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Chapter 13 Confidence Intervals for Full Second-Order Polynomial Models
279
CHAPTER 13
Confrdence Intewals for Full Second-Order Polynomial Models The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surjiuce itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models.
13.1 The model and a design In two-factors, the full second-order polynomial (FSOP) model is Y l , = Po
+ PlXll+
P2X2I
+ PI Ix:, + p22x:/ + P12Xl,X2/ + r l l
(13.1)
At least six distinctly different factor combinations (f=6) are required to fit the six parameters of this model ( p = 6). To provide three degrees of freedom for lack of fit, fmust be increased to 9. To provide three degrees of freedom for purely experimental uncertainty, n must be increased to 12. One experimental design that can be used to provide data to fit the two-factor FSOP model is the central composite design with four replicates at the center point. This design was introduced in Chapter 12 and is shown in Figure 12.12. A sums of squares and degrees of freedom tree for the design with four center points is given in Figure 13.1.
280
3
3
Figure 13.1 Sums of squares and degrees of freedom tree for a two-factor full second-order polynomial model fitted to a central composite design with a total of four center point replicates.
13.2 Normalized uncertainty and normalized information The confidence interval (C.I.) for one estimate of the true response (q) at a given factor combination is given by
or
The confidence interval for one estimate of a measured response 6 )must also take into account the uncertainty of that one measurement, and the confidence interval then becomes
or
Half the width of the confidence interval is given by
28 1
J { q l , n - p & u + [~o(~x)-'xOl)) (13.6) The width of the confidence interval depends on both F(l,n-p)and sf. But F(l,n-p)is a function of n, p , and the level of confidence the experimenter chooses to set for the particular confidence interval. And sf depends on both the lack of fit of the model to the data and the repeatability of experimentation. Because the values of these quantities depend on the experimenter, the model, and the system, F(l,n-p)and sf can be removed to give a normalized confidence interval half width that depends on the design only: J{l+[xO(xx)-lxbl)
(13.7)
This normalized half width will be called the normalized uncertainty in the predicted response. It is bounded between 1 and 00. Information theory states that uncertainty and information are related reciprocally.
-+d -4
-2
Factor X I
Figure 13.2 Central composite design. Square points +2, star points k4, DF,or= 3, DFp, = 3 .
282
Thus, we will define the normalized information as the reciprocal of the normalized uncertainty. The normalized information is bounded between 1 and 0:
I/&+
[Xo(~X)-’Xbl)
(13.8)
The normalized uncertainty and normalized information are related to the variance function and information function, respectively, defined by Box and Draper (1987). One purpose of a good design is to minimize uncertainty and maximize information over the region of interest. We will use both normalized uncertainty and normalized information to discuss the effect of experimental design on the quality of information obtained from a two-factor FSOP model.
13.3 The central composite design A central composite design is constructed from a two-level factorial design (a so-called “square”) and a multi-dimensional univariate design (a so-called ‘‘star”). A central composite design (or “star-square” design) is illustrated in Figure 13.2. This figure contains four smaller figures or panels. The lower lefi panel in Figure 13.2 shows the central composite design in the two factors x1 and x2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45” to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (xl = 0, x2 = 0). The factorial points are located +2 units from the center. The star points are located i4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is -2 -2
D=
2 2 -4 4 0 0 0
0 0 0
-2 2 -2 2 0 0 -4
4 0 0 0 0
( 13.9)
283
For a two-factor FSOP model fitted to this experimental design, n = 12, f = 9, and p = 6 . The corresponding sums of squares and degrees of freedom tree is shown in Figure 13.1. The upper left panel shows a surface of normalized uncertainty (defined in Equation 13.7 above) as a function of factors x , and x,. The normalized uncertainty is relatively small in the center (approximately 1.1) and relatively large at the comers (approximately 4.0). Note that this surface generally reflects the underlying design: the uncertainty surface is relatively low in those regions where experiments have been carried out and is relatively high in those regions where experiments have not been carried out. It is important to note that the normalized uncertainty surface shown in the upper left panel is not the response surface generated by the FSOP model itself. Instead, this upper left panel is a measure of how much the response surface might “flap around” in different regions of the factor space. Experiments serve to anchor the underlying model, to “pin it to the data,” and thereby reduce the amount of uncertainty in the model at those points. The large amount of uncertainty at the comers of this upper left panel is a reflection of the freedom the model has to move up and down in those regions where experiments have not been performed. The upper right panel shows a surface of normalized information (defined in Equation 13.8 above) as a function of factors x1 and x2. The normalized information is relatively large in the center (approximately 0.91) and relatively small at the comers (approximately 0.25). Note that this surface also reflects the underlying design: the information is relatively high in those regions where experiments have been performed and is relatively low in those regions where experiments have not been carried out. Again, it is important to note that the normalized information surface shown in the upper right panel is not the response surface generated by the FSOP model itself. Instead, this upper right panel is a reflection of how “tight” the model is in different regions of factor space. Experiments serve to give information, to provide rigidity, and thereby to provide precision. The large amount of information at the center of this upper right panel is a reflection of the tightness of the model in this region. The lower right panel plots normalized information as a function of factor x1 for x, = -5, -4, -3, -2, -1, and 0. These lines show the left front edge (x2 = -5) and parallel slices through the normalized information surface in the panel above. (For this design which is symmetric about the x1 axis, the graph lines for x2 = 1, 2, 3 , 4, and 5 are identical to lines that are already present.) One of the striking features of this central composite design is the flatness of the normalized uncertainty and normalized information surfaces near the center of the design. In Figure 13.2, the experimental design, the normalized uncertainty surface, and the normalized information surface each have four planes of mirror-image symmetry, all of which are perpendicular to the x1 - x, plane. One reflection plane contains the
284
x1 axis; a second plane contains the x2 axis; the third and fourth planes contain the
+45" and -45" diagonals.
13.4 A rotatable central composite design Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower lefi panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x1and x,. The upper right panel shows the normalized information as a function of factors x1 and x,. The lower right panel plots normalized information as a function of factor x1 for x, = -5, -4, -3, -2, -1, and 0. The experimental design matrix is
- 1
~,
-4
,
-2
,
,
,
0 Fartor X I
,
2
,
,
J
4
Figure 13.3 Rotatable central composite design. Square points 22, star points 2242, DF,,, = 3, DFp, = 3.
285
D=
-2 -2 2 2 -2J2 2J2 0 0 0 0 0 0
-2 2 -2 2 0 0 -2J2 2J2 0 0 0 0
(13.10)
This design is similar to the design in Figure 13.2, but the star points are located *242 from the center, not +4. This design is sometimes called a “circumscribed central composite design.” All of the peripheral points lie on a circumference equidistant from the center and makes the design rotatable: the uncertainty depends only on the distance from the center and not on the direction (see Section 12.9). Although the experimental design still has the four planes of mirror-image symmetry discussed in the previous section, the uncertainty and information contours each have an infinite number of planes of mirror-image symmetry passing through the origin. That is, they each have a C, axis of rotational symmetry at the origin and perpendicular to the xI-x2 plane. The normalized information at the center (and at the edges) of the factor space in Figure 13.3 is less than the normalized information at the center (and at the edges) in Figure 13.2. These effects are a result of the relative compactness of the star points in this rotatable design which allows the FSOP model to flex more at the corners of the factor space and, consequently, at the center as well.
13.5 An orthogonal rotatable central composite design
Figure 13.4 shows four panels for still another central composite design. The lower leji panel shows the experimental design itself. The upper le) panel shows the normalized uncertainty associated with this design. The upper right panel shows the
286
normalized information. The lower right panel plots normalized information as a function of factor x1 for fixed values of x2. The experimental design matrix is
D=
-2 -2 2 2 -2J2 2J2 0 0 0
-2 2 -2 2 0 0 -2J2 2J2 0
0 0 0 0 0
0 0 0 0 0 0
0 0
(13.11)
0
The design in Figure 13.4 is similar to the design in Figure 13.3, but the center point has been replicated a total of eight times, not four. This makes the design not only rotatable but also orthogonal in the coded factor space: that is, the estimate of one factor effect (i.e., p;, pi. p;l, p;*, or p;,) is independent of the estimates of all other factor effects (see Section 12.10). The normalized information at the center of the design in Figure 13.4 is somewhat greater than the normalized information at the center of the design in Figure 13.3. This is because of the additional information supplied by the extra four center-point replicates in the orthogonal design and because the additional experiments decrease the amount the FSOP model can flex. Note that orthogonality and greater information have been achieved at a relatively high cost: 16 instead of 12 (or 113 more!) experiments.
13.6 A three-level full factorial design Figure 13.5 shows still another central composite design. The experimental design matrix is
287
Figure 13.4 Orthogonal rotatable central composite design. Square points -e2, star points +2./2, DF,, = 3, DFF = 7.
288
Figure 13.5 Face centered central composite design. Square points t 2 , star points t 2 , DF, = 3.
= 3,
289
D=
-2 -2 2 2 -2 2 0 0 0 0 0 0
-2 2 -2 2 0 0 -2
(13.12)
2
0
0 0 0
This design is similar to the designs in Figures 13.2 and 13.3, but the star points are located *2 from the center, not *4 or *2d2. In effect, the star points have been brought in to the faces of the square. This design is sometimes called a “face centered central composite design”. This design is also equivalent to a two-factor three-level full factorial design. (The equivalence between a face centered central composite design and a three-level full factorial design does not hold for factor spaces of dimension greater than two, however.) A comparison of the three-level full factorial design shown in Figure 13.5 with the rotatable central composite design shown in Figure 13.3 reveals that the normalized uncertainty and normalized information at the center (and at the comers) of the factor space are approximately equal for the two designs. However, both the normalized uncertainty and normalized information surfaces in Figure 13.5 are pushed in from the sides toward the center compared with the corresponding surface in Figure 13.3. The “shaped” contours in Figure 13.5 are not circularly symmetrical but are “squared off”.
13.7 A small star design within a larger factorial design Figure 13.6 continues the sequence of Figures 13.2, 13.3, and 13.5. The experimental design matrix is
w
0
Normallzed Information 0 5 075
025 1 0
Normalized Information 0 025 0 5 075 1 0
V
I
N o r m a l i z e d Uncertainly 2 4 6 6 10
0
29 1
D=
-2 -2 2 2 4 2 J2 0
0 0 0 0 0
-2 2 -2 2 0 0 -J2 J2 0
(13.13)
0 0 0
Figure 13.7 Rotatable central composite design. Square points k2d2, star points k4, DF,or= 3, OF, = 3.
292
This design is similar to previous designs, but the star points are located 4 2 from the center, not *4 or 5~242or *2. The star points have been brought inside the faces of the square. This design is sometimes called an “inscribed central composite design’’. Note that the “sides” of the rectangular normalized information surface have been pinched inward. The shape of this surface is clearly related to the placement of the experiments in factor space as shown in the lower left panel. A constant theme of experimental design is that generally in those regions where experiments have been carried out, there is superior information; in those regions where experiments have not been carried out, there is inferior information.
13.8 A larger rotatable central composite design The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of 42: the square points expand from k 2 to k242 from the center; the star points expand from *242 to +4 from the center. The experimental design matrix is
-
-2J2 -2J2 2J2 2J2 -4 4
D=
-
0 0 0 0 0 0
-
-2J2
2J2 -2J2 2J2
0 0
-4 4
(13.14)
0 0 0 0
The design in Figure 13.7 is thus the same design as in Figure 13.3, but larger. As expected, the normalized uncertainty surface and the normalized information surface also expand in all directions from the center by the same amount (42). The result is that the normalized uncertainty is generally lower (and the normalized information is generally higher) over the whole factor domain. Further comparison of Figure 13.7 with Figure 13.3 suggests that over a given region of factor space, a broader design gives less uncertainty. This is another way of saying that if you want to find out what is happening in a certain region of factor
293
space, the most informative method is to carry out an experiment there. Extrapolation of a narrower design is done with greater uncertainty. However, as the experimental design becomes broader, there will be a greater likelihood that the empirical FSOP model will fit less well and might exhibit a greater amount of lack of fit.
13.9 A larger three-level full factorial design Figure 13.8 is related to Figure 13.5 through expansion by a factor of 2: the factorial and star points are all on the square, *4 units from the center of the design. Again, this design is equivalent to a two-factor three-level full factorial design. The experimental design matrix is
-4
-2
0
2
4
Factor X I
Figure 13.8 Face centered central composite design. Square points -c4, star points 24, DF,,, = 3, DFpc= 3.
294
-
D=
-
-4
-4
-4
4
4 4
-4
-4 4 0
0 0 -4
0 0 0 0 0
4 0 0 0 0
-
4 (13.15)
In Figure 13.8, a high degree of information is provided over most of the bounded factor domain. However, as before, the possibility of lack of fit of this broader design is greater than for the narrower design of Figure 13.5.
13.10 The effect of the distribution of replicates Replicates don’t all have to be carried out at the center point. Using the experimental design of Figure 13.2 as a basis, Figures 13.9 and 13.10 show the effects of different distributions of replicates. In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is
-
D=
-2 -2 -2 -2 2 2 2 2
-4 4 0 0
-2 -2 2 2 -2 -2 2 2 0 0 -4
4
-
(13.16)
295
-
I
,
-4
,
,
-2
,
,
,
0 Factor x i
,
2
,
,
4
'
Figure 13.9 Central composite design. Square points +4, star points +2, DF,,,, = 2, DFF = 4.
This allocation of experiments has the effect of making the normalized uncertainty and normalized information contours more axially symmetric (the design isn't quite rotatable; there are still only four mirror-image planes of reflection symmetry). However, because no experiments are now being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there). The overall effect is to provide a normalized information surface that looks like a slightly square-shaped volcano. Because there are now only eight distinctly different factor combinations (f = 8; the center point is not present), there are four degrees of freedom for purely experimental uncertainty (n - f = 12 - 8 = 4) and only two degrees of freedom for lack of fit ( f - p = 8 - 6 = 2). Figure 13.10 shows the effect of placing the four replicates at each of the star points. The experimental design matrix is
296
-
D=
-2 -2 2 2 -4 -4
4 4
0 0 0 0
-2 2
-
-2 2 0
0 0
(13.17)
0 -4 -4 4 4
v
, -4
,
,
-2
,
,
0 F a c t o r XI
,
,
2
,
,
4
Figure 13.10 Central composite design. Square points +4, star points +2, DF,,, = 2, DFF = 4.
297
This allocation of experiments has the effect of emphasizing the star points in the normalized uncertainty and normalized information contours. The contours are “bumpier” now, with the bumps occurring at the star points. Because no experiments are being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there) than in the original design of Figure 13.2. As in Figure 13.9, there are only eight distinctly different factor combinations (f= 8). Thus, there are four degrees of freedom for purely experimental uncertainty and only two degrees of freedom for lack of fit.
13.11 Non-central composite designs Figure 13.1 1 is a non-central composite design - the center of the square design and the center of the star design do not coincide. Thus, the two individual designs which make up the combined design are not centered. It is a composite design, but it is non-central. The experimental design matrix is
-
D=
-2 -2 2 0 0 0 0 0 0
-2 0
0 -2 2
(13.18)
0 0 0 0-
There are many ways to view the construction of the experimental design shown in Figure 13.11. Of these, perhaps the most straightforward is to note that the design in Figure 13.11 can be derived from the design in Figure 13.5 by removing three of the comer points. This leaves a smaller factorial design in the lower left quadrant of the original design while still retaining the original star design. The design in Figure 13.11 has only nine experiments (n = 9) and only six distinctly different factor combinations (f= 6). Thus, there are still three degrees of freedom for purely experimental uncertainty (n -f= 9 - 6 = 3). However, the FSOP model has six parameters ( p = 6 ) , so there are now no degrees of freedom for lack of fit ( f - p = 6 - 6 = 0). In the upper left panel of Figure 13.1 1, standard uncertainties greater than 10 have been truncated to 10. The experimental design in Figure 13.11 is more efficient than the experimental design in Figure 13.5 in the sense that fewer experiments are used to estimate the parameters of the model ( E = p / f = 6/6 = I.OO), but the quality of information (as
298
shown in the normalized uncertainty and normalized information surfaces) suffers as a result. Figure 13.12 is another non-central composite design. The experimental design matrix is
-2 -2 2
-2
D=
0 0 0 0 0
-2 2 -2 0 -2 0 0 0
(13.19)
0
a
u
,
-4
,
-2
,
,
0
,
,
2
~
I
,
4
Factor X 1
Figure 13.11 Non-central composite design. Square point -2, star points +2, DF,,, = 0, DFpe= 3.
299
The design in Figure 13.12 can be derived from the design in Figure 13.5 by removing one of the comer points and two of the edge points. This leaves a small factorial design in the lower left quadrant of the original design. Now, however, the arms of the star have been pulled apart and placed along two perpendicular edges of the small factorial design. The degrees of freedom are the same as for the design in Figure 13.11. The design in Figure 13.12 is similar to the mixture design shown in the lower right panel of Figure 12.28.
13.12 A randomly generated design The nine distinctly different factor combinations in Figure 13.13 were obtained from a random number generator. In this sense, the experimental design in Figure 13.13 is a “totally random design.” One factor combination was chosen (again at random) and three additional experiments were carried out there to provide three degrees of freedom for purely experimental uncertainty. There are three degrees of freedom for lack of fit. The experimental design matrix is
-
3. I2920 0.56375 -2.94542 -2.86738 - 3.73338 -0.1 1982 D= 2.45737 - 3.3465 1 4.15284 4.15284 4.15284 - 4.15284
2.56303
- 4. I3827
1.23608 0.62983 4.95222 - 1.48349 -0.453 19 - 1.53869 0.25859 0.25859 0.25859 0.25859
( 13.20)
Note that information is greatest in regions where experiments have been carried out. In the lower left panel of Figure 13.13 there is an open area where few experiments have been carried out (between xI = -3 and x, = 3, and at x, > 0). In the upper right figure this barren region appears as a valley sloping downward toward the back right. Even so, this randomly generated experimental design provides fairly good information across the factor domain. Occasionally, random number generators will give a poor design (e.g., all experiments might end up in a very small fraction of the desired factor domain). However, as long as the resulting design spans the desired factor domain, has enough factor combinations to determine lack of fit, and has replicates to determine purely
300
-4
-2
0 Factor
2
4
xi
Figure 13.12 Non-central composite design. Square points *2, star point -2, DF,,,, = 0, DFpc= 3.
experimental uncertainty, random processes seem to generate ‘‘pretty good” experimental designs. This is not a recommendation of random or haphazard designs, especially pseudo-random designs generated by researchers. Such designs frequently do not span the factor space, have far too many degrees of freedom for lack of fit, and have no degrees of freedom for purely experimental uncertainty. Using standard experimental designs (such as the central composite design) or creating new experimental designs from sound statistical principles is almost always more efficient and informative than any randomly generated design. Of course, the design in Figure 13.13 could be improved by moving one of the two closely placed experiments near ( x , = -2.5, x2 = 1) into the open space discussed above. And the point near (xl = -3, x2 = -1) might be moved a bit lower in x2. And then if ....
301
13.13 A star design Figure 13.14 shows a star design that can be used to fit a two-factor FSOP model. The experimental design matrix is
-
D=
-4.0 -2.5 -2.5 -1.0 0.5 1.0 3.5 4.0
1.5 2.0 2.0 1.5 1.0 -1.0 -1.0 1.0
(13.21)
1
"
-
I
, -4
,
,
-2
,
, 0
,
FaCt3r X I
Figure 13.13 A totally random design. DF,,, = 3, DFpe= 3.
,
2
,
, 4
'
302
-
D
7
-
1
Factor X 1
Figure 13.14 A star design. (The replicatcs are Mizar and Alcor in Ursa Major.) DF,,, = 1, DFF = 1.
Because the number of distinctly different factor combinations is seven ( f = 7), and because the number of experiments is eight (n = 8), there is only one degree of freedom for lack of fit (f- p = 7 - 6 = 1) and only one degree of freedom for purely experimental uncertainty (n - f = 8 - 7 = 1).
13.14 Rotatable polyhedral designs Figure 13.15 shows a design based on a regular polyhedron, the pentagon [Himmelblau (1970)l. The experimental design matrix is
303
D=
2.000 0.618 -1.618 - 1.618 0.618 0.000 0.000 0.000 0.000
0.000 1.902 1.176 - 1.176 - 1.902 0.000
(13.22)
0.000
0.000 0.000
Because each of the pentagonal points is equidistant from the center of the design, the design is rotatable. This rotatability is seen in the axially symmetric surfaces for normalized uncertainty and normalized information.
-4
-2
0
2
Factor X I
Figure 13.15 A pentagonal rotatable design with center point. DF,, = 0, DFp = 3.
4
304
In this design there are only six distinctly different factor combinations. Thus, there are no degrees of freedom for lack of fit when fitting a two-factor FSOP model with six parameters. There are three degrees of freedom for purely experimental uncertainty because of the four replicate experiments at the center point. Figure 13.16 is a design based on a regular hexagon [Himmelblau (1970)l. The experimental design matrix is
D=
2.000 1.000 - 1.000 -2.000 - 1.OOO 1.000 0.000 0.000 0.000 0.000
0.000 1.732 1.732 0.000 - 1.732 - 1.732 0.000 0.000 0.000 0.000
( 13.23)
Because each of the hexagonal points is equidistant from the center of the design, the design is rotatable. There are seven distinctly different factor combinations. Thus, there is one degree of freedom for lack of fit when fitting a two-factor FSOP model with six parameters. There are three degrees of freedom for purely experimental uncertainty because of the four replicate experiments at the center point.
13.15 The flexing geometry of full second-order polynomial models Figure 13.17 shows a design similar to the hexagonal design in the previous Figure 13.16, but in this new design only one experiment has been carried out at the center. The experimental design matrix is
D=
2.000 1.000 - 1.000 -2.000
- 1.OOO
1.000 0.000
0.000 1.732 1.732 0.000 - 1.732 - 1.732 0.000
( 13.24)
305
" I
01
,
-4
,
,
-2
,
,
,
0 Factor X1
,
2
,
,
4
Figure 13.16 A hexagonal rotatable design with center point. DF,,, = 1, DFpe = 3 .
A comparison of the lower right panels of Figures 13.16 and 13.17 shows that at the center of the factor domain, the normalized information is less for the design with only one experiment at the center (Figure 13.17) than for the design with four replicates at the center (Figure 13.16). This is confirmed in the upper right panels of both figures: the design with fewer experiments at the center (Figure 13.17) has a depression in the center of its normalized information surface; the design with more experiments at the center (Figure 13.16) has a smoothly sloped dome at the center of its normalized information surface. A moment's thought suggests that the decreased information at the center of the design in Figure 13.17 is reasonable and expected - the depression is a result of the principle that if many experiments give more information, then fewer experiments give less information. But this simple explanation becomes inadequate when it is realized that the information is greater on the ring of hexagonal points that surround the center point
306
than it is at the center point itself. After all, the six circumferential points and the center point are laid out on an equilateral triangular grid: why should one of them (especially the center point) provide less information than the other six? The answer to this seeming conundrum lies in the geometry of the fitted model. In Chapter 12, Figures 12.18-12.22 show some of the possible response surfaces that can be represented by the two-factor FSOP model. (Here it is important to note that the response surfaces shown in Figures 12.18-12.22 are generated by the FSOP model and are not the normalized uncertainty or normalized information surfaces shown in the upper panels of Figures 13.16 or 13.17.) Consider one of these response surfaces, the parabolic bowl (paraboloid) opening downward, shown in Figure 12.19. In this canonical form, the maximum of the dome corresponds to the center of the factor domain and the sides of the surface slope downward away from the center. The paraboloid is symmetrical about an axis at the origin and perpendicular to the x1-x2 plane. The discussion that follows will use the geometry of this somewhat unique
Factor x i
Figure 13.17 A hexagonal rotatable design with center point. DF,or= 1, DFpe= 0.
307
response surface to rationalize the shapes of the normalized uncertainty and normalized information surfaces in Figure 13.17. (The results are identical for developments that involve other forms of the FSOP model, but the explanations are less straightforward.) The six outer experiments of the hexagonal design are like a pair of hands clasping the paraboloid around its sides, much as an American football player might catch a well-thrown pass. Even though the response surface might flex and writhe over other regions of factor space, it will be held rigidly around this circumference and won't move much at all. Thus, the information content will be high over this circular region. Geometrically, there are an infinite number of paraboloids that can pass through a circle (e.g., the circle of equal responses at the hexagonal points in this example). Some of the paraboloids will be tall and elongated, some will be short and compressed, some will point up, some will point down, one of them will even be a degenerate flat plane [Rider (1947)l. Although the hexagonal points hold the sides
_7_
-4
0
Factor X 1
Figure 13.18 A hexagonal design with no center point. DF,,, = 0,DFF = 0.
7
4
308
Factor X I
Figure 13.19 A hexagonal design with extra outcr point. DF,,, = I , DFpe= 0.
tightly, the tip (or apex) of the paraboloid is defined by the response at the center point only. Any noise or uncertainty at the center point will move the apex up or down without any resistance from the circumferential points. Any uncertainty in the single center point will not be averaged by the other points, and the uncertainty there will remain relatively large. This is why there is a depression at the top of the normalized information surface in Figure 13.17. The hexagonal ring of points represents a circular node in one of the normal vibrations of a paraboloid. If the paraboloid is held rigidly around this circle, then pressing down on the top of the paraboloid will cause the sides of the paraboloid to flair outward below the circle. Pulling up on the top of the paraboloid will cause the sides of the paraboloid to squeeze inward below the circle. The greater the variation at the center point inside the circular node of points, the greater will be the variation in regions outside this node. This effect can be seen by comparing Figures 13.16 and 13.17. The upper left panels of the two figures show that, as expected, the normalized
309
uncertainty for the design with only one center point (Figure 13.17) is greater at the comers of the factor domain than for the design with four center points (Figure 13.16). At a very basic level, the shapes of the normalized uncertainty and normalized information surfaces for a given model are a result of the location of points in factor space simply because carrying out an experiment provides information - that is, information is greatest in the vicinity of the design. But at a more sophisticated and often more important level, the shapes of the normalized uncertainty and normalized information surfaces are caused by the geometric vibrations of the response surfaces themselves - the more rigidly the model is “pinned down” by the experiments and the less it can squirm and thrash about, then the less will be the uncertainty and the greater will be the information content.
13.16 An extreme effect illustrated with the hexagonal design
The lower left panel of Figure 13.18 shows a hexagonal design without a center point. Considerations of degrees of freedom suggest that it should be possible to fit a FSOP model to data from these six experiments: n = 6 , f = 6, p = 6; thus, although the degrees of freedom for residuals, lack of fit, and purely experimental uncertainty are all equal to zero, they are not negative, and the model would be expected to fit perfectly. Even so, the determinant of the (X’X)matrix is zero, the matrix cannot be inverted, and the two-factor FSOP model cannot be fitted to the data in Figure 13.18. Geometrically, the zero determinant arises because the number of FSOP models that can be made to pass through the responses above the hexagonal factor combinations is now truly infinite (see Section 5.6). There is no center point to define the apex and prevent the model from fluttering about. For purposes of illustration only, to circumvent the problem of a zero determinant but still show the distributions of uncertainty and information in this design, a seventh experiment was added at a factor combination just slightly removed from one of the hexagonal points (at x1 = 2.000, x2 = 0.001). The hexagonal points were also adjusted somewhat to coincide with the grid lines in the pseudo-three-dimensional plots (this is equivalent to a minor adjustment of scale in the x, dimension). Neither of these modifications significantly affects the overall conclusions to be drawn from this example. The actual design is
310
D=
2.000 1.000 - 1.000 -2.000 - 1.000 1.000 2.000
0.000 1.750 1.750 0.000 - 1.750 - 1.750 0.001
( 13.25)
The results from this seven-experiment design are shown in Figure 13.18. The striking feature of this design is the set of six spikes in both the normalized uncertainty and normalized information surfaces. These spikes are an extreme expression of the basic idea that experiments provide information. Even if the experimental design is not a good match for the model; even if the (X’X) matrix is ill conditioned; even if the model doesn’t fit the data very well, there is still highquality information at the points where experiments have been carried out. Figure 13.19 shows the effect of moving the seventh experiment farther away from one of the hexagonal points. In this example, the seventh experiment is at n, = 2.000, x2 = 0.500. This experiment gives enough leverage that the response surface becomes better defined around the ring of hexagonal points. Because the resolution of the plotting grid is not sufficient to resolve all of the sharp detail in the cylindrical surfaces, the normalized uncertainty was truncated at 4.0 and the normalized information was correspondingly truncated at 0.25; the actual surfaces extend below and above the capped surfaces shown in Figure 13.19. Figure 13.20 shows the effect of moving the seventh experiment still farther away from the hexagonal points (to x , = 4.000, x2 = 4.000). The more distant point has good leverage, and the fitted response surface becomes much more rigid (as indicated by the normalized uncertainty and normalized information surfaces in this figure). Note also that the information is high above the distant point.
13.17 Two final examples Figure 13.21 shows the effect of adding an extra, remote star point to a small face centered central composite design. The experimental design matrix is
I
I
o\
h,
w
> 0 0 0 0 0 0 + e - - - -
II
b
I
I
P
II
G
2
w f:
z
c
3
09 0
e, X
z
P
0
L
z
025
4
0 5
6
8
1
075
0
1 0
Normalized Information 0
A
2
~ o r m a l i r e dUncertainty
0
312
_
I
,
-4
,
,
-2
,
,
0
,
~,
2
,
4
F a c t o r XI
Figure 13.21 An inscribed central composite design with distantly located extra star point. DF,or = 4, DFF = 3.
The elephant-like contours result from the stabilizing leverage of the distant factor combination. Note that this additional point stabilizes the fitted model in the x,-direction (the dimension in which it was extended), but has little effect in the x,-direction. The coded design shown in Figure 13.22 is from a clinical chemical study investigating the interference of magnesium in the analytical chemical determination of calcium [Olansky, Parker, Morgan, and Deming (1977), Deming and Morgan (1979)l. The uncoded factors represent the concentrations of calcium and magnesium in human blood serum. The experimental design matrix is
3
B It
b
b
Y
a
e,
N o r m a l i z e d Information
4
6
8
1
0
Y
Y
I
Normalized Information 0 025 0 5 0 7 5 1 0
2
Normalized Uncertainty 0
314
D=
-3 -3 -2 -2 -1 -1 -1 -1 0 0 0 0 0 0 1 1 1 1 2 2
0 0 0 0
-1 -1
1 1
0 0 0 0 2 3
( 13.27)
-1 -1
1 1 0 0
It could be argued statistically that the number of replicates in Figure 13.22 is excessively large (the person carrying out these experiments would also argue that the number of replicates is excessively large!): the relatively small improvement in the quality of information in the region of the design has been gained at the relatively large expense of the additional experiments. It might also be argued statistically (indeed, it is perhaps one of the major points of this chapter) that the domain of the experimental design represents only a small fraction of the factor space shown: a broader design would have given smaller uncertainties and more precise information over the whole factor space (see, for example, Figure 13.8). However, in the example of Figure 13.22, the factorial part of the design adequately covers the combinations of calcium and magnesium found in living humans. The extended star points were used to obtain precise estimates of curvature in xI and x2.There is no practical reason for investigating combinations of calcium and magnesium in the unexplored regions: serum samples with these combinations of concentrations could only have come from the morgue.
315
Exercises 13.1 Minimalist design. Add one design point to a two-factor star design to generate a design that is sufficient to fit a full second-order polynomial model ( yli = Po + plxli+ p~~~+ p1& + p,&, + plgl$z + rli).Hint: see Figure 13.11. 13.2 Optimal design.
Assume a constrained factor space of -5 I x, I +5, -5 I x, I +5. Assume the full two-factor model with interaction, y l i = Po + pixli + pzxzi + plzx l ixzi + rli. Assume a 22 factorial design. How should the four design points be placed to maximize the determinant of the (X'X)matrix? Demonstrate with a few calculations.
13.3 Existing designs. Find a report of a two-factor experimental design. Speculate about the shape of the normalized uncertainty and normalized information surfaces for the design. Sketch their shape.
CHAPTER 13
Confrdence Intewals for Full Second-Order Polynomial Models The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surjiuce itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models.
13.1 The model and a design In two-factors, the full second-order polynomial (FSOP) model is Y l , = Po
+ PlXll+
P2X2I
+ PI Ix:, + p22x:/ + P12Xl,X2/ + r l l
(13.1)
At least six distinctly different factor combinations (f=6) are required to fit the six parameters of this model ( p = 6). To provide three degrees of freedom for lack of fit, fmust be increased to 9. To provide three degrees of freedom for purely experimental uncertainty, n must be increased to 12. One experimental design that can be used to provide data to fit the two-factor FSOP model is the central composite design with four replicates at the center point. This design was introduced in Chapter 12 and is shown in Figure 12.12. A sums of squares and degrees of freedom tree for the design with four center points is given in Figure 13.1.
280
3
3
Figure 13.1 Sums of squares and degrees of freedom tree for a two-factor full second-order polynomial model fitted to a central composite design with a total of four center point replicates.
13.2 Normalized uncertainty and normalized information The confidence interval (C.I.) for one estimate of the true response (q) at a given factor combination is given by
or
The confidence interval for one estimate of a measured response 6 )must also take into account the uncertainty of that one measurement, and the confidence interval then becomes
or
Half the width of the confidence interval is given by
28 1
J { q l , n - p & u + [~o(~x)-'xOl)) (13.6) The width of the confidence interval depends on both F(l,n-p)and sf. But F(l,n-p)is a function of n, p , and the level of confidence the experimenter chooses to set for the particular confidence interval. And sf depends on both the lack of fit of the model to the data and the repeatability of experimentation. Because the values of these quantities depend on the experimenter, the model, and the system, F(l,n-p)and sf can be removed to give a normalized confidence interval half width that depends on the design only: J{l+[xO(xx)-lxbl)
(13.7)
This normalized half width will be called the normalized uncertainty in the predicted response. It is bounded between 1 and 00. Information theory states that uncertainty and information are related reciprocally.
-+d -4
-2
Factor X I
Figure 13.2 Central composite design. Square points +2, star points k4, DF,or= 3, DFp, = 3 .
282
Thus, we will define the normalized information as the reciprocal of the normalized uncertainty. The normalized information is bounded between 1 and 0:
I/&+
[Xo(~X)-’Xbl)
(13.8)
The normalized uncertainty and normalized information are related to the variance function and information function, respectively, defined by Box and Draper (1987). One purpose of a good design is to minimize uncertainty and maximize information over the region of interest. We will use both normalized uncertainty and normalized information to discuss the effect of experimental design on the quality of information obtained from a two-factor FSOP model.
13.3 The central composite design A central composite design is constructed from a two-level factorial design (a so-called “square”) and a multi-dimensional univariate design (a so-called ‘‘star”). A central composite design (or “star-square” design) is illustrated in Figure 13.2. This figure contains four smaller figures or panels. The lower lefi panel in Figure 13.2 shows the central composite design in the two factors x1 and x2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45” to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (xl = 0, x2 = 0). The factorial points are located +2 units from the center. The star points are located i4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is -2 -2
D=
2 2 -4 4 0 0 0
0 0 0
-2 2 -2 2 0 0 -4
4 0 0 0 0
( 13.9)
283
For a two-factor FSOP model fitted to this experimental design, n = 12, f = 9, and p = 6 . The corresponding sums of squares and degrees of freedom tree is shown in Figure 13.1. The upper left panel shows a surface of normalized uncertainty (defined in Equation 13.7 above) as a function of factors x , and x,. The normalized uncertainty is relatively small in the center (approximately 1.1) and relatively large at the comers (approximately 4.0). Note that this surface generally reflects the underlying design: the uncertainty surface is relatively low in those regions where experiments have been carried out and is relatively high in those regions where experiments have not been carried out. It is important to note that the normalized uncertainty surface shown in the upper left panel is not the response surface generated by the FSOP model itself. Instead, this upper left panel is a measure of how much the response surface might “flap around” in different regions of the factor space. Experiments serve to anchor the underlying model, to “pin it to the data,” and thereby reduce the amount of uncertainty in the model at those points. The large amount of uncertainty at the comers of this upper left panel is a reflection of the freedom the model has to move up and down in those regions where experiments have not been performed. The upper right panel shows a surface of normalized information (defined in Equation 13.8 above) as a function of factors x1 and x2. The normalized information is relatively large in the center (approximately 0.91) and relatively small at the comers (approximately 0.25). Note that this surface also reflects the underlying design: the information is relatively high in those regions where experiments have been performed and is relatively low in those regions where experiments have not been carried out. Again, it is important to note that the normalized information surface shown in the upper right panel is not the response surface generated by the FSOP model itself. Instead, this upper right panel is a reflection of how “tight” the model is in different regions of factor space. Experiments serve to give information, to provide rigidity, and thereby to provide precision. The large amount of information at the center of this upper right panel is a reflection of the tightness of the model in this region. The lower right panel plots normalized information as a function of factor x1 for x, = -5, -4, -3, -2, -1, and 0. These lines show the left front edge (x2 = -5) and parallel slices through the normalized information surface in the panel above. (For this design which is symmetric about the x1 axis, the graph lines for x2 = 1, 2, 3 , 4, and 5 are identical to lines that are already present.) One of the striking features of this central composite design is the flatness of the normalized uncertainty and normalized information surfaces near the center of the design. In Figure 13.2, the experimental design, the normalized uncertainty surface, and the normalized information surface each have four planes of mirror-image symmetry, all of which are perpendicular to the x1 - x, plane. One reflection plane contains the
284
x1 axis; a second plane contains the x2 axis; the third and fourth planes contain the
+45" and -45" diagonals.
13.4 A rotatable central composite design Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower lefi panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x1and x,. The upper right panel shows the normalized information as a function of factors x1 and x,. The lower right panel plots normalized information as a function of factor x1 for x, = -5, -4, -3, -2, -1, and 0. The experimental design matrix is
- 1
~,
-4
,
-2
,
,
,
0 Fartor X I
,
2
,
,
J
4
Figure 13.3 Rotatable central composite design. Square points 22, star points 2242, DF,,, = 3, DFp, = 3.
285
D=
-2 -2 2 2 -2J2 2J2 0 0 0 0 0 0
-2 2 -2 2 0 0 -2J2 2J2 0 0 0 0
(13.10)
This design is similar to the design in Figure 13.2, but the star points are located *242 from the center, not +4. This design is sometimes called a “circumscribed central composite design.” All of the peripheral points lie on a circumference equidistant from the center and makes the design rotatable: the uncertainty depends only on the distance from the center and not on the direction (see Section 12.9). Although the experimental design still has the four planes of mirror-image symmetry discussed in the previous section, the uncertainty and information contours each have an infinite number of planes of mirror-image symmetry passing through the origin. That is, they each have a C, axis of rotational symmetry at the origin and perpendicular to the xI-x2 plane. The normalized information at the center (and at the edges) of the factor space in Figure 13.3 is less than the normalized information at the center (and at the edges) in Figure 13.2. These effects are a result of the relative compactness of the star points in this rotatable design which allows the FSOP model to flex more at the corners of the factor space and, consequently, at the center as well.
13.5 An orthogonal rotatable central composite design
Figure 13.4 shows four panels for still another central composite design. The lower leji panel shows the experimental design itself. The upper le) panel shows the normalized uncertainty associated with this design. The upper right panel shows the
286
normalized information. The lower right panel plots normalized information as a function of factor x1 for fixed values of x2. The experimental design matrix is
D=
-2 -2 2 2 -2J2 2J2 0 0 0
-2 2 -2 2 0 0 -2J2 2J2 0
0 0 0 0 0
0 0 0 0 0 0
0 0
(13.11)
0
The design in Figure 13.4 is similar to the design in Figure 13.3, but the center point has been replicated a total of eight times, not four. This makes the design not only rotatable but also orthogonal in the coded factor space: that is, the estimate of one factor effect (i.e., p;, pi. p;l, p;*, or p;,) is independent of the estimates of all other factor effects (see Section 12.10). The normalized information at the center of the design in Figure 13.4 is somewhat greater than the normalized information at the center of the design in Figure 13.3. This is because of the additional information supplied by the extra four center-point replicates in the orthogonal design and because the additional experiments decrease the amount the FSOP model can flex. Note that orthogonality and greater information have been achieved at a relatively high cost: 16 instead of 12 (or 113 more!) experiments.
13.6 A three-level full factorial design Figure 13.5 shows still another central composite design. The experimental design matrix is
287
Figure 13.4 Orthogonal rotatable central composite design. Square points -e2, star points +2./2, DF,, = 3, DFF = 7.
288
Figure 13.5 Face centered central composite design. Square points t 2 , star points t 2 , DF, = 3.
= 3,
289
D=
-2 -2 2 2 -2 2 0 0 0 0 0 0
-2 2 -2 2 0 0 -2
(13.12)
2
0
0 0 0
This design is similar to the designs in Figures 13.2 and 13.3, but the star points are located *2 from the center, not *4 or *2d2. In effect, the star points have been brought in to the faces of the square. This design is sometimes called a “face centered central composite design”. This design is also equivalent to a two-factor three-level full factorial design. (The equivalence between a face centered central composite design and a three-level full factorial design does not hold for factor spaces of dimension greater than two, however.) A comparison of the three-level full factorial design shown in Figure 13.5 with the rotatable central composite design shown in Figure 13.3 reveals that the normalized uncertainty and normalized information at the center (and at the comers) of the factor space are approximately equal for the two designs. However, both the normalized uncertainty and normalized information surfaces in Figure 13.5 are pushed in from the sides toward the center compared with the corresponding surface in Figure 13.3. The “shaped” contours in Figure 13.5 are not circularly symmetrical but are “squared off”.
13.7 A small star design within a larger factorial design Figure 13.6 continues the sequence of Figures 13.2, 13.3, and 13.5. The experimental design matrix is
w
0
Normallzed Information 0 5 075
025 1 0
Normalized Information 0 025 0 5 075 1 0
V
I
N o r m a l i z e d Uncertainly 2 4 6 6 10
0
29 1
D=
-2 -2 2 2 4 2 J2 0
0 0 0 0 0
-2 2 -2 2 0 0 -J2 J2 0
(13.13)
0 0 0
Figure 13.7 Rotatable central composite design. Square points k2d2, star points k4, DF,or= 3, OF, = 3.
292
This design is similar to previous designs, but the star points are located 4 2 from the center, not *4 or 5~242or *2. The star points have been brought inside the faces of the square. This design is sometimes called an “inscribed central composite design’’. Note that the “sides” of the rectangular normalized information surface have been pinched inward. The shape of this surface is clearly related to the placement of the experiments in factor space as shown in the lower left panel. A constant theme of experimental design is that generally in those regions where experiments have been carried out, there is superior information; in those regions where experiments have not been carried out, there is inferior information.
13.8 A larger rotatable central composite design The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of 42: the square points expand from k 2 to k242 from the center; the star points expand from *242 to +4 from the center. The experimental design matrix is
-
-2J2 -2J2 2J2 2J2 -4 4
D=
-
0 0 0 0 0 0
-
-2J2
2J2 -2J2 2J2
0 0
-4 4
(13.14)
0 0 0 0
The design in Figure 13.7 is thus the same design as in Figure 13.3, but larger. As expected, the normalized uncertainty surface and the normalized information surface also expand in all directions from the center by the same amount (42). The result is that the normalized uncertainty is generally lower (and the normalized information is generally higher) over the whole factor domain. Further comparison of Figure 13.7 with Figure 13.3 suggests that over a given region of factor space, a broader design gives less uncertainty. This is another way of saying that if you want to find out what is happening in a certain region of factor
293
space, the most informative method is to carry out an experiment there. Extrapolation of a narrower design is done with greater uncertainty. However, as the experimental design becomes broader, there will be a greater likelihood that the empirical FSOP model will fit less well and might exhibit a greater amount of lack of fit.
13.9 A larger three-level full factorial design Figure 13.8 is related to Figure 13.5 through expansion by a factor of 2: the factorial and star points are all on the square, *4 units from the center of the design. Again, this design is equivalent to a two-factor three-level full factorial design. The experimental design matrix is
-4
-2
0
2
4
Factor X I
Figure 13.8 Face centered central composite design. Square points -c4, star points 24, DF,,, = 3, DFpc= 3.
294
-
D=
-
-4
-4
-4
4
4 4
-4
-4 4 0
0 0 -4
0 0 0 0 0
4 0 0 0 0
-
4 (13.15)
In Figure 13.8, a high degree of information is provided over most of the bounded factor domain. However, as before, the possibility of lack of fit of this broader design is greater than for the narrower design of Figure 13.5.
13.10 The effect of the distribution of replicates Replicates don’t all have to be carried out at the center point. Using the experimental design of Figure 13.2 as a basis, Figures 13.9 and 13.10 show the effects of different distributions of replicates. In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is
-
D=
-2 -2 -2 -2 2 2 2 2
-4 4 0 0
-2 -2 2 2 -2 -2 2 2 0 0 -4
4
-
(13.16)
295
-
I
,
-4
,
,
-2
,
,
,
0 Factor x i
,
2
,
,
4
'
Figure 13.9 Central composite design. Square points +4, star points +2, DF,,,, = 2, DFF = 4.
This allocation of experiments has the effect of making the normalized uncertainty and normalized information contours more axially symmetric (the design isn't quite rotatable; there are still only four mirror-image planes of reflection symmetry). However, because no experiments are now being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there). The overall effect is to provide a normalized information surface that looks like a slightly square-shaped volcano. Because there are now only eight distinctly different factor combinations (f = 8; the center point is not present), there are four degrees of freedom for purely experimental uncertainty (n - f = 12 - 8 = 4) and only two degrees of freedom for lack of fit ( f - p = 8 - 6 = 2). Figure 13.10 shows the effect of placing the four replicates at each of the star points. The experimental design matrix is
296
-
D=
-2 -2 2 2 -4 -4
4 4
0 0 0 0
-2 2
-
-2 2 0
0 0
(13.17)
0 -4 -4 4 4
v
, -4
,
,
-2
,
,
0 F a c t o r XI
,
,
2
,
,
4
Figure 13.10 Central composite design. Square points +4, star points +2, DF,,, = 2, DFF = 4.
297
This allocation of experiments has the effect of emphasizing the star points in the normalized uncertainty and normalized information contours. The contours are “bumpier” now, with the bumps occurring at the star points. Because no experiments are being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there) than in the original design of Figure 13.2. As in Figure 13.9, there are only eight distinctly different factor combinations (f= 8). Thus, there are four degrees of freedom for purely experimental uncertainty and only two degrees of freedom for lack of fit.
13.11 Non-central composite designs Figure 13.1 1 is a non-central composite design - the center of the square design and the center of the star design do not coincide. Thus, the two individual designs which make up the combined design are not centered. It is a composite design, but it is non-central. The experimental design matrix is
-
D=
-2 -2 2 0 0 0 0 0 0
-2 0
0 -2 2
(13.18)
0 0 0 0-
There are many ways to view the construction of the experimental design shown in Figure 13.11. Of these, perhaps the most straightforward is to note that the design in Figure 13.11 can be derived from the design in Figure 13.5 by removing three of the comer points. This leaves a smaller factorial design in the lower left quadrant of the original design while still retaining the original star design. The design in Figure 13.11 has only nine experiments (n = 9) and only six distinctly different factor combinations (f= 6). Thus, there are still three degrees of freedom for purely experimental uncertainty (n -f= 9 - 6 = 3). However, the FSOP model has six parameters ( p = 6 ) , so there are now no degrees of freedom for lack of fit ( f - p = 6 - 6 = 0). In the upper left panel of Figure 13.1 1, standard uncertainties greater than 10 have been truncated to 10. The experimental design in Figure 13.11 is more efficient than the experimental design in Figure 13.5 in the sense that fewer experiments are used to estimate the parameters of the model ( E = p / f = 6/6 = I.OO), but the quality of information (as
298
shown in the normalized uncertainty and normalized information surfaces) suffers as a result. Figure 13.12 is another non-central composite design. The experimental design matrix is
-2 -2 2
-2
D=
0 0 0 0 0
-2 2 -2 0 -2 0 0 0
(13.19)
0
a
u
,
-4
,
-2
,
,
0
,
,
2
~
I
,
4
Factor X 1
Figure 13.11 Non-central composite design. Square point -2, star points +2, DF,,, = 0, DFpe= 3.
299
The design in Figure 13.12 can be derived from the design in Figure 13.5 by removing one of the comer points and two of the edge points. This leaves a small factorial design in the lower left quadrant of the original design. Now, however, the arms of the star have been pulled apart and placed along two perpendicular edges of the small factorial design. The degrees of freedom are the same as for the design in Figure 13.11. The design in Figure 13.12 is similar to the mixture design shown in the lower right panel of Figure 12.28.
13.12 A randomly generated design The nine distinctly different factor combinations in Figure 13.13 were obtained from a random number generator. In this sense, the experimental design in Figure 13.13 is a “totally random design.” One factor combination was chosen (again at random) and three additional experiments were carried out there to provide three degrees of freedom for purely experimental uncertainty. There are three degrees of freedom for lack of fit. The experimental design matrix is
-
3. I2920 0.56375 -2.94542 -2.86738 - 3.73338 -0.1 1982 D= 2.45737 - 3.3465 1 4.15284 4.15284 4.15284 - 4.15284
2.56303
- 4. I3827
1.23608 0.62983 4.95222 - 1.48349 -0.453 19 - 1.53869 0.25859 0.25859 0.25859 0.25859
( 13.20)
Note that information is greatest in regions where experiments have been carried out. In the lower left panel of Figure 13.13 there is an open area where few experiments have been carried out (between xI = -3 and x, = 3, and at x, > 0). In the upper right figure this barren region appears as a valley sloping downward toward the back right. Even so, this randomly generated experimental design provides fairly good information across the factor domain. Occasionally, random number generators will give a poor design (e.g., all experiments might end up in a very small fraction of the desired factor domain). However, as long as the resulting design spans the desired factor domain, has enough factor combinations to determine lack of fit, and has replicates to determine purely
300
-4
-2
0 Factor
2
4
xi
Figure 13.12 Non-central composite design. Square points *2, star point -2, DF,,,, = 0, DFpc= 3.
experimental uncertainty, random processes seem to generate ‘‘pretty good” experimental designs. This is not a recommendation of random or haphazard designs, especially pseudo-random designs generated by researchers. Such designs frequently do not span the factor space, have far too many degrees of freedom for lack of fit, and have no degrees of freedom for purely experimental uncertainty. Using standard experimental designs (such as the central composite design) or creating new experimental designs from sound statistical principles is almost always more efficient and informative than any randomly generated design. Of course, the design in Figure 13.13 could be improved by moving one of the two closely placed experiments near ( x , = -2.5, x2 = 1) into the open space discussed above. And the point near (xl = -3, x2 = -1) might be moved a bit lower in x2. And then if ....
301
13.13 A star design Figure 13.14 shows a star design that can be used to fit a two-factor FSOP model. The experimental design matrix is
-
D=
-4.0 -2.5 -2.5 -1.0 0.5 1.0 3.5 4.0
1.5 2.0 2.0 1.5 1.0 -1.0 -1.0 1.0
(13.21)
1
"
-
I
, -4
,
,
-2
,
, 0
,
FaCt3r X I
Figure 13.13 A totally random design. DF,,, = 3, DFpe= 3.
,
2
,
, 4
'
302
-
D
7
-
1
Factor X 1
Figure 13.14 A star design. (The replicatcs are Mizar and Alcor in Ursa Major.) DF,,, = 1, DFF = 1.
Because the number of distinctly different factor combinations is seven ( f = 7), and because the number of experiments is eight (n = 8), there is only one degree of freedom for lack of fit (f- p = 7 - 6 = 1) and only one degree of freedom for purely experimental uncertainty (n - f = 8 - 7 = 1).
13.14 Rotatable polyhedral designs Figure 13.15 shows a design based on a regular polyhedron, the pentagon [Himmelblau (1970)l. The experimental design matrix is
303
D=
2.000 0.618 -1.618 - 1.618 0.618 0.000 0.000 0.000 0.000
0.000 1.902 1.176 - 1.176 - 1.902 0.000
(13.22)
0.000
0.000 0.000
Because each of the pentagonal points is equidistant from the center of the design, the design is rotatable. This rotatability is seen in the axially symmetric surfaces for normalized uncertainty and normalized information.
-4
-2
0
2
Factor X I
Figure 13.15 A pentagonal rotatable design with center point. DF,, = 0, DFp = 3.
4
304
In this design there are only six distinctly different factor combinations. Thus, there are no degrees of freedom for lack of fit when fitting a two-factor FSOP model with six parameters. There are three degrees of freedom for purely experimental uncertainty because of the four replicate experiments at the center point. Figure 13.16 is a design based on a regular hexagon [Himmelblau (1970)l. The experimental design matrix is
D=
2.000 1.000 - 1.000 -2.000 - 1.OOO 1.000 0.000 0.000 0.000 0.000
0.000 1.732 1.732 0.000 - 1.732 - 1.732 0.000 0.000 0.000 0.000
( 13.23)
Because each of the hexagonal points is equidistant from the center of the design, the design is rotatable. There are seven distinctly different factor combinations. Thus, there is one degree of freedom for lack of fit when fitting a two-factor FSOP model with six parameters. There are three degrees of freedom for purely experimental uncertainty because of the four replicate experiments at the center point.
13.15 The flexing geometry of full second-order polynomial models Figure 13.17 shows a design similar to the hexagonal design in the previous Figure 13.16, but in this new design only one experiment has been carried out at the center. The experimental design matrix is
D=
2.000 1.000 - 1.000 -2.000
- 1.OOO
1.000 0.000
0.000 1.732 1.732 0.000 - 1.732 - 1.732 0.000
( 13.24)
305
" I
01
,
-4
,
,
-2
,
,
,
0 Factor X1
,
2
,
,
4
Figure 13.16 A hexagonal rotatable design with center point. DF,,, = 1, DFpe = 3 .
A comparison of the lower right panels of Figures 13.16 and 13.17 shows that at the center of the factor domain, the normalized information is less for the design with only one experiment at the center (Figure 13.17) than for the design with four replicates at the center (Figure 13.16). This is confirmed in the upper right panels of both figures: the design with fewer experiments at the center (Figure 13.17) has a depression in the center of its normalized information surface; the design with more experiments at the center (Figure 13.16) has a smoothly sloped dome at the center of its normalized information surface. A moment's thought suggests that the decreased information at the center of the design in Figure 13.17 is reasonable and expected - the depression is a result of the principle that if many experiments give more information, then fewer experiments give less information. But this simple explanation becomes inadequate when it is realized that the information is greater on the ring of hexagonal points that surround the center point
306
than it is at the center point itself. After all, the six circumferential points and the center point are laid out on an equilateral triangular grid: why should one of them (especially the center point) provide less information than the other six? The answer to this seeming conundrum lies in the geometry of the fitted model. In Chapter 12, Figures 12.18-12.22 show some of the possible response surfaces that can be represented by the two-factor FSOP model. (Here it is important to note that the response surfaces shown in Figures 12.18-12.22 are generated by the FSOP model and are not the normalized uncertainty or normalized information surfaces shown in the upper panels of Figures 13.16 or 13.17.) Consider one of these response surfaces, the parabolic bowl (paraboloid) opening downward, shown in Figure 12.19. In this canonical form, the maximum of the dome corresponds to the center of the factor domain and the sides of the surface slope downward away from the center. The paraboloid is symmetrical about an axis at the origin and perpendicular to the x1-x2 plane. The discussion that follows will use the geometry of this somewhat unique
Factor x i
Figure 13.17 A hexagonal rotatable design with center point. DF,or= 1, DFpe= 0.
307
response surface to rationalize the shapes of the normalized uncertainty and normalized information surfaces in Figure 13.17. (The results are identical for developments that involve other forms of the FSOP model, but the explanations are less straightforward.) The six outer experiments of the hexagonal design are like a pair of hands clasping the paraboloid around its sides, much as an American football player might catch a well-thrown pass. Even though the response surface might flex and writhe over other regions of factor space, it will be held rigidly around this circumference and won't move much at all. Thus, the information content will be high over this circular region. Geometrically, there are an infinite number of paraboloids that can pass through a circle (e.g., the circle of equal responses at the hexagonal points in this example). Some of the paraboloids will be tall and elongated, some will be short and compressed, some will point up, some will point down, one of them will even be a degenerate flat plane [Rider (1947)l. Although the hexagonal points hold the sides
_7_
-4
0
Factor X 1
Figure 13.18 A hexagonal design with no center point. DF,,, = 0,DFF = 0.
7
4
308
Factor X I
Figure 13.19 A hexagonal design with extra outcr point. DF,,, = I , DFpe= 0.
tightly, the tip (or apex) of the paraboloid is defined by the response at the center point only. Any noise or uncertainty at the center point will move the apex up or down without any resistance from the circumferential points. Any uncertainty in the single center point will not be averaged by the other points, and the uncertainty there will remain relatively large. This is why there is a depression at the top of the normalized information surface in Figure 13.17. The hexagonal ring of points represents a circular node in one of the normal vibrations of a paraboloid. If the paraboloid is held rigidly around this circle, then pressing down on the top of the paraboloid will cause the sides of the paraboloid to flair outward below the circle. Pulling up on the top of the paraboloid will cause the sides of the paraboloid to squeeze inward below the circle. The greater the variation at the center point inside the circular node of points, the greater will be the variation in regions outside this node. This effect can be seen by comparing Figures 13.16 and 13.17. The upper left panels of the two figures show that, as expected, the normalized
309
uncertainty for the design with only one center point (Figure 13.17) is greater at the comers of the factor domain than for the design with four center points (Figure 13.16). At a very basic level, the shapes of the normalized uncertainty and normalized information surfaces for a given model are a result of the location of points in factor space simply because carrying out an experiment provides information - that is, information is greatest in the vicinity of the design. But at a more sophisticated and often more important level, the shapes of the normalized uncertainty and normalized information surfaces are caused by the geometric vibrations of the response surfaces themselves - the more rigidly the model is “pinned down” by the experiments and the less it can squirm and thrash about, then the less will be the uncertainty and the greater will be the information content.
13.16 An extreme effect illustrated with the hexagonal design
The lower left panel of Figure 13.18 shows a hexagonal design without a center point. Considerations of degrees of freedom suggest that it should be possible to fit a FSOP model to data from these six experiments: n = 6 , f = 6, p = 6; thus, although the degrees of freedom for residuals, lack of fit, and purely experimental uncertainty are all equal to zero, they are not negative, and the model would be expected to fit perfectly. Even so, the determinant of the (X’X)matrix is zero, the matrix cannot be inverted, and the two-factor FSOP model cannot be fitted to the data in Figure 13.18. Geometrically, the zero determinant arises because the number of FSOP models that can be made to pass through the responses above the hexagonal factor combinations is now truly infinite (see Section 5.6). There is no center point to define the apex and prevent the model from fluttering about. For purposes of illustration only, to circumvent the problem of a zero determinant but still show the distributions of uncertainty and information in this design, a seventh experiment was added at a factor combination just slightly removed from one of the hexagonal points (at x1 = 2.000, x2 = 0.001). The hexagonal points were also adjusted somewhat to coincide with the grid lines in the pseudo-three-dimensional plots (this is equivalent to a minor adjustment of scale in the x, dimension). Neither of these modifications significantly affects the overall conclusions to be drawn from this example. The actual design is
310
D=
2.000 1.000 - 1.000 -2.000 - 1.000 1.000 2.000
0.000 1.750 1.750 0.000 - 1.750 - 1.750 0.001
( 13.25)
The results from this seven-experiment design are shown in Figure 13.18. The striking feature of this design is the set of six spikes in both the normalized uncertainty and normalized information surfaces. These spikes are an extreme expression of the basic idea that experiments provide information. Even if the experimental design is not a good match for the model; even if the (X’X) matrix is ill conditioned; even if the model doesn’t fit the data very well, there is still highquality information at the points where experiments have been carried out. Figure 13.19 shows the effect of moving the seventh experiment farther away from one of the hexagonal points. In this example, the seventh experiment is at n, = 2.000, x2 = 0.500. This experiment gives enough leverage that the response surface becomes better defined around the ring of hexagonal points. Because the resolution of the plotting grid is not sufficient to resolve all of the sharp detail in the cylindrical surfaces, the normalized uncertainty was truncated at 4.0 and the normalized information was correspondingly truncated at 0.25; the actual surfaces extend below and above the capped surfaces shown in Figure 13.19. Figure 13.20 shows the effect of moving the seventh experiment still farther away from the hexagonal points (to x , = 4.000, x2 = 4.000). The more distant point has good leverage, and the fitted response surface becomes much more rigid (as indicated by the normalized uncertainty and normalized information surfaces in this figure). Note also that the information is high above the distant point.
13.17 Two final examples Figure 13.21 shows the effect of adding an extra, remote star point to a small face centered central composite design. The experimental design matrix is
I
I
o\
h,
w
> 0 0 0 0 0 0 + e - - - -
II
b
I
I
P
II
G
2
w f:
z
c
3
09 0
e, X
z
P
0
L
z
025
4
0 5
6
8
1
075
0
1 0
Normalized Information 0
A
2
~ o r m a l i r e dUncertainty
0
312
_
I
,
-4
,
,
-2
,
,
0
,
~,
2
,
4
F a c t o r XI
Figure 13.21 An inscribed central composite design with distantly located extra star point. DF,or = 4, DFF = 3.
The elephant-like contours result from the stabilizing leverage of the distant factor combination. Note that this additional point stabilizes the fitted model in the x,-direction (the dimension in which it was extended), but has little effect in the x,-direction. The coded design shown in Figure 13.22 is from a clinical chemical study investigating the interference of magnesium in the analytical chemical determination of calcium [Olansky, Parker, Morgan, and Deming (1977), Deming and Morgan (1979)l. The uncoded factors represent the concentrations of calcium and magnesium in human blood serum. The experimental design matrix is
3
B It
b
b
Y
a
e,
N o r m a l i z e d Information
4
6
8
1
0
Y
Y
I
Normalized Information 0 025 0 5 0 7 5 1 0
2
Normalized Uncertainty 0
314
D=
-3 -3 -2 -2 -1 -1 -1 -1 0 0 0 0 0 0 1 1 1 1 2 2
0 0 0 0
-1 -1
1 1
0 0 0 0 2 3
( 13.27)
-1 -1
1 1 0 0
It could be argued statistically that the number of replicates in Figure 13.22 is excessively large (the person carrying out these experiments would also argue that the number of replicates is excessively large!): the relatively small improvement in the quality of information in the region of the design has been gained at the relatively large expense of the additional experiments. It might also be argued statistically (indeed, it is perhaps one of the major points of this chapter) that the domain of the experimental design represents only a small fraction of the factor space shown: a broader design would have given smaller uncertainties and more precise information over the whole factor space (see, for example, Figure 13.8). However, in the example of Figure 13.22, the factorial part of the design adequately covers the combinations of calcium and magnesium found in living humans. The extended star points were used to obtain precise estimates of curvature in xI and x2.There is no practical reason for investigating combinations of calcium and magnesium in the unexplored regions: serum samples with these combinations of concentrations could only have come from the morgue.
315
Exercises 13.1 Minimalist design. Add one design point to a two-factor star design to generate a design that is sufficient to fit a full second-order polynomial model ( yli = Po + plxli+ p~~~+ p1& + p,&, + plgl$z + rli).Hint: see Figure 13.11. 13.2 Optimal design.
Assume a constrained factor space of -5 I x, I +5, -5 I x, I +5. Assume the full two-factor model with interaction, y l i = Po + pixli + pzxzi + plzx l ixzi + rli. Assume a 22 factorial design. How should the four design points be placed to maximize the determinant of the (X'X)matrix? Demonstrate with a few calculations.
13.3 Existing designs. Find a report of a two-factor experimental design. Speculate about the shape of the normalized uncertainty and normalized information surfaces for the design. Sketch their shape.