Blocking & Confounding in the 2 k Factorial Design

1. Blocking & Confounding in the 2k Factorial Design • Text reference, Chapter 7 • Blocking is a technique for dealing with controllable nuisance variables • Two cases are considered • Replicated designs • Unreplicated designs

2. Blocking a Replicated Design • This is the same scenario discussed previously (Chapter 5, Section 5-6) • If there are n replicates of the design, then each replicate is a block • Each replicate is run in one of the blocks (time periods, batches of raw material, etc.) • Runs within the block are randomized

3. Blocking a Replicated Design Consider the example from Section 6-2; k = 2 factors, n = 3 replicates This is the “usual” method for calculating a block sum of squares Section 6-2

4. ANOVA for the Blocked Design Page 267 Section 6-2 Analysis of variance table [Partial sum of squares] Sum ofMeanFSourceSquaresDFSquareValueProb > FA208.331208.3353.19< 0.0001B75.00175.0019.150.0024AB8.3318.332.130.1828 Pure Error 31.33 8 3.92 Cor Total 323.00 11

5. Confounding in Blocks • Now consider the unreplicated case • Clearly the previous discussion does not apply, since there is only one replicate • To illustrate, consider the situation of Example 6-2, Page 228 • This is a 24, n = 1 replicate

6. Confounding in Two Blocks • A single replicate 22 design • Each raw material is only enough for two runs – needs two materials (blocks) A possible design

7. Confounding in Two Blocks • Estimating effects • A = ½[ab + a – b – (1)] • B = ½[ab + b – a – (1)] • AB = ½[ab + (1) – a – b]

8. Example of Confounding for a 23 design in Two Blocks

9. Other Methods of Constructing the Blocks • Use of a defining contrast • L = a1x1 + a2x2 + a3x3 + …+ akxk • xi: level of the ith factor in a particular treatment combination (0 or 1) • ai: exponent appearing on the ith factor in the effect to be confounded (0 or 1) • Treatment combinations that produce the same value of L (mod 2) will be placed in the same block

10. Other Methods of Constructing the Blocks • Example: a 23 design with ABC confounded with blocks • a1 = 1; a2 = 1; a3 = 1 • x1 A; x2 B; x3 C; • L = x1 + x2 + x3 • (1) 000: L = 0 = 0 a 100: L = 1 = 1 • ab 110: L = 2 = 0 b 010: L = 1 = 1 • ac 101: L = 2 = 0 c 001: L = 1 = 1 • bc 011: L = 2 = 0 abc 111: L = 3 = 1

11. Estimation of Error • Example: a 23 design, must be run in two blocks with ABC confounded, four replicates

12. It would be better if blocks are designed differently in each replicate, to confound a different effect in each replicate – partial confounding

13. Example of Unreplicated Design (Ex. 7-2) • Response: filtration rate of a resin • Factors: A = temperature, B = pressure, C = mole ratio/concentration, D= stirring rate • One batch of raw material is only enough for 8 runs. Two materials are required. • ABCD is chosen for confounding. • L = x1 + x2 + x3 + x4

14. Example 6-2 Suppose only 8 runs can be made from one batch of a raw material

15. The Table of + & - Signs, Example 6-2 Construction and analysis of the 2k factorial design in 2p incomplete blocks (p<k).

16. ABCD is Confounded with Blocks (Page 272) Observations in block 1 are reduced by 20 units…this is the simulated “block effect”

17. Effect Estimates Term Effect SumSqr A 21.625 1870.56 B 3.125 39.0625 C 9.875 390.062 D 14.625 855.563 AB 0.125 0.0625 AC -18.125 1314.06 AD 16.625 1105.56 BC 2.375 22.5625 BD -0.375 0.5625 CD -1.125 5.0625 ABC 1.875 14.0625 ABD 4.125 68.0625 ACD -1.625 10.5625 BCD -2.625 27.5625 ABCD 1.375 7.5625 Example 6-2

18. The ANOVA The ABCD interaction (or the block effect) is not considered as part of the error term The rest of the analysis is unchanged from Example 6-2