1 Chapter 7 Blocking and Confounding in the 2 k Factorial Design

2 7.2 Blocking a Replicated 2 k Factorial Design Blocking is a technique for dealing with controllable nuisance variables A 2 k factorial design with n replicates. 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 Example 7.1 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

4 The ANOVA table of Example 7.1

5 7.3 Confounding in the 2 k Factorial Design Confounding is a design technique for arranging a complete factorial experiment in blocks, where block size is smaller than the number of treatment combinations in one replicate. Cause information about certain treatment effects to be indistinguishable from (confounded with) blocks. Consider the construction and analysis of the 2 k factorial design in 2 p incomplete blocks with p < k

6 7.4 Confounding the 2 k Factorial Design in Two Blocks For example: Consider a 2 2 factorial design in 2 blocks. –Block 1: (1) and ab –Block 2: a and b –AB is confounded with blocks! –See Page 289 –How to construct such designs??

7 Defining contrast: –x i is the level of the ith factor appearing in a particular treatment combination –  i is the exponent appearing on the ith factor in the effect to be confounded –Treatment combinations that produce the same value of L (mod 2) will be placed in the same block. –See Page 290 Group: –Principal block

8 Estimation of error: See Page 292