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A graphical approach for theconstruction of constrained a and l-optimal designs using efficiency plotsWeng. Kee Wong aa Department of Biostatistics , University of California at LosAngeles , Los Angeles, CA, 90024, U.S.APublished online: 20 Mar 2007.

To cite this article: Weng. Kee Wong (1995) A graphical approach for the construction ofconstrained a and l-optimal designs using efficiency plots, Journal of Statistical Computation andSimulation, 53:3-4, 143-152, DOI: 10.1080/00949659508811702

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J. Statist. Comput. Simul., 1995, Vol. 53, pp. 143-152 Reprints available directly from the publisher Photocopying permitted by license only

0 1995 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers SA

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A GRAPHICAL APPROACH FOR THE CONSTRUCTION OF CONSTRAINED D AND

L-OPTIMAL DESIGNS USING EFFICIENCY PLOTS

WENG KEE WONG

Department of Biostatistics, University of California at Los Angeles, Los Angeles, CA 90024, U.S.A.

(Received October 7, 1994; in final form May 1, 1995)

In this paper, a graphical approach for computing constrained optimal designs in regression experiments is presented. The method is simple to implement and can be used in design problems where there are two optimality criteria, and these criteria are expressible as convex functionals of information matrices.

The method is applied to seek constrained L and D-optimal designs in quadratic and cubic poly- nomial models. Several interesting features are noted, including (i) if 5, and 5, are optimal designs for two optimality criteria 4, and 4, and the two designs have the same support, the optimal design for a convex combination of 4, and 4, can have a different set of support points, and (ii) D-restricted extrapolation designs are generally more efficient than A-restricted extrapolation designs for estimating the model parameters.

KEY WORDS: Approximate designs; constrained optimal designs; design efficiencies; extrapolation; information matrices.

1. INTRODUCTION

Many experiments are designed with more than one objective in mind. For example, the experimenter may be interested in estimating the model parameters and at the same time, in making inference on the response at some point outside the design space. Alternatively, the experimenter may want to estimate the response surface but is concerned about the validity of the model. Since the optimal design under one criterion is usually different under another, the experimenter in confronted with the problem of finding a design which balances the need of the two competing criteria. The goal then is to find a design which is deemed efficient under both criteria.

There are two common approaches for finding an optimal design in this situation: (i) find a design which is deemed best under a weighted average of the two criteria, and (ii) find a best design for one objective function subject to the requirement that the design meets a minimum level of performance in terms of the other criterion. Method (i) is essentially the approach adopted by Lauter (1974,1976), Giovagnoli and Sebastiani (1989), Pecar (1989), Dette (1990) among many others, while the second method was used in Lee (1987,1988), and Huang and Chang (1993). The latter method is a constrained optimization problem which can be difficult to solve analytically even for relatively simple problems. Method (i) on the other hand is

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144 W. K. WONG

straightforward since the optimality criterion is convex (it being a weighted average of two convex functionals), and there are algorithms for generating an optimal design (Pazman, 1986, pg. 146).

It is generally known that these two seemingly divergent approaches are equival- ent in the sense that the set of solutions generated from one approach coincides with the other. This paper exploits this result further by identifying the corresponding solutions from the two approaches by means of efficiency plots. As an application, constrained optimal designs using the D and L-optimality criteria are constructed and their properties studied for polynomial models.

Optimal design theory is discussed in Section 2 and the constrained optimal designs are constructed in Section 3. An example in Lee (1987) is reworked to demonstrate that the results presented here coincide with his analytical results. In addition, the results in Lee (1987) for quadratic models are extended to cubic models.

2. BACKGROUND

Assume that a closed and bounded design space x is given from which the experi- menter may select the levels of the independent variables. The statistical model is

where y(x) is the response at the x-level of the independent variables, fl is a k x 1 vector of unknown parameters, f (x) is a k x 1 vector of known regression functions, and E is an error normally distributed with mean zero and constant variance. All experimental designs 5 in this paper are approximate designs; this means they are viewed as probability measures on X. Consequently, if a total of N independent observations are to be taken from a design 5 and t has mass mi at xiex, then approximately Nm, observations are observed at xi.

Following common practice, the worth of a design < is measured by its informa- tion matrix defined by

Many objective functions or optimality criteria are convex functions &(M(t)), or simply 4(5) of the information matrices, (Pazman, 1986, pg. 75). Designs which minimize 4(5) over the set of all designs on x are &optimal. A popular choice is D-optimality defined by $(5)= -In lM(<)l. Another is the L-optimality criterion defined by 4(e) = tr B M - '(e) for some non-negative definite matrix B. For simplic- ity, two specific choices of B in the L-optimality criterion are considered:

a) B = k x k identity matrix (A-optimality) b) B = f ( 2 ) f T(z) for some point z not in the design space X.

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A GRAPHICAL APPROACH 145

The optimal designs under (a) and (b) are called respectively A-optimal and Extrapolation optimal designs at the point z (in terms of smallest variance). For additional discussion of the setup and motivation of these criteria, see Pazman (1986 Chapter 4).

Denote the two convex objective functions by 4,(() and 4,((). Method (i) calls for finding a design 5, to minimize

for a fixed 3, E [O, 11, whereas method (ii) solves the problem:

minimize 4,(5) subject to < c (2.2)

for some user-selected constant c. In practice, both 4, and 4, are expressed in terms of design efficiencies so that c may be appropriately chosen, (see the examples in Section 3) and the results do not depend on the design space X.

Cook and Wong (1994) showed that under certain regularity assumptions that are not restrictive in practice, the optimal design 5, found from (2.1) for a given A is also an optimal design for (2.2) for some constant c, and likewise, the optimal design found from (2.2) for a given c is also an optimal design for (2.1) for some A. The essential idea then is to solve the easier problem (2.1) for a range of values of i~ [O,l] and afterwards by means of efficiency plots, identify the right i that corre- sponds to the c specified in (2.2). The last step cannot be over-emphasized since it is often overlooked resulting in many misinterpretations of A in the literature. For example, assigning I = 0.5 in (2.1) does not imply equal interest in both the optimal- ity criteria because the resulting design (,=,,, does not guarantee equal efficiency under both criteria, see the example in Cook and Wong (1994). Thus, the graphical approach here has the distinct advantage of being easy to implement and at the same time provides a meaningful correct interpretation for i in (2.1).

The aforementioned efficiency plots are graphs of the efficiencies of 5, under both the criteria 4, and 4, versus values of 2 E [0, I]. With the exception of D-optimality, the efficiency of a design ( is defined by E,(() = 4(5*)/4(5), where 5* is the optimal design under 4 . In practice, E&<) denotes reciprocally the number of times the design ( needs to be replicated for it to do as well as (*. In order to maintain this nice interpretation of design efficiency, the D-efficiency of a design is given by {IM(5)1/IM(5*)l)'1k = exp{($(<*) - + ( ( ) ) / k ) . By studying the slopes and other charac- teristics of the efficiency plots, the experimenter can carefully assess the potential tradeoff between the competing objectives, and thus obtain useful information in selecting a design.

3. ILLUSTRATIVE EXAMPLES

In this section, the above procedure is used to construct constrained optimal designs for the quadratic model, f,T(x) = (1, x, x2), and the cubic model f: (x) = (1, x, x2, x3). Corresponding results for the simple linear model are omitted because they are easy

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146 W. K. WONG

to obtain analytically. Suppose the experimenter has more interest in $,-the primary criterion, and 4, is the secondary criterion. The goal here is to find a best or optimal design under 4, which guarantees a minimum level of efficiency under 4, at the same time. Let ti denote the optimal design for $i and let Ei(5) denote the efficiency of a design 5 under 4 , i = 0, 1. By construction, 5,=, = 5, and ( A = , = t o .

The computation to follow uses MathematicaTM (Wolfram, 1988), and the optimal designs t, displayed in Tables 1, 3 and 4 are either optimal or nearly 100% efficient. Their optimality is verified using tools from convex analysis, see Fedorov (1980), and all 4, satisfy the checking conditions for optimality to within 0.00001.

Let d (x, z, 4 ) = f T(~)M-l(() f (z) and note that the quantity t.(x, 5) = d(x, x, 5) is proportional to the variance of the response at the point x using design 4. For expository purposes, we assume x = [- 1, I] in all the examples.

3.1. A-Restricted D-Optimal Designs

Set $,(t) = trM-'(t)/tr M-'(t,) and $,(<) = - (10g{lM(t)I/IM(5~)l})/3. It is readily checked that tr M-'(t,) = 8 and IM(to)I =4/27 for the quadratic model. In the terminology of Lee (1987), this type of design is called an A-restricted D-optimal design. Arguing as in Lee (1987), 5, is a symmetric design supported on - 1, 0 and 1 for every I. E [O,l]. Applying Fedorov (1980, theorem 1) to the convex functional $(51A) = A$ , ( t l ) + (1 - A)$,(<), it follows that 5, is optimal if and only if for all x EX,

Using this checking condition, the optimality of the design in Table 1 can be readily checked. For this example, El(() = $,(5) and Eo(5) = exp{ - $,(<)). The effi- ciency plot is shown in Figure l(a) with the plain line, here and throughout, denot- ing E,((). Generally, steep graphs mean that the two goals are quite incompatible and much has to be sacrificed by one at the expense of doing well in terms of the other. On the other hand, if the graphs are relatively flat, the goals are not competi- tive and a design that does well in terms of one criterion also performs well in terms of the other. In this example, the graphs show that both the goals are comparable and the A and D-efficiencies of 5, are more than 89% for any 0 < I. < 1. Clearly for this plot, if c < 819, to would be the A-restricted D-optimal design.

Once the efficiency plots are constructed, solution to (2.2) can be found. For the specified value of c in (2.2), a horizontal line is drawn at E l = c in Figure l(a) and the corresponding value for 3. is read off. The required design is the solution to (2.1) for this value of i. If desired, the exact relationship between c and 3, can be obtained from standard optimal design theory by replacing the inequality in (3.1) with an equality at the support points of the optimal design. Doing so for this example yields, for c 2 819

d = (1 - (1 - c)'I2)/2 and A = 4d (1 - 3d)(l- d)/{12d3 - 16d2 + 10d- 31,

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A GRAPHICAL APPROACH 147

(a) Quadratic model (b) Cubic model

Figure 1 Efficiency plots for the A-restricted D-optimal designs.

Table 1 A-restricted D-optimal designs and their efficiencies for the quadratic (cubic) model.

i. mass of 5 , ar the interior support E , (ti) Eo(5,) interior support point point of ti

where d is the mass of [, at the point x = 0. For example, the correspondence between c and h for some values of c are:

It is apparent the results here are identical to the analytical results derived in Lee (1987) for the quadratic model. Corresponding results for the cubic model are shown in Table 1 in parentheses. For all k [ O , 11, the optimal design [, is supported at - 1,

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148 W . K . WONG

- s, s and 1 and their masses at these points are (1 - 2d)/2, d, d and (1 - 2d)/2 respectively. The efficiency plot is exhibited in Figure l(b). Note that in both plots, El({,) increases but Eo(t,) decreases as /Z increases. This is a general property of these plots, see Cook and Wong (1994) for technical details and other properties of these plots.

3.2. A and D-Restricted Extrapolation-Optima1 Designs

The interest now is to make inference at a given point z for a quadratic model, and at the same time ensure the design has a minimum level of A-efficiency. The cases when (i) z = 2 and (ii) z = 1.1 to investigate the effect of z on estimating the model parameters are considered. The A-restricted Extrapolation optimal design 5, is found by first setting q5 (< / I ) = A tr M - l (<)/tr M - (5,) + (1 - A) v (z, 5)/v ( z , 5,) and applying Fedorov (1980, theorem 1). The checking condition is: 5, is optimal if it

satisfies for all XEX,

A straightforward calculation shows v(2,5,) = 49 and ~(1.1, to) = 2.0164. Some of the optimal designs t,, k[O, 11 are shown in Table 2 for z = 2 and 1.1 along with their efficiencies. Corresponding efficiency plots for z = 2 and z = 1.1 are shown in

Table 2 A-restricted Extrapolation optimai designs and their efficiencies with z = 2 (1.1)

I mass of 5 , at 0 mass of 5, at 1 - & ( < A ) Eo(Sn)

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A GRAPHICAL APPROACH 149

0 D.2 0 . 4 A Od 0.0 1 0 0.2 0.4 2 0 6 D.0 1

(a) z = 2 (b) z = 1.1

Figure 2 Efficiency plots for A-restricted Extrapolation designs at the point z.

Table3 Interior support point of the D-restricted extrapolation design at.z = 1.1 for the quadratic model.

i. s (interior point) i. s (interior point)

Figures 2(a) and 2(b) respectively. In all cases, the A-restricted Extrapolation designs are supported at - 1, 0 and 1.

Analogous results for D-restricted Extrapolation-optima1 designs are shown in Table 4. The inequaIity corresponding to the above one is

Interestingly, unlike all previous cases, the optimal designs here can be supported at a nonzero interior point even though both c, and 5, has zero as the only interior support point. The theoretical support points for the D-restricted extrapolation designs are 1, s and - 1, where s varies continuously with 1. For z = 1.1, s varies from 0 to about - 0.072 as ;t increases from 0 to 0.35, and thereafter s returns to 0 as ;t varies from 0.35 to 1. The table below reports some values of s for the case when z = 1.1.

The same phenomenon takes place when z = 2, but to a smaller extent with s varying from 0 to - 0.035 and then back to 0 again. Figures 3(a) and 3(b) display the efficiency plots of the constrained optimal designs for these two cases.

4. DISCUSSION

The strategy used here for seeking constrained optimal designs applies to other combinations of optimality criteria so long as the regularity conditions in Cook and

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Table 4 D-restricted Extrapolation optimal designs and their efficiencies with z = 2(1.1).

L mass of 5, at the mass of 5, at 1 EI (5,) Ed[,) interior support point

(a) z = 2 (b) z = 1.1

Figure 3 Efficiency plots for D-restricted Extrapolation designs at the point z.

Wong (1994) are met. The main requirements are convexity and differefntiability of both the objective functions. While the computation assumes 4, as the primary objective function and 4, as secondary, their roles are interchangeable in the sense if they are switched, corresponding results could be obtained readily from the effi- ciency plots. For example, if a D-restricted A-optimal design is desired for the quadratic model, one simply plots the efficiencies obtained from Table 1 versus 1 - ,I and the resulting graph is the desired one. There is thus flexibility in this approach permitting the experimenter to assess the efficiency of a design under two criteria.

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A GRAPHICAL APPROACH 151

Note that for A-restricted Extrapolation optimal designs with z = 1.1(2), the efficiency of 5 , varies greatly from 0.30(0.37) to 1.00 depending on the value of 1. This suggests that the two criteria are very competitive and much has to be traded for an increase in efficiency for the other criterion. In this case, the researcher has to decide carefully on a compromise between the efficiency of the design and the goals of the experiment. Fortunately, when z is further away from the design space, this problem is alleviated as can be seen in Tables 2 and 4. The D and A-efficiencies can drop to as low as 0.5011 and 0.2958 respectively when z = 1.1 but these increase dramatically to 0.8571 and 0.89 15 respectively when z = 2.

The numerical results here show both A-restricted Extrapolation optimal designs and D-restricted Extrapolation optimal designs are not symmetric in general. In addition, columns 4 and 5 of Tables 2 and 4 suggest that D-restricted extrapolation designs are generally more efficient than A-restricted extrapolation designs for esti- mating the model parameters.

For more complex models, the graphical method here still applies except that the checking conditions become more involved and it may be desirable to uses a com- puter-aided type of algorithm to generate an optimal design (Pazman, 1980, pg. 146). However, extension of this methodology to problems involving more than two criteria is not straightforward. It is not clear how the efficiency plots could be modified to accommodate the increase in the number of optimality criteria and simultaneously maintain a useful interpretation for the 3,'s in (2.1). Current research in this direction is underway, including the use of dynamic graphics to study this problem. Applications of this methodology to the design problems considered in Atkinson (1972), Herzberg and Cox (1972), Huang and Chang (1993), and Pukelsheim and Rosenberger (1993) are also possible.

References

Atkinson, A. C. (1972). Planning Experiments to Detect Inadequate Regression Models. Biometrika, 59, 275-293.

Cook, R. D. and Wong, W. K. (1994). On the Equivalence of Constrained and Compound Optimal Designs. Journal o f the American Statistical Association. 89, #426, 687-692.

Dette, ~ .71990) . A ~eneralization of D and Dl-optimal Designs in Polynomial Regression. The Annals of Statistics. 18. No. 4. 1784-1804.

Fedorov, V. V. ('1980). Convex Design Theory. Math. Operationsforsch Statistics, Ser. Statistics, 11, 401-41 1 . - - . - - .

Giovagnoli, A. and Sebastiani, P. (1989). Experimental Designs for Mean and Variance Estimation in Variance Components Models. Computational Statistics &Data Analysis, 8, 21-28.

Herzberg, A. M. and Cox, D. R. (1972). Some Optimal Designs for Interpolation and Extrapolation. Biometrika, 59> 551-561.

Huang, M. N. L and Chang, H. F. (1993). Marginally Restricted Constrained Optimal Designs. Submit- ted.

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Lauter, E. (1976). Optimal Multipurpose Designs for Regression Models. Operatlonsforsh und Statistics, 7, 51-67.

Lee, M. S. (1987). Constra~ned Optimal Designs for Regression Models. Communications rn Statistics, Part A-Theor): and Methods, 16, 765-783.

Lee, C. M. S. (1988). Constrained Optimal Designs. Journal of Statistical Planning and Inference, 18, 377-389.

Pazman, A. (1986). Foundations of Optimum experimental Design. D. Reidel Publishing Company.

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