Introduction to Response Surface Experiments
Many industrial experiments are conducted to discover which values
of given factor variables optimize a response. If each factor is
measured at three or more values, a quadratic response surface can
be estimated by least-squares regression. The predicted optimal
value can be found from the estimated surface if the surface is
shaped like a simple hill or a valley. If the estimated surface
is more complicated, or if the predicted optimum is far from the
region of experimentation, then the shape of the surface can be
analyzed to indicate the directions in which new experiments
should be performed.
Suppose that a response variable y is measured at combinations
of values of two factor variables, x1 and x2. The quadratic
response-surface model for this variable is written as
The steps in the analysis for such data are
- model fitting and analysis of
variance to estimate parameters
- canonical analysis to investigate the
shape of the predicted response surface
- ridge analysis to search for the region of optimum response
Model Fitting and Analysis of Variance
The first task in analyzing the response surface is to estimate
the parameters of the model by least-squares regression and to
obtain information about the fit in the form of an analysis of
variance. The estimated surface is typically curved: a
"hill" whose peak occurs at the unique estimated point of
maximum response, a "valley," or a "saddle-surface"
with no unique minimum or maximum. Use the results of this phase
of the analysis to answer the following questions:
- What is the contribution of each type of effect -linear,
quadratic, and crossproduct -to the statistical fit?
The ANOVA table with sources labeled "Regression"
addresses this question.
- What part of the residual error is due to lack of fit?
Does the quadratic response model adequately represent the
true response surface? If you specify the LACKFIT option in
the MODEL statement, then the ANOVA table with sources
labeled "Residual" addresses this question.
- What is the contribution of each factor variable to the
statistical fit? Can the response be predicted as well if
the variable is removed? The ANOVA table with sources
labeled "Factor" addresses this question.
- What are the predicted responses for a grid of factor
values? (See the section "Plotting the Surface"
and the "Searching for Multiple Response Conditions" section.)
The test for lack-of-fit compares the variation around the model with
"pure" variation within replicated observations. This measures the
adequacy of the quadratic response surface model.
In particular, if there are ni replicated observations
Yi1, ... ,Yini of the response all at the same values xi of the factors, then we can predict the true response at xi either by using the predicted value based on the
model or by using the mean of the replicated values. The
test for lack-of-fit decomposes the residual error into a component
due to the variation of the replications around their mean value (the
"pure" error), and a component due to the variation of the mean
values around the model prediction (the "bias" error):
If the model is adequate, then both components estimate the nominal
level of error; however, if the bias component of error is much larger
than the pure error, then this constitutes evidence that there is
significant lack of fit.
observations in your design are replicated, you can test for
lack of fit by specifying the LACKFIT option in the MODEL statement.
Note that, since all other tests use
total error rather than pure error, you may want to hand-calculate the
tests with respect to pure error if the lack-of-fit is significant.
On the other hand, significant lack-of-fit indicates the quadratic model
is inadequate, so if this is a problem you can also try to refine the
model, possibly using PROC GLM for general polynomial modeling; refer
to Chapter 30, "The GLM Procedure," for more information.
Example 56.1 illustrates the use of the
The second task in analyzing the response surface is to examine
the overall shape of the curve and determine whether the estimated
stationary point is a maximum, a minimum, or a saddle point.
The canonical analysis can be used to answer the following questions:
- Is the surface shaped like a hill, a valley,
a saddle surface, or a flat surface?
- If there is a unique optimum combination
of factor values, where is it?
- To which factor or factors are the
predicted responses most sensitive?
The eigenvalues and eigenvectors in the matrix of second-order
parameters characterize the shape of the response surface.
The eigenvectors point in the directions of principle orientation
for the surface, and the signs and magnitudes of the associated
eigenvalues give the shape of the surface in these directions.
Positive eigenvalues indicate directions of upward curvature, and
negative eigenvalues indicate directions of downward curvature.
The larger an eigenvalue is in absolute value,
the more pronounced is the curvature of the
response surface in the associated direction.
Often, all of the coefficients of an eigenvector except
for one are relatively small, indicating that the vector
points roughly along the axis associated with the
factor corresponding to the single large coefficient.
In this case, the canonical analysis can be used to
determine the relative sensitivity of the predicted
response surface to variations in that factor.
(See the "Getting Started" section for an example.)
If the estimated surface is found to have a simple optimum
well within the range of experimentation, the analysis
performed by the preceding two steps may be sufficient.
In more complicated situations, further search
for the region of optimum response is required.
The method of ridge analysis computes the
estimated ridge of optimum response for increasing
radii from the center of the original design.
The ridge analysis answers the following question:
You can use the RIDGE statement to compute the ridge of
maximum or minimum response.
- If there is not a unique optimum of the response
surface within the range of experimentation,
in which direction should further searching
be done in order to locate the optimum?
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.