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Compound D- and Ds-Optimum Designs for Determining the Order of a Chemical ReactionAuthor(s): Anthony C. Atkinson and Barbara BogackaSource: Technometrics, Vol. 39, No. 4 (Nov., 1997), pp. 347-356Published by: American Statistical Association and American Society for QualityStable URL: http://www.jstor.org/stable/1271499 .Accessed: 07/06/2011 01:55

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Compound D- and Ds-Optimum Designs for Determining the Order of a Chemical Reaction

Anthony C. ATKINSON

Department of Statistics London School of Economics

London WC2A 2AE United Kingdom

Barbara BOGACKA

Department of Mathematical and Statistical Methods Agricultural University of Poznan

60-637 Poznan Poland

Estimation of the order of a chemical reaction is often at least as important as estimation of the rate of the reaction. Locally optimum experimental designs are found for the order and the rate, sep- arately and together. Compound D-optimum designs provide a method for designing experiments with specified efficiency for rate or order determination. A relationship between the compound designs and D-optimum designs for both rate and order aids interpretation of the plots of design efficiencies. Bayesian designs incorporating prior uncertainty are exemplified. Monte Carlo sampling of the prior is used to design an experiment for the esterification of acetic anhydride.

KEY WORDS: Acetic anhydride; Bayesian design; Chemical kinetics; Monte Carlo integration; Multipurpose design.

Methods for the construction of optimum experimental designs for the rate parameters in kinetic models of known form were introduced by Box and Lucas (1959). Locally D- optimum designs are found for the parameters of the model after linearization by Taylor series expansion. Several addi- tional examples were given by Atkinson and Donev (1992, chap. 18), who also described designs incorporating prior information about the parameters of the models. In all of this work, it is assumed that, although the rates of reaction are unknown, the orders of the various reactions are known. In chemical practice, however, determination of the order of reaction is often at least as important as determination of rate constants. Yet we have been unable to find any references in the chemical and statistical literature to optimum experimental design procedures for determination of the order of reactions. It is the purpose of this article to develop optimum experimental designs when the order of reaction is of primary experimental concern and also when both the reaction rate and the order matter.

In Section 1 we introduce the general decay model, which will be used to exemplify our procedures. We find D- optimum designs both for estimation of the rate when the order is assumed known and for the estimation of rate and reaction order together. Ds-optimum designs for the order of reaction when the rate is treated as a nuisance parameter are also found. Section 2 is concerned with compound optimum designs that show the relationship between the designs of Section 1 and provide plots for the selection of multipurpose designs of specified efficiency. These designs all depend on point prior estimates of the nonlinear parameters. Bayesian designs incorporating prior uncertainty are introduced in Section 3. Monte Carlo sampling of the prior is used to design an experiment for the esterification of acetic anhydride. The article closes in Section 4 with some suggestions for further developments.

1. D-OPTIMUM DESIGNS

1.1 The General Decay Model Reviews of optimum experimental design for nonlinear

models in general include those of Ford, Titterington, and Kitsos (1989), Ford, Torsney, and Wu (1992), and Atkin- son and Haines (1996). For models arising specifically in chemical kinetics, both Box and Lucas (1959) and Atkin- son and Donev (1992) found locally D-optimum designs for the nonlinear response model resulting from first-order decay

A - B, in which the concentration of chemical A at time t is given by the nonlinear function

[A] = r1(t, 0) =e-Ot O,t > o, (1)

if it is assumed that the initial concentration of A is 1. The model comes from solution of the differential equation

d[A] _ d -O[A], dt (2)

which assumes first-order kinetics. More generally, the order of the reaction may be represented by the parameter A, when the differential equation becomes

[A - [A]A dt (3)

so that, for (1) and (2), A = 1. In a simple case such as (3) the more general differential

equation can also be integrated to yield a new model in which the expected value of concentration at time t is

[A] = 77(t, 'b) = {1 - (1 - A)t}l/('-),

A,,t > 0; A 1, (4)

where , = [, A]T. As A -* 1, (4) reduces to (1). For A < 1, [A] = 0 for t > 1/{(1 - A)0}. For many kinetic

? 1997 American Statistical Association and the American Society for Quality

TECHNOMETRICS, NOVEMBER 1997, VOL. 39, NO. 4

347

ANTHONY C. ATKINSON AND BARBARA BOGACKA

Response for General Decay 0

co r

CD 0

cD

0. C) (1) (1) L- I_

N

0V

6 -

5 10 15 0 5 10 15

time

Figure 1. General Decay Model: Response as a Function of Time: Reading Upward, A = .5, 1.0, 1.5, and 2.

models, however, it is only possible to obtain explicit expressions like (4) for a few values of A. The implications for experimental design of the lack of analytical solutions are discussed in Section 4.

Figure 1 shows plots of [A] as a function of t for four values of A (.5, 1, 1.5, and 2) all for the same value of 0. For illustrative purposes we take 0 = .5. The actual values of rate constants for chemical reactions depend on the units in which time is measured. See, for example, Pilling and Seakins (1995, pp. 10-20). As (4) demonstrates, however, it is the quantity Ot that determines how far the reaction has progressed. As the figure shows, when A = .5, [A] becomes 0 at t = 4. For larger values of A and fixed 0, the reaction proceeds more slowly. It is therefore to be expected that experiments at relatively high values of t will be required to provide information about the value of A.

1.2 Experimental Design

1.2.1 Theory. The experiment consists of measuring the concentration of A after the reaction has been running for a time t. One experimental run yields one observation yi, and the experimental design is a list of the n times, ti, i = 1,..., n, not necessarily distinct, at which measure- ments are to be made. Some comments on experimental design when observations can be made at several time points during one run are given in Section 4.

We consider the more general definition of an experimental design in which ~ is a continuous design specifying a set of p distinct points in a design region T and the proportions, Wi, of observations taken at these points

i= r ::j...: p Wl,...,Wp Wp

The times ti are the points of support of ~ and wi the design weights. In practice, when only n observations can be taken, an exact design will be required. Often the optimum exact


design is approximated by a design with the number of trials at ti, the integer closest to nwi.

We assume that the response, y, for a nonlinear regression model has mean E(Y) = rj(t, 0) and dispersion matrix D(y) = a21, a2 > 0, and I the n x n identity matrix. Here rj is a function nonlinear in at least one of the k parameters Vb, and t is a vector of m explanatory variables defined on T. The information matrix of a design S for the k parameters b is given by M((, b) = FTWF, where

fT(tl, )

fT(tp, ) the vector fT(ti, b) has jth element

fji (ti,) = ( forj = ,...

and W = diag{wi,..., wp}. The information matrix thus depends on the unknown parameter, b, and a natural way of accommodating the obvious problems, which follow from this dependence, is to adopt a best guess for the parameters, say 0?, and to consider designs that maximize an appropriate function of M(<, p) evaluated at L = 0? (Chernoff 1953). Such designs are termed locally optimum.

D-optimum designs maximize the logarithm of the determinant of the information matrix, log|M((, b) 1, or, equiva- lently, minimize the asymptotic generalized variance of the parameter estimators. It follows from Caratheodory's theorem (Silvey 1980, p. 72) that a continuous D-optimum design is based on at most k(k + 1)/2 design points. Often locally D-optimum designs for nonlinear models are based on exactly k points of support when the weights associ- ated with the support points are equal to 1/k (Silvey 1980, p. 42).

The general equivalence theorem of Kiefer and Wol- fowitz (1960) makes it possible to check the optimality of a candidate continuous locally D-optimum design. With the standardized variance of the prediction at t defined by

(5)

the relevant part of the equivalence theorem states that, for the optimum design, J*, the maximum value of d(t, *, ) over the design region T is k, the number of parameters in the model, and further that this maximum value is attained at the support points ~*. The theorem also provides a basis for algorithmic construction of locally D-optimum designs (Wynn 1970).

If only an s subset of the parameters b2 is of interest, let the parameters be partitioned as -T = (l'T /T) with the information matrix M(,, q;) partitioned so that the information for b1 is M1(J, 4). Then the Ds-optimum design for qb2 maximizes log{IM( )|, )/IMl1(,, )l|}. The equivalence theorem for Ds-optimum designs states that, for the optimum measure ,*,

d(t, , g) = fT(t, ?)M-1(), ()f(t, ?)

- f(T) (t, I)M1 (~, b)f(1) (t, /) < s, (6)

348

d(t, 1, Ob) = f T(t, O)M-1 (, ? )f (t, ?),

COMPOUND D- AND Ds-OPTIMUM DESIGNS

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

0 5 10 15

Time Figure 2. Derivative Ifi = I9[A]/90l and Locally D-Optimum Designs When A Is Assumed Known.

where f() (t, ,) relates to the information matrix M11 (1, l)). An early example of the use of Ds-optimum designs in the study of kinetic models is that of Hunter, Hill, and Henson (1969).

Calculation of locally D- or D,-optimum designs for the general decay model (4) requires values of the derivatives

fl(t, I) = rl(t,)/00

f2(t, V) = 9r7(t, )/OA, (7)

which can be found analytically to be:

f (t, b) = -t{1 - (1 - A)0t}A/(1-) (8)

and

f2(t, $) = {1/(1 - A)2}[log{ ( - A)0t} + (1 - AX)t/{1 - (1 - A)0t}]

x{1 - (1 - A)Ot}l/(l-A) for t < 1/{(1 - A)0}. (9)

The case of exponential decay, A = 1, requires special attention. Differentiation of the response (1) yields

fl(t, f) = -t exp(-0t), (10)

which can also be found as the limit of (8) as A --, 1. The same limiting operation on (9) yields

lim f2(t, )b) = 1

(0t)2exp(-Ot). A->-1 2 (11)

Although (10) is found directly with A = 1, calculation of the derivative (11) requires the derivative (9) for general A. So, even to find designs for testing whether A = 1, we need (9). The importance of this point for more complicated models is discussed in Section 4.

We now use these derivatives to find some D- and Ds- optimum designs for 0 and for A.

1.2.2 D-Optimum Designs for the Rate 0. For the mo- ment we assume that the order of the reaction, A, is known. Then the D-optimum design for 0 puts all the trials where loglM11(,00?)I is a maximum. In the case of one-point designs, IMl (,0 , ?) reduces to If?( , 0?)l or the absolute value lfi (,0?)l. Differentiation of (10) with respect to t, followed by equating the derivative to 0, shows that the optimum design concentrates all trials at t = 1/0?. For general A, differentiation of (8) again surprisingly yields the same result so that there is nothing special about the design for estimating the rate of reaction when A = 1.

This result is important for our comparisons of more complicated designs involving both A and 0 in that it jus-


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fl Figure 3. Design Loci and Locally D-Optimum Designs for the Order A and the Rate of Reaction 0: p0O = [5, .5]T, t* = [1.27, 3.09]T; p?O = [.5,

1]T, t* = [1.27, 4.73]T;,4O =[5, 1.5]T, t* = [1.26, 7.23]T;10? = [.5, 2]T, t* = [1.24, 11.03]T.

tifies taking a constant value for 0, as we did in Figure 1, rather than making 0 a function of A.

When A = 2, the model becomes

T7(t, 0) = 1/(1 + Ot), (12)

a simple inverse polynomial. For the numerical examples, we take 0? = .5 so that the optimal design puts all trials at t = 2. Figure 2 gives plots of If (t, .5)1 for the four values of A? (.5, 1, 1.5, and 2) which show that the extreme value of fi is indeed at t = 2. For A = .5, fi is a parabola for t < 4. We take it as equal to 0 for t > 4, as the concentration of A is 0 for such values of time. For the higher values of A, the figure shows fi approaching 0 more slowly, a reflection of the behavior of the response shown in Figure 1.

1.2.3 D-Optimum Designs for Both Parameters. The preceding designs were based on the assumption that the order of reaction, A, was known. If, however, both parameters are of interest, the D-optimum design maximizing logjM((, 0?)1 is more appropriate.

For our two-parameter model, the D-optimum design has two points of support, with weight 1/2 at each. The values of the design points, however, depend on the prior values of the parameters. Figure 3 shows plots of the space of the derivatives fi(t, ?0) and f2(t, 0?) (called the "design locus"


by Box and Lucas 1959). The points of the D-optimum design, shown by heavy dots, form, with the origin, a triangle of maximum area. The area of this triangle is proportional to iM(, o0)l1/2. The decreasing size of the triangles as A increases shows that, for a fixed number of readings, the variance of the parameter estimators increases with A.

To show how the designs depend on the prior value A?, we give in Figure 4 a plot of the two design points t1 and t2 as a function of A? when 0? = .5. For all values of A?, the time points are either side of the value of 2 for the D-optimum designs for 0 when A is known of Subsection 1.2.2. As A? increases, t1 remains sensibly unchanged, decreasing from 1.27 to 1.24, but t2 increases rapidly from 3.09 reaching 11.03 for A? = 2. This emphasis on high values of t when A? is large is in line with the plots of the response in Figure 1 and of the design loci in Figure 3.

We return in Section 2 to consideration of the sense in which the D-optimum design reflects equal interest in the parameters 0 and A.

1.2.4 Ds-Optimum Designs for the Order of Reaction A. If the main purpose of the experimenter is to determine the order of the reaction, with the actual value of the rate constant 0 of secondary importance, the Ds-optimum design for A is appropriate. To find this Ds-optimum design, we

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D-optimum Designs

r-

0-

C cl

m- (O -

- 0-

Cb (t 40

0

x 0, C'4-

a -

0.5 1.0 1.5 2.0

lambda

Figure 4. Points of the Locally D-Optimum Design for A and 0 as a Function of the Prior Value of A When 0? = .5.

partition the information matrix M((, /) so that the information for 0 is Ml (, 4). The required design then maximizes log(lM({, ) I/Mil (, I7)).

Like the preceding D-optimum designs, the Ds-optimum designs have two points of support. Unlike them, however, the design weights are not equal. Figure 5 is a plot of the Ds-optimum designs as a function of A?, as Figure 4 was for the D-optimum design. Because the design weight is not one-half, the weight w2 for the upper time point, t2 is

also included in Figure 5. For A? = .5, the weight is .570, increasing to .693 when A? = 2. Over the same range, t, decreases from .980 to .849, whereas t2 increases from 3.33 to 17.38. These designs, as the figures show, require more extreme values of time than do the D-optimum designs.

2. COMPOUND OPTIMUM DESIGNS Three different designs were described in Section 1. The

D-optimum design for 0, maximizing loglMll(H,0?)l, is appropriate if the order of the reaction is known without doubt and interest is solely in estimating the rate of reaction. The D-optimum design for 0 and A, maximizing loglM((, 0?)|, minimizes the asymptotic generalized variance of the estimators of these two parameters. Finally, if determination of the order of the reaction is of primary importance, the D,-optimum design is found maximizing log{|M(~, ?)l/lMil(s,0?)|}. We now use the theory of compound optimum designs to establish a relationship between these different designs, which leads to a practically useful balance between the various experimental objectives.

Examples of compound D-optimum designs for linear models were given by Atkinson and Donev (1992, chap. 21) and by Cook and Wong (1994). Suppose that there are H models of interest that are to be included in the design of an experiment on the design region T. Let the jth model have information matrix M(/(j, j), where 0j is the kj-dimensional vector of parameters of the jth model, and let interest in this model be expressed by the nonnegative weight aj. Then the compound D-optimum design is found by maximization of the criterion

Ds-optimum Designs

CO

o) 4--

o) 0_ E CL

x 0)

1.0 1.5 2.0

lambda

Design Weight

CD (0 - d

fO- d

00 () - d o 6O

0.5 10 1.5 2.0 0.5 1.0 1.5 2.0

lambda Figure 5. Points of the Locally Ds-Optimum Design for A and the Design Weight for t2 as Functions of the Prior Value of A When 0? = .5.


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H

@(I() = E aj log IM(, ?)1. (13) j=l

An equivalence theorem applies to this compound design criterion. The analog of the standardized variance (5) for a combination of linear predictors at t is

H

d(t, ) = a.Jf )(t)Mfl1(, ' )?f(J)(t). (14) j=l

The equivalence theorem then states that, for the optimum design J*, the maximum value of d(t, J*) over T is

,H a jkj. The theory can be extended straightforwardly to include Ds-optimum designs and D optimum designs for nonlinear models.

For the problem of determining the order of general decay, the most extreme designs are the D-optimum designs for 0 alone, which are completely uninformative, and the Ds-optimum designs for A, which are the most informa- tive. Combining these two yields the compound design criterion

(I(, 0)= (1 - a) log M(l ,)

+ e log{IM(, M )|/Ml(, )}. (15)

In (15), a (0 < a < 1) expresses the experimenter's relative interest in determination of the order of the reaction, with a = 1 corresponding to interest solely in order determination.

The surprising relationship with D optimality emerges if (15) is rewritten as

<(I(S, ) = (1 - 2a) logMl1(, ) + alog IM(, ).

(16)

When a = .5, the criterion reduces to a multiple of that for D optimality. Because multiplication of the design criterion does not affect the optimum design, we have the unexpected result that, for this design problem, the idea of D optimality as intermediate between the other two criteria has a precise interpretation. Although we have written (16) for scalar 0, the result extends to vector 0, when M1l (~, 0) is replaced by the determinant IM11n (, X)l. The value of .5 for a still corresponds to D optimality.

Despite the precise meaning to be given to a = .5, it is unlikely that an experimenter will be able to specify a numerical value of a. To obtain a suitable design we find, by numerical optimization, the designs maximizing the compound criterion (16) for a series of values of a. For each we calculate the efficiencies of the design relative to the D- optimum design for the estimation of rate, the Ds-optimum design for estimation of order, and the D-optimum design for both. A plot of these efficiencies against a makes it possible to choose a design with a balance of efficiencies for all aspects of the problem.

Plots of these efficiencies are given in Figure 6. The compound designs themselves are not given. For .5 < a < 1, however, the designs consist of experiments at two values


of time with unequal weights, both the weights and times being between those for the D- and D8-optimum designs of Section 1. As a -- 0, the design problem approaches that of D optimality for 0 when A is known, the one-point design at t = 2 of Section 1. The compound design reflects this because tl -? 2 as a -> 0, while w2 decreases toward 0.

The efficiencies are calculated using the optimum designs for the particular aspect of interest. Let the optimum compound design be ( and the D-optimum design for estimating 0 be ~3. Then the efficiency of the compound design if only 0 is of interest is

(17)

Likewise, if the Ds-optimum design for estimating A is *, the relevant efficiency is

EA =100IM(~%, ?)l/Mnl(C(, 00) EM, = 100 c (I ,o) ' (18)

If the D-optimum design for 0 and A is (D, however, the efficiency is

ED = 100{lM(c*, ~?)I/lM(*D, )0)11/2, (19)

the square root being required as the determinant is for a model with two parameters.

As an example, suppose A = 2. The compound design for a = .5 has, of course, ED = 100%. The other efficiencies are Eo = 58.2% and Ex = 79.0%. For a = .375, the values are ED = 97.9%, E = 67.2%, and EA = 65.5%, high efficiencies for any way in which the data may be analyzed.

These compound designs have been found by maximizing the criterion (15). The results of Cook and Wong (1994) give an alternative interpretation of the designs as satisfying a second optimality criterion including a constraint. Sup- pose that a minimum value of the efficiency Eo is specified: Subject to this constraint being satisfied, it is required to find the most efficient D,-optimum design for A. This, they show, is the compound optimum design giving the specified value of Eo.

3. BAYESIAN DESIGNS

3.1 Theory The compound optimum designs of Section 2 provide a

flexible means of designing experiments to determine the order and estimate the rate of a chemical reaction. The designs are, however, only locally optimum, being based on a point prior estimate ?0. Fuller information concerning uncertainty about O can be incorporated using Bayesian D optimality, for which the expectation, with respect to the prior parameter distribution p(i), of the logarithm of the determinant of the information matrix,

E, log M(, f) 1 = log IM(~, /) Ip(4) do, (20)

is maximized. A discussion of this form of Bayesian experimental design was given by Chaloner and Larntz (1989). Chaloner and Verdinelli (1995) gave a full review. Sev- eral examples were given by Atkinson and Donev (1992, chap. 19).

352

Ee = 100M1 ((c, ?)/M1 ((a, 00).


0.0 0.2 0.4 0.6 0.8 1.0

100

80

60

40

20

0

100

80

60

40

20

0

0.0 0.2 0.4 0.6 0.8 1.0

Alpha Figure 6. Efficiencies of Compound Optimum Designs: D-Optimum, ED, Continuous Line; Ds-Optimum for A, E,, Dashed Line; D-Optimum for

0, Eo, Dotted Line.

There is clearly a mathematical relationship between this Bayesian criterion and the criterion for compound D optimality (15). The connection becomes clearest when the in- tegral in (20) is calculated numerically, when the criterion reduces to a weighted sum of the logarithms of determinants for the various values of qb.

An equivalence theorem applies to Bayesian D optimality. For the D-optimum design ~*, we must have

d(t, M*) = f fT(t, )M-I ((*, )f(t, )p(o) do < k,

(21)

where k is again the number of parameters in the model. The maxima of (21) are once more at the points of support of the design, a feature that is useful in constructing and checking optimum designs.

Table 1. Bayesian Ds-Optimum Design for A With Prior Putting Weights .25 on the Values .5, 1, 1.5, and 2 for A, the Order of the Reaction

Time t Weight w

.81 .379 3.30 .211 7.32 .410

The locally optimum designs of previous sections put trials at k design points, usually 2. A feature of Bayesian designs is that, as prior uncertainty about b increases, the number of design points increases. We illustrate this prop- erty with two examples.

3.2 Ds -Optimum Designs for A To demonstrate the effect of prior uncertainty about A

on the design for Model (4), we extend the Ds-optimum design for A to the case in which we put equal weight of .25 on the four A values .5, 1, 1.5, and 2. The criterion to be maximized is thus

4

() = E .25log{fM(, 0?, Aj)I/M1 (, 00, Aj)}, j=l

(22)

an extension of (20) to Ds optimality. The resulting three-point design is summarized in Table

1. The support points range from .81 to 7.32, whereas for the equivalent designs for individual A the range is from .85 to 17.38: Readings at such a high value of t are uninformative for the smaller values of A and, of course, completely uninformative for A = .5, as are those at t = 7.32. That the


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Figure 7. Variance of Prediction Function for the Bayesian Ds- Optimum Design for A With Uniform Prior Over Four Values of A.

design is indeed optimum is shown by the plot of the standardized variance d(t, *) in Figure 7. The three maxima each correspond to a function value of 1.

3.3 The Esterification of Acetic Anhydride In the preceding example it was straightforward to cal-

culate the expectation of the design criterion over the four- point prior. But, in general, numerical integration will be required, often in several dimensions. As a last example, we describe and use a form of Monte Carlo integration as an alternative to traditional quadrature.

In chemical work the prior estimates of the parameters often come from previous experiments. If these are large enough for the estimates to be approximately normally distributed, we can use the method of Atkinson, Demetrio, and Zocchi (1995) to generate a sample of values for <.

over which the design criterion (20) can be summed. Let the prior estimates have mean p and asymptotic

variance-covariance matrix Q, and let SST = Q, where S is a triangular matrix that can be found by the Choleski decomposition. Then, if Z is a vector of standard normal random variables, the random variables X given by

X = + SZ (23)

will have the required distribution. It is possible to sample from other multivariate priors either by transforming the

Table 2. Esterification of Acetic Anhydride: Bayesian D-Optimum Design for 0 and A With Monte Carlo Prior

2 =1 r2 = 4 T2 = 9

Time t S(t) Time t J(t) Time t ,(t)

.902 .5 .920 .493 .915 .478 8.280 .5 8.328 .473 5.171 .117

18.099 .034 7.602 .165 10.600 .240

NOTE: Increasing values of T2 correspond to increasing prior uncertainty.


distributions to near-normality using Box-Cox transforma- tions (Andrews, Gnanadesikan, and Warner 1971) or, more importantly, by invoking Markov-chain Monte Carlo methods (Gilks, Richardson, and Spiegelhalter 1995) to generate the sample of variables b.

As an example we use some data given by Menschutkin (1887) on the esterification of acetic anhydride, which reacts with isobutanol to form isobutyl acetate and acetic acid:

(CH3CO)20 + C4H9OH - CH3COOC4H9

+ CH3COOH. (24)

In the presence of excess isobutanol, this reaction is second- order with respect to acetic anhydride, the concentration of which was measured at eight time points. Derived from Menschutkin's results, the following parameter values were obtained by nonlinear least squares:

( .0609 .0234 ) .0234 .01225 J

and , = (0? AO)T = (.681 1.969)T. These data certainly agree with the hypothesis that the reaction is second-order.

For calculation of the Bayesian D-optimum design for both parameters, a prior was formed by sampling 100 pairs of values of 0 and A. The resulting two-point design is given in Table 2. A three-point Bayesian design was found when the prior variance was increased on replacing Q by T2Q, with T2 = 4. A four-point design resulted when r2 = 9. These designs are also given in Table 2. The variance of prediction functions for the three designs, together with one for a locally D-optimum design, are plotted in Figure 8.

These results illustrate several properties of Bayesian D- optimum designs. Not only does the number of support points of the design increase with increasing prior uncertainty, as measured by r2, but the standardized variance d(t, E*) becomes flatter. As r2 increases, it becomes increas- ingly difficult to find the precise optimum design, but the flatness of the variance function is an indication that many really rather different designs may yield virtually indistin- guishable values of IM(()l-changes in the design make compensating changes in the determinants for the various pairs of prior parameter values. As an example, the design for T2 = 4 has a value of .3834E-03 for IM((*) . The design is very similar, however, to the two-point design for T2 = 1, but with a weight of 3.4% at the surprisingly high t value of 18.1. The best two-point design for r2 = 4 has trials at .942 and 8.878 and JM(((2))} = .3796E-03, where ~(2) is the best two-point design. The efficiency of F(2) relative to the three- point design is thus 100(3.796/3.834)1/2 = 99.5%. The three-point design, with its highly unbalanced number of trials, has therefore little advantage over the two-point design. The standardized variance function for the two-point design, however, has a maximum around t = 23.3, where d(t, (2)) = 3.257, a clear indication that the design is not optimum. This is a reminder that the methods of optimum design, such as those of this article provide both algorithms for the generation of designs and means of comparison. For example Welch (1982) suggested that his branch-and-bound

354


0 5 10 15 20

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Time Figure 8. The Esterification of Acetic Anhydride: Variance of Prediction Function for the Bayesian D-Optimum Design for \ and 0 With Priors Drawn by Monte Carlo Sampling With Increasing Variance -r2Q.

algorithm can be used to produce a list of D-optimum and near D-optimum designs, the choice between which can be made on a secondary criterion such as the average of the variance of prediction (5) over the design region. Numerous references to comparisons were given by Borkowski (1995), such as those, for response surface designs, of Lucas (1976).

4. DISCUSSION There are two very different ways in which this work can

be extended. One concerns calculation of the derivatives of the function r7 with respect to the parameters for general A. Our method for determining the order of a kinetic model depends on embedding the kinetic equations in more general ones of unknown order. In our simple example we were able to obtain explicit expressions for these derivatives because the kinetic differential equations can be solved analytically. Introduction of the general order parameter A into kinetic equations renders this uncommon. For example, for the two successive reactions

A B C, (25) in which the concentration of B is measured, if B is formed at rate 01[A]Xl, given by (3), and itself reacts with order A2, analytical expressions for [B] can only be found for a few

values of A1 and A2, the case of both parameters equal to 1 being given both by Box and Lucas (1959) and by Atkinson and Donev (1992, chap. 18). To find optimum designs for such models will require use of a program such as that of Valko and Vajda (1984) to provide a grid of numerical values of the derivatives, or first-order sensitivities as they are called in the chemical literature, which will then have to be interpolated. Although the theory of the construction and properties of these designs is similar to that for the general decay model, which is the subject of this article, the computational requirement will be much heavier.

A second extension is that we have assumed that a set of single independent observations is taken at a specified set of time points. But sometimes in kinetic experiments the data arise as a series from observations at a specified number of distinct time points. An example was given by Bohachevsky, Johnson, and Stein (1986), who distributed points in time by assuming that there is a specified minimum time interval between successive observations. They find the optimum design by numerical optimization: No mention is made of equivalence theory, neither for generation of designs nor for checking their properties.

ACKNOWLEDGMENTS

We are grateful to Dr. Mariusz Bogacki of the Techni-


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cal University, Poznan, for stimulating conversations about chemical kinetics and for providing the reference to the data on acetic anhydride. Doug Bates of the University of Wisconsin provided patient and much-needed help on the generation of trellis plots. Financial support of the Staff Re- search Fund of the London School of Economics is thank- fully acknowledged.

[Received April 1996. Revised April 1997.]

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Atkinson, A. C., Demetrio, C. G. B., and Zocchi, S. (1995), "Optimum Dose Levels When Males and Females Differ in Response," Applied Statistics, 44, 213-226.

Atkinson, A. C., and Donev, A. N. (1992), Optimum Experimental Designs, Oxford, U.K.: Oxford University Press.

Atkinson, A. C., and Haines, L. M. (1996), "Designs for Nonlinear and Generalized Linear Models," in Handbook of Statistics (Vol. 13), eds. S. Ghosh and C. R. Rao, Amsterdam: Elsevier, pp. 437-475.

Bohachevsky, I. O., Johnson, M. E., and Stein, M. L. (1986), "General- ized Simulated Annealing for Function Optimization," Technometrics, 28, 209-217.

Borkowski, J. J. (1995), "Spherical Prediction-Variance Properties of Cen- tral Composite and Box-Behnken Designs," Technometrics, 37, 399- 410.

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Inference, 21, 191-208.

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Ford, I., Torsney, B., and Wu, C. F. J. (1992), "The Use of a Canonical Form in the Construction of Locally Optimal Designs for Non-linear Problems," Journal of the Royal Statistical Society, Ser. B, 54, 569-583.

Gilks, W. R., Richardson, S., and Spiegelhalter, D. (1995), Markov Chain Monte Carlo in Practice, London: Chapman and Hall.

Hunter, W. G., Hill, W. J., and Henson, T. L. (1969), "Designing Experi- ments for Some or all of the Constants in a Mechanistic Model," Cana- dian Journal of Chemical Engineering, 47, pp. 76-80.

Kiefer, J., and Wolfowitz, J. (1960), "The Equivalence of Two Extremum Problems," Canadian Journal of Mathematics, 12, 363-366.

Lucas, J. M. (1976), "Which Response Surface Design Is Best," Techno- metrics, 18, 411-417.

Menschutkin, N. (1887), "Reaktionen der Acetik Anhydrid," Zeitschriftfiir Physikalische Chemie, 1, 611.

Pilling, M. J., and Seakins, P. W. (1995), Reaction Kinetics, Oxford, U.K.: Oxford University Press.

Silvey, S. D. (1980), Optimum Design, London: Chapman and Hall. Valko, P., and Vajda, S. (1984), "An Extended ODE Solver for Sensitivity

Calculations," Computers and Chemistry, 8, 255-271. Welch, W. J. (1982), "Branch and Bound Search for Experimental Designs

Based on D-Optimality and Other Criteria," Technometrics, 24, 41-48. Wynn, H. P. (1970), "The Sequential Generation of D-Optimal Experimen-

tal Designs," The Annals of Mathematical Statistics, 41, 1055-1064.


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