Applied Statistics and Probability for Engineers

14-8 BLOCKING AND CONFOUNDING IN THE 2kDESIGN 547

It is possible to confound the 2kdesign in four blocks of 2k^2 observations each. To con-

struct the design, two effects are chosen to confound with blocks, and their defining contrasts

are obtained. A third effect, the generalized interactionof the two effects initially chosen, is

also confounded with blocks. The generalized interaction of two effects is found by multiply-

ing their respective letters and reducing the exponents modulus 2.

For example, consider the 2^4 design in four blocks. If ACand BDare confounded with

blocks, their generalized interaction is (AC)(BD) = ABCD. The design is constructed by using

Table 14-22 Minitab Effect Estimates for

Example 14-6

Estimated Effects and Coefficients for Distance

Term Effect Coef

Constant 6.938

Block 0.063

A 2.625 1.312

B 0.625 0.313

C 0.875 0.438

D 1.875 0.938

AB 0.125 0.063

AC 2.375 1.187

AD 1.625 0.813

BC 0.375 0.188

BD 0.375 0.187

CD 0.125 0.062

ABC 0.125 0.063

ABD 0.875 0.438

ACD 0.375 0.187

BCD 0.375 0.187

_ 2

_ 1

0

1

02

A

D

AD

AC

Effect

Normal score

Figure 14-27 Normal probability plot of the

effects from Minitab, Example 14-6.

Table 14-23 Analysis of Variance for Example 14-6

Source of Sum of Degrees of Mean

Variation Squares Freedom Square f 0 P-Value

Blocks (ABCD) 0.0625 1 0.0625 0.06 —

A 27.5625 1 27.5625 25.94 0.0070

B 1.5625 1 1.5625 1.47 0.2920

C 3.0625 1 3.0625 2.88 0.1648

D 14.0625 1 14.0625 13.24 0.0220

AB 0.0625 1 0.0625 0.06 —

AC 22.5625 1 22.5625 21.24 0.0100

AD 10.5625 1 10.5625 9.94 0.0344

BC 0.5625 1 0.5625 0.53 —

BD 0.5625 1 0.5625 0.53 —

CD 0.0625 1 0.0625 0.06 —

Error (ABC ABD ACD BCD) 4.2500 4 1.0625

Total 84.9375 15

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