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 calcu- lating 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 calcu- lating 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.