Blocking in the 2 k Design Blocking may be required because: we - - PowerPoint PPT Presentation

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 + · · · + αkxk, 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. A factor (here Block) whose levels are labeled the same, but with different meanings, across levels of another factor (here Rep), is said to be nested within that factor: “Blocks are nested within Reps.” The main effect of a nested factor has no meaning, and should be left out of the analysis; only the interaction of the nested factor with the outer factor has any meaning.

19 / 23 Blocking in the 2k Design Partial Confounding

ST 516 Experimental Statistics for Engineers II

summary(aov(Rate ~ Block:Rep + A * B * C, partialConfounding))

Output

Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 16.1941 0.010079 * B 1 218 218 0.0853 0.781987 C 1 374850 374850 146.9446 6.75e-05 *** Block:Rep 3 4333 1444 0.5662 0.660744 A:B 1 3528 3528 1.3830 0.292529 A:C 1 94403 94403 37.0066 0.001736 ** B:C 1 18 18 0.0071 0.936205 A:B:C 1 6 6 0.0024 0.962816 Residuals 5 12755 2551

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

20 / 23 Blocking in the 2k Design Partial Confounding

ST 516 Experimental Statistics for Engineers II

If blocks are given unique labels, the analysis is simpler:

partialConfounding$Blocks <- factor(paste(partialConfounding$Rep, partialConfounding$Block, sep = "-"))

Output

partialConfounding Rep Block A B C Rate Blocks 1 I 1 - - - 550 I-1 2 I 1 + + - 642 I-1 3 I 1 + - + 749 I-1 4 I 1 - + + 1075 I-1 5 I 2 + - - 669 I-2 6 I 2 - + - 633 I-2 7 I 2 - - + 1037 I-2 8 I 2 + + + 729 I-2 9 II 1 - - - 604 II-1 10 II 1 - - + 1052 II-1 . . . 16 II 2 - + + 1063 II-2

21 / 23 Blocking in the 2k Design Partial Confounding

ST 516 Experimental Statistics for Engineers II

summary(aov(Rate ~ Blocks + A * B * C, partialConfounding))

Output

Df Sum Sq Mean Sq F value Pr(>F) Blocks 3 4333 1444 0.5662 0.660744 A 1 41311 41311 16.1941 0.010079 * B 1 218 218 0.0853 0.781987 C 1 374850 374850 146.9446 6.75e-05 *** A:B 1 3528 3528 1.3830 0.292529 A:C 1 94403 94403 37.0066 0.001736 ** B:C 1 18 18 0.0071 0.936205 A:B:C 1 6 6 0.0024 0.962816 Residuals 5 12755 2551

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

22 / 23 Blocking in the 2k Design Partial Confounding

ST 516 Experimental Statistics for Engineers II

Note larger standard errors for partially confounded effects (factor

• f

√ 2):

summary(lm(Rate ~ Blocks + coded(A) * coded(B) * coded(C), partialConfounding)) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 753.125 30.929 24.350 2.18e-06 *** BlocksI-2 14.750 50.507 0.292 0.78199 BlocksII-1 55.625 43.740 1.272 0.25942 BlocksII-2 21.375 43.740 0.489 0.64575 coded(A)

• 50.812

12.627

• 4.024

0.01008 * coded(B) 3.688 12.627 0.292 0.78199 coded(C) 153.062 12.627 12.122 6.75e-05 *** coded(A):coded(B)

• 21.000

17.857

• 1.176

0.29253 coded(A):coded(C)

• 76.812

12.627

• 6.083

0.00174 ** coded(B):coded(C)

• 1.062

12.627

• 0.084

0.93621 coded(A):coded(B):coded(C)

• 0.875

17.857

• 0.049

0.96282

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

23 / 23 Blocking in the 2k Design Partial Confounding