Experimental Design for the Weibull Function as a Dose Response Model
by
Karen A. Dassel
A thesis submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the Degree of
Doctor of Philosophy
Department of Statistics
Raleigh
1 9 8 7
Approved By:
DASSEL, KAREN ANN. Experimental Design for the Weibull
Function as a Dose Response Model. (Under the direction of
J. O. Ra~lings)
The objective of this research ~as to determine design
points that ~ould maximize the precision of the estimated
relative yield losses over a range of 03 pollutant levels
~hen the dose-response curve ~as characterized by the
Weibull. An "H-equivalent" transformation ~as developed
such that, ~ith the appropriate transformation, the optimum
design (by the determinant criterion) for one choice of the
model parameters could be translated to any choice of the
parameters. This permitted the search for an optimum design
to be restricted to one set of parameter values. Several
different optimization criterion, allocation strategies and
amount of point replication ~ere considered. A design was
found by each of the criteria. A robustness study was
con-ducted to investigate the performance of the designs with
regard to the research objective in situations where the
values of the model parameters had been estimated
incor-rectly and the design space shifted. The impact of
deviat-ing from the optimum design assumdeviat-ing known parameter values
was also studied. A Monte Carlo simulation was performed to
another check, a second order adjustment was calculated and
its relative magnitude noted.
It was found that the design space should have the left
end point as near zero as possible and the right end point
extended to approximately w (1.7)1/1.. The remaining design
points (preferably six) should be evenly dispersed through
the dose interval. A first-order approximation of the
vari-ance expressions is adequate for selection among
Biography
Karen A. Dassel was born in Evansville, Indiana on October
24, 1957. She graduated from Evansville's Central High
School in 1975. That fall she enrolled at the University of
Evansville. In May, 1979, she received a Bachelor of Arts
degree in Secondary Math Education, followed in August,
1979, by a Bachelor of Science degree in COml)uter Science.
From March, 1976, to May, 1979, she was employed
part-time in the offices of Sears Roebuck
&
Co., Evansville,Indiana. In June, 1979, she began work for Mead Johnson
&
Co., Pharmaceutical Research Division, Evansville, Indiana.
Hired as a programmer in the Data Services Group for the
Clinical Information and Statistic Department, she later
became group supervisor.
In August, 1981, she left Mead Johnson to attend
gradu-ate school at North Carolina Stgradu-ate University, Department of
Computer Studies, Raleigh, North Carolina. She worked in
the Computer Studies Department first as a Teaching
Assis-tant, and then as a Research Assistant. Upon receipt of the
Master of Science degree in Computer Studies in August,
1983, she continued her studies in the Department of
Statis-tics, North Carolina State University, Raleigh, North
Car-olina. She worked in the Statistic Department as a Teaching
her Master of Statistics degree in December, 1984, and upon
receipt of the Doctor of Philosophy degree in Statistics,
she plans to work as a Post-Doctoral Research Associate in
the Department of Statistics, North Carolina State
ACKHOWLEDGllfENTS
I wish to thank the many people who have encouraged,
advised, and assisted me during the preparation of this
dissertation. In particular I would like to thank my
com-mittee chairman Dr. John Rawlings for his patience,
under-standing, and always ready assistance while advising this
research., Special thanks are also extended to the other
members of my committee, Dr. Francis Giesbrecht and Dr.
Sastry Pantula. And finally, I am grateful for a research
grant supported by an Interagency Agreement between the
En-vironmental Protection Agency and the USDA, Interagency
agreement number AD-12-F-1-490-2, and specific Cooperative
Agreement ~58-43YK-6-0041between the USDA and the North
Page
1. Introduction.. . . .. 1
2. Experimental Design for the Weibull Function as a Dose Response Model Assuming an Unconstrained Dose
Scale 2
2.1 Introduction... 2
2.2 Literature Review 3
2.3 The Weibull Model 7
2.4 Variances of Estimates 8
2.5 Estimated Relative Yield Loss (RYL) 9
2.6 H-equivalent Transformation and Optimization
Invariance 10
2.7 Design Development 16
2.7.1 Parameter Estimation (Determinant
Criter-ion) 17
2.7.1.1 2.7.1.2
2.7.1.3
2.7.1.4
2.7.1.5
2.7.1.6
Results from Optimum Three-point Design 17 Results from Unconstrained Four-point
Search. FDET Criterion 18
Results from Unconstrained Four-point
Search. SDET Criterion 20
Results from Search for Six-point Design.
SDET Criterion 21
Results from Constrained Four-point
Search 22
Conclusion 23
2.7.2 2.7.3
Estimation of Relative Losses 25
Comparison of Designs from Parameter Estimation and Estimation of Relative
Losses 26
2.8 Conclusion 27
2. 9 References... 31
Tables for Chapter 2 '" 34
3. Experimental Design for the Weibull Model as a Dose Response Function. Assuming a Constrained Dose
Scale 39
3. 1 Introduction... 39
3.2 The Weibull Model 41
3.3 Estimated Relative Yield Loss (RYL) 42
3.4 H-equivalent Transformation 43
3.5 Design Development 44
3.6.1 FDET and SDET Criteria 49 3.6.2 Alternative Allocation Strategies: EQX,
EQH, GEO 50
3.6.3 Effect of changing YMIN keeping YMAX equal
to 0.98 53
3.6.4 Effect of changing YMAX keeping YMIN equal
to O.25 55
3.6.5 Effect of Unequal Replication 56
3. 7 Corlclusion... 57 3 . 8 Refererlces... 61
Tables for Chapter 3 63
Figures for Chapter 3 73
4J Robustness of Experimental Design for the Weibull
Function as a Dose Response Model 76
4. 1 Introduction... 76
4.2 Literature Review 77
4.3 The Weibull Model 79
4.4 Estimated Relative Yield Loss (RYL) 81
4.5 H-equivalent Transform~tion 81
4.6 The Design Strategies 82
4.7 Robustness of the Design 84
4.7.1 Incorrectly Guessing One Parameter 86
4.7.2 EQH Strategy, Incorrectly Guessing Both
Parameters 88
4.7.3 EQX Strategy, Incorrectly Guessing Both
Parameters 90
4.7.4 Effect of CNT=6, 8 91
4.7.5 Fixing the Estimation Interval 93
4.8 Conclusion 99
4.9 References 103
Tables for Chapter 4 104
Figures for Chapter 4 112
5. Wald Variances as a Method for Comparing Designs Assuming the Weibull Function as a Dose Response
Mode 1 121
5.1 Introduction 121
5. 2 Literature Review 122
5.3 The Weibull Model 127
5.4 Estimated Relative Yield Loss (RYL) 128
5.5 H-equivalent Transformation 129
5.6 Des ign Strategies 130
5.7.1 5.7.2 5.7.3
5.7.4
Model Parameter Estimation 134
Behavior of the Variance Approximations 135
Effect of Variance Approximation on Optimum
Design Choice 137
Simulation of a Non-optimal Design (NOPT) .. 142
5.8 Conclusion... 145
5.9 R e f e r e n c e s . . . 149
Tables for Chapter 5 152
Figures for Chapter 5 164
6. C o n c l u s i o n . . . 197
7 . A p p e n d i x . . . 204
7.1 Robustness Plots from Chapter 4 204
7.2 Simulation Plots from Chapter 5 225
7.3 Expressions for Parameter Variances and
Covariance in terms of Xl, X2 and X3 238
7.3.1 7.3.2 7.3.3 7.3.4
Variance of cO ••••••••••••••••••••••••••••• 238
Variance of t 239
Covariance between cO and .t : • • . • . • • • • • • • . • • 239
Variance of the Relative Yield Loss 240
7.4 Prograu'! Listings 241
7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.4.6 7.4.7 7.4.8 7.4.9 7.4.10 .7.4.11
RANGEN2 241
SIMLINn, n=O, 1, , 9 243
EDITSIM 24 4
ANLYZE 245
MPAPER 247
CONF 25t
CONFCNT 253
VARASSIG 254
ROBUSTX 256
ROBUST 259
This research will address the problem of optimum allocation
of design points when the Weibull model is assumed to be the
appropriate model. The motivation for this investigation
was to efficiently estimate the relative crop yield loss
over a range of ozone pollutant levels. Three criteria will
be used to compare designs: (1) the minimization of the
generalized variance of the three model parameters (FDET),
(2) the minimization of the generalized variance of a subset
of the model parameters (SDET) and (3) the minimization of
the variance of the estimated relative yield loss.
The dissertation takes the form of four independent
papers to be submitted for publication (Chapters 2 thru 5).
Chapter 2 examines the problem of experimental design
assum-ing a Weibull model with known parameter values when the
dose scale is unrestricted. Chapter 3 re-examines the
prob-lem under the constraint of a restricted dose scale. The
robustness of the design strategies developed in Chapters 2
and 3 is considered in Chapter 4. The model parameters are
now subject to being incorrectly guessed. Finally, Chapter
5 is a simulation study to investigate the behavior of the
Wald estimates of the parameter variances and in particular,
to determine the validity of using the Wald estimates as a
2 Experimental Design for the Weibull Function as a Dose Response Model Assuming an Unconstrained Dose Scale
2.1 Introduction
The National Crop Loss Assessment Network (NCLAN), a
consor-tium of research teams, has the goal of developing ozone
dose-plant response relationships for economically important
crop species with the specific purpose of estimating the
reduction in crop yields due to ozone (03) pollution (Heck,
et al., 1984). Models relating 03 concentration to crop
yields are required to quantify the impact of present 03
concentrations and to assess the benefits of alternative
national 03 pollution standards. NCLAN has found the
Weibull nonlinear model to be a useful functional form for
characterizing the response of crop species to 03 (Rawlings
and Cure, 1985).
This paper addresses the problem of optimum allocation
of design points when the Wei bull model is assumed to be the
appropriate model and an unconstrained dose scale is used.
Primary interest is in choosing design points that will
maximize the precision of the estimated relative yield
losses over a range of 03 pollutant levels, to simulate
different 03 pollution standards. Alternatively, designs
which are optimum (with regard to some criterion) for model
2.2 Literature Review
Several criteria have been used for choosing optimum
experi-mental designs with respect to parameter estimation for
non-linear models. Minimizing the determinant of the
variance-covariance (V-C) matrix, that is, D-optimality (see
Stein-berg and Hunter, 1984), is common. A justification for the
use of the criterion can be found in (Atkinson and Hunter,
1968; Box, 1971; Box and Draper, 1971; Box and Lucas, 1959;
Cochran, 1973; Nalimov, Golikova and Mikeshina, 1970).
Properties of the criterion are that it minimizes the
gener-alized variance of the parameter estimates, it is invariant
to changes of scale of the parameters, and it minimizes the
volume of the approximate confidence region for the
param-eter estimates assuming the response functions are
approx-imately linear in the vicinity of the least square
esti-mates. Under a Bayesian framework, the determinant
crite-rion maximizes the joint posterior probability density of
the estimates assuming a joint locally uniform prior
distri-bution for the parameters and the response function is
locally linear in the vicinity of the estimates (Cochran,
1973). Kiefer (1961), using the concept of a design
mea-sure, proved a general equivalence theorem in which a design
which minimizes the determinant of the V-C matrix will also
minimize the maximum variance of any predicted value along
Chernoff (1953) used as a criterion the minimum average
of the asymptotic variances of the maximum likelihood
esti-mates of the parameters of primary interest. He showed
there is an upper limit on the distinct number of design
points needed. This criterion is not invariant under scale
changes of the parameters (Box and Draper, 1971; Hamilton
and Watts, 1985).
The criterion of Ehrenfeld (1955) minimizes the maximum
variance of the parameter estimates. This is accomplished
by finding a design which maximizes the smallest nonzero
characteristic root of the inverse of the
variance-covariance matrix. This criterion is also not invariant
under scale changes of the parameters.
Box and Lucas (1959) looked for an optimum design when
the number of design points, N, equals the number of
param-eters, p. Using the determinant criterion they noted the
fact that this allows no testing of the fit of the model and
does not provide any "insurance" against poor guesses of the
parameter values.
Atkinson and Hunter (1968) looked at an experiment
including N>p design points. They established conditions
for the determinant criterion under which, when N is a
mul-tiple of p, replication of the optimum design using p points
Box (1971) considered finding an optimum design when a
subset, s, of the parameters are of interest, and the
remaining (p-s) are nuisance parameters. His criterion can
be interpreted as minimizing the generalized variance of the
parameters of interest. Box notes this may be more readily
obtained by first determining the V-C matrix of all
param-eters being estimated, partitioning the matrix to give the
V-C matrix of the subset of parameters of interest and then
minimizing the determinant of the submatrix.
M. J. Box and Draper (1971) and Box and Lucas (1959)
give some disadvantages of using the determinant of the V-C
matrix as a criterion. They point out this approach is a
"variance criterion" and effectively assumes the model
con-sidered is the true model. When observations are subject to
error, discrepancies between the fitted model and true
re-sponses occur because of sampling or "variance" error and
due to the inadequacy of the fitted model to represent the
true model, that is, "bias error" (G. E. P. Box and Draper,
1959). G. E. P. Box and Draper (1959) and M. J. Box and
Draper (1971) point out that bias error from using an
incor-rect model can be expected to have a greater effect on the
choice of design points than does variance error caused by
an unrestricted design. However, in situations where the
design is physically restricted to a small region of
the best design protecting against incorrect specification
of the model, and the best design for protecting against
variance error is often not large. They state the
determi-nant criterion is not unrealistic when the model is correct
or when the design is restricted to the region of interest,
or both.
Box (1971) and Cook, Tsai and Wei (1986) consider
parameter estimate bias in nonlinear regression. Parameter
estimate bias is not to be confused with model error or
bias. Box (1971) gives a way to quantitatively estimate the
expected biases in parameter estimates for a nonlinear
model. The expression he derives for the bias is a function
of the V-C matrix of the parameter estimates. He concludes
a design which minimizes the generalized variance will' tend
to minimize the expression of bias. His view is that a
design criterion minimizing the generalized variance is
preferable to one minimizing a measure of the bias. Since
the biases can be computed prior to running the experiment,
the parameter estimates can be corrected for their biases.
All criteria discussed assume a local linear
approx-imation to the nonlinear model. The appropriateness and/or
consequences of the assumption for a general nonlinear model
are discussed in Bates and Watts (1980, 1981), Box (1971),
Box and Hunter (1962, 1965), Clarke (1980), Cook and Tsai
(1985), and Vila (1987). In what follows it will be assumed
the locally linear assumption is appropriate. This is
investigated more fully in Chapter 5 for the Weibull model
of interest in this study.
2.3 The Weibull Model
The basic Weibull response model has the form (Rawlings and
Cure, 1985)
Y(x; a.w.1) - aexp(-(x/w).\) + E - aH(x; w.1)+ E
(2.1 )
where Y is the yield and x an appropriate measure of Os
exposure. The exponential part of the model, H(x; w ,.\),
characterizes the proportional yield remaining at dose x. A
parameter ~ allows for actual rather than proportional yield
levels; ~ is the hypothetical maximum yield at zero 03 dose.
The parameter w represents the dose at which the yield is
reduced to O.37~. The parameter .\ is a dimensionless
param-eter controlling the rate at which damage is incurred, that
is, the shape of the response curve. For example, if .\=1,
the Weibull function is the exponential decay curve with the
relative rate of yield loss being 1/w for all x. Values of
.\>1 cause the relative rate of loss to increase with x, with
the relative rate of increase in the relative rate of loss
being (.\-l)/x. This causes the impact of the pollutant to'
be increasingly concentrated at x= w as A. increases; the
of ~ (~>1) give an increasingly distinct plateau appearance
to the response curve. For a more thorough discussion of
the Weibull model see Rawlings and Cure (1985) or Johnson
and Kotz (1970).
2.4 Variances of Estimates
The Weibull model is nonlinear in its parameters and
requires the use of nonlinear least squares to estimate the
parameters. Estimates of the parameters and their
variance-covariance matrix can be characterized as linear and
quadratic forms which are similar in appearance to those
occurring in linear regression to an error of approximation
which becomes negligible in large s~mples (Gallant, 1975;
Gallant, 1987; Malinvaud, 1966). Let v be the parameter
vector of true values (~, w ,~) and denote the matrix of
partial derivatives of (2.1) with respect to the j t h
param-eter and evaluated at the i th experimental point and v by
(2.2)
i
=
1, 2, ... , nj
=
1, 2, 3where i is the row index and j is the column index. It has
been shown (Gallant, 1975; Gallant, 1987) in large samples
that the random vector Y, the estimated parameter vector,
where q2 is the error variance. The V-C matrix is estimated
by substitution of
v
for v and an estimate of q2.The derivatives of the given Weibull model are:
OY /
oa"
H(x;w,A.)lJY /lJw -a[H(x;w,A.)](A./w)(x/w)A.
lJY / lJA. - -a[H(x; w, A.)](x /w)A.[ln(x / w)]
(2.3)
2.5 Estimated Relative Yield Loss (RYL)
Let XO be the base level of 03 from which the yield losses
are to be measured, and let Xr be the postulated new level
'of 03. Then the estimated relative yield loss (RYL) is
(2.4)
where the H() functions are evaluated for a given estimate
of the parameters wand A.. The asymptotic variance of the
estimated relative yield loss is given by
V(RYL)
=
(H(x r)/H(
x o))2{D 2 V (X) + E2V(w) - 2DECov(X.w)}
(2.5)
where
D=A(xr)-A(xo )
A(x()= (x/w )1
1n (x/w)
E=B(xr)-B(x
o )This can be written as
V(RYL) - a "V(a.w.l)a
where
and
2.6 H-equivalent Transformation and Optimization Invariance
Primary interest is in determining optimum design points so
as to efficiently estimate V(RYL) assuming the Weibull model
is the appropriate dose response model. Conceptually this
entails finding optimum dose levels (x) for all combinations
of the parameters. In this section i t is shown that, with
the appropriate transformation, the optimum design using
either determinant criterion, for one choice of parameters,
can be translated to any choice of parameters. In addition,
optimization with respect to either determinant criterion is
invariant with respect to scale transformations on a, and a
is a nuisance parameter with respect to the subdeterminant
criterion of Box (1971).
The H-equivalent transformation of H(x; w ,~) (see
scale x to z and parameters v to v* such that H(x; v)=H(z; v'~
From the functional form of H(x; w ,~) i t is evident that
H(X;W,A)- H(Z;W-,A-)
if
P - p
-z - k X . W - k W • and A - AI P
(2.6)
for k>O, p>O. Examples of H-equivalent transformations are
given in Table 2.1.
For generality, a scale transformation was allowed on
4. 4*=d4. This is interpreted as a scale transformation on
Y to Y*=dY so that there is a change in error variance ~*2
::: d2 (la, Under H-equivalent transformation and scale
trans-formation on 4, the transformed model is
_
_
((
8)1.·.
Yt - a exp - Ztlw + E t •
The asymptotic variance matrix for a*,
w*,
1* is(-- -- --) [ (
-) (
_)]-1_2
V
a
, ( 0 ,A = F' Z,V F Z,V (Jwhere
is the matrix of partial derivatives of Y* with respect to
each of the parameters evaluated at v*' = (4*, W *J 1.*) and
Zi·. i=l, 2, ... , n. Each of the partial derivatives with
respect to G*, w*, and 1.* can be expressed in terms of the
o
Y[d
.(!--l)(
1)
I IPJ
oy·/cw··-
-w p-oW
p . kThus,
v(
a
a,
W
a
,1*) •
[F'(
z. va)
F
(z.
Va)
rid
2(J2-KV(a.w.X)Kd2
where
Therefore,
V(a:.w·
.X·).
(
k ) lip
dpw· w. C(a..w)
( )
2/P
P2W ·2 ; . V
Cw)
~C(a..X)
p
(
k ) lip
w· w.
CCw.X)
~V(X)
p
(2.7)
Note that the elements of K do not depend on the choice of
design points. It follows that the choice of design points
determinant criterion will generate the optimum design for
any 4*, W*, ~* by using H-equivalent transformations.
Thus, finding the optimum design for one choice of
param-eters suffices. The preceding shows an invariance with
respect to optimum design by the determinant criterion
ex-ists for the Weibull model for the power and scale
transfor-mations on w.
This (equation 2.7) shows that rescaling 4 does not
affect the variances and covariance of cD and t. Therefore,
4 can be treated as a nuisance parameter with arbitrary
choice of 4 for cri teria involving only variances of riJ, X.
Finally, under H-equivalent transformation, the
first-order approximation of the variance of the estimated
relative yield loss, given by (2.5), remains constant.
First, recall that
Therefore,
H(x r ) H(zr)
-H(x o) H(zo)
Secondly, i t will be shown that the first-order
(x)A. (x)
[(Z/k)l/PJPA.'
[(Z/k)l/PJ
A (x ) ...
w
Inw
...
(w .. /
k )1/ P In (w .. /
k)1/P1.'
-
~(;.)
In(;.)-
~A(Z)
Therefore,
Also,
Therefore,
B(
x
r) -
B(
x
0) -
pk
IIpw
*(l-l/P )[B(
Zr) -
B(
Z0)]
Substituting the above relationship between H(x) and H(z),
A(x) and A(z), and B(x) and B(z) and variances (2.7) into
(2.5)
-2[ A (zr) - A (z .)][B(zr) - B(z.)] Cov(cD0 •
x.
oJR OO 0)
* V( YL;z.a. .w .Ao
This implies that the H-equivalent transformation
leaves the variance of the estimated relative yield
un-changed as long as the transformed values are at the same
locations along the Weibull curve with respect to H(x; w ,~).
All results on variances of
a,
w,
1 and RYL can bedeter-mined from the results for anyone choice of a, w, and ~ as
long as the set of H(Xii W )~) are the same for the different
designs. Further, the comparison of design point
alloca-tions need to be done only for one choice of a, w, and ~ as
long as the point specifications and estimation interval are
in terms of Hex). Once an optimum allocation of design
para-metric situation, ie.,the design minimizes V(RYL), the
results can be translated to other parametric cases via
H-equivalent transformations.
2.7 Design Development
All design strategies in this Chapter assume the true
param-eter values are known. For the initial phase of
investiga-tion the experimental designs were explored for parameter
values of ~=w=~=l (the H-equivalent transformation of the
x's would be used to find the optimum design for other
choices of ~ and w). Since only relative variances are
considered, i t was assumed q2=1. Primary interest was in
choosing a design that would maximize the precision of the
estimated relative yield losses over a range of dose levels.
Two approaches were considered: (1) using the determinant
criterion to obtain a design which minimizes the generalized
variance of the parameter estimates, (2) using numerical
methods to find a design which minimizes V(RYL) for a given
estimation interval.
The grid search used to find the minimizing design
restricted x=.OOOl as the minimum value allowed to avoid
using In(O) in the evaluation of the determinants. No upper
limit was placed on the possible values of x. Four
estima-tion intervals (PY) were used for which V(RYL) were
deter-mined. Each represents a change in H(x) of 0.20. The
0.80 to 0.60 (PY86), from 0.65 to 0.45 (PY64), and from 0.55
to 0.35 (PY53). A total of 24 design points were used so
that criteria are directly comparable over choices of
num-bers of distinct design points (GNT).
2.7.1 Parameter Estimation (Determinant Criterion)
Initially a design was determined which was optimum with
respect to the determinant criterion when none of the three
parameters was considered to be a nuisance parameter. A
grid search procedure was used to find an optimum three
point design. The Weibull model satisfied the sufficient
conditions in Atkinson and Hunter (1968) guaranteeing that
replication (r) for the optimal three point design by the
determinant criterion would be optimal when the total number
of design points is 3r.
When ~ was treated as a nuisance parameter, the method
of Box (1971) was used and a design found to minimize the
determinant of the sub-matrix corresponding to wand t of
the full V-C matrix. For both, the results were rounded--to
two decimal places for the x-values and four decimal places
for V(RYL) and values of the determinants--before recording
them in the appropriate table.
2.7.1.1 Results from Optimum Three-point Design
The grid search, gave the optimum three-point design for the
corre-sponding to H(x)=0.9999, 0.703 and 0.192, respectively. The
first section of Table 2.2 gives the value of the
determi-nant of the V-C matrix (FDET) as well as the determidetermi-nant of
the submatrix (SDET) corresponding to wand X.. The vahl.ee
of V(RYL) are shown for each of the four estimation
inter-vals.
Treating « as a nuisance parameter, a grid saarch was
used to find the design points which would minimize SDET,
Box's (1971) criterion. Since three parameters are being
estimated the minimum number of distinct points needed is
three. The three-point design that was found to minimize
SDET was 0.0001, 0.42 and 1.74 corresponding to H(x)=0.9999.
0.657 and 0.1755, respectively. Comparing the best
three-point design for the SDET criterion to the best
three-point design for the FDET criterion (Table 2.2), two
things have occurred. First, minimizing SDET has slightly
increased FDET (2%). Secondly, V(RYL) has decreased
slight-,
ly (5%) for the two PY of prim3ry interest and trivially
increased (.4% and 2.5%, respectively) for the other two.
2.7.1.2 Results from Unconstrained Four-point Search, FDET Criterion
Two four-point designs were sought, one to minimize FDET and
a second to minimize SDET, among all four-point designs.
Since replicating a three-point design is optimum when N=24
four-point designs will measure the loss of efficiency
resulting from adding another distinct x-value. A grid
search was used to find an allocation of two design points
between the end points of the optimum three-point design
minimizing FDET. The grid search always converged to a
design in which one of the two intermediate points coincided
with the center point for the optimum three-point design
(0.35) and the other duplicated one of the three points
depending on the initial starting values (Table 2.3). Thus,
attempts to find optimum four-point designs always led to
unequal replication of the optimum three-point design. The
upper half of Table 2.3 summarizes the results for the three
three-point designs with unequal replication found from
attempts to find results for the four-point designs that
minimized FDET. While the three three-point, unequal
repli-cate designs are equally efficient with respect to FDET,
they are not as efficient as the optimum three-point design
with equal replication. The three-point unequal replicate
designs are not equally efficient with respect to SDET and
V(RYL) for each PY.
For NCLAN, interest would be in PY97, implying that
replication of x=0.35 would be best with regard to V(RYL).
This three-point, unequal replicate design has increased
FDET by 20%. However, there is still no way of testing
The design also shows that unequal replication of the three
optimum design points is not as efficient with respect to
FDET as equal replication of the same points.
2.7.1.3 Results from Unconstrained Four-point Search. SDET
Criterion
A grid search was also used to find an allocation of two
design points between the ends of the three-point design
minimizing SDET, such that the four-point design would
mini-mize SDET among such designs. As for the FDET criterion,
the result from the search for a four-point design with
respect to SDET consisted of the three original points and
one duplicate (Table 2.3). However, only one
combina-tion,(Xl,X2,X3,X4)=(O.OOOl,O.42,1.74,1.74), corresponding to
H(x)=O.9999, O.657,and 0.176, respectively, gave a minimum
value of SDET. The bottom half of Table 2.3 summarizes this
three-point, unequal replicate design, as well as the
results of duplicating one of the other two points. While
replicating x=1.74 is optimum with respect to minimizing
SDET, replicating x=0.42 gives smaller variance of RYL for
the PY of interest to NCLAN than does either the optimum
three-point or three-point, unequal replicate design. Also,
while equal replication of the three-point design was
opti-mum when estimating all three parameters, unequal
replica-tion of the three-point design by the SDET criterion gave a
equal replicate design, obtained by treating " as a nULsance
parameter, is not as efficient with respect to the SDET
criterion as the three-point, unequal replicate design by
the- SDET criterion.
2.7.1.4 Results from Search for Six-point Design, SDET Criterion
A grid search was then used to find the optimum six-point
design to determine whether i t would be an improvement on
the optimum three-point, unequal replicate design with
respect to SDET. Because of the magni tude I~f the search i t
was not as thorough as those for the three- and four-point
designs. Initial coarse searches seemed to indicate that
the optimum settings occurred in an area containing the
original optimum three-point design. SDET was evaluated for
the design . 0001, . 0001, .42, .42, 1.74 and 1.7-1 and the
location of individual points systematically shifted. In
this way i t was concluded the optimum six-point design was
the starting design. The results are not given separately
in Table 3 since they are the same as for the three-pc.int
design. Given the designs investigated, the design
millimiz-ing SDET for estimatmillimiz-ing wand 4 while treating CI as a
nuisance parameter was an unequal replication of the betit
2.7.1.5 Results from Constrained Four-point Search
Two different allocation strategies were used to force four
distinct design points. The "Equal X" (EQX) strategy placed
the x-values at equally spaced intervals on the dose scale
between the end points from the optimum three-point designs
by the FDET and SDET criterion. The "Equal H" (EQH)
strat-egy allocated the design points by finding the values of x
such that the H(x) values were equally spaced between
H(O.OOOl) and H(1.65) (FDET criterion), or between H(O.OOOl)
and H(1.74) (SDET criterion). For example, the design
points might be at the dose levels giving
H(Xl)=H(O.OOOl)=O.9999, H(X2)=O.73, H(X3)=O.46 and
H(x4)=H(1.65)=O.lg.
The results are given in Table 2.4. The EQH strategy
with the end points determined by the FDET criterion
increased the value of FDET 6% over the value obtained using
the three-point, unequal replicate designs. V(RYL) for the
intervals more typical of NCLAN research, PY97 and PY86 , has
decreased by 4% and 27%, respectively, relative to the
design replicating x=O.35. When compared against the
opti-mum three-point design, FDET has increased 25%, but the
values of V(RYL) have decreased 8% and 15%. Despite the
four-point design using the EQH strategy being less
effi-cient for estimating ~, w, and 1 than the optimum
designs, it allows for a lack-of-fit test, which the
three-point designs did not, and the distribution of the
design points is more efficient for estimating V(RYL) in the
region of ~nterest.
Repeating the use of the EQH and EQX strategies when
the end points are determined by the SDET criterion, the EQH
strategy was more efficient than the EQX with respect to
values of SDET and V(RYL) for PY97. Compared against the
optimum three-point, unequal replicate design by the SDET
criterion (Table 2.3), the value of SDET has increased 21%,
while V(RYL) for PY97 and PY86 have decreased by 24% and
10%, respectively. The results obtained using the EQH
strategy or the EQX strategy between the two sets of end
points (that is, (.0001,1.65) and (.0001,1.74», differ
trivially.
2.7.1.6 Conclusion
In Section 2.7.1 different ways of finding experimental
designs were used to estimate the parameters of the
nonlin-ear Weibull model. As a by-product, designs were compared
with respect to V(RYL) for PY of interest. In Section 2.6
it was proved that if an optimal design (by the FDET or SDET
criterion) is found given any values of a, w, and ~, with
be optimal (with respect to the same criterion) for all
values of 4 , wand 1 both in terms of estimating the
param-eters and maintaining a constant value of V(RYL).
The results of Tables 2.2-2.4 re-emphasize the point
that finding the optimum design with regard to parameter
estimation need not give the optimum design for predicting
responses along the curve. The two are not unrelated. In
scanning the tables designs which do a poor job of
estimat-ing the parameters tend to be deficient in estimatestimat-ing
rela-tive yield loss. Whether considering all three parameters
to be of interest or only wand A., the optimum design for
estimating V(RYL) (relative to Tables 2.2-2.4) compared to
the optimum design for estimating the parameters caused a
25% and 21% increase, respectively, in the value of the
relevant determinant (that is, FDET and SDET). Comparing
the values of x from the two determinant criteria seems to
indicate a shift to the right in the design points has
improved V(RYL) for the two rightmost PY, PY64 and PY53,
while increasing V(RYL) for the other two. Likewise, a
shifting of the design points to the left causes the reverse
to occur. As the design points are shifted toward or away
from a particular PY we would expect V(RYL) to decrease or
increase accordingly. Given the four PY and the results in
all four simultaneously. Therefore, the concern is to focus
on the two PY more relevant to NCLAN while trying to prevent
V(RYL) for the other two from becoming "too unreasonable."
2.1.2 Kstimation of Relative Losses
Having verified that the criterion of Atkinson and Hunter
(1968) is satisfied by the Weibull model, equal replication
of the three-point design satisfying the FDET determinant
criterion is optimal for 24 points. For this reason, the
search for designs to minimize V(RYL) for a given PY were
restricted to designs containing three points so that they
would be comparable to the optimum designs of the previous
section. Denote this three-point design by the generic sets
of points X=(Xl,X2,X3). Using the algebraic programming
system REDUCEl to perform calculations with symbolic
matri-ces, the estimated V-C matrix from Section 2.4 was expressed
in terms of (Xl,X2,X3). It was apparent from the
expres-sions for V(
w ),
V(1) and Covlw ,.1) that they could not beindividually minimized by the same design. The components
corresponding to V(
w),
V( t) and Cov(w
,1) were substitutedinto (2.5) giving V(RYL) expressed as a function of
(Xl,X2,X3). Using a general optimization scheme, the
Sim-plex Method (NeIder and Mead, 1965), the three-point design
which minimized V(RYL) for a given PY was found. This was
repeated for each PY. Designs providing efficient estimates
of V(RYL) for the four PY were (1) PY97, x=(.OOOl,.5,.69),
(2) PY86, x=(.OOOl,.85,1.1), (3) PY64, x=(.05,l.23,1.72),
and (4) PY53, x=(.16,1.53.2.0). Slight perturbations from
the optimum design points resulted in small changes in the
value of V(RYL). (This is important because of calibration
restrictions on the equipment used in the ozone field
stud-ies.)
2.7.3 Comparison of Designs from Parameter Estimation and Estimation of Relative Losses
The values of V(RYL) for the PY of interest were evaluated
at each "optimum" design and the results summarized in Table
2.5. For the specific estimation interval being
investi-gated, the variance could be approximately halved by using
the design obtained for interval estimation compared to the
variances obtained for the PY using the parameter estimation
optimum design. However, the variances of the other PY
increased sharply; they were much worse than the values
obtained using either determinant criterion design. In each
case the variance for the PY, evaluated at the estimation
interval design, was approximately .5 to .6 the size of the
variance obtained using the optimum three-point design for
the estimation of the parameters of the Weibull (FDET
crite-rion). The variances for the other PY, however, are many
three-point design. Thus, the preferred strategy when there
is interest in a range of PY, is to use a design point
allocation based on optimum estimation of the parameters of
the response function rather than on estimation of loss for
a specific interval.
2.8 Conclusion
This paper has addressed the problem of determining an
allo-cation of experimental design points when the Weibull model
is assumed to be the appropriate model. The goal of the
experiment (with respect to NCLAN) is to efficiently
esti-mate the relative yield loss over a range of 03 pollutant
levels. Three criteria were used to compare designs: (1)
the minimization of the generalized variance of the three
model parameters (FDET), (2) the minimization of the
gener-alized variance of the parameters w and ~ (SDET) and (3)
the minimization of V(RYL) for a given estimation interval.
The work of Atkinson and Hunter (1968) allowed the scope of
the FDET criterion to be restricted to investigating equal
replication of three-point designs. The H-equivalent
trans-formation was developed such that a design optimum by either
the FDET or SDET criterion for one choice of parameters can
be translated to an optimum design for any choice of
Designs were investigated which were optimum for model
parameter estimation (FDKT and SDKT criteria) and as a
by-product, values of V(RYL) recorded. These designs were
compared against designs which minimized V(RYL) for a given
estimation interval. The dose scale was assumed to be
unre-stricted, and particular attention given to PY in an area
along the response curve relevant to NCLAN.
While three-point designs were shown to be
theoretical-ly optimal for parameter estimation, the desire to test for
model lack-of-fit requires a minimum of four distinct points
be used. The search for four-point designs minimizing FDKT
and SDKT, and using the left and right end points from the
corresponding optimum three-point designs, led to unequal
replicates of the three-point designs. A three-point,
equally replicated design minimized FDET, however, among the
designs considered, a three-point, unequally replicated
design minimized SDKT. Two strategies, EQX and KQH, were
then used to force the allocation of four distinct design
points. These designs were not as efficient as the optimum
three-point designs with respect to values of FDET and SDKT,
however, the EQX and KQH designs provided more efficient
estimates of RYL for the low dose Py's, PY97 and PY86.
A search was also made for designs which would
effi-ciently estimate V(RYL) for a given PY. The resulting
approximately one half as large as the corresponding
vari-ance obtained from using a design chosen for its ability to
estimate the model parameters. However, the variances of
the other PY increase sharply (Table 2.5) and are less
effi-cient than the values obtained using either determinant
criterion design. The preferred strategy when there is
interest in a range of PY, is to use a design point
allo-cation strategy based on estimating the parameters of the
response function rather than for estimation of loss for a
specific interval. The estimated variances will not be
min-imized, but acceptable values can be obtained across
estima-tion intervals. Finding an optimum design with respect to
parameter estimation need not give an optimum design for
predicting a response along the curve. For a specific
esti-mation interval, the variance could be approximately halved
by using the design obtained for interval estimation
com-pared to the variance obtained using the parameter
estima-tion optimum design. However, the variances of other
esti-mation intervals increase sharply, to values larger than
obtained using either determinant criterion design. Given
the four PY and the results in the tables, no design will
minimize V(RYL) for all four simultaneously. Therefore,
concern should be focused on the PY of primary interest,
while trying to prevent V(RYL) for the other estimation
In determining the allocation of design points this
work indicates that the left and right end points should be
chosen so that Xmi n is as c lose to zero and Xma x
=
w ( 1. 7)1/1-as is practically possible. A value of 1.7 was chosen as a
compromise between the right end points found for the
opti-mum FDET and SDET designs, 1.65 and 1.74, respectively. Two
methods, EQX and EQH, were used to allocate design points
between the end points. Although estimates of V(RYL) for
PY97 were more efficient using EQH (19%), EQX is more
effi-cient for all other PY and its strategy is less dependent on
having correctly specified the model parameters. For these
reasons, EQX would be the recommended strategy.
This paper has only considered designs containing three
or four distinct points and assumes the parameters are
known. Chapter 4 examines the robustness of the EQH and EQX
strategies to mistakes in the model parameters as well as
the effect of increasing the number of distinct design
2.9 REFERENCES
Atkinson, A. C. and W. G. Hunter. (1968). The design of experiments for parameter estimation. Technometrics 10, 271-289.
Bates, D. M. and D. G. Watts. measures of nonlinearity. tical Society B 42, 1-25.
(1980). Relative curvature Journal of the Royal
Statis-Bates, D. M. and D. G. Watts. (1981). Parameter transfor-mations for improved approximate confidence regions in nonlinear least squares. Annals of Statistics 9,
1152-1167.
Box, G. E. P. and N. R. Draper. (1959). A basis for the selection of a response surface design. Journal of the American Statistical Association 54, 622-654.
Box, G. E .P. and W. G. Hunter. (1962). A useful method for model-building. Technometrics 4, 301-318.
Box, G. E. P. and W. G. Hunter. (1965). The experimental study of physical mechanisms. Technometrics 7, 23-42.
Box, G. E. P. and H. L. Lucas. (1959). Design of experi-ments in.nonlinear situations. Biometrika 46, 77-90.
Box, M. J. (1971). An experimental design criterion for precise estimation" of a subset of the parameters in a nonlinear model. Biometrika 58, 149-153.
Box, M. J. (1971). Bias in nonlinear estimation. Journal of the Royal Statistical Society B 33, 171-201.
Box, M. J. and N. R. Draper. (1971). Factorial designs, the :X'X: criterion, and some related matters. Techno-metrics 13, 731-742.
Chernoff, H. (1953). Locally optimal designs for estimat-ing parameters. Annals of Mathematical Statistics 23, 586-602.
Clarke, G. P. Y. (1980). Moments of the least squares es-timators in a non-linear regression model. Journal of Royal Statistical Society B, 42, 227-237.
Cook, R. D. and C.-L. Tsai. (1985). Residuals in nonlinear
regression. Biometrika 72, 23-29.
Cook, R. D., C.-L. Tsai and B. C. Wei. (1986). Bias in
nonlinear regression. Biometrika 73, 615-623.
Ehrenfeld, S. designs.
(1955). On the efficiency of experimental
Annals of Mathematical Statistics 26, 247-255.
Gallant, A. R. (1975). Nonlinear regression. The American
Statistician 29, 73-81.
Gallant, A. R. (1987). Nonlinear statistical models. John
Wiley
&
Sons, Inc. New York.Hamilton, D. C. and D. G. Watts. (1985). A quadratic
design criterion for precise estimation in nonlinear
regression models. Technometrics 27, 241-250.
Heck, W. W., W. W. Cure, J. O. Rawlings, L. J. Zaragoza, A. S. Heagle, H. E. Heggestad, R. J. Kohut, L. W. Kress and
P. J. Temple. (1984). Assessing impacts of ozone on
agricultural crops: I. Overview. Journal of the Air
Pollution Control Association 34, 729-735.
Hougaard, P. (1985). The appropriateness of the asymptotic
destribution in a nonlinear regression model in relation
to curvature. Journal of the Royal Statistical Society
B 47, 103-114.
Johnson, N. L. and S. Kotz. (1970). Continuous univariate
distributions-1. Houghton Mifflin Co. Boston
Kiefer, J. (1961). Optimum designs in regression problems,
II. Annals of Mathematical Statistics 32, 298-325.
Malinvaud, E. (1966). Statistical methods of econometrics.
Rand McNally and Company. Chicago.
Nalimov, V. V., T. I. Golikova and N. G. Mikeshina. (1970).
On practical use of the concept of D-optimality.
Tech-nometrics 12, 799-812.
Rawlings, J. O. and W. W. Cure. (1985). The Weibull
func-tion as a dose-response model to describe ozone effects
on crop yields. Crop Science 25, 807-814.
REDUCE. (1984). The Rand Corporation. Copyright 1984.
Steinberg, D. M. and W. G. Hunter. (1984). Experimental
Table 2.1. Translation of design points, x, on the standard-ized scale of ~-w-~-l to design points, z, on the scale /1··1, (,0. and ~.. The translation is
z.w -(xlW )AJ'A·
~·l,w·l,A·l
Xl X2 X3
.0001 .35 1.65
.01 .42 1. 74
.02 .60 1. 38
Zl Z2 Z3
-
- LA - - 2 (.0001)1/2 • •01 r35 ~l .65w
•
1 • .5(.01)1/3 • •108 .37 .60w - -,A - 3 2
•- 2 tA·. 4- 2(.02)1/• • . 75 1. 76 2.17
Table 2.2. Optimum choices of design points for ~-w-~-l
using the minimization of the determinant of the variance-covariance matrix (FDKT) and submatrix of the variance of cD and 1 (SDET) as the design criterion for N=24 experimental units. The vari-ances of the estimated relative yield losses
(V(RYL» are also given.
Criterion x-values FDET SDET PY V(RYLl
min(FDKT) .0001, .35,1.65 .1334 1.6334 PY91 .0603
PY86 .0411
PY64 .0834
PY53 .1522
min(SDET) .0001, .42,1. 14 .1366 1.5966 PY91 .0569
PY86 .0446
PY64 .0831
Table 2.3. Unequal replication of the optimum three-point designs using the determinant of the variance-covariance (V-C) matrix (FDET) and the determi-nant of the submatrix of the V-C matrix for
w
and 1 (SDET) as the design criterion for N=24 experimental units. The variances of the esti-mated relative yield losses (V(RYL» are also given. II
CriterionI
min(FDET) x-values .0001, .35,1.65, 1.65 FDET .1581 SDET 1.4870-IT-
VCR'LL)I
PY97 .07671
PY86 .0462 PY64 .0576 PY53 .1057
min(SDET)
.0001, .35, .35, 1. 65
.0001,.0001, .35,1.65
.0001, .42,1. 74, 1. 74
Table 2.4. Allocating N=24 experimental units to 4 distinct design points between end points determined by the EDET criterion and between end points deter-mined by the SDET criterion using the Equal X
(EQX) and Equal H (EQH) strategies. Values of the determinant of the variance-covariance matrix
(FDET), the determinant of the submatrix of the V-C matrix for co and .\ (SDET) and the variance
of the estimated yield loss (V(RYL» are also given.
Criterion = FDET
...fL YCRYL)
Design Strategy
EQX
EQY
x-values
.0001, .55,1.10, 1. 65
.0001, .31, .77, 1.65
FDET
.2217
.1672
SDET
1.9695
1.7894
PY97 PY86 PY64 PY53
PY97 PY86 PY64 PY53
.0658 .0333 .0768 .1780
.0553 .0402 .0877 .1764
Criterion
=
SDETDesign
Strategy x-values FDET SDET ...fL YCRYL)
EQX .0001, . 58, 1. 16, .2250 1. 9470 PY97 .0693
1. 74 PY86 .0341
PY64 .0721 PY53 .1675
EQY .0001, .32, .80, .1657 1. 7490 PY97 .0558
1. 74 PY86 .0405
Table 2.5. Variance of the estimated relative yield loss (V(RYL» using three-point designs which mini-mized, in turn, the "generalized variance" of the three parameters (FDET), the "generalized vari-ance .. of the parameters wand A. (SDET), or gave
near-minimal values of V(RYL) for each of the specified estimation intervals (PY).
Minimized with respect to:
....fL []lEI. SDEI. fYll eYa6.. fIll f.Y5.a
PY97 .060 .057 .037 .287 1.009 4.489
PY86 .047 .045 .281 .028 .338 2.152
PY64 .083 .084 1.567 .54.7 .045 .420
PY53 .152 .156 3.314 1.634 .277 .076
The design points which gave the above variances are:
FDET: X
=
(.0001, .35,1.65)SDKT: X
=
(.0001,.42,1.74)PY97: X
=
( .0001, .50,0.69)PY86: X
=
(.0001, .85,1.10)PY64: X
=
( . 05, 1. 23 , 1. 72)3 Experimental Design for the Weibull Hodel as a Dose Response Function, Assuming a Constrained Dose Scale
3.1 Introduction
The National Crop Loss Assessment Network (NCLAN), a
consor-tium of research teams, has the goal of developing ozone
dose-plant response relationships for economically important
crop species with the specific purpose of estimating the
reduction in crop yields due to ozone 03 pollution
(Heck) et al.) 1984). Data from field experiments indicates
that 03 can reduce yields of important crop species and has
been estimated as being responsible for up to 90% of the
crop losses from air pollutants in the U.S. (Heagle, et al.,
1984; Heagle) et al.) 1979; Heck, et al., 1983; Heck, et
al., 1982). Models relating 03 concentration to crop yields
are required to quantify the impact of present 03
concentra-tions and to assess the benefits of alternative national 03
standards. NCLAN has found the Weibull nonlinear model to
be a useful functional form for characterizing the response
of crop species to 03.
The question of experimental design for nonlinear
models relevant to the biological sciences has had some
investigation. Currie (1982) considered experimental design
as it pertained to estimating the parameters of the
Michaelis-Menten function; Cobby, Chapman and Pike (1986)
investigated designs for estimating inverse quadratic
allo-cation of design points when the Weibull model is assumed to
be the appropriate model and a constrained dose scale is
used. Primary interest is in choosing design points that
will maximize the precision of the estimated relative yield
losses over a range of Os pollutant levels, to simulate
different 03 pollution standards. Alternatively, designs
which are optimum (with regard to some criterion) for model
parameter estimation can be considered.
The criterion used in this paper for choosing an
exper-imental design was the minimization of the determinant of
the variance-covariance (V-C) matrix of the model parameter
estimates (FDET). A discussion of the use of the criterion
can be found in (Atkinson and Hunter, 1968; Box, 1971; Box
and Draper, 1971; Box and Lucas, 1959; Cochran, 1973). For
designs in which the number of design points (N) is a
multi-ple of the number of parameters (p), Atkinson and Hunter
(1968) have established conditions under which equal
replication of the best p point design by the FDET criterion
is optimum for designs containing N points. Box (1971)
con-sidered finding an optimum design when a subset, s, of the
parameters are of interest, and the remaining (p-s) are
nuisance parameters. His criterion can be interpreted as
minimizing the generalized variance of the parameters of
matrix of all the parameters estimated and then minimizing
the determinant of the submatrix corresponding to the subset
of parameters of interest (SDRT).
3.2 The Meibull Model
The Basic Weibull response model has the form (Rawlings and
Cure, 1985)
Vex;
a, w,
1) -aexp(
-(x/w)"')+ € - aH(x;w.
1)+€(3.1 )
where Y is the yield and x an appropriate measure of 03
exposure. The exponential part of the model, Hlx;w.~),
char-acterizes the proportional yield remaining at dose x. A
parameter " allows for actual rather than proportional yield
levels; « is the hypothetical maximum yield at zero 03 dose.
The parameter tV represents the dose at which the yield is
reduced to 0.37". The parameter 1 is a dimensionless
param-eter controlling the rate at which damage is incurred, that
is, the shape of the response curve. Uor example, if "'=1,
the Weibull function is the exponential decay curve with the
relative rate of yield loss being l/w for all x. Values of
~>l cause the relative rate of loss to increase with x, with
the relative rate of increase in the relative rate of loss
being 1~-l)/x. This causes the impact of the pollutant to be
increasingly concentrated at x= tV as A. increases; the
rela-tive rate of loss at x= w being 1t.Iw. Increasing values of
the response curve. For a more thorough discussion of the
Weibull model see Rawlings and Cure (1985) or Johnson and
Kotz (1910).
The Weibull model is nonlinear in its parameters and
requires the use of nonlinear least squares to estimate the
parameters. In nonlinear least squares, the n x p matrix F
of partial derivatives of the model with respect to each
parameter and evaluated at each data point in turn plays the
role of the X matrix in linear least squares. Let v be the
parameter vector of true values (~.w.~I. In large samples,
the estimated parameter
v
has approximatevariance-covariance (V-C) matrix (Gallant, 1915j Gallant, 1981)
yea,
cD, 1).
a2(F '(v)F(V»-1where D2 is the error variance. The V-C matrix is
approx-imated by substitution of
v
for v and q2 for D2(3.2)
3.3 Estimated Relative Yield Loss CRYL)
Let xo be the base level of 03 from which the yield losses
are to be measured, and let Xr be the postulated new level
of 03. Then, the estimated relative yield loss (RYL) is
of the parameters wand A.. The estimated approximate
vari-ances of the estimated relative yield loss is given by
V(IfYL).
(H(x,.)/H(x
o))2{D 2V(l)+E2Yew) - 2DECov(1.w)}
(3.4)
where
D· A(x,,)- A(x
o)A(x,)- (x/w
)1
1n (xc/W)
E-B(xr)-B(xo)
B(xc)- (X/w)(xc
/W)l
3.4 B-Equivalent Transformation
In Chapter 2 it was shown that V(RYL), for a given
estima-tion interval, is not affected by scale transformaestima-tions on l
nor by scale and power transformations on w when the
x-values are transformed by the H-equivalent transformation
of the Weibull function. H-equivalent transformation of
H(~:w.~) is the simultaneous transformation of the dose scale
x to z and parameters v to v' such that H(x;v):H(z;II·). This
requires z=kx P, w*=kw P and A.*=l/p for k>O, p>O. For
gen-erality, a scale transformation on a is allowed, ~*=da.
This is interpreted as a scale transformation on Y to Y*=dY
60 that there is a corresponding change in error variance 0':1
=d2a:l. Under H-equivalent transformation and scale
All results on variances of
a.
wand 1 and RYL can bedetermined from the results for anyone choice of ~. wand 1
as long as the set of H(Xliw.~) are the same for the
differ-ent designs. Further, the comparison of design point
allo-cations needs to be done for one choice of ~. ~. w as long as
the point specifications and estimation intervals are in
terms of H(x). Once an optimum allocation of design points
is found for one parametric situation, that is, the design
minimizes V(RYL), the results can be translated to other
parametric cases via the H-equivalent transformation.
3.5 Design Development
Optimum experimental design for an unrestricted dose scale
was investigated in Chapter 2. It was shown the preferred
design strategy for estimating yield loss when there is
interest in several prediction intervals, was to use a
design point allocation scheme based on optimum estimation
of the parameters of the response function rather than on
estimation of loss over a specific dose interval.
Minimiza-tion of the determinant of the V-C matrix (to be referred to
as the determinant criterion) was used for choosing
experi-mental designs with respect to parameter estimation
(Atkin-son and Hunter, 1968; Box, 1971; Box and Draper, 1971; Box
and Lucas, 1959; Cochran, 1973; Nalimov, Golikova and
This paper is concerned with designs that take into
account a restricted dose scale more consistent with the
NCLAN experiments. The lowest concentration of ozone that
can be attained under field conditions of the NCLAN
experi-ments is approximately 0.02 ppm. The highest seasonal
aver-age concentrations attained under the most severe treatments
were approximately 0.15 ppm; the maximum concentration
var-ied over experiments from approximately 0.10 to 0.18 ppm.
The upper limits were restricted to these levels by the
increased cost of maintaining higher ozone concentrations,
and the natural reluctance of the biologists to use
experi-mental treatment levels that were well above environexperi-mentally
realistic levels. The maximum yield losses experienced in
the NCLAN experiments were as high as approximately 60% and
the estimates of UJ were in the vicinity of 0.12. This
consideration led to the somewhat arbitrary choice of end
points for the dose range of interest as the concentrations
at which H(xmin)=YMAX=0.98 and H(xmax)=YMJN=0.25. This
implies that the dose range, and the assigned design points,
are dependent on the presumed form of the true Neibull
response. If .lo-J, this translates to Xmi n=O. 02 tD and
Xmax= 1.386 ( 0 , which is not as rest.ricted as most of the
NCLAN studies. If .lo-3, Xmin=O.27(O and xmax=1.115(O, more
like the NCLAN studies. The H-equivalent. transformation
one choice of parameters to be translated to any choice of
parameters. Therefore, without loss of generality, the
parameters ~, ~ and 1 were set to a value of one.
Several strategies were investigated for allocating the
distinct design points to the region of interest. The
num-ber of distinct design points is denoted .by the variable
CNT. In all cases, N=24 experimental units were used for
each experiment, and all the strategies used the two end
points of the region of interest as the two extreme levels,
Xl=Xmin and XCNT=Xmax on the standardized scale of ~·w-1-1 .
Strategies for allocation of four or more distinct points
were investigated and values of FDET, SDET and V(RYL)
com-pared. The different strategies determined the allocation
of the (CNT-2) distinct design points to the interval. The
five strategies used were:
(1) Minimize FDKT: CNT=4 distinct dose levels were used
with six replicates of each. The two intermediate
design points, X2 and X3, were allocated so as to
mini-mize FDET of the V-C matrix of the parameter estimates.
The invariance of V(RYL) with respect to H-equivalent
transformations insures that the optimum allocation on
the standardized scale can be translated to the optimum
allocation for the same criterion for any set of
(2) Minimize SDET: The same strategy as (1) except the
sub-determinant (SDIT) criterion was used to allocate X2 and
X3. Again, H-equivalent transformations are used to
limit the search for an optimum design to the case tD =1
and 1=1.
(3) Equal X Strategy (EQX): The (CNT-2) intermediate
dis-tinct design points, with CNT=4, 6, and 8, were assigned
to dose levels equally spaced between the minimum and
maximum dose levels on the original x-scale. Each
choice of 1 for this design strategy gives different
allocations of the dose levels when transformed to the
standardized scale. When used to define the allocation
of design points, 1 will be denoted "ALLOC" in order to
avoid confusion with the use of 1 as the parameter of
the true Weibull response. Design points were allocated
for the EQX strategy for ALLOC=1, 2, 3 and 5. For
exam-ple, if ALLOC=1 and CNT=4, the four design points are
xl=0.02, x2=0.475, x3=0.931 and x4=1.386. If ALLOC=2
and CNT=4, the design points are equally spaced between
xl*=0.142=-ln(.98)1/2 and x4*=1.177=-ln(.25)1/2 to give
x*'=(0.142, 0.487, 0.832, 1.177). Transforming these
points to the standardized scale of ~-l gives x'=(0.02,
0.237, 0.692, 1.385). Table 3.1 gives the dose levels
scale, assuming 1=ALLOC, and transformed to the
stan-dardized scale assuming 1=1. The same transformation is
applied when CNT=6 and 8.
(4) Equal H Strategy (KQH): The (CNT-2) intermediate
dis-tinct desian points, with CNT=4,6 and 8 are allocated 60
as to provide equal spacing with respect to y=H(x)
between YHAX=0.98 and YHIN=0.25. If CNT=4, for example,
the design points would occur at the dose levels giving
H(Xl)=0.98, H(X2)=0.74, H(xs)=0.49 and H(x4)=0.25.
Transformation of the dose levels determined by the EQH
strategy to the standardized scale gives the same values
of x regardless of the choice of ALLOC.
(5) Geometric Spacing (GEO): The design points were
allo-for i=l, 2, ... , CNT, a=xmin=
O.02 an~ d ar·[eNT-I] =x.ax=1.386.
V(RYL) was evaluated for each design for each of four
estimation intervals (PY). Each interval represents a
change in H(x) of 0.20. The intervals were H(x) changir~
from 0.90 to 0.70 (PY97), from 0.80 to 0.60 (PY86), from
0.65 to 0.45 (PY64) and from 0.55 to 0.35 (PY53). Also
examined was the effect on V(RYL) of changing the values of
YHIN and YHAX and using unequal replication of design
3.6 Resul~8
In Chapter 2 designs were investigated assuming an
uncon-strained dose scale. The results of this section are
pre-sented in the framework of a constrained dose scale.
A
gridsearch procedure was used to find four-point designs
mini-mizing FDET and SDET. The EQX, EQH and GEO strategies were
investigated as alternative methods of allocating the design
points.
3.6.1 FORT and SDIT Criteria
A grid search to find the four-point design minimizing FDET
gave the x-values .02, .35, 1.386 and 1.386. The four-point
design minimizing SDET gave the x-values .02, .42, 1.386 and
1.386. The results for these two designs are given in Table
3.2. The table also contains results for the designs which
minimized FDET and SDET in Chapter 2, when an unconstrained
dose scale was used. In each of the four cases in the
table, the search to find the optimum allocation of four
design points minimizing either determinant, converged to
three distinct points with the last point replicated. The
values of FDET and SDET, when using a restricted dose scale,
increase 51% and 54%, respectively, above the values
at-tained in the unrestricted case. Comparing results, V(RYL)
has increased 30% from PY97 and on average approximately 25%