Page 369 - Six Sigma Demystified

Part 3 S i x S i g m a To o l S 349

Multiple Regression Higher-Order Models
Higher- order terms, such as squares (the second power), also may be added.
These higher-order terms generate curvilinear response surfaces, such as peaks
(maximums) and valleys (minimums). To estimate these higher-order terms, we
need at least three or more experimental levels for the factor because two-level
designs can estimate only linear terms.
The full model includes p parameters for k main factors, where p is calcu-
lated as
p = 1 + 2k + k(k – 1)/2

For example, a model for two factors includes one constant term plus four
2
2
main factor and second-order terms (β X + β X + β X + β X ) plus one
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1
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11
1
1
2
2
interaction term (β X X ). A three-factor model includes one constant term
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2
12
plus six main factor and second-order terms (β X + β X + β X + β X +
2
1
2
1
3
1
3
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2
2
β X + β X ) plus three interaction terms (β X X + β X X + β X X ). To
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2
1
12
3
3
3
22
1
13
2
33
2
estimate these terms, at least p distinct parameter conditions are needed.
The effect of higher-order terms is generally small, unless the main factors
themselves are important. In this way, initial experiments seek to discover the
significance of only the main factors (X , X , X , etc.) and their interactions
3
1
2
(X , X , X , etc.) and ignore the effect of the higher-order terms. If main fac-
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tors or interactions appear significant, higher-order effects may be investigated
with additional experimental trials.
The shape of the resulting response surface is highly dependent on the region
analyzed and the signs and magnitudes of the coefficients in the model, particu-
larly the pure quadratic (the squared terms) and the interaction terms. The
shape of the surface is estimated subject to errors inherent in the model-build-
ing process.
Despite their limitations, second-order models are used widely because of
their flexibility in mapping a variety of surfaces, including simple maximums
(peaks) or minimums (valleys), stationary ridges (such as in a mountain
range), rising ridges (a ridge that increases in height), and saddles (also known
as minimax, where a minimum in one factor meets a maximum in the other).
Second-order models are used in response surface analysis to define the sur-
face around a stationary point (a maximum, minimum, or saddle) and to
predict the response with better accuracy than first-order models near the
optimal regions.
In investigative analysis, second-order models are also used to understand the
effect of current operating parameters on the response.