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(1)

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:

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

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

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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

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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

(10)

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

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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

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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

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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

(14)

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

(15)

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

(16)

(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

(17)

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, ... , n

j

=

1, 2, 3

where 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,

(18)

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 )

(19)

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

(20)

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 (J

where

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

(21)

o

Y[d

.(!--l)(

1)

I I

PJ

oy·/cw··-

-w p

-oW

p . k

Thus,

v(

a

a,

W

a

,1*) •

[F'(

z. va)

F

(z.

V

a)

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

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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

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(x)A. (x)

[(Z/k)l/P

JPA.'

[(Z/k)l/P

J

A (x ) ...

w

In

w

...

(w .. /

k )1/ P In (

w .. /

k)1/P

1.'

-

~(;.)

In(;.)-

~A(Z)

Therefore,

Also,

Therefore,

B(

x

r) -

B(

x

0) -

pk

IIp

w

*(l-l/P )[

B(

Z

r) -

B(

Z

0)]

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)

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-2[ A (zr) - A (z .)][B(zr) - B(z.)] Cov(cD0 •

x.

oJ

R 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 be

deter-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

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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

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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

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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

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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

(29)

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

(30)

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

(31)

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

(32)

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

(33)

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

(34)

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 be

individually minimized by the same design. The components

corresponding to V(

w),

V( t) and Cov(

w

,1) were substituted

into (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

(35)

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

(36)

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

(37)

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

(38)

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

(39)

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

(40)

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.

(41)

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.

(42)

Steinberg, D. M. and W. G. Hunter. (1984). Experimental

(43)

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 .65

w

1 • .5(.01)1/3 • •108 .37 .60

w - -,A - 3 2

- 2 tA·. 4- 2(.02)1/• • . 75 1. 76 2.17

(44)

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

(45)

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. I

I

Criterion

I

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

(46)

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

=

SDET

Design

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

(47)

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)

(48)

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

(49)

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

(50)

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

(51)

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 approximate

variance-covariance (V-C) matrix (Gallant, 1915j Gallant, 1981)

yea,

cD, 1).

a2(F '(v)F(V»-1

where 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

(52)

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

(53)

All results on variances of

a.

wand 1 and RYL can be

determined 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

(54)

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

(55)

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

(56)

(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

(57)

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

(58)

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

grid

search 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%

References

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