
−
+
(1)
a
b
ab
c
ac
bc
abc
B
−
−
+
+
−
−
+
+
C
−
−
−
−
+
+
+
+
ABC
−
+
+
−
+
−
−
+
a
b
c
abc
(1)
ab
ac
bc
A
+
−
−
+
−
+
+
−
B
−
+
−
+
−
+
−
+
C
−
−
+
+
−
−
+
+
ABC
Block
1
1
1
1
2
2
2
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).
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
4.4.2
A
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
B
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
C
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
D
−
−
−
−
−
−
−
−
+
+
+
+
+
+
+
+
ABCD
+
−
−
+
−
+
+
−
−
+
+
−
+
−
−
+
(1)
ab
ac
bc
ad
bd
cd
abcd
a
b
c
abc
d
abd
acd
bcd
A
−
+
+
−
+
−
−
+
+
−
−
+
−
+
+
−
B
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
C
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
D
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
ABCD
Block
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
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).
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
A
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
B
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
C
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
D ABC BCD
−
−
−
−
+
−
−
+
+
−
+
−
−
+
+
−
+
−
−
−
−
−
+
−
+
−
+
+
+
+
+
+
−
+
−
−
+
+
−
+
−
−
+
−
+
+
+
+
b
c
ad
abcd
a
abc
bd
cd
ab
ac
d
bcd
(1)
bc
abd
acd
A
−
−
+
+
+
+
−
−
+
+
−
−
−
−
+
+
B
+
−
−
+
−
+
+
−
+
−
−
+
−
+
+
−
C
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
D ABC BCD Block
−
1
−
1
+
1
+
1
−
2
−
2
+
2
+
2
−
3
−
3
+
3
3
+
−
4
−
4
+
4
+
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
2
2
even power (e.g. B or C ) we get a column of all ones, and for an odd power we get the
original column back (e.g. B = B 3 = B 5 ).
– In the
column we get all + or − within a block.
– If you select any two of the three interactions (ABC, BCD,
the same blocks.
), then you would form
• 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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
B
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
C
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
ABC
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
ACD
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
BD
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
BLOCK
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
ANALYSIS WITH BLOCKS
The GLM Procedure
Dependent Variable: YIELD
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
Error
Corrected Total
12
3
15
283.2500000
8.5000000
291.7500000
23.6041667
2.8333333
8.33
0.0534
R-Square
0.970865
Source
DF
BLOCK
A
B
C
D
A*B
A*C
A*D
B*C
C*D
3
1
1
1
1
1
1
1
1
1
Coeff Var
9.687775
Type III SS
4.250 <-81.000
1.000
16.000
42.250
2.250
72.250
64.000
0.250
0.000
Level of
BLOCK
1
2
3
4
Root MSE
1.683251
Mean Square
1.4167
81.0000
1.0000
16.0000
42.2500
2.2500
72.2500
64.0000
0.2500
0.0000
YIELD Mean
17.37500
F Value
Pr > F
0.50
28.59
0.35
5.65
14.91
0.79
25.50
22.59
0.09
0.00
0.7082
0.0128
0.5943
0.0979
0.0307
0.4385
0.0150
0.0177
0.7858
1.0000
<- SS = 4.250
<
<
<
<
<
No B*D
interaction
in the
model.
----------YIELD---------N
4
4
4
4
Mean
17.000
18.000
16.750
17.750
Std Dev
5.09901951
5.29150262
4.99165971
4.11298756
61
< Relatively small
< block-to-block
< variability.
<
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
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
Error
Corrected Total
12
3
15
283.2500000
8.5000000
291.7500000
23.6041667
2.8333333
8.33
0.0534
R-Square
0.970865
Source
DF
A
B
C
D
A*B
A*C
A*D
B*C
C*D
B*D
A*B*C
A*C*D
BLOCK
1
1
1
1
1
1
1
1
1
1
1
1
0
Coeff Var
9.687775
Type I SS
Root MSE
1.683251
Mean Square
81.000
1.000
16.000
42.250
2.250
72.250
64.000
0.250
0.000
0.000 <4.000 <0.250 <0.000
81.0000
1.0000
16.0000
42.2500
2.2500
72.2500
64.0000
0.2500
0.0000
0.0000
4.0000
0.2500
.
YIELD Mean
17.37500
F Value
Pr > F
28.59
0.35
5.65
14.91
0.79
25.50
22.59
0.09
0.00
0.00
1.41
0.09
.
0.0128
0.5943
0.0979
0.0307
0.4385
0.0150
0.0177
0.7858
1.0000
1.0000
0.3203
0.7858
.
<
<
<
<
Combined SS
= 4.250
= Block SS in
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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
B
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
C
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
BCD
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
ABD
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AC
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
BLOCK
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
ANALYSIS WITH BLOCKS
The GLM Procedure
Dependent Variable: YIELD
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
Error
Corrected Total
12
3
15
283.5000000
8.2500000
291.7500000
23.6250000
2.7500000
8.59
0.0512
Root MSE
1.658312
YIELD Mean
17.37500
R-Square
0.971722
Source
Coeff Var
9.544244
DF Type III SS
BLOCK
A
B
C
D
A*B
A*D
B*C
B*D
C*D
3
1
1
1
1
1
1
1
1
1
Mean Square
76.750 <-- 25.5833
81.000
81.0000
1.000
1.0000
16.000
16.0000
42.250
42.2500
2.250
2.2500
64.000
64.0000
0.250
0.2500
0.000
0.0000
0.000
0.0000
Level of
BLOCK
1
2
3
4
F Value
9.30
29.45
0.36
5.82
15.36
0.82
23.27
0.09
0.00
0.00
Pr > F
0.0498 <-- Is this due
0.0123
to block
0.5890
effects or to
0.0948
confounded
0.0295
effects?
0.4324
0.0170
0.7827
1.0000
1.0000
----------YIELD---------N
4
4
4
4
Mean
15.250
18.750
20.250
15.250
Std Dev
4.03112887
4.19324854
2.62995564
5.56027577
<
<
<
<
Much larger
block-to-block
variability than
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
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
Error
Corrected Total
12
3
15
283.5000000
8.2500000
291.7500000
23.6250000
2.7500000
8.59
0.0512
Root MSE
1.658312
YIELD Mean
17.37500
R-Square
0.971722
Coeff Var
9.544244
63
Source
DF
Type I SS
A
B
C
D
A*B
A*D
B*C
B*D
C*D
A*C
B*C*D
A*B*D
BLOCK
1
1
1
1
1
1
1
1
1
1
1
1
0
81.000
1.000
16.000
42.250
2.250
64.000
0.250
0.000
0.000
72.250
2.250
2.250
0.000
Mean Square
81.0000
1.0000
16.0000
42.2500
2.2500
64.0000
0.2500
0.0000
0.0000
72.2500
2.2500
2.2500
.
F Value
29.45
0.36
5.82
15.36
0.82
23.27
0.09
0.00
0.00
26.27
0.82
0.82
.
Pr > F
0.0123
0.5890
0.0948
0.0295
0.4324
0.0170
0.7827
1.0000
1.0000
0.0144
0.4324
0.4324
.
<
<
<
<
Combined SS
= 76.750
= Block SS in
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).
(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
A
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
B
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
C
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
D
−
−
−
−
−
−
−
−
+
+
+
+
+
+
+
+
−
−
−
−
−
−
−
−
+
+
+
+
+
+
+
+
E ABE BCE CDE
−
−
−
−
−
+
−
−
−
+
+
−
−
−
+
−
−
+
+
−
−
+
+
+
−
+
−
+
−
−
−
+
−
−
−
+
−
+
−
+
+
+
+
−
−
−
+
+
−
−
+
−
−
+
+
−
+
−
−
−
−
−
−
−
+
+
+
+
+
−
+
+
+
−
−
+
+
+
−
+
+
+
−
−
+
−
−
−
+
−
+
−
+
+
+
−
+
+
+
−
+
−
+
−
+
−
−
−
+
+
−
−
+
+
−
+
+
−
−
+
+
−
+
+
+
+
+
+
66
The blocking structure (plus the generalized interactions):
e
ac
bd
abcde
b
acd
abce
de
bc
ad
abe
cde
a
bcd
ce
abde
c
abd
ae
bcde
ab
cd
bce
ade
abc
d
be
acde
(1)
abcd
ace
bde
A
−
+
−
+
−
+
+
−
−
+
+
−
+
−
−
+
−
+
+
−
+
−
−
+
+
−
−
+
−
+
+
−
B
−
−
+
+
+
−
+
−
+
−
+
−
−
+
−
+
−
+
−
+
+
−
+
−
+
−
+
−
−
+
−
+
C
−
+
−
+
−
+
+
−
+
−
−
+
−
+
+
−
+
−
−
+
−
+
+
−
+
−
−
+
−
+
+
−
D
−
−
+
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
E ABE BCE CDE Block
+ +
+
1
− +
+
1
+
1
− +
+ +
+
1
− +
+
2
− +
+
2
+
2
+ +
+ +
+
2
− +
−
3
−
3
− +
+ +
−
3
−
3
+ +
− +
−
4
−
4
− +
+ +
−
4
+ +
−
4
− −
+
5
− −
+
5
+ −
+
5
+ −
+
5
− −
+
6
+
6
− −
+
6
+ −
+ −
+
6
− −
−
7
− −
−
7
+ −
−
7
−
7
+ −
− −
−
8
− −
−
8
+ −
−
8
−
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) = AB 2 CE 2 =
(ABE)(CDE) = ABCDE 2 =
(BCE)(CDE) = BC 2 DE 2 =
(ABE)(BCE)(CDE) = AB 2 C 2 DE 3 = =
• 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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
BLOCK
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
X1
LOW
LOW
LOW
LOW
HIGH
HIGH
HIGH
HIGH
LOW
LOW
LOW
LOW
HIGH
HIGH
HIGH
HIGH
LOW
LOW
LOW
LOW
HIGH
HIGH
HIGH
HIGH
LOW
LOW
LOW
LOW
HIGH
HIGH
HIGH
HIGH
X2
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
TOP
TOP
BOT
BOT
X3
100
100
0
0
0
0
100
100
0
0
100
100
100
100
0
0
0
0
100
100
100
100
0
0
100
100
0
0
0
0
100
100
X4
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
70
X5
NEW
OLD
NEW
OLD
OLD
NEW
OLD
NEW
NEW
OLD
NEW
OLD
OLD
NEW
OLD
NEW
OLD
NEW
OLD
NEW
NEW
OLD
NEW
OLD
OLD
NEW
OLD
NEW
NEW
OLD
NEW
OLD