Applied Statistics and Probability for Engineers

548 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

EXERCISES FOR SECTION 14-8

14-23. Consider the data from the first replicate of Exercise

14-13. Suppose that these observations could not all be run

under the same conditions. Set up a design to run these obser-

vations in two blocks of four observations each, with ABC

confounded. Analyze the data.

14-24. Consider the data from the first replicate of Exercise

14-14. Construct a design with two blocks of eight observa-

tions each, with ABCDconfounded. Analyze the data.

14-25. Repeat Exercise 14-24 assuming that four blocks

are required. Confound ABDand ABC(and consequently CD)

with blocks.

14-26. Construct a 2^5 design in two blocks. Select the

ABCDEinteraction to be confounded with blocks.

14-27. Construct a 2^5 design in four blocks. Select the ap-

propriate effects to confound so that the highest possible in-

teractions are confounded with blocks.

14-28. Consider the data from Exercise 14-18. Construct the

design that would have been used to run this experiment in two

blocks of eight runs each. Analyze the data and draw conclusions.

14-29. An article in Industrial and Engineering Chemistry

(“Factorial Experiments in Pilot Plant Studies,” 1951,

pp. 1300–1306) reports on an experiment to investigate the

effect of temperature (A), gas throughput (B), and concentra-

tion (C) on the strength of product solution in a recirculation

unit. Two blocks were used with ABCconfounded, and the ex-

periment was replicated twice. The data are as follows:

Replicate 1

Block 1 Block 2

(1)  99 a 18

ab 52 b 51

ac 42 c 108

bc 95 abc 35

Replicate 2

Block 3 Block 4

(1)  46 a 18

ab 47 b 62

ac 22 c 104

bc 67 abc 36

the defining contrasts for ACand BD:

It is easy to verify that the four blocks are

L 2 x 2 x 4

L 1 x 1 x 3

Block 1 Block 2 Block 3 Block 4

L 1  0, L 2  0 L 1  1, L 2  0 L 1  0, L 2  1 L 1  1, L 2  1

112 abab

ac c abc bc

bd abd d ad

abcd bcd acd cd

This general procedure can be extended to confounding the 2kdesign in 2pblocks, where

p k. Start by selecting peffects to be confounded, such that no effect chosen is a general-

ized interaction of the others. Then the blocks can be constructed from the pdefining contrasts

L 1 , L 2 ,... , Lpthat are associated with these effects. In addition to the peffects chosen to be

confounded, exactly 2p p 1 additional effects are confounded with blocks; these are the

generalized interactions of the original peffects chosen. Care should be taken so as not to

confound effects of potential interest.

For more information on confounding in the 2kfactorial design, refer to Montgomery

(2001, Chapter 7). This book contains guidelines for selecting factors to confound with blocks

so that main effects and low-order interactions are not confounded. In particular, the book

contains a table of suggested confounding schemes for designs with up to seven factors and a

range of block sizes, some of which are as small as two runs.

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