Chapter 7 Blocking and Confounding in the 2k Factorial Design

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Presentation transcript:

Chapter 7 Blocking and Confounding in the 2k Factorial Design

7.2 Blocking a Replicated 2k Factorial Design Blocking is a technique for dealing with controllable nuisance variables A 2k 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

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

The ANOVA table of Example 7.1

7.3 Confounding in the 2k 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 2k factorial design in 2p incomplete blocks with p < k

7.4 Confounding the 2k Factorial Design in Two Blocks For example: Consider a 22 factorial design in 2 blocks. Block 1: (1) and ab Block 2: a and b AB is confounded with blocks! See Page 275 How to construct such designs??

Defining contrast: xi 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 277 Group: Principal block: Contain the treatment (1)

Estimation of error:

Example 7.2

7.6 Confounding the 2k Factorial Design in Four Blocks Two defining contrasts: Consider 25 design.

The generalized interaction: (ADE)(BCE) = ABCD ADE, BCE and ABCD are all confounded with blocks.

7.7 Confounding the 2k Factorial Design in 2p Blocks Choose p independent effects to be confounded. Exact 2p -p -1 other effects will be confounded with blocks.