1. 1 Blocking & Confounding in the 2 k 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. 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. 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. 4 ANOVA for the Blocked Design Page 267 Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F A208.331208.3353.19< 0.0001 B75.00175.0019.150.0024 AB8.3318.332.130.1828 Pure Error31.3383.92 Cor Total323.0011 Section 6-2

5. 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 2 4, n = 1 replicate

6. 6 Confounding in Two Blocks A single replicate 2 2 design Each raw material is only enough for two runs – needs two materials (blocks) A possible design

7. 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. 8 Example of Confounding for a 2 3 design in Two Blocks

9. 9 Other Methods of Constructing the Blocks Use of a defining contrast L =  1 x 1 +  2 x 2 +  3 x 3 + …+  k x k x i : level of the ith factor in a particular treatment combination (0 or 1)  i : 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. 10 Other Methods of Constructing the Blocks Example: a 2 3 design with ABC confounded with blocks  1 = 1;  2 = 1;  3 = 1 x 1  A; x 2  B; x 3  C; L = x 1 + x 2 + x 3 (1)000:L = 0 = 0 a100:L = 1 = 1 ab110:L = 2 = 0 b010:L = 1 = 1 ac101:L = 2 = 0 c001:L = 1 = 1 bc011:L = 2 = 0 abc111:L = 3 = 1

11. 11 Estimation of Error Example: a 2 3 design, must be run in two blocks with ABC confounded, four replicates

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

13. 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 = x 1 + x 2 + x 3 + x 4

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

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

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

17. 17 Effect Estimates TermEffectSumSqr A21.6251870.56 B3.12539.0625 C9.875390.062 D14.625855.563 AB0.1250.0625 AC-18.1251314.06 AD16.6251105.56 BC2.37522.5625 BD-0.3750.5625 CD-1.1255.0625 ABC1.87514.0625 ABD4.12568.0625 ACD-1.62510.5625 BCD-2.62527.5625 ABCD1.3757.5625 Example 6-2

18. 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