Chapter 7 Blocking and Confounding in the 2 k Factorial Design

1

Chapter 7 Blocking and Confounding in the 2k Factorial Design

2

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

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

2 23...

1 4 12

6.50

iBlocks

i

B ySS

4

• The ANOVA table of Example 7.1

5

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

6

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??

7BlockbaabAB

ababB

baabA

2/])1([

2/)]1([

2/)]1([

8

• 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)

kk xxxL 2211

9

10

• Estimation of error:

• Example 7.2

11

12

13

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

14

532

541

2:

1:

xxxLBCE

xxxLADE

• The generalized interaction:

(ADE)(BCE) = ABCD– ADE, BCE and ABCD are all confounded with

blocks.

15

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.

16

17