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L1592_Frame_C30 Page 278 Tuesday, December 18, 2001 2:49 PM

2 2
X X X X X X X
0 1 2 1 2 1 2
+a
1
+1 - -1 +++1
1
1
¥
1
1+ +1-1+ 1
-
1+
+1 ¥ ¥
1-1+
+1 - + 1 1 1+
1+
1+ 1+ + 1
+1+
1
0
X ¥ ¥ ¥
2 2
+1+ a a 0
0 0
X=
2
+1 - a 0 0 a
0
-1 ¥ ¥
2
+1 + a
0 0 a
0
-a ¥
2
+ 1 - a
0 0 0 a
+ 1 0 0 0 0 0
+ 1 0 0 0 0 0
-a -1 a + 01 +
X
1
FIGURE 30.2 Experimental design and matrix of independent variables for a composite design with star points and center
points. This design allows a quadratic model to be fitted by regression.
Another common situation results from our inability to set the independent variables at the levels
called for by the design. As an example of this, suppose that a design specifies four runs at pH 6 and
four at pH 7.7, but the actual pH values at the low-level runs were 5.9, 6.0, 6.1, 6.0, and similar variation
existed at the high-level runs. These give a design matrix that is not orthogonal; it is fuzzy. The data
can be analyzed by regression.
Another situation, which is discussed further in Chapter 43, is when the two-level design is augmented
by adding “star points” and center points. Figure 30.2 shows an augmented design in two factors and
the matrix of independent variables. This design allows us to fit a quadratic model of the form:
y = b 0 + b 1 x 1 + b 2 x 2 + b 11 x 1 + b 22 x 2 + b 12 x 1 x 2
2
2
The matrix of independent variables is shown in Figure 30.2. This design is not orthogonal, but almost,
because the covariance is very small.
The center points are at (0, 0). The star points are a distance a from the center, where a > 1. Without
the center points there would be an information hole in the center of the experimental region. Replicate
center points are used to improve the balance of information obtained over the experimental region, and
also to provide an estimate of the experimental error.
How do we pick a? It cannot be too big because this model is intended only to describe a limited
region. If a = 1.414, then all the corner and star points fall on a circle of diameter 1.414 and the design
is balanced and rotatable. Another common augmented design is to use a = 2.
References
Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design,
Data Analysis, and Model Building, New York, Wiley Interscience.
Draper, N. R. and H. Smith, (1998). Applied Regression Analysis, 3rd ed., New York, John Wiley.
Milliken, G. A. and D. E. Johnson (1992). Analysis of Messy Data, Vol. I: Designed Experiments, New York,
Van Nostrand Reinhold.
Exercises
3
30.1 Nitrate Measurement. A 2 factorial experiment with four replicates at a center point was run
to compare two methods for measuring nitrate and the use of a preservative. Tests were done
on two types of wastewater. Use the log-transformed data and evaluate the main and interaction
effects.
© 2002 By CRC Press LLC