ST 516 Experimental Statistics for Engineers II Blocking in the 2k Design Blocking may be required because: we cannot perform all required runs under homogeneous conditions; e.g. raw material comes in limited batches; we want to carry out runs under a variety of conditions; e.g. to use material from different batches. 1 / 23 Blocking in the 2k Design Introduction ST 516 Experimental Statistics for Engineers II Blocking a Replicated Design If blocks are large enough for 2k runs, we can carry out each replicate in a single block. E.g. 2 × 2 yield example, in 3 blocks each of 4 runs (yield.txt): summary(aov(Yield ~ Rep + A * B, yield)) Output Df Sum Sq Mean Sq F value Pr(>F) Rep 2 6.500 3.250 0.7852 0.4978348 A 1 208.333 208.333 50.3356 0.0003937 *** B 1 75.000 75.000 18.1208 0.0053397 ** A:B 1 8.333 8.333 2.0134 0.2057101 Residuals 6 24.833 4.139 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 2 / 23 Blocking in the 2k Design Blocking a Replicated Design ST 516 Experimental Statistics for Engineers II Analysis as a replicated design This data set was previously analyzed as a replicated design, not blocked: summary(aov(Yield ~ A * B, yield)) Output Df Sum Sq Mean Sq F value Pr(>F) A 1 208.333 208.333 53.1915 8.444e-05 *** B 1 75.000 75.000 19.1489 0.002362 ** A:B 1 8.333 8.333 2.1277 0.182776 Residuals 8 31.333 3.917 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 3 / 23 Blocking in the 2k Design Blocking a Replicated Design ST 516 Experimental Statistics for Engineers II Differences 2(= n − 1) degrees of freedom are broken out from \Residuals" into \Rep" (= blocks). Residual mean square changed, and also F-ratios and P-values. 4 / 23 Blocking in the 2k Design Blocking a Replicated Design ST 516 Experimental Statistics for Engineers II Confounding If blocks are not large enough for 2k runs, we must use an incomplete block design. Simplest case: 2 × 2 design in 2 blocks of size 2. Treatment Effect Combination I A B AB Block (1) + - - + 2 a + + - - 1 b + - + - 1 ab + + + + 2 5 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Note that each block has both levels of A, and also both levels of B, so main effects can be estimated from within-block differences. But the AB interaction is the difference between block averages, and is confounded with blocks. We could also use either the A or B column to assign runs to blocks, but in this case a main effect would be confounded; we usually choose to confound the interaction. 6 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II The general 2k design can be carried out in 2 blocks each of 2k−1 runs in the same way: use the signs in the column for the highest-order interaction. Note: runs are sometimes assigned to blocks using a defining contrast L = α1x1 + α2x2 + ··· + αk xk ; where: each αi is 1 if factor i is in the interaction to be confounded, and 0 otherwise; xi is 0 for the low level of factor i and 1 for the high level; L is evaluated modulo 2. 7 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Example: a 23 design with ABC confounded with blocks Block assignments: Treatment Effect Block Combination I A B AB C AC BC ABC (1) + - - + - + + - 1 a + + - - - - + + 2 b + - + - - + - + 2 ab + + + + - - - - 1 c + - - + + - - + 2 ac + + - - + + - - 1 bc + - + - + - + - 1 abc + + + + + + + + 2 8 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Terminology: the block containing (1) is the principal block. In this case, the principal block is (1), ab, ac, and bc. These form a group: the product of any pair of elements is another element in the principal block (recall that e.g. a2 = (1)). You can form the other block by multiplying these by any run not in the principal block, e.g. a or abc. 9 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Example: filtration rate data filtration.txt, 4 factors in 2 blocks: # factors have already been converted from "-","+" to -1, +1 coding filtration$Block <- filtration$A * filtration$B * filtration$C * filtration$D summary(lm(Rate ~ Block + A * B * C * D, filtration)) Note that ABCD cannot be estimated, because it is confounded with blocks. Output Call: lm(formula = Rate ~ Block + A * B * C * D, data = filtration) Residuals: ALL 16 residuals are 0: no residual degrees of freedom! 10 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Output, continued Coefficients: (1 not defined because of singularities) Estimate Std. Error t value Pr(>|t|) (Intercept) 60.0625 NA NA NA Block 0.6875 NA NA NA A 10.8125 NA NA NA B 1.5625 NA NA NA C 4.9375 NA NA NA D 7.3125 NA NA NA A:B 0.0625 NA NA NA A:C -9.0625 NA NA NA B:C 1.1875 NA NA NA A:D 8.3125 NA NA NA B:D -0.1875 NA NA NA C:D -0.5625 NA NA NA A:B:C 0.9375 NA NA NA A:B:D 2.0625 NA NA NA A:C:D -0.8125 NA NA NA B:C:D -1.3125 NA NA NA A:B:C:D NA NA NA NA 11 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Reduced model summary(aov(Rate ~ Block + A + C + D + A * C + A * D, filtration)); Output Df Sum Sq Mean Sq F value Pr(>F) Block 1 7.56 7.56 0.3629 0.5617799 A 1 1870.56 1870.56 89.757 5.600e-06 *** C 1 390.06 390.06 18.717 0.0019155 ** D 1 855.56 855.56 41.053 0.0001242 *** A:C 1 1314.06 1314.06 63.054 2.349e-05 *** A:D 1 1105.56 1105.56 53.049 4.646e-05 *** Residuals 9 187.56 20.84 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Note that all effects in the reduced model can be estimated and tested. 12 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Confounding in More Than 2 Blocks Suppose that blocks hold only 2k−p runs ) we need 2p blocks. Choose p effects to be confounded with blocks, where no effect is the product of others. Use the combination of signs in those p columns (or p defining contrasts) to assign runs to blocks. 13 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II E.g. a 25 design in 4 blocks: we decide to confound ADE and BCE; Block assignments: Run ADE BCE Block (0) - - Block−− a + - Block+− b - + Block−+ . abcde + + Block++ and so on; the four combinations of + and − are used to make four block labels (distinct, but otherwise arbitrary; could be Larry, Moe, Curly, and Shemp). 14 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II The 4 blocks are: Block−−: (1); ad; bc; abcd; abe; ace; cde; bde; Block+−: a; d; abc; bcd; be; abde; ce; acde; Block−+: b; abd; c; acd; ae; de; abce; bcde; Block++: e; ade; bce; abcde; ab; bd; ac; cd. Note: the 4 blocks will remove 3 degrees of freedom; in addition to ADE and BCE, one other effect must be confounded. It is their product ADE × BCE = ABCDE 2 = ABCD. Note that I , ADE, BCE, and ABCD also form a group. 15 / 23 Blocking in the 2k Design Confounding ST 516 Experimental Statistics for Engineers II Replication and Partial Confounding Suppose that a design is replicated and confounded by blocking. If the same confounding structure is used in each replicate, the confounded effects are not estimable; they are said to be completely confounded. If different effects are confounded in each replicate, the design gives some information about all effects; they are said to be partially confounded. 16 / 23 Blocking in the 2k Design Partial Confounding ST 516 Experimental Statistics for Engineers II E.g. 23 in 2 replicates each in 2 blocks, with ABC confounded with blocks in Rep I, and AB confounded in Rep II (plasma etching tool data): plasmaLongRep1 <- plasmaLong[plasmaLong$Rep == 1,] A <- coded(plasmaLongRep1$A) B <- coded(plasmaLongRep1$B) C <- coded(plasmaLongRep1$C) plasmaLongRep1$Block <- ifelse(A * B * C < 0, 1, 2) plasmaLongRep1 <- plasmaLongRep1[order(plasmaLongRep1$Block),] plasmaLongRep2 <- plasmaLong[plasmaLong$Rep == 2,] A <- coded(plasmaLongRep2$A) B <- coded(plasmaLongRep2$B) plasmaLongRep2$Block <- ifelse(A * B > 0, 1, 2) plasmaLongRep2 <- plasmaLongRep2[order(plasmaLongRep2$Block),] partialConfounding <- rbind(plasmaLongRep1, plasmaLongRep2) partialConfounding$Rep <- factor(partialConfounding$Rep) partialConfounding$Block <- factor(partialConfounding$Block) 17 / 23 Blocking in the 2k Design Partial Confounding ST 516 Experimental Statistics for Engineers II partialConfounding A B C Rep Rate id Block 1.1 - - - 1 550 1 1 4.1 + + - 1 642 4 1 6.1 + - + 1 749 6 1 7.1 - + + 1 1075 7 1 2.1 + - - 1 669 2 2 3.1 - + - 1 633 3 2 5.1 - - + 1 1037 5 2 8.1 + + + 1 729 8 2 1.2 - - - 2 604 1 1 4.2 + + - 2 635 4 1 5.2 - - + 2 1052 5 1 8.2 + + + 2 860 8 1 2.2 + - - 2 650 2 2 3.2 - + - 2 601 3 2 6.2 + - + 2 868 6 2 7.2 - + + 2 1063 7 2 18 / 23 Blocking in the 2k Design Partial Confounding ST 516 Experimental Statistics for Engineers II Note The Blocks are labeled 1 and 2 in both Reps, but Block 1 in Rep I is not the same as Block 1 in Rep II.