ioumal of and inference
Journal of Statistical Planning and Inference 44 (1995) 385-397
I
.
Bayesian
D-optimal designs for the exponential growth model
Saurabh aISDS,
”
Duke
Mukhopadhyay”,*, University.
Bow 90251,
’ Unicersity
Received
Linda
14 September
of Nutul,
Durham, South
M. Hainesb
NC 277084251,
C’SA
Africa
1992; revised 31 January
1994
Abstract Bayesian optimal designs for nonlinear regression models are of some interest and importance in the statistical literature. Numerical methods for their construction are well-established. but very few analytical studies have been reported. In this paper. we consider an exponential growth model used extensively in the modelling of simple organisms, and examine the explicit form of the Bayesian D-optimal designs. In particular, we show that D,-optimal designs for this model are balanced two-point designs for all values of the parameters. We further derive explicit expressions for Bayesian D-optimal designs which are based on exactly two points of support, and provide necessary and sufficient conditions for such designs to exist. We illustrate our results by means of two examples.
AM.7 Szthjrct Classification:
Primary
Key words: Nonlinear regression; designs; Bayesian design.
62KOS; Secondary
Exponential
growth
62F15 model; D-optimality;
Locally optimal
1. Introduction Many authors have dealt with the problem of finding optimal designs for nonlinear models. For example, Tsutakawa (1972, 1980) and Abdelbasit and Plackett (1983) considered only designs comprising observations at equally spaced points and with equal numbers of observations at each point. More recently, Chaloner and Larntz (1989) provided a powerful framework for constructing Bayesian optimal designs,
*Corresponding
author.
Tel.: 919-681-8442.
037%3758/95/$09.50 :Q l995SSDI 0378-3758(94)00056-5
Fax: 919-684-8594.
E. mail: saurabh@,isds.
Elsevier Science B.V. All rights reserved
duke.edu.
386
S. Mukhopadhyay, L.M. Haines/Journal
which are those designs
of Statistical
that optimize
criteria
Planning and Inference 44 (1995) 385-397
averaged
over a prior distribution
on
the parameters of the model, and implemented their approach numerically but not algebraically. In the present paper, we derive some explicit expressions for Bayesian optimal
designs
for a specific
nonlinear
regression
model,
in so doing,
we aim to
provide a deeper understanding of the nature of these designs. In particular, we consider the exponential growth model which is commonly used to describe the growth of simple organisms (see Richards, 1969; Sandland, 1983;
Seber
attendant
and
optimal
Wild, designs
1989,
pp.
327-328).
are of interest
While
in their
this
growth
own right,
curve
and
the techniques
here for design construction can be readily extended to other nonlinear We use as our design criterion the log of the determinant of the information
its used
models. matrix
averaged over a prior distribution on the unknown parameters of the model. This criterion is related to the D-optimality criterion used extensively for design in linear models, and for that reason it is termed Bayesian D-optimality. For a prior degenerate
at a fixed parameter
value, 0, the corresponding
criterion
depends
on 0,
and is referred to as DO-optimality. The criterion of Bayesian D-optimality can be regarded, quite simply, as reflecting the experimenter’s prior belief in the parameter values with respect to design (Lauter, 1974, 1976), or, as suggested by Chaloner and Larntz (1989), it can be interpreted as an approximation to the expected increase in the Shannon information provided by the experiment (see also Lindley, 1956; Bernardo, 1979). We use the general Equivalence Theorem, introduced by Whittle (1973) for linear models, and modified by Chaloner and Larntz (1989) to accommodate nonlinear models, to verify the Bayesian D-optimality of a given design. In Section 2, we formally introduce the exponential growth model, and give details of the criterion used in obtaining optimal designs. In Section 3, we obtain an explicit expression for DO-optimal designs for this model, and, in particular, we show that such designs are based on exactly two points for all values of the parameters. It is natural, therefore, to investigate the circumstances under which Bayesian D-optimal designs for the exponential growth model are also based on two points, and this is done in Section 4. In particular, in Section 4.1, we derive explicit expressions for Bayesian D-optimal designs with exactly two points of support, and furthermore we provide
a necessary
and sufficient condition
for the existence
of such designs. We note
that no closed-form expressions for Bayesian D-optimal designs based on more than two support points are, as yet, available, and that in such cases recourse must be made to numerical methods for design construction (see Chaloner and Larntz, 1989). Finally in Section 4.2, we consider two specific classes of prior distribution on the parameters of the exponential growth model, namely those of the gamma and the uniform on the scale parameter, and investigate the resultant Bayesian D-optimal designs in some detail. These examples illustrate the theory developed in Section 4.1, and in addition demonstrate the effect of the prior on the nature of the designs.
S. Mukhopadhyay,
L.M. Haines; Journal ofStatistical
Planning and Inference 44 119951 385-397
387
2. The model and the criterion 2.1. Exponential We consider
growth model modelling
the growing
y= 1 _epY/2(x-d) +&,
of an organism
y>o, 6<0;
as
x>o.
(1)
where y is the response to a control variable, X, y and 6 are unknown scale and location parameters, respectively, and E is a random error with mean 0 and variance cr2. We assume that the design space, x, is a compact interval, [a, a + b], for some a >, 0 and h > 0, and that the errors are uncorrelated and normally distributed. Then the Fisher information matrix of the unknown parameters, 0 = (6, y)‘, for a single design point,
X, can be expressed
r(e,~,)cw(e,X)
as
y2 --y(x--6)
(
-y(x-6)
(x--q2
1 ’
where w(U,X)=~-Y’“-~) and 5, denotes the one point probability measure with support x. Further, we consider approximate designs, which are probability measures, 5, on the Bore1 field of the design space, x, as candidate optimal designs (see Kiefer and Wolfowitz, 1959), and denote the set of all such designs, r, by E. Then the information matrix of 6’for a design, [EE, is given by I(& 5)=
sx
(3)
I(& LX(d.4.
2.2. The criterion To ensure
that 8 is estimated
in an optimal
way, it is usual to construct
designs
which maximize some concave function, q, of the Fisher information matrix, termed the criterion function. However, the problem in adopting this procedure for nonlinear models is that the information matrix, and hence the criterion function, depend on the unknown parameters, 9. One approach to this problem is to adopt a nominal value for 8 and to determine the locally rpoptimal or qO-optimal design for that particular value of l3 (Chernoff, 1953). However, it is quite possible for a design which is optimal for some fixed value of 8 to be completely unreasonable for other values of 8. A natur.al approach to overcoming this difficulty is to introduce a prior distribution on the unknown parameters, and to construct designs which optimize criterion averaged over this distribution. Such optimal designs are termed Bayesian yFoptima1 designs. We consider here the log of the determinant of the information matrix averaged over a given prior on the parameters as our criterion function. Then for any prior, n. and for any design, 5~2, the criterion function is defined by ~(5) = & ~ns(
388
S. Mukhopadhyay, L.M. Haines/ Journal qfStatistica1 Planning and Inference 44 (1995) 385-397
where
%(5) = The design
log det Z(0, 5)
if det Z(0, 5) #O,
-CX
otherwise.
which maximizes
~(5) is termed
the Bayesian
D-optimal
design,
and is
denoted by 5:. The directional derivative of the criterion function, p, which is the derivative of v, at any design,
is given, for Bayesian
D-optimality,
by
d(5,x)=E,CtrZ(e,5,)Z(e,~)-‘l-2
(4)
for some prior distribution, x (see Silvey, 1980, pp. 20-21). It then follows from the Equivalence Theorem for nonlinear models given by Chaloner and Larntz (1989), that if and only if for any fixed prior, 71,a design, rx, is Bayesian D-optimal
E,CtrI(e,~,)l(O,5,*)-‘1~2
(5)
for all x in the design space [a, a + b], with equality
in (5) occurring
if x belongs
to the
support of t:. We observe, from the above definition, that a D,-optimal design can be regarded as a Bayesian D-optimal design for a degenerate prior distribution on 8. For such designs, we consider maximizing the log of the determinant of the information matrix, (3) for the appropriate simplifies to
and it further
8 vlaue.
follows that a necessary
The associated
and sufficient
directional
condition
derivative,
for a design,
(4)
5, to be
D,-optimal is that the inequality de(<, x) d 0 holds for all x in the design space. We note that D,-optimal designs are particularly useful when they are relatively insensitive to changes in the parameter values, 8. If the measure, r, has support at only one point, then the matrix, Z(0, 0, is singular, detz(0, c)=O for all 0, and &()=-a for any prior, 7~. It therefore follows that a one-point design cannot be Bayesian D-optimal, and that it is sufficient to consider candidate optimal designs with at least two points of support.
3. DO-optimal designs We begin considering D,-optimal designs for the exponential growth model, (1). We introduce the term balanced two-point design and denote it by t* to describe a twopoint design which puts equal masses on each of its support points, and present our results in the following theorem.
S. Mukhopadhyay,
L.M.
Haines / Journul
Theorem 3.1. For the exponential 5; is a balanced
of Statistical
Planning
und I~f>rence
44 11995) 385JY7
growth model (l), the D,-optimal
389
design, denoted by
two-point design with unique support at the points, XT and xs, where
and .~*= 2
a+217
if b>2/y,
a+b
if Q
Proof. Consider the information matrix, Z(0, <,), for a single design point, x, given in (2); then the set, { 1, w(~,x), x ~(0, w)}, forms a Tchebycheff system on any fixed interval, [a, a + b], in ‘93. It thus follows from the results of Karlin and Studden (1966, p. 333) and of Fedorov (1972, pp. 85-86) that, for any fixed 0, the D,-optimal design is based on exactly two points of support, and further that it puts equal masses on those points, i.e. it is a balanced two-point design (see also Haines, 1992). Now for any balanced
two-point
design,
t*,
a
with support points, calculations show that
.yl
and
.x2 such
that
and that (7)
Thus the necessary the inequality,
and sufficient
dJt*,x)
holds
condition
for such a design to be D,-optimal
for all x in the interval,
[a,a+
b], with
is that equality
occurring at the points of support. The expression, (7), for d,(t*,x) is free of the parameter, 6. Thus it follows that the optimal design points, denoted by XT and xs, respectively, and hence that the D,-optimal design itself, do not depend on 6. Furthermore, suppose that 5 @ L’ denotes a design obtained by translating each of the support points of < by an amount, c. Then we observe that de(t* @ c, x + c) E d,([*, x) for any CE%, and thus it follows that a design, t$, is D,-optimal for the design space x = [0, b], if and only if the design, 5: @ a, is D,-optimal for design space, x = [a, a + b]. Therefore, without any loss of generality, we assume that a=O, i.e. that the design space is [0, b], throughout the remainder of this theorem. Note that the above conclusions can also be deduced from a consideration of the form of the information matrix, (6). We now derive explicit expressions for the support points, XT and x:, of the D,-optimal design in the following three steps. Step I: Suppose that XT >O; i.e. XT is an interior point of the design space, [O.h]. Then, since de(4$, x) must have a local maximum of 0 at x = xf , it follows that
390
S. Mukhopadhyay, L.M. Haines/ Journal of Statistical Planning and Inference 44 (1995) 385-397
dh(<$, x) = 0 at x=x T, where d@(l, x) is the derivative of de(& x) with respect to x. Now for any balanced two-point design, it is straightforward to show that
&(5*,x)=
e-Y”‘(x1-x)[-2y(x1-x)-4]+e-Y”2(x,-x)[-2y(x,-x)-4] e-rc~l+X2-X)(X*_X1)2
and hence that db(t*,xr)=
-2y-4/(x2-x1).
But, since -2y-44/(x2-x,)
choice of x1, where O< x1
be zero, and in particular Thus XT cannot be an
(8) 9 any for the interior
i.e. x: =O.
Step II: Suppose now that xz is an interior point of [0, b]. Then, by the same arguments as in Step I, &(lt, x) must have a local maximum, and, more particularly, for x1 =x7 =0 in (8), and simplifying, &(5$,x) must equal 0, at x=x z. Subsituting gives &(4*,x,)=
-2,+4 x2’
and this derivative is zero when x2 =2/y. Thus if b > 2/y, then 2/y, is an interior point, and xz =2/y. Suppose however that b <2/y. Then the point, 2/y, falls outside the design space, [0, b], and xz cannot be an interior point, but must equal b. Step III. From Step I, it follows that x: is always zero. For b < 2/y, we have shown in Step II that x$ must equal b; thus the balanced design with support at 0 and b is unique. Consider now b > 2/y. Then it follows from Step II that, if x; is an interior point of the design space, then xg =2/y. However, in order to prove the uniqueness of this design, we must exclude the possibility that the optimal design point, xt, can be the boundary point, b. In fact, for the balanced design, <*, with support points, xf =0 and xf = 6, do(<*, b) = - 2y +4/b, which is negative for b > 2/y. This in turn implies that 0 there exists some x in [0, b] with de(& x) >O, which is a contradiction.
4. Bayesian D-optimal designs 4.1. Main results We now consider Bayesian D-optimal designs for the exponential growth model, (l), with nondegenerate priors on the parameters. Two results from the previous section pertaining to D,-optimal designs are of particular importance in this regard, namely (1) The D,-optimal design, ?jg, does not depend on the location parameter, 6. (2) If b < 2/y, then the D,-optimal design, r$, is a balanced two-point design with support on the boundary points, a and a + b, of the design space, [a, a + b], and is thus insensitive to changes in scale parameter, y.
S. Mukhopadhyay, L.M. Hainesj Journal of Statistical Planning and Injkrence 44 II995
We exploit D-optimal changes
these two features
designs.
Firstly,
in 6 values and obtain
the following
some robustness
the insensitivity
an analogous
properties
for Bayesian
of the DO-optimal
result for Bayesian
D-optimal
designs
to
designs in
theorem.
Theorem 4.1. Zf the prior distributed,
to obtain
we consider
39 I
i 385-397
on the parameters
then the Bayesian
D-optimal
is such
design,
that y and 6 are is independent
c:,
independentl?
of the distribution
on 6.
Proof. Consider a design, t, defined on the design space, [a, a + b], and assume, as 1s usual, that the measure, 5, has finite support on this interval. Let Ci= jx [eeyX xi] [(ds). i = 0, 1,2. Then a straightforward calculation gives det I(O, 5) cc e-26yyZ [cOcZ -cf], and thus the criterion
function,
q(t),
is equal to
&logCGJc,-c:l, where the equality holds up to an additive constant. For this latter expression, the term inside the expectation, E,, is independent of 6, and hence the required result follows. 0 We note that similar results, but for parameters which enter models linearly, been reported by Pronzato and Walter (1988) and by Atkinson et al. (1993). Secondly DO-optimal
we introduce designs
on a subset
-aj<6<0,
then the Bayesian the boundary
the exponential
is supported
S,={@y)‘:
under
which the Bayesian
D-optimal
and the
coincide.
Theorem 4.2. For parameters
conditions
have
D-optimal
points,
growth
model,
(l), [f the prior
distribution
on the
of the region,
O
{,*, is the balanced
a and a + b, of the design
space,
two-point
design with support
at
[a, a + b].
Proof. This result follows immediately from the fact that, provided b 6 2/y, then the DO-optimal design is the same for all parameter values in SO, and puts equal masses on n the boundary points, a and a + b, of the design space. In general, however, Bayesian D-optimal designs for the exponential growth model are not necessarily based on exactly two points of support, and this is demonstrated numerically in the examples of Section 4.2. In contrast, DO-optimal designs for this model are always based on two points. It is natural, therefore, to investigate conditions under which Bayesian D-optimal designs are based on exactly two points, and to
392
S. Mukhopadhyay,
L.M. Haines/ Journal of Statistical PIanning and Inference 44 (I995)
characterize such two-point designs when they exist. We present regard in the following theorem and its corollary.
385-397
our results
in this
Theorem 4.3. For the exponential growth model, (I), andfor any prior distribution, 71,on the parameters, y and 6, zfthe Bayesian D-optimal design is based on exactly two points, then it is a balanced design with support at the points, XT and x;. where x;=a, and x”= 2
a+2/E,y
if E,y<
a+b
otherwise.
Proof. If a two-point that given in Theorem
Bayesian
co and E,y>2/b,
D-optimal
3.1 for D,-optimal
design exists, then, by reasoning designs, it must be balanced.
similar to
It then follows
immediately from condition, (5) and from expression, (7), that the necessary and sufficient condition for such a balanced two-point design, t*, to be Bayesian D-optimal is that the inequality, e-Y”‘(x1-x)2+e-y”z(x2-x)2 ,-rcx1
+xqx2
_x1)2
holds for all x in the interval, [a, a + b]. Thus, assuming regularity conditions prior distribution which permit the interchange of the differentiation,
on the (d/dx),
and the expectation, E,, operators, the remainder of the proof then follows that of Theorem 3.1, but with certain obvious modifications. 0 Corollary.
The Bayesian D-optimal design is a balanced two-point design ifand only if
E,[x2e-Y’“-x~)+(x-x~)2e-Y”]
<(xf)’
(9)
for all x in [0, b], where x5 = 2/E,y tj”b > 2/E,y and x9 = b if b < 2/E,y. Proof. The design space, [0, b], is adopted
without
loss of generality,
and the values
for x1 and x2 in the necessary and sufficient condition of the theorem are replaced the optimal values, x7 =0 and xf, respectively, to give the modified condition, (9).
by 0
Remarks: 1. The necessary and sufficient condition, (9), presented in the above corollary, is of considerable practical importance, since it provides a mechanism for checking whether or not the Bayesian D-optimal design for a particular value of b and for a particular prior, rc, is based on exactly two points. Its use will be demonstrated in the examples of the next subsection. 2. Condition (9) can be expressed as
S. Mukhopadhyay, L.M. Haines / Journal of Stc~tisticul Planning and Irzference
where the function,
M,(t), is the MGF
a necessary condition
for the Bayesian
points is that the prior MGF the mean, E,y, is finite.
of the marginal D-optimal
44 (199.5) 385-397
prior distribution
19:!
of 7. Thus
design to be based on exactly two
of y exists and is finite. Note that this in turn implies that
3. The problem of characterizing Bayesian D-optimal designs, which are based on three or more points, remains. This would seem to be extremely difficult to solve analytically,
and is not discussed
further
here.
4.2. Applications We now introduce two examples in which parameters are adopted. In each case we assume
different classes of priors on the (i) that the parameters, y and 6, are
independently distributed, and (ii) that the design space, x, is the interval, [O,b]. i.e. a = 0. The former assumption is somewhat stringent, but allows us to consider priors on y only, i.e. to ignore the distribution of 6, while the latter assumption can be made without any loss of generality. The aim of these examples is firstly to illustrate the results in the previous section, and secondly to examine, albeit for only two cases, the effect of the prior on the nature
of the Bayesian
D-optimal
Example 4.1. Since y is a scale parameter for the exponential natural to adopt a gamma distribution for its prior, i.e.
y>o, c>o
ir(p)=~yc~le~y,
=0
designs. growth
model, (1). it is
( 10)
elsewhere.
The MGF for the gamma distribution, M,(r), exists and is finite for all t < 1, and thus it follows that the left-hand side (1.h.s.) of the inequality, (9) converges if and only if x$ < 1. Then E,y = c and the necessary and sufficient condition, (9) for the Bayesian D-optimal design to be a balanced two-point design is given by X2(1 +x-xq)-‘+(x-.X~)z(l
+x)-c<(x;)2
(1 I)
for all x in [0, b]. Suppose that h>,2/E,y, i.e. h > 2/c, and that c> 2. Then it is straightforward to show that a necessary and a conjectured sufficient condition for the inequality, (1 l), to hold for all XE[O, h], i.e. for the Bayesian D-optimal design to be a balanced two-point design, with support at 0 and 2/c, is that c >c:, where CT E 2.58809. For b < 2/c, with b < 1, extensive algebraic and numerical investigations of (11) suggest that the necessary and sufficient conditions for the Bayesian D-optimal design to be a balanced two-point design, in this case with support at 0 and h, is that the pair (c, h) belongs to the set, Sr={(c,h):
O
O
394
S. Mukhopadhyay, L.M. Haines/ Journal of Statistical Planning and Inference 44 (1995) 385-397
where b,=l for O
of the latter,
design,
cl,*, puts
we observe
masses
that for c=2.4
0.497, 0.258,
and
and b= 1, the Bayesian
0.245 on the points
D-optimal
0, 0.706, and
1,
respectively.
Example 4.2. We also consider the interval, always
a uniform prior on the parameter, y, with support on 1, c+ 11, where c> 1. Since the MGF of the uniform distribution
[c-
exists and is finite, the 1.h.s. of (9) always
condition
(9) can be expressed
converges.
Thus
E,y=c<
00 and
as
(12) for all x in [0, b]. For b > 2/c, numerical studies of (12) together with some algebraic manipulations, suggest that the necessary and sufficient conditions for the Bayesian D-optimal design to be a balanced two-point design, with support at 0 and 2/c, are either that c>cz, where ~~~1.13212, or that the pair, (c, b), belongs to the set, S,={(c,b):
2/cdb6b,},
ldc
.............. ............... .............. .......... ...... ......... ..... ... *:: .......... *::: ............... ......
2.0
1.5
t b
1.0
. . . . . .
0.5
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
.:::. .:::. .. .. .... .. .. .... *:::. .:::. . . . . :::: :::: ..... .. . 1:: . .. . .. . .. . .. .. .. .... *::. . .:::. -:::. *:::. . . . . . . . . . . .A f2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..... L :$:c>:z . . ....... .......... :. :. *:: . . . *:. . . ..... - * ::. .:.::. . * . . . *. ..:.
c
0
. . :.. . .. . . :. *. :.. ‘. -. *: ... ... ... . . :. :. :. :. :. *. .L . . . . . . . I
1
1
2
c; z 2.58809
4
3
cFig.
1 Plot of the points (c. b) where a two-point
Bayes D-optimal
design is available
in Example
4.1.
S. Mukhopadhyay,
L.M. Haines iJournal
qf Statistical Planniny crnd Ittference 44 i 1995 i 36
395
397
where
the h, values are determined numerically, and for example, h, ~2.48829, h i,es E 3.01895, bl,l z 3.97439, and h, T x 6.88258. For the pairs, (c, b), with c >, 1, not
in SZ, the Bayesian D-optimal designs are based on more than two points, and, for example, for c = 1.05 and b = 20, this design tt, puts masses 0.489,0.478, and 0.033 on the points
0, 1.786, and
c + 1<2/b,
i.e. b < 2/(c + l), then the interval,
For b<2/c, we firstly observe
11.377, respectively.
[c - 1, c + 11, which supports
that
if
the uniform
prior, is contained in the interval [0,2/h], and it follows from Theorem 4.2 that the Bayesian D-optimal design is always the balanced two point design with support at 0 and b. In fact numerical studies indicate that this is true for all c and h values fcfr which b <2/c. These results are represented graphically in Fig. 2, in the same way a.s those for the gamma prior on y.
Remarks.
1. A number signs, the more diffuse
of authors have observed that for Bayesian the prior on the parameters of the model,
D-optimal the larger
dethe
required number of support points for the optimal design (see Chaloner and Larntz, 1989; Atkinson and Donev, 1993, pp. 219-220). However, our results for the exponential growth model, (1) are not in strict accord with this. In particular, for the unifortn prior on the parameter, y, with support on the interval, [c- 1, c+ 11, the variance is constant, but it is clear from Fig. 2 that the number of support points for the Bayesian D-optimal design depends on the mean, c, of the prior, and on the length, b. of the design space. For the gamma prior on y, the mean and the variance both equal c, and the situation is necessarily more complicated; nevertheless, the dependence of the number of support points of the optimal design on both c and b is apparent. Overall
....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... .......
,.. l.Oo ..
.
.
.
.
. .
.bc .._
...... ....... ........................ .................
1 .oo
.
-ri..
(.z.
.
.
.
.
.
.
.
.
.
.
.
.:
............. ................
1.05
Fig. 2. Plot of the points (c, h) where a two-pomt
1.10
......... I c; z 1.13212 1.15
Bayes D-optimal
design is available
. 1.20
in Example
4.2
396
S. Mukhopadhyay,
the most striking
L.M. Haines/ Journal
feature
ofStatisticalPlanning
of these dependencies
and Inference 44 11995) 385-397
is that, for prior means, the Bayesian
D-optimal design is based on the minimal support of two points. It is tempting to relate these observations to certain underlying factors, such as the nonlinearity of the model,
but this is beyond
the scope of the present
paper.
2. There are some broad similarities in approach between our work and that of Chaloner (1992), in that both studies rely on directional derivatives to suggest and to confirm Bayesian D-optimal designs. 3. In a recent report, Dette and Neugebauer (1993) extended our study to encompass two-parameter gamma tion for the Bayesian D-optimal
priors on y, and formally design to be a balanced
derived a sufficient conditwo-point design. For the
one-parameter gamma prior, (1 l), their sufficient condition reduces to c > 3, and, while this has been proved to hold true, it is, nevertheless, weaker than our conjectured condition, c 3 CT z 2.58809. 4. Full details of the numerical investigations underpinning the above examples are available
from the authors
upon
request.
Acknowledgements This study forms part of the Ph.D. thesis of SM. presented through the Department of Statistics, Purdue University, USA and was completed while one of the authors (L.M.H.) was visiting that department. The work was sponsored by NSF Grant DMS 89-230-71 at Purdue University (S.M.), and by funds from the University of Natal and the Foundation for Research Development, South Africa, (L.M.H.).
References Abdelbasit, KM. and R.L. Plackett (1983). Experimental design for binary data. J. Amer. Statist. Assoc. 78, 9C-98. Atkinson, A.C., K. Chaloner, A.M. Hertzberg and J. Juritz (1993). Optimum experimental designs for properties of a compartmental model, Biometrics 49, 325-338. Atkinson, A.C. and A.N. Donev (1992). Optimum Experimenta Designs. Oxford University Press, New York. Bernardo, J.O. (1979). Expected information as expected utility, Ann Stat&. 7, 686690. Chaloner, K. (1993). A note on optimal Bayesian design for nonlinear problems. J. Statist. Plann. Inference 37, 229-235. Chaloner, K. and K. Larntz (1989). Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference 21, 191-208. Chernoff, H. (1953). ‘Locally optimal designs for estimating parameters, Ann. Math. Statist. 24, 586602. Dette, H. and H.-M. Neugebauer (1993). Bayesian D-optimal designs for exponential regression models. Technical Report. Fedorov, V.V. (1972). Theory of Optimal Experiments. Academic Press. New York. Haines, L.M. (1992). Optimal design for inverse quadratic polynomials. South African Statist. J. 26, 2541. Karlin, S. and Studden, W.J. (1966). TchebyheflSystems: with Applications in Analysis and Statistics. Wiley, New York.
S. Mukhopadhyay,
L.M.
Haines / Journal
of Statistiud
Planniq
and l&v-ewe
44 (1YY.F) 385-.W7
39;
Kiefer, J. and J. Wolfowitz(l959). Optimum designs in regression problems. Ann. Math. Statist. 30,271-~294 Liiuter. E. (1974). Experimental design in a class of models. Math. Oper. Statist. 5, 379-396. Liiuter, E. (1976). Optimal multipurpose designs for regression models. Math. Oper. Statist. 7, 5 l-68. Lindley, D.V. (1956). On a measure of information provided by an experiment. Ann. Math. Statist. 27. 986-1005.
Pronzato.
L. and E. Walter
Workshop
on Mode/-oriented
(1988). Robust Data
Analysis,
experiment design for nonlinear regression models. Prm. Lecture Notes in Economics and Mathematical System,;
No. 297, Springer, Berlin, 77-86. Richards, F.J. (1969). The quantitative analysis of growth. in: F.C. Stewart, ed., P/ant Physiolog!,, li~/ww V A: Analysis of’Growth: Behavior oj Plants and their Oryans, Academic Press, New York. 3-76. Sandland. R.L. (1983). Mathematics and the growth of organisms some historical impressions. .‘M~th. Scientist 8, I I-30. Seber. G.A.F. and C.J. Wild (1989). Nonlinear Rryression. Wiley, New York. Silvey, S.D. (1980). Optimul Design: An Introduction to the Theory /or Parameter Estimation, Chapman & Hall. London. Studden. W.J. (1982). Some robust type D-optimal designs in polynomial regressions. J. Amer Stcui\t. 4~0~. 77. 916-921.
Tsutakawa, R.K. (1972). Design of experiment for bioassay. J. Amer Sttrtist. Assoc. 67. 584-590. Tsutakawa. R.K. (1980). Selection of dose levels for estimating a percentage point of a logistic quanta1 response curve. Appl. Statist. 29, 25-33. Whittle. P. (1973). Some general points in the theory of and constructlon of D-optimum experimental design. J. Roy. Statist. Sot. B 35. 123-130