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Home , Blocking (statistics)

Chapter 7 and Systems for Two-Level Factorials

許湘伶

Design and Analysis of (Douglas C. Montgomery)

hsuhl (NUK) DAE Chap. 7 1 / 28 Introduction

Sometimes, it is impossible to perform all 2k factorial experiments under homogeneous condition.

I a batch of raw material: not large enough for the required runs

Blocking technique: making the treatments are equally effective across many situation

hsuhl (NUK) DAE Chap. 7 2 / 28 Blocking a Replicated 2k Factorial Design

2k factorial design, n replicates

Example 7.1: chemical process

22 factorial design: A-concentration; B-catalyst 4 trials; 3 replicates

hsuhl (NUK) DAE Chap. 7 3 / 28 Blocking a Replicated 2k Factorial Design (cont.)

n replicates a block: each set of nonhomogeneous conditions each replicate is run in one of the blocks

3 2 2 X Bi y··· SSBlocks= − (2 d.f .) 4 12 i=1 = 6.50

The block effect is small.

hsuhl (NUK) DAE Chap. 7 4 / 28 Confounding

Confounding(干擾；混雜；交絡) the block size is smaller than the number of treatment combinations impossible to perform a complete replicate of a factorial design in one block confounding: a design technique for arranging a complete in blocks causes information about certain treatment effects(high-order interactions) to be indistinguishable(難以辨別的) from, or confounded with blocks

hsuhl (NUK) DAE Chap. 7 5 / 28 Confounding the 2k Factorial Design in Two Blocks

a single replicate of 22 design two batches of raw material are required 2 factors with 2 blocks

hsuhl (NUK) DAE Chap. 7 6 / 28 Confounding the 2k Factorial Design in Two Blocks (cont.)

1  A = 2 [ab + a − b−(1)] 1 (any difference between block 1 and 2 will cancel out) B = 2 [ab + b − a−(1)] 1 AB = [ab+(1) − a − b] 2 (block effect and AB are identical; confounded with blocks)

hsuhl (NUK) DAE Chap. 7 7 / 28 Confounding the 2k Factorial Design in Two Blocks (cont.)

all treatment combinations with

 “+” on AB ⇒ block 1 “-” on AB ⇒ block 2

This approach van be used to confound any effect. can be used to confound any 2k design in 2 blocks

hsuhl (NUK) DAE Chap. 7 8 / 28 Confounding the 2k Factorial Design in Two Blocks (cont.)

Which effect is confounded with block?

hsuhl (NUK) DAE Chap. 7 9 / 28 defining

defining contrast(定義對比): method for constructing the block using the linear combination

L = α1x1 + α2x2 + α3x3 + ··· + αkxk

xi: the level of the ithe factor (xi = 0 or 1)

αi: the exponent(冪) appearing on the ith factor in the effect to be confounded (αi = 0 or 1)

hsuhl (NUK) DAE Chap. 7 10 / 28 defining contrast (cont.)

L = α1x1 + α2x2 + α3x3 + ··· + αkxk

23 design with ABC(= Aα1 Bα2 Cα3 ) confounded with blocks

defining contrast to ABC: L = x1 + x2 + x3 (α1 = α2 = α3 = 1)

each treatment combination:

(1): L = 1(0) + 1(0) + 1(0) = 0 = 0 (mod 2) ⇒ Block 1 a : L = 1(1) + 1(0) + 1(0) = 1 = 1 (mod 2) ⇒ Block 2 b : L = 1(0) + 1(1) + 1(0) = 1 = 1 (mod 2) ⇒ Block 2 ab : L = 1(1) + 1(1) + 1(0) = 2 = 0 (mod 2) ⇒ Block 1

hsuhl (NUK) DAE Chap. 7 11 / 28 defining contrast (cont.)

principal block(主集區): containing the treatment combination (1) group-theoretic property: 1 any element (except (1)) can be generated by multiplying two other elements in the principal block (Ex: ab · ac = a2bc = bc) 2 Treatment combinations in the other block can be generating by multiplying one element in the new block by each element in the principal block modulus 2. (Ex: b· (1)=b)

hsuhl (NUK) DAE Chap. 7 12 / 28 defining contrast (cont.)

Example 7.2: modified Example 6.2 4 factors; unreplicated A-temperature; B-pressure; C-concentration of formaldehyde(甲醛); D-stirring rate determine their effect on product filtration(過濾) rate run 8 treatment combinations from a single batch of material confound the highest order interaction ABCD with blocks

L = x1 + x2 + x3 + x4

hsuhl (NUK) DAE Chap. 7 13 / 28 defining contrast (cont.)

Observations in block 1 are reduced by 20 units compared with Example 6.2 (the simulated “block effect”)

hsuhl (NUK) DAE Chap. 7 14 / 28 defining contrast (cont.)

Block effect = ¯yBlock 1 − ¯yBlock 2 = −18.625 = 1.375 − 20 = origin ABCD − 20 (Estimates Blocks+ABCD)

hsuhl (NUK) DAE Chap. 7 15 / 28 defining contrast (cont.)

The ABCD interaction (or the block effect) is not regarded as part of the error term.

hsuhl (NUK) DAE Chap. 7 16 / 28 Why Blocking Is Important?

Blocking is a very useful and important design technique. noise reduction technique

MSE ≈ 109

hsuhl (NUK) DAE Chap. 7 17 / 28 Why Blocking Is Important? (cont.)

hsuhl (NUK) DAE Chap. 7 18 / 28 Why Blocking Is Important? (cont.)

If we don’t block, then the added variability from the nuisance variable effect ends up getting distributed across the other design factors.

hsuhl (NUK) DAE Chap. 7 19 / 28 Confounding the 2k Factorial Design in Four Blocks

construct 2k factorial design confounded in four blocks of 2k−2 observations each #{factors} moderately large; block size is small four blocks with 3 d.f. Select two factors to be confounded with block; a third factor will automatically be confounded with block (generalized interaction) Ex: 2k factorial design

ADE: L1 = x1 + x4 + x5

BCE: L2 = x2 + x3 + x5 ⇒four blocks confound with ADE, BCE, and ABCD = ADE · BCE(mod 2)

hsuhl (NUK) DAE Chap. 7 20 / 28 Confounding the 2k Factorial Design in Four Blocks (cont.)

the pair of L1, L2: (0,0)(principal block), (0,1), (1,0), (1,1)

Treatment Combinations in Sign on ADE Sign on BCE Sign on ABCD Block 1 - - + Block 2 + - - Block 3 - + - Block 4 + + +

hsuhl (NUK) DAE Chap. 7 21 / 28 Confounding the 2k Factorial Design in Four Blocks (cont.)

In selecting effects to be confounded with blocks, care must be exercised to obtain a design that does not confound effects that may be of interest. It is preferable to sacrifice(犧牲) information on the three-factor interactions instead of the two-factor interaction.

hsuhl (NUK) DAE Chap. 7 22 / 28 Confounding the 2k Factorial Design in 2p Blocks

Extend to the construction of a 2k factorial design confounded in 2p blocks (p < k) each block: 2k−p runs select p independent factors: no effect chosen is the generalized interaction of the others

generating the blocks by use of the p defining contrasts L1,..., Lp associated with these effects Care: information on effects that may be of potential interest is not sacrificed

hsuhl (NUK) DAE Chap. 7 23 / 28 Confounding the 2k Factorial Design in 2p Blocks (cont.)

hsuhl (NUK) DAE Chap. 7 24 / 28 Partial Confounding

Example 7.2: completely confounded partially confounded:

information on ABC can be obtained from the in replicates II, III, IV the ratio 3/4 the relative information for the confounded effects

hsuhl (NUK) DAE Chap. 7 25 / 28 Partial Confounding (cont.)

hsuhl (NUK) DAE Chap. 7 26 / 28 Partial Confounding (cont.)

Example 7.3: partial confounding (Example 6.1) 3 factors: A-gap; B-gas flow; C-RF power

[a + b + c + abc − ab − ac − bc − (1)]2 SS = = 6.1250 ABC n2k [(1) + abc − ac + c − a − b + ab − bc]2 SS = = 3528.0 AB n2k n X R2 y2 SS = h − ··· = 3875.0625 Rep 2k N h=1

Rh: the total of the observations in the hth replicate

hsuhl (NUK) DAE Chap. 7 27 / 28 Partial Confounding (cont.)

hsuhl (NUK) DAE Chap. 7 28 / 28