# Statistics for Environmental Engineers

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(a)

Time

Pressure

Temperature

FIGURE 22.5 Graphical demonstration of why one-factor-at-a-time (OFAT) experiments cannot estimate the two-factor interaction between temperature and pressure that is revealed by the two-level, two-factor design.

a>

0.

Yield = 11

10

7

Temperature

12

optional 7

Temperature

Yield = 11

2

>-

New

pressure

n

TO

New pressure

Standard

pressure

>-

Standard

pressure

_i_i_

Standard New Temperature

Standard New Temperature

One-Factor-at-a Time Experiment

Two-level Factorial Design Experiment

FIGURE 22.6 Four possible experimental designs for studying three factors. The worst is (a), the one-factor-at-a-time design (top left). (b) is a two-level, three-factor design in eight runs and can describe a smooth nonplanar surface. The Box-Behnken design (c) and the composite two-level, three-factor design (d) can describe quadratic effects (maxima and minima). The Box-Behnken design uses 12 observations located on the face of the cube plus a center point. The composite design has eight runs located at the corner of the cube, plus six “star” points, plus a center point. The corner and star points are equidistant from the center (i.e., located on a sphere having a diameter equal to the distance from the center to a corner).

It is generally true that (1) the factorial design gives better precision than the OFAT design if the factors do act additively; and (2) if the factors do not act additively, the factorial design can detect and estimate interactions that measure the nonadditivity.

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