Optimum Dosage Allocation in Multiple-objective Quantal ... - CiteSeerX

Optimum Dosage Allocation in Multiple-objective Quantal Dose Response Experiments Wei Zhu, Weng Kee Wong Department of Biostatistics, University of California, Los Angeles, CA 90095-1772

Abstract

The toxicity of a potential drug must be studied before it can go on to the clinical trial stage. Quantal dose-response experiments are conducted to relate the drug dose level to the probability of a response. The binary response denotes whether the drug is toxic or not at the given dose level. The results of quantal dose-response experiments are often summarized by estimates of dose levels at which the probabilities of being poisoned are 0.25, 0.50 and 0.75. The corresponding dose levels are the three quartiles in the logit scale. One of the research questions addressed here is to nd the optimal dosage allocation scheme for the estimation of the second quartile, also called the \median lethal dose", subject to the eciencies for estimating the other two quartiles are not too low. A searching strategy for such an optimal design is proposed and several illustrative examples are given.

Key Words: Bayesian optimal design, Compound optimality, Constrained optimality, Eciency plot, LD50, Logit model.

1 Background

1.1 Quantal Dose-Response Experiment

A common purpose of a quantal dose-response experiment is to relate the dose level of a certain drug to the probability of a toxic response. If we assume that each subject has a speci c dose tolerance such that any dose level above the tolerance will poison the subject while any dose below it will 1

not, and the distribution of tolerances among subjects is Gaussian, then the probit model is appropriate (Kalish, 1990). Since it is well known that (after adjusting for location and scale) the probit and logit models agree very closely in all but the tails of the tolerance distribution (Cox, 1970), the logit model is often chosen because of its greater computational ease. The logit model is expressed as log 1 ? (x()x) = (x ? ) ;

(1)

where  (x) = 1= (1 + exp (? (x ? ))) is the probability of being poisoned at dosage x, and x 2 , is assumed to be continuous. The parameter is the slope in the logit scale. The other parameter  is the value of x at which the probability of being poisoned is 0:5. It is the median in the logit scale and is called the \median lethal dose". Frequently, this value is denoted by LD50. In general, we use LD100 to denote the dose level x0 corresponding to a probability  of being poisoned. From (1), we have

 (x0) = 1= (1 + exp (? (x0 ? ))) ; therefore the LD100 is a function of the model parameters T = (; ) and the poison probability  , i.e:

LD100 = x0 =  + ;

(2)

where = log = (1 ?  ). As in the linear model situation, most design criterion for the logit model is based on the design information matrix. In this case, the design information matrix is simply the Fisher information matrix. Let the response data be denoted by y T = (y1;


; yN ), where yi = 1 or 0 depending on whether 2

the ith subject is poisoned or not, and suppose the contribution to the likelihood of a single observation at the design point x is  (yi j x; ), where  (1 j x; ) =  (x) and  (0 j x; ) = 1 ?  (x). The (i; j ) th element of the observed Fisher information matrix M (;  ) is given by (3) [M (;  )]ij = ? @@@ log ( (y j x; ))  (dx) : i j Now we are ready for a literature review on optimal designs for the logit model. Z

2

1.2 Overview of Optimal Designs for the Logit Model Research in the area of experimental design for the logistic regression model has been primarily restricted to D ? optimality (White, 1975; Kalish and Rosenberger, 1978; Abdelbasit and Plackett, 1983; Jain, 1985; Minkin, 1987; Myers, Myers and Carter Jr., 1994). The model usually assumed is the logit model in (1). In this work, unless otherwise speci ed, when we mention the logit model, we refer to the simple logit model de ned in (1). The name logistic regression model and logit model have been used interchangeably in research literatures (Agresti, 1990). We call the model a logistic regression model when we are more interested in its regression aspect, and we call it a logit model when we view it as a special general linear model with a logit link. The D ? optimal design for the logit model (1) allocates half the subjects to each of doses LD82.4 and LD17.6 (White, 1975; Kalish and Rosenberger, 1978). Scienti c investigators often try to characterize a pharmacological agent by determining its LD50. Kalish (1990) developed optimal designs for esti  d50 , as well as the designs that mating the LD50, i:e: minimizing Var LD   minimize Var b , i:e: slope optimal. Using the methods of Shenton and 3

Bowman (1977), Kalish derived expressions for the variance of the MLEs of parameters in the logit model, including terms to order 1=N 2. He considered a two or three-point design that allocates ?N subjects to LD50 and (1 ? ?) N=2 subjects to each of doses LD100 and LD100(1 ?  ) where 0  ?  1. The resulting LD50-optimal and Slope-optimal designs are functions of the design size N . He also performed an in depth comparison of these designs along with the D-optimal designs (for global estimation). As a personal favorite, he proposed allocating subjects equally to LD20, LD50 and LD80. For N  100, this design is almost D ? optimal and results in eciencies greater than 95% for estimating LD50 and greater than 81% for estimating the slope. In quantal dose-response experiment, interest is often focused on obtaining accurate estimates of percentiles (Wu, 1988; Chaloner and Larntz, 1989; Durham, Flournoy and Rosenberger, 1996). Kalish and Rosenberger (1978) developed a two-level G ? optimal design in which they sought to minimize 



(x) ; Maxx2R V ar d where R is the set of all real numbers. The optimal design places half the observations at LD23.2 and half at LD76.8. Wu (1985) developed some c ? optimal designs for estimating LD100 by minimizing the asymptotic   variance of c b , i.e: (rc ())T M ?1 (;  ) (rc ()) : Here T = (; ) ; c () = LD100 =  + = , and rc () is the vector of partial derivatives of c (). Here, M (;  ) is the Fisher information matrix as de ned in (3). 4

Notice that for the Frequentist's optimal designs above, we do not know the exact dosage level LD100 for a given  unless we know the unknown parameter values  and . One solution is to adopt a Bayesian framework and utilize a prior distribution for the unknown parameters. For instance, Chaloner and Larntz (1989) assumed that the LD50 and slope parameter have uniform and independent prior distributions on an interval. They proposed two optimality criteria: Bayesian D ? optimality and Bayesian A ? optimality, and derived some Bayesian D ? optimal and A ? optimal designs for the logit model. A detailed description of their results and a formal introduction of the Bayesian optimal design theory is given in the following section.

1.3 Bayesian Optimal Design Theory The Frequentist's strategy for designing a nonlinear model is to assume a best guess of the parameter values. This approach leads to what are termed `local optimal' designs. A natural generalization is to use a prior distribution on the unknown parameters rather than a best guess with a degenerate prior. Chaloner and Larntz (1989) argued that under mild conditions, the posterior distribution of  is approximately a multivariate normal distribution with mean equals to the maximum likelihood estimate, b, and variance covariance matrix equals to the inverse of the observed Fisher information matrix evaluated at b. Further, the prior distribution of  can be used as the predictive distribution of b. If  is a convex functional, a Bayesian optimality criterion given by Chaloner (1993) is  ( ) = E  (M (;  )) : In linear design problems the information matrix does not depend on 5

 and there is a compact set of possible information matrices to optimize

over. For the nonlinear problem we need to optimize over the set of design measures  directly. For a given design  , de ne

 = (1 ? )  + x (0    1) ; 0

and recall the directional derivative of  at the point  in the direction x is given by h   i   ?  ( ) : d (x; ) = lim !0+  0

(This limit always exists since  is assumed to be convex.) A general equivalence theorem for a Bayesian optimal design is as follows.

Lemma 1 (Chaloner and Larntz, 1989) Bayesian General Equivalence The-

orem The following three conditions are equivalent: 1. the design   minimizes  ( ) ; 2. the minimum of d (x;  )  0;

3. the derivative d (x;  ) achieves its minimum at the support points of the design   .

Chaloner and Larntz (1989) also proposed two optimality criteria analogous to the Frequestists' version, namely, the Bayesian D ? optimality given by  ( ) = ?E log det M (;  ) ; and Bayesian A ? optimality given by 



 ( ) = E trA () M (;  )?1 : 6

In the latter case, A () is some user selected nonnegative de nite matrix. For example, if A () = rc () (rc ())T ; we have the Bayesian c ? optimality criterion 



E (rc ())T M ?1 (; ) (rc ()) ; which will yield the Bayesian optimal design for estimating a function of the parameters  | c (). In all criteria, the expectation is over the prior distribution of ; and the optimal design is the one which minimizes the criterion. In this work, we will focus on nding the Bayesian A? and c ? optimal designs.

1.4 The LD100 Optimal Design It is often of interest, especially in reliability and dose-response experiments, to estimate the value of x at which the probability of success,  (x), is a particular value, say  . We call the corresponding x0 as LD100 , where LD100 =  + = and = log = (1 ? ). The optimal design for the estimation of LD100 is a Bayesian c ? optimal design with c () = LD100 , T = (; ) and (rc ())T = (1; ? ?2). Equivalently, if we would rather use the Bayesian A ? optimality criterion directly, we have  2 A () = rc () (rc ())T = 1 ? = :

? = 2 2= 4 For the logit model (1), assume the design region  is closed and bounded. Consider a design  on  putting pi weight at K distinct design points xi , i = 1;


; K , P pi = 1. De ne wi =  (xi) (1 ?  (xi)) ; t = 7

K X i=1

piwi;

x =

K

X t?1 piwixi ;

s=

i=1

K X i=1

piwi (xi ? x)2 :

The Fisher information matrix (3) can be shown to be #

"

2 M (; ) = ? t (xt? ) s?+ tt ((xx ?? ))2 ;

and its inverse is

M (; )?1 = ?2

"

#

1=t + (x ? )2 =s (x ? ) =s : 2=s (x ? ) =s

The Bayesian c ? optimality criterion function for the estimation of the LD100 is (Chaloner and Larntz, 1989) 



 (; ) = E (rc ())T M ?1 (;  ) (rc ())

n h io = E ?2 t?1 + ( ? (x ? ))2 ?2 s?1 ;

and its directional derivative is n

o

d (x; ; ) = ?E w (x; ) 2 st ?2 [t (x ? x) ( (x ? ) ? ) + s]2 + (; ) ; ?




where w (x; ) =  (x) (1 ?  (x)). Notice that we have added to the parameter lists of the functions  ( ) and d (x;  ) to facilitate the upcoming discussions. Substituting = 1 = 0 gives us the LD50 optimal design criterion function  (; 1) for estimating  alone. Substituting = 2 = ?3 gives us the LD25 optimal design criterion function  (; 2) for estimating the 25th percentile. Finally, substituting = 3 = 3 gives us the LD75 optimal design criterion function  (; 3) for estimating the 75th percentile. These three optimal designs for the estimation of the three quartiles separately shall be denoted by  i ; i = 1; 2; 3 respectively. 8

The design eciency of an arbitrary nonsingular design  relative to the LD100 optimal design   is de ned as:  E () = ((;;

)) ;

where = log = (1 ?  ). It can be viewed roughly as the ratio of the variances for the estimated LD100 given by the two designs.

2 A Constrained Optimal Design Problem A question of interest arises in neurology (Rosenberger and Grill, 1996), which can be described as follows. The results of quantal dose-response experiments are often summarized by an estimate of the LD50 (). Therefore, we want to estimate the LD50 as precisely as possible. At the same time, it is desirable that the eciencies for estimating the LD25 and the LD75 separately will not be lower than some given values. The research question of interest here is how to optimize the design under these conditions. We can formulate this problem as a three-objective constrained optimal design problem. The primary goal is to estimate the LD25 as precisely as possible, the secondary goal is to estimate the LD50 as precisely as possible and the tertiary objective is to estimate the LD75 as precisely as possible. The above question can be phrased as \Find the constrained optimal design such that we have the most eciency for goal 2 subject to the constraints that the eciencies of goal 1 and 3 will not be less than two given values." Using simple linear models, Lee (1987) demonstrated that it is not an easy task to nd a constrained optimal design analytically. Our models here are more complex and so all our optimal designs are found numerically. Justi cations for the approach used here are provided by Clyde and Chaloner 9

(1996) where they proved that the equivalence of the constrained and the compound optimal designs still hold when we have more than two objectives in a nonlinear model. Consequently, we will nd all possible compound optimal designs rst, then choose the desired constrained optimal design among the group. Since we are dealing with a nonlinear model, we will follow the popular practice and adopt a Bayesian approach. The readers can refer to Cook and Wong (1994), and Zhu (1996) for more information on the theory of the constrained and the compound optimal designs and their applications.

2.1 The Compound Optimal Designs 2.1.1 The De nition and A Coincidence By the de nition of the compound optimal designs, if the primary goal is to estimate the LD25, the secondary goal is to estimate the LD50 and the tertiary objective is to estimate the LD75 as precisely as possible. The compound optimal design criterion functional is  ( j ) =

3 X

i=1

i (; i) ;

(4)

P

where 0  i  1; i = 1; 2; 3; and 3i=1 i = 1. Its directional derivative is given by 3 X d (x;  j ) = id (x; ; i) : i=1

Given , the compound optimal design  is the design which minimizes the functional  ( j ). The optimality of the design can be checked by lemma 1. Observe that there are two terms that involve i ; i = 1; 2; 3; in P P the functional  ( j ). These two terms are 3i=1 i i and 3i=1 i i2. If we view as a random variable that takes on the value i with probability 10

i; i = 1; 2; 3; then these two terms can be written as E and E 2, and  ( j ) becomes  ( j ) = E [ (; )] : Here we have an interesting coincidence with a usual practice in the Bayesian framework when several percentile response points are of interest. The usual practice is to put a distribution on and average A () over this distribution. That is, the matrix 

2 ? E (

) = E A () = ?E ( ) = 2 E ( 2) = 4 is used to obtain the Bayesian A ? optimal design criterion functional

1



 ( ) = E tr [E A ()] M (;  )?1



for the estimation of several percentiles. If we assume the distribution of

is that will take on the values i with probability i; i = 1; 2; 3, it is straight forward to show that 

 ( j ) = E [ (; )] = E tr [E A ()] M (;  )?1



io n h = E ?2 t?1 + ?2s?1 E 2 ? 2 ?1s?1 (x ? ) E + s?1 (x ? )2 :

In other words, the compound optimal design for the estimation of several percentiles can be viewed as a special type of the Bayesian A ? optimal designs.

2.1.2 Some Symmetrical Properties of the Compound Optimal Designs Researchers usually have some idea about the magnitude of the two parameters  and at the onset of the trial. Suppose b is a prior point estimate of 11

 and we assume the prior distribution for  is symmetric about b. Without loss of generality, we may assume b = 0.

Lemma 2 Suppose T = (1; 2; 3), and the compound optimal design for (4) is given by



 x 1 x2


xK ?1 xK  = p p


p : 1 2 K ?1 pK Furthermore, assume the prior on  is symmetric about 0. Then we have

the following results. 



1. Given e T = e 1; e 2; e 3 = (3; 2; 1), the design which is symmetric to  is approximately a compound optimal design for e . That is 

 ? x ? x K K ? 1


?x2 ?x1 e  p pK?1


p2 p1 : K

2. The eciency of   relative to the primary objective is about equal to the eciency of e relative to the tertiary objective; the eciencies of   and e relative to the secondary objective are approximately equal; and nally, the eciency of   relative to the tertiary objective is approximately equal to the eciency of e relative to the primary objective. That is, ?








?








?








E1   E3 e ; E2   E2 e and E3   E1 e ; ?




where Ei  ; i = 1; 2; 3, represents the relative eciencies of the compound optimal designs  to each of the three objectives.

Corollary 1 The LD25 optimal and the LD75 optimal designs are symmetric to each other, and the following relations hold: ?




?




?




?




  1 ; 1    3 ; 3 and E2  1  E2  3 : 12

2.2 The Logit-Design Program Finding optimal designs is a numerical optimization problem. We need to nd the number of design support points K , the support points x1 ; x2;


; xK , and the design weights or proportions of subjects, p1; p2;


; pK , associated with each of the support points. Chaloner and Larntz (1989) developed a software named Logit-Design which implements the Bayesian methodology to nd the Bayesian D?; A? and c ? optimal designs for a class of prior distributions. This class of priors are independent beta priors on a bounded region for  and . Independent beta distributions cover a wide class of prior distributions on compact intervals. A special case are the independent and uniform prior distributions used in the examples in Chaloner and Larntz (1989). Due to the coincidence described in the previous section, the Logit-Design program is easily modi ed for nding the compound optimal designs to estimate the three quartiles. The optimal designs for estimating the LD25, the LD50 and the LD75 alone are special cases of this compound optimal design when T takes the form (1; 0; 0) ; (0; 1; 0) and (0; 0; 1). Logit-Design is based on the Nelder and Mead (1965) version of the simplex algorithm which was found to be eective in this situation. This method requires that the number of design support points be speci ed and so starting with 2 design points and increasing by 1 at each time, the design which minimizes the criterion functional on the smallest number of support was chosen. The menu of Logit-Design is: Available options: 1. enter a starting design 2. select a prior distribution 13

3. select a design criterion 4. evaluate the selected criterion for the selected prior 5. seek the optimal design for the criterion and prior 6. print out the current common values 7. quit The design criteria available in Logit-Design 1.0 is Bayesian D-optimal design and three Bayesian A-optimal designs. The three Bayesian A-optimal designs are: 1. LD50 optimal design 2. LD95 optimal design 3. Minimize the average variance of LD27 through LD73. By de nition, the Bayesian A-optimality design criterion functional can be expressed as a function of the three dierent elements in the inverse Fisher information matrix, M ?1 (;  ). These three elements were calculated in the program and denoted by var(1,1), var(1,2) and var(2,2). Therefore we modi ed the Logit-Design program easily to oer the criteria we desire, which are 1. LD25 optimal design 2. LD50 optimal design (which is available already) 3. LD75 optimal design 14

4. Compound optimal design (its criterion function is a weighted average of those of LD25, LD50 and LD75), for all possible combinations of 1, 2 and 3. The original program can also calculate the value of the criterion functional  ( ) for a given design  . Since the design eciency of an arbitrary nonsingular design  relative to the LD100 optimal design   is simply the ratio of the two criterion functions, i.e:  (  ) = ( ), we modi ed the program to calculate the design eciencies as well. Now if we choose the option of \compound optimal design", the program will ask for the three  values. The remaining steps are similar.

2.3 The Implementation Compound optimal designs are found for all combinations of  where i is a multiple of 0.1, i.e:, i = 0:1m; m = 0;


; 10 and i = 1; 2; 3. The design region  here is a suciently large closed and bounded interval including the origin. The priors we assumed are two independent uniform priors, where   U [?0:1; 0:1] and  U [6:9; 7:1]. These priors were also used by Chaloner and Larntz (1989). Due to the symmetric properties discussed in lemma 2, our work is now reduced by half. Table 1 lists the three optimal designs for estimating the LD25, the LD50 and the LD75 alone. The legends are: the rst column is the choice of ; the second column is the design support points; the third column gives the design weights associated with each support points; the rest of the three columns are relative eciencies for the three objectives respectively. From the table, we observe that  1 and  2 are approximately symmetric to each other, which agrees with our results in lemma 2. These two optimal designs 15

Table 1: LD25, LD50 and LD75 Optimal Designs;   U [?0:1; 0:1],  U [6:9; 7:1]. (1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:210; :093g (:8003; :1997) 1 .444 .202 (0; 1; 0) f?:127; :127g (:501; :499) .528 1 .528 (0; 0; 1) f:094 ? :210g (:1994; :8006) .202 .444 1 are not symmetric about 0, but the LD50 optimal design is symmetric about 0. Table 2 shows another interesting observation, that is when 1 = 3, the compound optimal designs are approximately symmetric about 0, and the design eciencies for the rst and the third objectives are about equal, i.e: e1  e3 . For this pair of priors, we have two design support points and the weights associated with each support points are about equal as well, that is p1  p3  0:5. These approximations are so close that usually the dierence is at the fth decimal place. An explanation for this observation is that when  is symmetric and the prior for  is symmetric about 0, both the criterion functional  ( j ) de ned in (4) and its directional derivative d (x;  j ) are also symmetric about 0. Therefore the solutions to d (x;  j ) = 0 are symmetric about 0. By lemma 1 (3), any two symmetrical solutions should have the same opportunity of being included as a design support point. This is why we end up with these symmetrical compound optimal designs. This observation is very helpful in nding the constrained optimal designs when we have equal interest in the rst and the third objectives. We will discuss more about the search procedure in the next section. Tables 3, 4 and 5 list the compound optimal designs when we have non-symmetrical . Notice that only a fraction of all possible compound 16

Table 2: Symmetrical Compound Optimal Designs;   U [?0:1; 0:1],  U [6:9; 7:1]. (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:144; :144g f?:155; :155g f?:164; :165g f?:172; :172g f?:178; :178g f?:184; :184g f?:187; :187g f?:189; :188g f?:193; :193g f?:197; :197g f?:200; :200g

e1

.587 .618 .637 .649 .656 .662 .664 .665 .666 .667 .668

e2

.989 .971 .953 .935 .917 .904 .894 .890 .876 .865 .853

e3

.587 .618 .637 .649 .656 .662 .664 .665 .666 .667 .668

optimal designs where 's are multiples of 0.1 are listed. The reason is that the rest of these compound optimal designs can be obtained by symmetry. For example, the compound optimal design when T = (0; :9; :1) is given in Table 3, which is  :138 :148  ;  = ?:4661 :5339

with eciencies e1 = :535; e2 = :982 and e3 = :633. The compound optimal design when T = (:1; :9; 0) is not listed in any of these tables, but can be readily obtained by symmetry as  :148 :138  ;   ?:5339 :4661 with eciencies e1  :633; e2  :982 and e3  :535. Lemma 1 (2) and (3) can be used to check whether a \compound optimal design" obtained from the Logit-Design program is indeed compound optimal. For example, for T = (:1; :42; :48), we run the Logit-Design program and obtained the \compound optimal design" as 17

0.030 0.025 0.020 0.015 0.0

0.005

0.010

Directional derivative

-1.0

-0.5

0.0

0.5

1.0

x

Figure 1: Plot of directional derivative function versus x  ? : 166 : 190  = :3963 :6037 : 

To check whether this design is compound optimal, we plot the direcP tional derivative of  ( j ), d (x;  j ) = 3i=1 i d (x; ; i), against x. The resulting plot (Figure 1) shows that the directional derivative is non-negative and reaches 0 at the two design support points. Therefore by lemma 1 (2) and (3), this design is indeed compound optimal for the given . Overall, for these pair of priors, the compound optimal designs have two support points. These two support points are always between -1 and 1, or more accurately, between -.211 and .211. Furthermore, they always have opposite signs. The design shifts to the negative direction when we put more weight on estimating the LD25, i.e: when 1 increases; and shifts to the positive direction when 3 increases. Generally speaking, objectives 1 18

Table 3: Non-symmetric Compound Optimal Designs (1);   U [?0:1; 0:1],

 U [6:9; 7:1].

(1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:121; :206g f?:134; :202g f?:142; :197g f?:146; :192g f?:148; :186g f?:148; :179g f?:146; :171g f?:143; :161g f?:138; :148g

(p1; p2) (:2350; :7650) (:2676; :7324) (:2969; :7031) (:3252; :6748) (:3526; :6474) (:3795; :6205) (:4073; :5927) (:4354; :5646) (:4661; :5339)

e1

.268 .318 .361 .399 .434 .465 .494 .519 .535

e2

.563 .647 .715 .776 .823 .868 .910 .949 .982

e3

.983 .954 .921 .886 .849 .807 .760 .704 .633

Table 4: Non-symmetric Compound Optimal Designs (2);   U [?0:1; 0:1],

 U [6:9; 7:1].

(1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:159; :211g f?:163; :206g f?:165; :202g f?:166; :197g f?:166; :191g f?:165; :184g f?:163; :177g f?:160; :168g

(p1; p2) (:2903; :7097) (:3165; :6835) (:3418; :6582) (:3670; :6330) (:3914; :6086) (:4164; :5836) (:4424; :5576) (:4702; :5298)

19

e1

.371 .407 .441 .473 .505 .535 .565 .593

e2

.686 .736 .780 .820 .856 .890 .920 .948

e3

.926 .899 .870 .839 .806 .769 .727 .678

Table 5: Non-symmetric Compound Optimal Designs (3);   U [?0:1; 0:1],  U [6:9; 7:1]. (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:179; :209g f?:179; :205g f?:179; :200g f?:178; :194g f?:177; :188g f?:175; :181g f?:189; :207g f?:188; :202g f?:187; :197g f?:186; :191g f?:196; :204g f?:194; :199g

(p1 ; p2) (:3547; :6453) (:3775; :6225) (:4004; :5996) (:4236; :5764) (:4474; :5526) (:4730; :5270) (:4076; :5924) (:4294; :5706) (:4517; :5483) (:4752; :5248) (:4550; :5450) (:4770; :5230)

e1

.468 .498 .5287 .558 .588 .618 .542 .571 .600 .630 .607 .636

e2

.777 .809 .840 .867 .893 .916 .823 .847 .868 .887 .846 .863

e3

.852 .825 .797 .767 .733 .694 .787 .760 .731 .698 .727 .698

and 3 are very competitive while the design is more robust for objective two, the estimation of the LD50. The more symmetric the design is, the better the eciency for estimating the LD50.

3 Searching for the Desired Constrained Optimal Designs 3.1 Some Guidelines

We use the Logit-Design program to nd all the compound optimal designs for the estimation of the three quartiles. The question now is how to nd the desired constrained optimal design among compound optimal designs. Our experience suggests that these compound optimal designs have the following properties which are helpful in nding the desired constrained optimal design. 20

 Rule 1. Generally, when i increases, ei increases; but exceptions do exist;

 Rule 2. If we x 1 and increase 2, then e1, e2 increases and e3 decreases. Similarly, if we x 3 and increase 2, then e3 , e2 increases and e1 decreases;  Rule 3. If we x 2, then e2 increases as j1 ? 3j decreases. When 1 = 3, e2 reaches its maximum for the given 2.  Rule 4 (Symmetry). By lemma 2, if T = (1; 2; 3) and eT =   e 1; e 2; e 3 = (3; 2; 1), then we have e1  ee3; e2  ee2 and e3  ee1 ,   ?
where ei = Ei  and eei = Ei e ; i = 1; 2; 3: Furthermore,  and e are symmetric with respect to 0.  Rule 5 (Symmetry). If 1 = 3, then e1  e3. The compound optimal

design is approximately symmetric about 0, meaning that the design support points are approximately symmetric about 0, and for each pair of symmetric support points, the associated design weights are also about equal.

Now we are ready for the search of the constrained optimal designs. Two scenarios will be studied separately depending on whether objectives 1 and 3 are of equal interest.

3.2 Equal Interest in the LD25 and the LD75 By Rule 5, when 1 = 3 = , the compound optimal design  is approximately symmetric about 0. Furthermore,  has the same eciency for the estimation of the LD25 and the LD75, i.e:, e1  e3. Therefore, if we 21

are equally interested in the estimation of the LD25 and the LD75, the desired constrained optimal design can be chosen from this group of symmetric designs  , where 0    1. We next refer to Table 2 to nd a design which has the closest e1 and e3 to the desired constrained optimal design. The design can be re ned by increasing or decreasing the value of  slightly, and rerun the Logit-Design program to obtain the new compound optimal design. By Rule 1, if we increase , the design eciencies for the LD25 and the LD75 increase, while the eciency for the LD50 decreases. The other way to nd the desired constrained optimal design is by means of an eciency plot. Since e1  e3 , we can plot e1 (or e3 ) and e2 versus . The following gure is the eciency plot of all symmetrical compound optimal designs for the two independent uniform priors   U [?0:1; 0:1] and  U [6:9; 7:1]. As proposed by Cook and Wong (1994) and Wong (1995), the desired constrained optimal designs can be found graphically from the eciency plot. In this case, we can nd all constrained optimal designs which have the same eciencies for the rst and the tertiary objectives from this plot.

Example 1 Find the constrained optimal design with e1  0:666 and e3  0:666 for the priors   U [?0:1; 0:1] and  U [6:9; 7:1]. From the eciency plot, draw a horizontal line when the eciency equals 0.666 and determine where it meets the graph of e1 at a certain point. Through the crossing point, draw a vertical line to intersect the  axis at   0:4. The eciency curve for e2 has value e2  0:876 at   0:4. Hence we declare that the compound optimal design 0:4 is the desired constraint optimal design. At this point, refer to Table 2 and conclude that the design 22

1.0 0.8 0.6 0.4

Efficiency 0.0

0.2

e1 e2

0.0

0.1

0.2

0.3

0.4

0.5

lambda

Figure 2: Eciency plot of symmetrical compound optimal designs when

  U [?0:1; 0:1] and  U [6:9; 7:1]. sought is

 ? : 193 : 193 0:4 = :500 :500 : 

If the desired value of  is not available from Table 2 (for example,  = :22), then we have to rerun the Logit Design program to nd the desired design.

3.3 Unequal Interest in the LD25 and the LD75 When we have unequal interest in estimating the LD25 and the LD75, we can nd the desired constrained optimal design according to the following procedures. If not speci ed otherwise, the priors used in the examples in this section are the two independent uniform priors   U [?0:1; 0:1] and  U [6:9; 7:1].

 Step1. Check the feasibility of such a design. If feasible, go to Step 2. 23

Some constrained optimal designs do not exist. For example, for the given priors, it is not possible to nd a constrained optimal design with e1  :8 and e3  :9. Here, we oer a rule of thumb to check the existence of a given constrained optimal design. Suppose the desired eciency for goal 1 is e1  e01 and for goal 3 is e3  e03 . Denote the eciencies of the symmetric compound optimal design =0:5 for goal 1 and 3 are e1 and e3 , (e1  e3 ). If (e01 + e03 ) > (e1 + e3 ), one has a good reason to suspect the existence of such a constrained optimal design. For our example, we have (:9 + :8) > (:668 + :668), and it turned out that we can not nd such a constrained optimal design. Of course, to be sure, we need to go through all the designs listed in those ve tables, check for their e1 and e3 in the hope of a close match.

 Step 2. Without loss of generality, suppose e01 < e03. Denote the group of compound optimal designs with T = (1; 2; 3) = (0; 1 ? 3 ; 3) as 3 , where 0  3  1. First, we look for the smallest 3 among 3 such that e3  e03 . Next, we check whether e1  e01. If the answer is armative, this design 3 is close to the constrained optimal design we are looking for, stop and go to the Re nement Step. Otherwise, go to Step 3.

 Step 3. Fix 3, decrease 2 and increase 1 by the same amount
(we suggest to take
= 0:1 for the crude adjustment and take
= 0:01 for the re nement). Now T = (1 +
; 2 ?
; 3). Check whether e1  e01 and e3  e03. If the answer is armative, stop and go to the Re nement Step. Otherwise, go to Step 4.

 Step 4. 24

1. If e3  e03 but e1 < e01 ; go to Step 3. 2. If e1  e01 but e3 < e03 , go to Step 5. 3. If e1 < e01 and e3 < e03 , if je1 ? e01 j  je3 ? e03j, go to Step 3; otherwise go to Step 5.

 Step 5. Fix 1, decrease 2 and increase 3 by the same amount
(we suggest to take
= 0:1 for the crude adjustment and take
= 0:01 for the re nement). Now T = (1; 2 ?
; 3 +
). Check whether e1  e01 and e3  e03. If the answer is armative, stop and go to the Re nement Step. Otherwise, go to Step 4.

 The Re nement Step. At this stage, both e1  e01 and e3  e03, the design is very close to the sought optimal design. The dierence of the i now with the true i sought should be less than 0.1.

1. If 3 > 1, x 1, decrease 3 and increase 2 by the same small amount
, say
= 0:01. Run the Logit-Design program and check for e3 . Repeat until e3 = e03 and stop. The current design is the desired constraint optimal design. 2. If 1 > 3, x 3, decrease 1 and increase 2 by the same small amount
, say
= 0:01. Run the Logit-Design program and check for e1. Repeat until e1 = e01 and stop The current design is the desired constraint optimal design. In the following, we will search for some constrained optimal designs for the given priors using this 6-step procedure.

Example 2 Find a constrained optimal design with e01 = 0:5 and e03 = 0:8 for the priors   U [?0:1; 0:1] and  U [6:9; 7:1]. 25

The searching stage. By Step 1, since (:5 + :8) < (:668 + :668), we suspect it is possible to nd such a constrained optimal design. Now we are in Step 2. From Table 3, we found that the design with T = (0; :6; :4) gives us the smallest 3 , with e3 = :807, but e1 = :465 < :5. Therefore we go to Step 3. Now we check for the design with T = (:1; :5; :4) from Table 4, and found that e1 = :535 > :5, but e3 = :769 < :8. Next we go to Step 4 which directed us to Step 5. Now we check for the design with T = (:1; :4; :5) from Table 4, and found that e1 = :505 > :5, and e3 = :806 > :8. The eciency for the LD50 now is e2 = :856. This design is very close to the desired constraint optimal design, we can now proceed to the re ning stage. The re ning stage. Now we have T = (:1; :4; :5) which is quite unbalanced. In order to increase e2, by Rule 3, we should decrease 3 and increase 2. We then run the Logit-Design program for T = (:1; :41; :49) and found the resulting eciencies are e1 = :508; e2 = :859 and e3 = :802. Again, by Rule 3, we should decrease 3 and increase 2. We rerun the LogitDesign program for T = (:1; :42; :48) and found the resulting eciencies are e1 = :511; e2 = :863 and e3 = :800. The corresponding design is  :166 :190  :  = ?:3963 :6037

This is the constrained optimal design we are looking for. At this point, one may wonder what happens if we decrease 1 and increase 2 by the same amount, will the design have a bigger e2? To answer this question, we rerun the Logit-Design program for T = (:09; :43; :48) and found that the resulting design has e1 = :505; e2 = :861 and e3 = :803. The eciency for the LD50 has decreased because this design is more unbalanced.

26

4 Other Priors

4.1 Three Other Combinations of Uniform Priors In the previous sections, we have studied the properties of the Bayesian compound optimal designs under a symmetric prior on . In particular, we have examined the situation where we have two independent uniform priors   U [?0:1; 0:1] and  U [6:9; 7:1]. At this point, one may wonder what will happen if we change the domain of these two priors. To answer this, we performed the same routine on three other combinations of independent uniform priors, namely:   U [?0:1; 0:1],  U [6; 8];   U [?1; 1] ;  U [6:9; 7:1]; and   U [?1; 1],  U [6; 8]. It was found that the designs appear to be much more sensitive to the prior distribution on  than on , and in general, as the uncertainty in the prior distribution increases so does the number of design support points. This result agrees with the ndings of Tsutakawa (1972, 1980), Abdelbasit and Plakett (1983), Chaloner and Larntz (1989) and King and Wong (1996). For the same prior on , given , the compound optimal designs under the two priors on , namely,   U [?0:1; 0:1] and   U [?1; 1], are very dierent. The former one with a more informative prior   U [?0:1; 0:1] has two support points while the latter one has six. However, the latter one is generally much more ecient for all three objectives than the former one, especially for objectives 1 and 3 which are the estimations of the LD25 and the LD75 respectively. For example, suppose we use the priors   U [?1; 1] and  U [6:9; 7:1] and are equally interested in estimating the LD25 and the LD75. Furthermore, suppose we require the design eciencies for these two objectives to be at least 0.9. It is easily seen that the LD50 optimal design is the constrained optimal design we sought. It is 100% ecient for 27

estimating the LD50, and 90.9% ecient for estimating the LD25 and the LD75. For the same prior on , given , the compound optimal designs under the two priors on , namely,  U [6:9; 7:1] and  U [6; 8], are very similar. They have the same number of support points with about the same values and the same design weights. The latter one with a less informative prior  U [6; 8] is generally slightly less ecient than the former one with  U [6:9; 7:1]. For detailed information, see tables and gures listed in Appendix B. The computational time for the compound optimal design increases as the number of design support points increases, or equivalently, as the prior for  becomes less informative. It does not change signi cantly as the prior on varies. To obtain the desired constrained optimal designs when there is unequal interest in estimating the LD25 and the LD75, the same 6-step searching procedure can be employed for any of these priors. Here is an example.

Example 3 Find a constrained optimal design with e01 = 0:85 and e03 = 0:95 for the priors   U [?1; 1] and  U [6; 8]. The searching stage. By Step 1, since (:85 + :95) < (:915 + :915), we deem it possible that such a constrained optimal design exists. Now we are in Step 2. From Table 18, we found that the design with T = (0; :7; :3) gives us the smallest 3 , with e3 = :954, and e1 = :858 < :85. Now we can go to the re nement step. The re ning stage. Now we have T = (0; :7; :3). Since 3 > 1, we decrease 3 and increase 2 by
= 0:01. Then we run the Logit-Design program for T = (0; :71; :29) and found the resulting eciencies are e1 = :860; 28

e2 = :989 and e3 = :953. We can decrease 3 and increase 2 further. Repeat this step two more times, we have T = (0; :73; :27), the corresponding

design is

 :938 ?:545 ?:185 :177 :551 :970  :  = ?:1519 :1587 :1594 :1640 :1722 :1939 Its eciencies are e1 = :863; e2 = :991 and e3 = :950. No further improvement on e2 can be made and we conclude that this is the constrained optimal

design we sought.

4.2 None-uniform Priors The beta distribution is one of the few common \named" distributions that give probability 1 to a nite interval (Casella and Berger, 1990, page 107109), for example (0; 1). The Beta(; ) pdf is 1 x?1 (1 ? x) ?1 ; 0 < x < 1;  > 0; > 0; f (x j ; ) = B (; ) where B (; ) denotes the beta function,

B (; ) =

Z 1 0

x?1 (1 ? x) ?1 dx:

The mean of the Beta(; ) de ned above is = ( + ). As the parameters  and vary, the beta distribution takes on many shapes. The pdf can be strictly increasing ( > 1; = 1), strictly decreasing ( = 1; > 1), Ushaped ( < 1; < 1)or unimodal ( > 1; > 1). The case  = yields a symmetric pdf. Finally, if  = = 1, the beta distribution reduces to the uniform distribution de ned in the same interval. Since we need a symmetrical prior on  to ensure the approximate symmetric properties of the desired optimal designs. We have chosen two none-uniform beta priors on , they are Beta(2; 2) de ned in the intervals 29

(?0:1; 0:1) and (?1; 1) respectively. The priors on , however, can be either symmetrical or unsymmetrical. We have chosen two symmetric beta priors on as Beta(2; 2) de ned in the intervals (6:9; 7:1) and (6; 8) respectively; and two unsymmetrical beta priors on as Beta(1; 2) de ned in the intervals (6:9; 7:2) and (6; 9) respectively. The intervals (6:9; 7:2) and (6; 9) are chosen such that the mean of the priors on will be 7. Assuming the priors on  nd are independent, we have derived the optimal designs for the estimation of the three quartiles under all 8 possible combinations of these none-uniform beta priors. The results proven to be very similar to those under the independent uniform priors. detailed results are given in tables 21 through 76 in Appendix B.

5 Constrained Optimal Designs for the LD50 and the Slope The results of quantal dose-response experiments are often summarized by an estimate of the LD50 (). At the same time, it is important to estimate the slope ( ) in order to study the overall shape of the dose-response curve. The importance of these two objectives might dier, but the researcher usually has an idea about their relative importance in the study. This can be formulated as a two-objective constrained optimization problem. Again, the desired constrained optimal design can be found graphically from the eciency plot of all Bayesian compound optimal designs for the LD50 and the slope. The same four combinations of independent uniform priors are used, namely   U [?0:1; 0:1] ;  U [6:9; 7:1];   U [?0:1; 0:1],  U [6; 8];   U [?1; 1] ;  U [6:9; 7:1]; and   U [?1; 1],  U [6; 8]. It was found 30

that the designs appear to be much more sensitive to the prior distribution on  than on , and in general, as the uncertainty in the prior distribution increases so does the number of design support points. For the logit model (1), log 1 ? (x()x) = (x ? ) ; recall that the inverse of the Fisher information matrix is

M (; )?1 = ?2

"

#

1=t + (x ? )2 =s (x ? ) =s ; (x ? ) =s 2=s

where

wi =  (xi) (1 ?  (xi)) ; t = x = t?1

K X i=1

piwixi ; s =

K X i=1

K X i=1

piwi;

piwi (xi ? x)2 :

The LD50 and the slope optimal designs alone are two Bayesian c ? optimal designs, with c () = , (rc ())T = (1; 0), and c () = , (rc ())T = (0; 1) respectively. Their design criterion functional are n

h

io

 ( ) = E ?2 1=t + (x ? )2 =s ; and

 ( ) = E (1=s) :

By the de nition of the compound optimal designs, given , the objective functional for the estimation of the LD50 and the slope is  (;  j ) =  ( ) + (1 ? )  ( ) : As we have argued in the three-quartile problem, if the prior on  is symmetric about 0, then the functions inside the expectation in the above 31

objective functional and its directional derivative are all symmetric about 0. Therefore each pair of symmetric points have the same chance to be the design support points. Furthermore, each pair of symmetric design support points appear to be of equal importance and hence the design weights associated with each of them should be about the same. This explained why all the compound optimal designs for the LD50 and the slope are all approximately symmetric about 0 under either priors on . For the same prior on , given , the compound optimal designs under the two priors on , namely,   U [?0:1; 0:1] and   U [?1; 1], are very dierent. The former one with a more informative prior   U [?0:1; 0:1] has two support points while the latter one has six. However, the latter one is generally more ecient for both objectives than the former one. In fact, all compound optimal designs with the prior   U [?1; 1] are more than 90% ecient for both objectives, while most compound optimal designs under the prior   U [?0:1; 0:1] are less than 50% ecient for the estimation of the LD50. At the same time, we observe that no matter which priors we put on  or , most compound optimal designs are nearly 100% ecient for estimating the slope. For the same prior on , given , the compound optimal designs under the two priors on , namely,  U [6:9; 7:1] and  U [6; 8], are very similar. They have the same number of support points with about the same values and the same design weights. Their design eciencies for the same objective are also about the same. Assuming the prior   U [?0:1; 0:1], if we desire the same eciency for both objectives, the design   :999  ?::522 ::22 5 ; is the solution with about 80% eciency for both objectives under either 32

priors on . Assuming the prior   U [?1; 1], if we desire the same eciency for both objectives, the design  1 ?:58 ?:19 :19 :58 1  ; :999  ? :15 :17 :18 :18 :17 :15

is the solution with about 98% eciency for both objectives under either priors on . As we have fully discussed before, any constrained optimal design, provided its existence, can be found easily from the eciency plot. The eciency plot for the four combinations of priors are provided in Appendix C.

6 Conclusion Bayesian compound optimal designs for the estimation of the three quartiles were derived for several combinations of priors. The properties of these designs were discussed. Searching procedures for the desired constrained optimal designs were proposed and some of these designs were found utilizing these procedures in the examples. However, there are some intriguing questions remain unanswered. For example, in practice, it is likely that no subjects will be allocated to dosages above a certain toxicity level, hence no information will be available on large dosage levels. In that situation, it is more reasonable to assume a non-symmetrical prior on , and it is likely that many properties of the compound optimal designs described here will change. Accordingly, the searching procedures have to be modi ed. Problems like this will keep motivating us, and hopefully many others, to continue work in this area. Bayesian compound optimal designs were found for the simultaneous estimation of the LD50 and the Slope. Corresponding constrained optimal 33

designs can be found easily via the eciency plot. Four combinations of independent uniform priors were studied. In the future, we shall generalize the results here to a broader class of priors, for example, the general beta priors. Finally, we thank Drs. Chaloner and Larntz for kindly sending us their Logit-Design program and its accompanying technical report. Without the help of their Logit-Design 1.0 program, many results in this paper would not be readily obtained.

Appendix A: Proofs of Results Proofs of Lemma 2 and Corollary 1: The Bayesian c ? optimality criterion function for the estimation of LD100 is (Chaloner and Larntz, 1989) h

n

 (; ) = E ?2 t?1 + ( ? (x ? ))2 ?2 s?1 = E (; ; ) ; where

io

i h (; ; ) = ?2 t?1 + ( ? (x ? ))2 ?2 s?1 ;

and its directional derivative is n

o

d (x; ; ) = ?E w (x; ) 2st ?2 [t (x ? x) ( (x ? ) ? ) + s]2 +  (; ) = E [ (x; ; ; ) + (; ; )] ; ?




?


?2

where

 (x; ; ; ) = ?w (x; ) 2 st

[t (x ? x) ( (x ? ) ? ) + s]2 ;

34

with T = (; ) ; = log = (1 ?  ) ;  (x) = 1= (1 + exp (? (x ? ))) ; w (x; ) =  (x) (1 ?  (x)) and

wi =  (xi) (1 ?  (xi)) ; t = x =

K

X t?1 piwixi ;

i=1

s=

K X i=1

K X i=1

piwi;

piwi (xi ? x)2 :

It is easily seen that the functions w (x; ), (x ? x)2 , (x ? )2 ; s and t are all symmetric with respect to , while the functions (x ? x) and (x ? ) are symmetric with respect to  in the origin. Recall that a function f (x) is symmetric with respect to  if f (2 ? x) = f (x), and f (x) is symmetric with respect to  in the origin if f (2 ? x) = ?f (x) : On the other hand, by de nition, given , the compound optimality criterion functional for estimating the three quartiles are  ( j ) =

3 X

i=1

i (; i) = E

3 X

i=1

i (; ; i) ;

with 1 = ?3; 2 = 0 and 3 = 3. Its directional derivative is

d (x;  j ) =

3 X

i=1

= E 

i d (x; ; i) 3 X

i=1

i [ (x; ; ; i) + (; ; i)] :



Let e T = e 1; e 2; e 3 = (3; 2; 1) ;    = xp1 xp2





xp K?1 xp K ; 1 2 K ?1 K

and

  e = 2 p? xK 2 p? xK?1





2 p? x2 2 p? x1 : K K ?1 2 1

35

Since 1 = ? 3 and 2 = 0, we have 



 (x; ; ; 1) =  2 ? x; ; e; 3 ;    (x; ; ; 2) =  2 ? x; ; e; 2 ;    (x; ; ; 3) =  2 ? x; ; e; 1 ; and 







(; ; 1) = ; e ; 3 ;   (; ; 2) = ; e ; 2 ; (; ; 3) = ; e ; 1 : It follows that 3 X

i=1

and

i (x; ; ; i) = 3 X

i=1

3 X

i=1

i (; ; i) =



e i 2 ? x; ; e; i

3 X







e i ; e ; i :

i=1

Now under the assumption that the prior on  is symmetric about 0, a good point estimate of  is b = 0. Therefore we have 3 X

i=1

and Hence where

i (x; ; ; i) =

3 X

i=1

i (; ; i) = 

3 X

i=1

3 X

i=1



e i ?x; ; e0 ; i







e i ; e0; i : 





 ( j )   e0 j e and d (x;  j )  d ?x; e0 j e ; 

 ? x ? x K K ? 1


?x2 ?x1 0 = p pK?1


p2 p1 : K e

36

By lemma 1 (2) and (3), if  =  is a compound optimal design for the given , then e = e0 must be a compound optimal design for the given e . Clearly, the LD25 optimal design  1 and the LD75 optimal design  3 are special cases of the compound optimal designs with T = (1; 0; 0) and eT = (0; 0; 1). Therefore by the above results, these two designs are approximately symmetric to each other. Furthermore, the following relations hold: ?




?




?




?




  1 ; 1    3 ; 3 and E2  1  E2  3 : Therefore, by the de nition of the design eciencies and the above results, the general  and e satisfy the following relations: ?








?








?








E1   E3 e ; E2   E2 e and E3   E1 e :

37

0.4

Efficiency

0.6

0.8

1.0

Appendix B: Figures and Tables for the Three Quartiles Estimation Problem

0.0

0.2

e1 e2

0.0

0.1

0.2

0.3

0.4

0.5

lambda

Figure 3: Eciency plot of symmetrical compound optimal designs when

  U [?0:1; 0:1] and  U [6; 8].

38

1.00 0.95 0.90 0.85

Efficiency

0.80

e1 e2

0.0

0.1

0.2

0.3

0.4

0.5

lambda

0.90 0.85

Efficiency

0.95

1.00

Figure 4: Eciency plot of symmetrical compound optimal designs when   U [?1; 1] and  U [6:9; 7:1].

0.80

e1 e2

0.0

0.1

0.2

0.3

0.4

0.5

lambda

Figure 5: Eciency plot of symmetrical compound optimal designs when

  U [?1; 1] and  U [6; 8].

39

Table 6: LD25, LD50 and LD75 Optimal Designs;   U [?0:1; 0:1], 

U [6; 8].

(1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:215; :085g (:7956; :2044) 1 .440 .201 (0; 1; 0) f?:127; :128g (:5015; :4985) .523 1 .523 (0; 0; 1) f:081 ? :215g (:2048; :7952) .201 .440 1 Table 7: Symmetrical Compound Optimal Designs;   U [?0:1; 0:1],  U [6; 8]. (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:145; :145g f?:157; :157g f?:166; :166g f?:174; :174g f?:180; :180g f?:186; :186g f?:189; :189g f?:191; :191g f?:195; :195g f?:199; :199g f?:203; :203g

e1

.584 .616 .636 .648 .656 .661 .663 .664 .666 .667 .668

e2

.989 .970 .950 .932 .915 .900 .890 .885 .872 .860 .849

e3

.584 .616 .636 .648 .656 .661 .663 .664 .666 .667 .668

Table 8: Non-symmetrical Compound Optimal Designs (1);   U [?0:1; 0:1],  U [6; 8]. (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:119; :210g f?:134; :206g f?:141; :201g f?:146; :195g f?:148; :189g f?:149; :182g f?:147; :174g f?:143; :161g f?:138; :149g

(p1; p2) (:2376; :7624) (:2690; :7310) (:2987; :7013) (:3263; :6737) (:3536; :6464) (:38025; :6200) (:4074; :5926) (:4354; :5646) (:4665; :5335) 40

e1

.267 .317 .360 .398 .432 .464 .492 .519 .532

e2

.559 .644 .712 .769 .819 .866 .908 .949 .981

e3

.984 .955 .922 .886 .848 .807 .759 .704 .630

Table 9: Non-symmetrical Compound Optimal Designs (2);  

U [?0:1; 0:1],  U [6; 8]. (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:160; :214g f?:164; :210g f?:166; :205g f?:167; :200g f?:167; :194g f?:166; :187g f?:165; :179g f?:162; :170g

(p1; p2) (:2906; :7094) (:3170; :6830) (:3425; :6575) (:3669; :6331) (:3916; :6084) (:4169; :5831) (:4425; :5575) (:4705; :5295)

e1

.370 .407 .441 .473 .504 .534 .564 .592

e2

.683 .732 .776 .816 .852 .886 .917 .946

e3

.926 .899 .870 .839 .806 .769 .727 .678

Table 10: Non-symmetrical Compound Optimal Designs (3);   U [?0:1; 0:1],  U [6; 8]. (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:180; :212g f?:181; :208g f?:181; :203g f?:180; :197g f?:179; :191g f?:177; :183g f?:191; :210g f?:190; :205g f?:189; :200g f?:188; :193g f?:198; :207g f?:197; :201g

(p1; p2) (:3551; :6449) (:3777; :6223) (:4004; :5996) (:4238; :5762) (:4476; :5524) (:4727; :5273) (:4077; :5923) (:4294; :5706) (:4517; :5483) (:4753; :5247) (:4550; :5450) (:4768; :5232)

41

e1

.467 .498 .527 .557 .587 .617 .542 .571 .600 .630 .607 .636

e2

.773 .805 .835 .864 .889 .912 .818 .842 .864 .883 .842 .858

e3

.852 .825 .797 .766 .732 .694 .786 .760 .730 .698 .726 .698

Table 11: LD25, LD50 and LD75 Optimal Designs;   U [?1; 1], 

U [6:9; 7:1].

(1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?:998 :2356 ?:943 :1814 ?:861 :1102

?:555 :1882 ?:533 :1610 ?:502 :1434



?:183 :1656 ?:183 :1532 ?:166 :1565

:166 :1569 :167 :1585 :181 :1658

:499 :1426 :531 :1651 :554 :1875

:859 :1111 :943 :1809 :999 :2365

e1

e2

1

e3

.863 .664

.909

1

.909

.666 .864

1

Table 12: Symmetrical Compound Optimal Designs;   U [?1; 1],  U [6:9; 7:1]. (1; 2; 3)

?:947 :1786 ?:951 (:1; :8; :1) :1762 ? :959 (:15; :7; :15) :1707 ? :963 (:2; :6; :2) :1686 ? :966 (:25; :5; :25) :1675 ? :971 (:3; :4; :3) :1639 :973 (1=3; 1=3; 1=3) ?:1637 :974 (:35; :3; :35) ?:1631 ?:977 (:4; :2; :4) :1620 :980 (:45; :1; :45) ?:1606 ?:983 (:5; 0; :5) :1595 (:05; :9; :05)

?:531 :1694 ?:530 :1744 ?:549 :1675 ?:551 :1695 ?:549 :1726 ?:561 :1689 ?:556 :1735 ?:559 :1717 ?:559 :1738 ?:563 :1726 ?:563 :1744



?:160 :1609 ?:150 :1628 ?:181 :1615 ?:180 :1627 ?:175 :1625 ?:188 :1659 ?:175 :1682 ?:182 :1671 ?:179 :1683 ?:184 :1687 ?:180 :1705 42

:194 :1541 :202 :1536 :179 :1612 :184 :1634 :187 :1623 :184 :1673 :192 :1624 :190 :1668 :195 :1672 :192 :1686 :195 :1666

:545 :1605 :553 :1614 :549 :1686 :555 :1680 :556 :1693 :559 :1698 :559 :1686 :564 :1692 :568 :1683 :567 :1697 :568 :1700

:950 :1765 :956 :1717 :959 :1705 :964 :1677 :967 :1658 :971 :1643 :972 :1637 :975 :1621 :979 :1605 :981 :1598 :983 :1590

e1

e2

e3

.911 1.00 .911 .913 .999 .913 .914 .999 .914 .915 .998 .915 .916 .997 .916 .916 .996 .916 .916 .996 .916 .916 .996 .916 .917 .994 .917 .917 .994 .917 .917 .993 .917

Table 13: Non-symmetrical Compound Optimal Designs (1);   U [?1; 1],  U [6:9; 7:1]. (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:879 :1136 ?:897 :1140 ?:910 :1195 ?:920 :1256 ?:926 :1333 ?:932 :1412 ?:936 :1496 ?:939 :1596 ?:940 :1710

?:505 :1527 ?:531 :1492 ?:532 :1554 ?:540 :1547 ?:541 :1560 ?:540 :1591 ?:541 :1596 ?:536 :1636 ?:527 :1676



?:155 :1587 ?:183 :1592 ?:174 :1614 ?:185 :1592 ?:186 :1582 ?:180 :1596 ?:183 :1574 ?:171 :1593 ?:160 :1573

43

:194 :1647 :175 :1687 :183 :1650 :174 :1669 :173 :1665 :177 :1626 :173 :1624 :184 :1588 :185 :1528

:563 :1822 :555 :1850 :556 :1809 :553 :1814 :551 :1795 :551 :1770 :547 :1759 :548 :1695 :541 :1667

:998 :2280 :994 :2238 :991 :2179 :987 :2122 :982 :2066 :978 :2005 :971 :1952 :964 :1892 :955 :1846

e1

e2

e3

.708 .897 .998 .729 .923 .994 .772 .942 .989 .798 .958 .982 .820 .971 .974 .841 .981 .965 .860 .989 .954 .877 .995 .942 .894 .998 .927

Table 14: Non-symmetrical Compound Optimal Designs (2);   U [?1; 1],  U [6:9; 7:1]. (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:917 :1097 ?:928 :1145 ?:934 :1223 ?:941 :1283 ?:946 :1358 ?:949 :1444 ?:951 :1535 ?:952 :1638

?:539 :1573 ?:549 :1558 ?:546 :1604 ?:554 :1582 ?:555 :1604 ?:553 :1619 ?:551 :1640 ?:544 :1673



?:180 :1629 ?:191 :1645 ?:183 :1629 ?:194 :1630 ?:191 :1626 ?:190 :1612 ?:185 :1611 ?:175 :1617

44

:179 :1672 :176 :1690 :180 :1662 :173 :1691 :174 :1666 :174 :1663 :178 :1640 :184 :1596

:557 :1841 :558 :1836 :557 :1810 :556 :1803 :554 :1790 :553 :1764 :552 :1734 :550 :1693

:997 :2188 :995 :2126 :991 :2072 :988 :2011 :983 :1956 :978 :1897 :971 :1840 :964 :1783

e1

e2

e3

.762 .933 .991 .788 .949 .986 .810 .962 .979 .830 .973 .972 .849 .982 .963 .866 .989 .953 .883 .994 .942 .898 .998 .929

Table 15: Non-symmetrical Compound Optimal Designs (3);   U [?1; 1],  U [6:9; 7:1]. (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:945 :1206 ?:949 :1284 ?:957 :1328 ?:959 :1413 ?:960 :1503 ?:962 :1590 ?:965 :1313 ?:968 :1389 ?:968 :1479 ?:970 :1561 ?:976 :1454 ?:977 :1533

?:547 :1667 ?:546 :1693 ?:562 :1621 ?:558 :1646 ?:556 :1662 ?:554 :1684 ?:566 :1635 ?:565 :1651 ?:559 :1682 ?:558 :1697 ?:564 :1698 ?:564 :1697



?:170 :1695 ?:163 :1717 ?:194 :1667 ?:189 :1654 ?:187 :1636 ?:182 :1633 ?:195 :1702 ?:194 :1678 ?:186 :1669 ?:185 :1648 ?:185 :1704 ?:189 :1681

45

:199 :1673 :205 :1630 :178 :1694 :180 :1668 :180 :1657 :182 :1641 :182 :1697 :180 :1695 :187 :1682 :183 :1663 :192 :1689 :186 :1685

:571 :1756 :572 :1730 :560 :1778 :559 :1764 :557 :1746 :555 :1714 :563 :1778 :562 :1767 :565 :1736 :560 :1729 :568 :1729 :564 :1731

:998 :2003 :994 :1947 :988 :1913 :983 :1854 :978 :1796 :971 :1739 :992 :1876 :988 :1820 :985 :1753 :978 :1703 :989 :1727 :984 :1674

e1

e2

e3

.819 .964 .976 .838 .974 .969 .855 .982 .961 .871 .988 .952 .886 .993 .941 .901 .996 .929 .859 .981 .959 .874 .987 .950 .889 .992 .940 .903 .995 .929 .891 .990 .939 .904 .993 .929

Table 16: LD25, LD50 and LD75 Optimal Designs;   U [?1; 1], 

U [6; 8].

(1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?1:00 :2376 ?:943 :1816 ?:860 :1105

?:558 :1860 ?:533 :1631 ?:500 :1437



?:183 :1683 ?:174 :1558 ?:164 :1545

:167 :1554 :175 :1550 :183 :1674

:502 :1427 :533 :1631 :557 :1860

:860 :1101 :943 :1814 1:00 :2379

e1

e2

1

e3

.862 .664

.908

1

.908

.663 .862

1

Table 17: Symmetrical Compound Optimal Designs;   U [?1; 1],  U [6; 8]. (1; 2; 3)

?:950 :1766 ?:954 (:1; :8; :1) :1742 ? :959 (:15; :7; :15) :1712 ? :964 (:2; :6; :2) :1690 ? :968 (:25; :5; :25) :1668 ? :971 (:3; :4; :3) :1649 :975 (1=3; 1=3; 1=3) ?:1630 :975 (:35; :3; :35) ?:1633 ?:978 (:4; :2; :4) :1620 :981 (:45; :1; :45) ?:1607 ?:984 (:5; 0; :5) :1594 (:05; :9; :05)

?:545 :1618 ?:545 :1660 ?:550 :1672 ?:553 :1688 ?:557 :1690 ?:560 :1695 ?:565 :1677 ?:562 :1697 ?:565 :1698 ?:566 :1704 ?:568 :1704



?:186 :1576 ?:178 :1602 ?:180 :1621 ?:181 :1624 ?:182 :1652 ?:184 :1661 ?:192 :1669 ?:187 :1669 ?:187 :1684 ?:187 :1700 ?:189 :1707 46

:170 :1594 :181 :1603 :183 :1618 :183 :1633 :186 :1642 :188 :1668 :182 :1676 :186 :1671 :188 :1687 :190 :1689 :191 :1700

:537 :1667 :547 :1654 :551 :1667 :554 :1677 :558 :1684 :562 :1681 :559 :1703 :562 :1697 :566 :1696 :567 :1693 :569 :1700

:949 :1779 :954 :1740 :959 :1710 :964 :1688 :968 :1665 :972 :1646 :973 :1645 :975 :1632 :979 :1616 :981 :1608 :984 :1595

e1

e2

e3

.910 1.00 .910 .912 .999 .912 .913 .999 .913 .914 .998 .914 .914 .997 .914 .915 .996 .915 .915 .996 .915 .915 .996 .915 .915 .994 .915 .915 .994 .915 .915 .993 .915

Table 18: Non-symmetrical Compound Optimal Designs (1);   U [?1; 1],  U [6; 8]. (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:883 :1099 ?:898 :1149 ?:911 :1186 ?:921 :1247 ?:927 :1330 ?:932 :1413 ?:936 :1502 ?:939 :1605 ?:941 :1707

?:522 :1462 ?:525 :1530 ?:540 :1506 ?:545 :1528 ?:543 :1556 ?:544 :1564 ?:542 :1591 ?:536 :1630 ?:534 :1638



?:174 :1615 ?:171 :1585 ?:188 :1605 ?:188 :1619 ?:185 :1596 ?:187 :1584 ?:182 :1581 ?:172 :1574 ?:172 :1566

47

:185 :1682 :182 :1656 :176 :1693 :178 :1680 :176 :1664 :174 :1656 :176 :1628 :181 :1588 :180 :1571

:559 :1831 :556 :1826 :556 :1815 :556 :1795 :553 :1779 :551 :1764 :550 :1737 :547 :1701 :542 :1665

:998 :2311 :995 :2254 :991 :2195 :988 :2130 :983 :2076 :978 :2019 :972 :1961 :965 :1902 :955 :1853

e1

e2

e3

.706 .897 .998 .741 .922 .994 .770 .942 .989 .796 .958 .982 .818 .970 .974 .839 .980 .964 .858 .989 .954 .876 .994 .941 .892 .998 .926

Table 19: Non-symmetrical Compound Optimal Designs (2);   U [?1; 1],  U [6; 8]. (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:916 :1106 ?:929 :1138 ?:936 :1206 ?:942 :1281 ?:946 :1360 ?:950 :1440 ?:952 :1531 ?:953 :1634

?:535 :1581 ?:554 :1535 ?:556 :1557 ?:557 :1569 ?:557 :1582 ?:557 :1608 ?:555 :1620 ?:550 :1642



?:168 :1692 ?:195 :1646 ?:196 :1637 ?:195 :1642 ?:195 :1634 ?:190 :1635 ?:189 :1625 ?:183 :1620

48

:200 :1670 :174 :1711 :173 :1702 :175 :1695 :175 :1686 :178 :1668 :178 :1654 :182 :1632

:570 :1779 :558 :1828 :557 :1813 :558 :1788 :556 :1773 :556 :1743 :553 :1722 :553 :1686

1:00 :2172 :995 :2142 :992 :2084 :988 :2024 :984 :1966 :978 :1906 :972 :1848 :965 :1786

e1

e2

e3

.759 .931 .991 .785 .948 .986 .808 .961 .979 .828 .972 .971 .847 .981 .963 .864 .989 .953 .881 .994 .941 .897 .997 .928

Table 20: Non-symmetrical Compound Optimal Designs (3);   U [?1; 1],  U [6; 8]. (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:945 :1207 ?:953 :1258 ?:957 :1331 ?:959 :1418 ?:962 :1499 ?:963 :1590 ?:967 :1310 ?:968 :1396 ?:970 :1471 ?:971 :1558 ?:977 :1454 ?:978 :1533

?:549 :1650 ?:564 :1586 ?:564 :1604 ?:560 :1629 ?:550 :1655 ?:558 :1663 ?:570 :1622 ?:564 :1658 ?:566 :1651 ?:563 :1678 ?:569 :1665 ?:567 :1683



?:170 :1725 ?:198 :1672 ?:197 :1667 ?:193 :1643 ?:187 :1662 ?:185 :1656 ?:197 :1703 ?:190 :1677 ?:195 :1669 ?:188 :1671 ?:194 :1694 ?:190 :1695

49

:207 :1690 :176 :1709 :179 :1708 :179 :1698 :185 :1667 :186 :1654 :181 :1710 :184 :1693 :180 :1694 :185 :1673 :185 :1714 :188 :1696

:577 :1724 :560 :1795 :561 :1767 :561 :1750 :561 :1717 :559 :1694 :564 :1768 :564 :1750 :562 :1744 :562 :1711 :567 :1734 :567 :1715

:999 :2004 :993 :1981 :989 :1923 :984 :1862 :979 :1800 :972 :1743 :993 :1887 :989 :1826 :984 :1771 :978 :1710 :989 :1739 :985 :1677

e1

e2

e3

.817 .964 .976 .836 .974 .969 .853 .982 .960 .869 .988 .951 .885 .993 .940 .900 .996 .928 .858 .981 .958 .873 .987 .949 .887 .991 .939 .901 .995 .928 .889 .990 .938 .902 .993 .927

Table 21: LD25, LD50 and LD75 Optimal Designs;  2 (?0:1; 0:1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:196; :097g (:8379; :1621) 1 .374 .160 (0; 1; 0) f?:112; :112g (:500; :500) .456 1 .456 (0; 0; 1) f:094 ? :210g (:1994; :8006) .160 .374 1 Table 22: Symmetrical Designs;  2 (?0:1; 0:1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:135; :134g f?:148; :148g f?:159; :159g f?:167; :167g f?:174; :174g f?:180; :180g f?:183; :183g f?:185; :185g f?:190; :190g f?:194; :194g f?:198; :198g

e1

.544 .584 .607 .620 .629 .635 .637 .638 .640 .641 .642

e2

.981 .955 .931 .910 .891 .874 .863 .858 .844 .831 .819

e3

.544 .584 .607 .620 .629 .635 .637 .638 .640 .641 .642

Table 23: Non-symmetric Designs (1);  2 (?0:1; 0:1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:129; :194g f?:141; :191g f?:147; :186g f?:149; :182g f?:149; :176g f?:147; :170g f?:144; :162g f?:138; :151g f?:129; :137g

(p1; p2) (:2089; :7911) (:2467; :7533) (:2798; :7202) (:3104; :6896) (:3394; :6606) (:3681; :6319) (:3971; :6029) (:4271; :5729) (:4608; :5392) 50

e1

.236 .289 .333 .370 .404 .434 .461 .482 .491

e2

.518 .610 .682 .741 .794 .843 .888 .932 .973

e3

.976 .941 .904 .866 .825 .781 .731 .670 .589

Table 24: Non-symmetric Designs (2);  2 (?0:1; 0:1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:167; :202g f?:168; :198g f?:168; :194g f?:168; :189g f?:166; :183g f?:164; :177g f?:160; :170g f?:156; :161g

(p1; p2) (:2780; :7220) (:3060; :6940) (:3329; :6671) (:3586; :6414) (:3839; :6161) (:4102; :5898) (:4376; :5624) (:4669; :5331)

e1

.347 .383 .416 .448 .478 .508 .536 .562

e2

.654 .705 .749 .790 .827 .862 .895 .927

e3

.906 .877 .847 .815 .781 .743 .700 .649

Table 25: Non-symmetric Designs (3);  2 (?0:1; 0:1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:182; :202g f?:181; :198g f?:179; :193g f?:177; :188g f?:175; :182g f?:171; :176g f?:189; :201g f?:188; :197g f?:186; :192g f?:183; :186g f?:194; :200g f?:192; :195g

(p1; p2) (:3485; :6515) (:3721; :6279) (:3956; :6044) (:4196; :5804) (:4447; :5553) (:4710; :5290) (:4042; :5958) (:4264; :5736) (:4496; :5504) (:4738; :5262) (:4534; :5466) (:4760; :5240)

51

e1

.444 .474 .503 .532 .561 .591 .518 .546 .575 .604 .582 .611

e2

.745 .778 .808 .837 .863 .888 .790 .814 .836 .856 .812 .829

e3

.827 .800 .771 .740 .706 .667 .760 .734 .704 .672 .700 .672

Table 26: LD25, LD50 and LD75 Optimal Designs;  2 (?0:1; 0:1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:200; :094g (:8329; :1671) 1 .374 .161 (0; 1; 0) f?:112; :112g (:5011; :4989) .453 1 .453 (0; 0; 1) f:092 ? :200g (:1677; :8323) .161 .374 1 Table 27: Symmetrical Designs (1);  2 (?0:1; 0:1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:135; :135g f?:149; :149g f?:160; :160g f?:168; :168g f?:175; :175g f?:181; :181g f?:185; :185g f?:186; :186g f?:191; :191g f?:195; :195g f?:199; :199g

e1

.543 .584 .607 .621 .630 .635 .638 .639 .641 .642 .642

e2

.981 .954 .930 .908 .889 .871 .861 .856 .842 .829 .817

e3

.543 .584 .607 .621 .630 .635 .638 .639 .641 .642 .642

Table 28: Non-symmetrical Designs (1);  2 (?0:1; 0:1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:127; :197g f?:140; :193g f?:146; :189g f?:149; :184g f?:149; :178g f?:147; :172g f?:144; :163g f?:139; :153g f?:130; :138g

(p1; p2) (:2114; :7886) (:2483; :7517) (:2812; :7188) (:3114; :6886) (:3403; :6597) (:3689; :6311) (:3977; :6023) (:4283; :5717) (:4612; :5388) 52

e1

.236 .289 .333 .371 .404 .435 .461 .482 .490

e2

.517 .609 .680 .740 .792 .841 .887 .931 .973

e3

.977 .941 .904 .866 .826 .782 .731 .670 .589

Table 29: Non-symmetrical Designs (2);  2 (?0:1; 0:1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:167; :204g f?:169; :200g f?:169; :195g f?:169; :191g f?:167; :185g f?:165; :179g f?:161; :171g f?:156; :162g

(p1; p2) (:2789; :7211) (:3065; :6935) (:3331; :6669) (:3587; :6413) (:3846; :6154) (:4107; :5893) (:4380; :5620) (:4672; :5328)

e1

.348 .383 .417 .448 .479 .508 .536 .563

e2

.653 .703 .747 .788 .825 .860 .894 .925

e3

.907 .878 .848 .816 .781 .743 .700 .649

Table 30: Non-symmetrical Designs (3);  2 (?0:1; 0:1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:183; :204g f?:182; :200g f?:181; :195g f?:179; :190g f?:176; :184g f?:173; :177g f?:191; :203g f?:189; :199g f?:187; :194g f?:184; :188g f?:196; :202g f?:194; :197g

(p1; p2) (:3487; :6513) (:3723; :6277) (:3956; :6044) (:4197; :5083) (:4445; :5555) (:4712; :5288) (:4043; :5957) (:4266; :5734) (:4496; :5504) (:4739; :5261) (:4535; :5465) (:4760; :5240)

53

e1

.445 .474 .504 .533 .562 .591 .519 .547 .575 .605 .583 .611

e2

.743 .775 .806 .834 .861 .886 .787 .811 .833 .854 .810 .827

e3

.827 .801 .773 .741 .707 .667 .761 .734 .705 .673 .701 .673

Table 31: LD25, LD50 and LD75 Optimal Designs;  2 (?1; 1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?:899 :2052 ?:865 :1487 ?:802 :0846

?:507 :2185 ?:482 :1804 ?:437 :1425



?:161 :1988 ?:140 :1819 ?:115 :1716

:139 :1543 :173 :1718 :191 :1895

:442 :1394 :495 :1709 :517 :2077

:804 :0837 :867 :1463 :900 :2042

e1 1 .914

e2

e3

.880 .691 1

.689 .879

.914 1

Table 32: Symmetrical Designs (1);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3)

?:867 :1485 ? :870 (:1; :8; :1) :1469 ? :874 (:15; :7; :15) :1438 :876 (:2; :6; :2) ?:1439 :877 (:25; :5; :25) ?:1433 :879 (:3; :4; :3) ?:1431 (:05; :9; :05)

?:477 :1864 ?:480 :1867 ?:497 :1737 ?:495 :1752 ?:496 :1746 ?:495 :1756



?:119 :1953 ?:121 :1940 ?:160 :1842 ?:157 :1841 ?:158 :1850 ?:156 :1850

54

:212 :1730 :204 :1701 :163 :1797 :165 :1802 :166 :1812 :167 :1815

:518 :1551 :511 :1592 :495 :1743 :497 :1730 :498 :1728 :499 :1722

:873 :1418 :873 :1430 :874 :1443 :876 :1437 :878 :1430 :8791 :1425

e1

e2

e3

.914 1.00 .914 .914 1.00 .914 .915 1.00 .915 .915 1.00 .915 .915 1.00 .915 .915 .999 .915

Table 33: Symmetrical Designs (2);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

?:880 :1426 ?:881 :1421 ?:882 :1419 ?:884 :1606 ?:886 :1404

?:496 :1751 ?:497 :1746 ?:497 :1750 ?:498 :1726 ?:501 :1736



?:157 :1856 ?:159 :1857 ?:158 :1865 ?:156 :1687 ?:162 :1878

:166 :1819 :165 :1824 :167 :1826 :170 :1686 :166 :1855

:499 :1725 :499 :1731 :500 :1725 :503 :1697 :502 :1721

e1

:880 :1423 :881 :1421 :883 :1416 :885 :1598 :886 :1406

e2

e3

.915 .999 .915 .915 .999 .915 .915 .999 .915 .915 .999 .915 .915 .999 .915

Table 34: Non-symmetrical Designs (1);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5)

?:819 :0869 ?:830 :0923 ?:836 :0988 ?:843 :1048 ?:848 :1118

?:458 :1405 ?:463 :1458 ?:464 :1514 ?:469 :1547 ?:470 :1603



?:137 :1711 ?:139 :1715 ?:132 :1774 ?:139 :1742 ?:133 :1795

55

:178 :1956 :175 :1925 :187 :1900 :175 :1868 :185 :1839

:514 :2073 :511 :2036 :516 :1954 :507 :1965 :511 :1882

:899 :1986 :896 :1943 :896 :1869 :892 :1829 :890 :1762

e1

e2

e3

.721 .905 .999 .750 .927 .995 .775 .944 .990 .799 .959 .983 .821 .972 .976

Table 35: Non-symmetrical Designs (2);  2 (?1; 1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:855 :1162 ?:859 :1236 ?:861 :1320 ?:863 :1400

?:486 :1567 ?:486 :1623 ?:484 :1683 ?:484 :1728



?:155 :1785 ?:150 :1818 ?:150 :1776 ?:146 :1828

:169 :1893 :177 :1864 :166 :1796 :175 :1782

:506 :1888 :508 :1816 :498 :1829 :502 :1747

:888 :1705 :884 :1643 :878 :1596 :875 :1516

e1

e2

e3

.841 .982 .967 .861 .989 .956 .879 .995 .944 .897 .999 .930

Table 36: Non-symmetrical Designs (3);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6)

?:837 :0936 ?:849 :0958 ?:853 :1028 ?:858 :1092

?:458 :1543 ?:486 :1431 ?:484 :1496 ?:485 :1555



?:118 :1801 ?:162 :1766 ?:155 :1788 ?:151 :1803

56

:195 :1867 :165 :1979 :172 :1944 :173 :1895

:518 :1959 :508 :2011 :510 :1949 :508 :1910

:899 :1893 :896 :1855 :895 :1796 :892 :1745

e1

e2

e3

.761 .934 .994 .784 .950 .988 .806 .963 .982 .827 .974 .974

Table 37: Non-symmetrical Designs (4);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:860 :1171 ?:863 :1244 ?:868 :1296 ?:868 :1385

?:482 :1604 ?:482 :1666 ?:492 :1661 ?:487 :1731



?:149 :1788 ?:144 :1810 ?:159 :1794 ?:150 :1813

57

:171 :1870 :177 :1834 :164 :1851 :169 :1802

:506 :1881 :507 :1821 :499 :1822 :500 :1760

:889 :1685 :886 :1626 :882 :1576 :878 :1510

e1

e2

e3

.846 .983 .965 .864 .990 .954 .881 .995 .943 .898 .999 .930

Table 38: Non-symmetrical Designs (5);  2 (?1; 1), 2 (6:9; 7:1);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3)

?:856 :1057 ?:863 :1103 ?:866 :1169 ?:869 :1227 ?:872 :1298 ?:873 :1368

?:472 :1592 ?:483 :1584 ?:484 :1628 ?:490 :1640 ?:490 :1688 ?:493 :1704



?:130 :1850 ?:149 :1778 ?:147 :1805 ?:153 :1816 ?:150 :1853 ?:158 :1825

:194 :1890 :170 :1890 :173 :1854 :169 :1857 :177 :1844 :165 :1835

:521 :1864 :506 :1913 :505 :1870 :505 :1848 :507 :1767 :500 :1775

:898 :1746 :893 :1731 :891 :1674 :888 :1612 :886 :1549 :881 :1493

e1

e2

e3

.812 .966 .979 .832 .976 .972 .850 .984 .963 .867 .991 .953 .884 .995 .942 .899 .999 .929

Table 39: Non-symmetrical Designs (6);  2 (?1; 1), 2 (6:9; 7:1);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:872 :1156 ?:875 :1211 ?:876 :1284 ?:878 :1353 ?:880 :1279 ?:882 :1347

?:492 :1601 ?:499 :1598 ?:499 :1668 ?:496 :1714 ?:499 :1646 ?:498 :1703



?:152 :1872 ?:166 :1832 ?:157 :1859 ?:154 :1885 ?:163 :1862 ?:160 :1849

58

:181 :1927 :164 :1920 :172 :1869 :176 :1858 :170 :1926 :166 :1867

:516 :1814 :505 :1844 :506 :1784 :509 :1727 :510 :1771 :503 :1762

:895 :1631 :891 :1595 :888 :1536 :886 :1463 :891 :1516 :887 :1472

e1

e2

e3

.853 .985 .961 .870 .991 .952 .885 .995 .941 .900 .998 .929 .887 .995 .940 .901 .998 .928

Table 40: LD25, LD50 and LD75 Optimal Designs;  2 (?1; 1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?:898 :2081 ?:867 :1477 ?:803 :0842

?:505 :2183 ?:488 :1774 ?:439 :1411



?:162 :1941 ?:150 :1792 ?:128 :1600

:137 :1573 :164 :1754 :172 :1936

:444 :1392 :493 :1729 :510 :2143

:805 :0829 :867 :1474 :900 :2068

e1 1 .913

e2

e3

.879 .688 1

.688 .879

.913 1

Table 41: Symmetrical Designs (1);  2 (?1; 1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3)

?:869 :1464 ? :872 (:1; :8; :1) :1450 ? :874 (:15; :7; :15) :1448 :876 (:2; :6; :2) ?:1441 :878 (:25; :5; :25) ?:1431 :880 (:3; :4; :3) ?:1430 (:05; :9; :05)

?:488 :1799 ?:497 :1727 ?:496 :1741 ?:495 :1751 ?:499 :1734 ?:498 :1738



?:138 :1907 ?:161 :1838 ?:158 :1841 ?:157 :1840 ?:161 :1846 ?:160 :1852

59

:188 :1744 :165 :1820 :167 :1818 :167 :1813 :164 :1827 :166 :1834

:507 :1645 :498 :1716 :499 :1711 :499 :1722 :498 :1730 :500 :1720

:873 :1441 :872 :1449 :874 :1441 :877 :1434 :878 :1433 :880 :1426

e1

e2

e3

.913 1.00 .914 .914 1.00 .914 .914 1.00 .914 .914 1.00 .914 .914 1.00 .914 .914 1.00 .914

Table 42: Symmetrical Designs (2);  2 (?1; 1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

?:880 :1421 ?:882 :1419 ?:883 :1416 ?:884 :1415 ?:886 :1411

?:500 :1734 ?:501 :1732 ?:500 :1741 ?:500 :1737 ?:501 :1738



?:161 :1867 ?:161 :1866 ?:160 :1871 ?:159 :1888 ?:159 :1890

:167 :1839 :167 :1844 :168 :1846 :171 :1860 :171 :1862

:502 :1721 :502 :1719 :503 :1712 :505 :1697 :506 :1695

e1

:881 :1419 :882 :1419 :883 :1415 :885 :1408 :887 :1404

e2

e3

.914 .999 .914 .914 .999 .914 .914 .999 .914 .914 .999 .914 .914 .999 .914

Table 43: Non-symmetrical Designs (1);  2 (?1; 1), 2 (6; 8);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5)

?:817 :0882 ?:828 :0933 ?:840 :0964 ?:846 :1033 ?:851 :1101

?:450 :1461 ?:459 :1475 ?:480 :1428 ?:479 :1496 ?:480 :1554



?:119 :1757 ?:133 :1711 ?:157 :1734 ?:151 :1754 ?:147 :1775

60

:194 :1896 :178 :1893 :165 :1951 :171 :1916 :175 :1886

:519 :2015 :510 :2036 :507 :2030 :508 :1965 :509 :1911

:900 :1989 :897 :1952 :895 :1892 :893 :1836 :891 :1771

e1

e2

e3

.720 .905 .999 .748 .926 .995 .774 .944 .990 .798 .959 .984 .820 .972 .976

Table 44: Non-symmetrical Designs (2);  2 (?1; 1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:857 :1156 ?:859 :1241 ?:861 :1322 ?:863 :1403

?:488 :1569 ?:484 :1638 ?:484 :1691 ?:484 :1744



?:156 :1780 ?:148 :1803 ?:147 :1789 ?:143 :1822

:168 :1885 :175 :1839 :174 :1827 :176 :1767

:505 :1889 :506 :1826 :506 :1786 :501 :1736

:887 :1720 :884 :1653 :880 :1584 :874 :1529

e1

e2

e3

.840 .981 .966 .860 .989 .956 .878 .995 .943 .899 .999 .929

Table 45: Non-symmetrical Designs (3);  2 (?1; 1), 2 (6; 8);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6)

?:835 :0943 ?:846 :0979 ?:855 :1017 ?:859 :1091

?:459 :1509 ?:477 :1476 ?:489 :1483 ?:488 :1536



?:126 :1787 ?:150 :1738 ?:160 :1776 ?:156 :1792

61

:194 :1927 :169 :1943 :165 :1937 :171 :1930

:523 :1954 :509 :2000 :507 :1976 :510 :1904

:901 :1881 :896 :1864 :894 :1811 :893 :1748

e1

e2

e3

.759 .933 .993 .783 .950 .988 .805 .963 .981 .826 .974 .973

Table 46: Non-symmetrical Designs (4);  2 (?1; 1), 2 (6; 8);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:861 :1168 ?:864 :1241 ?:867 :1306 ?:870 :1378

?:486 :1591 ?:485 :1652 ?:488 :1696 ?:488 :1751



?:151 :1813 ?:144 :1840 ?:144 :1838 ?:144 :1855

62

:175 :1885 :182 :1856 :178 :1823 :179 :1792

:510 :1854 :512 :1790 :507 :1772 :506 :1723

:890 :1689 :888 :1621 :883 :1566 :879 :1501

e1

e2

e3

.845 .983 .964 .863 .990 .954 .881 .995 .942 .897 .999 .929

Table 47: Non-symmetrical Designs (5);  2 (?1; 1), 2 (6; 8);   Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3)

?:858 :1041 ?:866 :1083 ?:868 :1150 ?:871 :1216 ?:873 :1293 ?:873 :1376

?:484 :1504 ?:496 :1512 ?:496 :1554 ?:496 :1611 ?:493 :1683 ?:492 :1717



?:157 :1779 ?:166 :1798 ?:163 :1822 ?:158 :1858 ?:152 :1847 ?:153 :1841

:173 :1985 :164 :1955 :168 :1933 :173 :1890 :175 :1851 :172 :1835

:517 :1937 :507 :1923 :508 :1870 :509 :1818 :507 :1774 :505 :1739

:899 :1754 :894 :1730 :892 :1671 :889 :1608 :886 :1552 :882 :1492

e1

e2

e3

.811 .966 .979 .830 .976 .971 .849 .984 .963 .866 .991 .953 .883 .996 .941 .899 .999 .928

Table 48: Non-symmetrical Designs (6);  2 (?1; 1), 2 (6; 8);  

Beta (2; 2) ;  Beta (2; 2). (1; 2; 3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:870 :1173 ?:876 :1210 ?:877 :1286 ?:878 :1362 ?:881 :1277 ?:882 :1346

?:486 :1630 ?:501 :1588 ?:497 :1662 ?:495 :1712 ?:503 :1632 ?:501 :1691



?:145 :1837 ?:167 :1844 ?:158 :1849 ?:154 :1871 ?:166 :1861 ?:161 :1871

63

:176 :1846 :165 :1920 :170 :1876 :174 :1844 :166 :1915 :169 :1878

:507 :1856 :506 :1837 :506 :17864 :507 :1733 :507 :1785 :506 :1743

:894 :1658 :891 :1601 :888 :1542 :885 :1478 :890 :1531 :888 :1470

e1

e2

e3

.852 .985 .961 .868 .991 .951 .884 .996 .940 .900 .998 .928 .886 .995 .939 .900 .998 .928

Table 49: LD25, LD50 and LD75 Optimal Designs;  2 (?0:1; 0:1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:197; :097g (:8376; :1624) 1 .374 .160 (0; 1; 0) f?:112; :112g (:501; :499) .456 1 .456 (0; 0; 1) f:097 ? :196g (:1622; :8378) .160 .374 1 Table 50: Symmetrical Designs;  2 (?0:1; 0:1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:134; :134g f?:148; :148g f?:159; :159g f?:167; :167g f?:174; :174g f?:180; :180g f?:183; :183g f?:185; :185g f?:190; :190g f?:194; :194g f?:198; :198g

e1

.544 .584 .607 .620 .629 .635 .637 .638 .640 .641 .642

e2

.981 .955 .931 .910 .891 .874 .863 .858 .844 .831 .819

e3

.544 .584 .607 .620 .629 .635 .637 .638 .640 .641 .642

Table 51: Non-symmetric Designs (1);  2 (?0:1; 0:1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:129; :194g f?:141; :191g f?:147; :187g f?:149; :182g f?:149; :176g f?:147; :170g f?:144; :162g f?:138; :152g f?:129; :137g

(p1; p2) (:2089; :7911) (:2467; :7533) (:2801; :7199) (:3104; :6896) (:3394; :6606) (:3683; :6317) (:3972; :6028) (:4276; :5724) (:4609; :5391) 64

e1

.236 .289 .333 .370 .404 .434 .461 .482 .491

e2

.518 .610 .682 .741 .794 .843 .889 .932 .973

e3

.976 .941 .904 .866 .825 .781 .731 .670 .589

Table 52: Non-symmetric Designs (2);  2 (?0:1; 0:1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:166; :202g f?:168; :198g f?:168; :194g f?:168; :189g f?:166; :183g f?:164; :177g f?:161; :170g f?:155; :161g

(p1; p2) (:2784; :7216) (:3061; :6939) (:3328; :6672) (:3584; :6416) (:3842; :6158) (:4103; :5897) (:4375; :5625) (:4671; :5329)

e1

.347 .383 .416 .448 .478 .508 .536 .562

e2

.654 .705 .749 .790 .827 .862 .896 .927

e3

.906 .877 .847 .815 .780 .743 .700 .649

Table 53: Non-symmetric Designs (3);  2 (?0:1; 0:1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:182; :202g f?:181; :198g f?:179; :194g f?:178; :188g f?:175; :182g f?:171; :176g f?:189; :202g f?:188; :197g f?:186; :192g f?:183; :187g f?:194; :200g f?:192; :195g

(p1; p2) (:3485; :6515) (:3721; :6279) (:3956; :6044) (:4194; :5806) (:4445; :5555) (:4709; :5291) (:4043; :5957) (:4265; :5735) (:4494; :5506) (:4739; :5261) (:4536; :5464) (:4759; :5241)

65

e1

.444 .474 .503 .532 .561 .591 .518 .546 .575 .604 .582 .611

e2

.745 .778 .808 .837 .863 .888 .790 .814 .836 .856 .813 .829

e3

.827 .800 .771 .740 .706 .667 .760 .734 .705 .672 .700 .672

Table 54: LD25, LD50 and LD75 Optimal Designs;  2 (?0:1; 0:1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) fx1; x2g (p1; p2) e1 e2 e3 (1; 0; 0) f?:205; :085g (:8256; :1744) 1 .373 .161 (0; 1; 0) f?:112; :113g (:5011; :4989) .449 1 .449 (0; 0; 1) f:085 ? :205g (:1746; :8354) .161 .373 1 Table 55: Symmetrical Designs (1);  2 (?0:1; 0:1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:05; :9; :05) (:1; :8; :1) (:15; :7; :15) (:2; :6; :2) (:25; :5; :25) (:3; :4; :3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

fx1; x2g f?:136; :136g f?:151; :151g f?:161; :161g f?:170; :170g f?:177; :177g f?:183; :183g f?:187; :187g f?:188; :188g f?:193; :193g f?:197; :197g f?:201; :201g

e1

.542 .584 .607 .621 .630 .636 .638 .639 .641 .642 .643

e2

.980 .952 .927 .905 .886 .869 .858 .853 .838 .825 .813

e3

.542 .584 .607 .621 .630 .636 .638 .639 .641 .642 .643

Table 56: Non-symmetrical Designs (1);  2 (?0:1; 0:1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

fx1; x2g f?:124; :201g f?:138; :197g f?:146; :192g f?:148; :187g f?:149; :181g f?:148; :174g f?:145; :166g f?:139; :155g f?:130; :140g

(p1; p2) (:2147; :7853) (:2507; :7493) (:2825; :7175) (:3129; :6871) (:3417; :6583) (:3700; :6300) (:3988; :6012) (:4291; :5709) (:4623; :5377) 66

e1

.236 .289 .333 .370 .404 .434 .461 .481 .488

e2

.515 .606 .677 .737 .790 .839 .885 .929 .972

e3

.977 .942 .905 .866 .826 .782 .731 .670 .587

Table 57: Non-symmetrical Designs (2);  2 (?0:1; 0:1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

fx1; x2g f?:167; :207g f?:169; :203g f?:170; :198g f?:169; :193g f?:168; :188g f?:166; :181g f?:162; :174g f?:158; :164g

(p1; p2) (:2801; :7199) (:3074; :6926) (:3337; :6663) (:3596; :6404) (:3850; :6150) (:4110; :5890) (:4384; :5616) (:4672; :5328)

e1

.348 .384 .417 .449 .479 .508 .536 .562

e2

.651 .700 .744 .785 .822 .858 .891 .923

e3

.907 .878 .848 .816 .781 .744 .701 .649

Table 58: Non-symmetrical Designs (3);  2 (?0:1; 0:1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

fx1; x2g f?:184; :207g f?:183; :202g f?:182; :198g f?:180; :192g f?:177; :186g f?:174; :179g f?:193; :205g f?:191; :201g f?:189; :196g f?:186; :190g f?:198; :204g f?:195; :199g

(p1; p2) (:3490; :6510) (:3725; :6275) (:3961; :6039) (:4200; :5800) (:4449; :55515) (:4714; :5286) (:4043; :5957) (:4266; :5734) (:4497; :5503) (:4741; :5259) (:4535; :5465) (:4761; :5239)

67

e1

.445 .475 .504 .533 .562 .592 .519 .547 .576 .605 .583 .612

e2

.740 .772 .803 .831 .858 .883 .784 .808 .830 .851 .807 .823

e3

.828 .801 .772 .741 .707 .667 .761 .735 .705 .673 .701 .673

Table 59: LD25, LD50 and LD75 Optimal Designs;  2 (?1; 1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?:900 :2053 ?:864 :1496 ?:803 :0838

?:511 :2136 ?:475 :1859 ?:442 :1402



?:172 :1982 ?:123 :1873 ?:123 :1696

:143 :1686 :191 :1665 :188 :1960

:457 :1330 :501 :1650 :520 :2073

:808 :0813 :868 :1456 :901 :2031

e1 1 .914

e2

e3

.880 .690 1

.690 .879

.914 1

Table 60: Symmetrical Designs (1);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3)

?:868 :1477 ? :871 (:1; :8; :1) :1456 ? :874 (:15; :7; :15) :1442 :876 (:2; :6; :2) ?:1437 :877 (:25; :5; :25) ?:1433 :879 (:3; :4; :3) ?:143 (:05; :9; :05)

?:480 :1844 ?:487 :1811 ?:493 :1776 ?:497 :1729 ?:496 :1747 ?:496 :1747



?:128 :1907 ?:138 :1894 ?:153 :1833 ?:163 :1829 ?:158 :1846 ?:158 :1850

68

:195 :1714 :182 :1714 :166 :1768 :160 :1823 :166 :1820 :166 :1824

:509 :1634 :500 :1683 :494 :1734 :496 :1744 :498 :1722 :498 :1721

:873 :1423 :873 :1441 :874 :1446 :876 :1438 :877 :1431 :879 :1428

e1

e2

e3

.915 1.00 .914 .914 1.00 .914 .915 1.00 .914 .915 1.00 .915 .915 1.00 .915 .915 .999 .915

Table 61: Symmetrical Designs (2);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

?:880 :1423 ?:881 :1422 ?:882 :1417 ?:884 :1414 ?:886 :1406

?:498 :1746 ?:497 :1752 ?:498 :1744 ?:498 :1746 ?:501 :1731



?:160 :1853 ?:157 :1863 ?:159 :1864 ?:159 :1864 ?:163 :1869

:165 :1830 :168 :1833 :167 :1840 :167 :1848 :164 :1862

:498 :1722 :501 :1713 :501 :1721 :502 :1718 :501 :1727

e1

:880 :1427 :881 :1417 :882 :1414 :884 :1409 :886 :1405

e2

e3

.915 .999 .915 .915 .999 .915 .915 .999 .915 .915 .999 .915 .915 .999 .915

Table 62: Non-symmetrical Designs (1);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5)

?:817 :0880 ?:830 :0916 ?:838 :0981 ?:845 :1037 ?:845 :1138

?:451 :1450 ?:467 :1440 ?:469 :1500 ?:473 :1542 ?:460 :1672



?:124 :1740 ?:139 :1769 ?:135 :1796 ?:137 :1794 ?:106 :1910

69

:189 :1912 :183 :1940 :186 :1891 :182 :1867 :221 :1810

:517 :2034 :515 :2005 :514 :1956 :512 :1943 :529 :1741

:899 :1983 :897 :1930 :895 :1877 :893 :1816 :893 :1729

e1

e2

e3

.722 .905 .999 .750 .927 .995 .776 .945 .990 .799 .959 .984 .821 .972 .976

Table 63: Non-symmetrical Designs (2);  2 (?1; 1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:854 :1177 ?:858 :1243 ?:859 :1332 ?:863 :1405

?:482 :1566 ?:483 :1639 ?:478 :1712 ?:482 :1750



?:157 :1730 ?:149 :1781 ?:140 :1797 ?:141 :1830

:161 :1883 :172 :1852 :176 :1780 :179 :1770

:501 :1924 :505 :1840 :502 :1789 :503 :1726

:886 :1719 :884 :1646 :879 :1590 :875 :1519

e1

e2

e3

.841 .982 .967 .861 .989 .956 .879 .995 .944 .897 .999 .930

Table 64: Non-symmetrical Designs (3);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6)

?:837 :0937 ?:849 :0958 ?:853 :1032 ?:857 :1098

?:458 :1534 ?:483 :1471 ?:481 :1517 ?:483 :1564



?:118 :1825 ?:147 :1844 ?:150 :1786 ?:148 :1804

70

:199 :1873 :184 :1954 :175 :1931 :177 :1912

:520 :1941 :517 :1935 :511 :1940 :512 :1883

:899 :1889 :897 :1838 :894 :1793 :892 :1739

e1

e2

e3

.760 .934 .994 .784 .950 .988 .806 .963 .981 .827 .974 .974

Table 65: Non-symmetrical Designs (4);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:860 :1168 ?:863 :1237 ?:866 :1313 ?:868 :1389

?:484 :1597 ?:484 :1663 ?:484 :1705 ?:484 :1760



?:152 :1780 ?:145 :1811 ?:143 :1835 ?:140 :1851

71

:168 :1872 :176 :1845 :178 :1811 :180 :1774

:504 :1891 :507 :1817 :506 :1775 :505 :1725

:889 :1692 :886 :1627 :883 :1563 :879 :1501

e1

e2

e3

.846 .983 .965 .864 .990 .954 .881 .995 .943 .898 .999 .930

Table 66: Non-symmetrical Designs (5);  2 (?1; 1), 2 (6:9; 7:2);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3)

?:859 :1035 ?:865 :1088 ?:867 :1162 ?:870 :1225 ?:871 :1299 ?:874 :1368

?:486 :1505 ?:491 :1534 ?:489 :1588 ?:490 :1642 ?:491 :1681 ?:492 :1717



?:158 :1780 ?:160 :1796 ?:154 :1824 ?:153 :1819 ?:151 :1849 ?:156 :1821

:172 :1996 :165 :1929 :173 :1900 :170 :1865 :175 :1846 :166 :1830

:514 :1922 :506 :1926 :5085 :1860 :505 :1835 :506 :1776 :500 :1768

:897 :1762 :893 :1727 :892 :1665 :888 :1613 :885 :1549 :881 :1496

e1

e2

e3

.812 .966 .979 .832 .976 .972 .850 .984 .963 .867 .991 .953 .884 .996 .942 .899 .999 .929

Table 67: Non-symmetrical Designs (6);  2 (?1; 1), 2 (6:9; 7:2);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:870 :1172 ?:875 :1215 ?:876 :1287 ?:879 :1349 ?:880 :1284 ?:881 :1349

?:485 :1642 ?:497 :1615 ?:495 :1666 ?:498 :1698 ?:496 :1675 ?:497 :1714



?:141 :1866 ?:163 :1819 ?:158 :1840 ?:161 :1850 ?:155 :1882 ?:158 :1851

72

:186 :1875 :164 :1912 :170 :1889 :168 :1873 :175 :1877 :170 :1878

:516 :1821 :505 :1841 :507 :1783 :504 :1753 :509 :1761 :506 :1743

:896 :1625 :891 :1597 :888 :1534 :885 :1476 :891 :1520 :887 :1465

e1

e2

e3

.853 .985 .961 .870 .991 .951 .885 .996 .941 .900 .998 .929 .887 .995 .940 .901 .998 .928

Table 68: LD25, LD50 and LD75 Optimal Designs;  2 (?1; 1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (1; 0; 0) (0; 1; 0) (0; 0; 1)

?:904 :2028 ?:868 :1467 ?:807 :0814

?:532 :1973 ?:499 :1690 ?:453 :1364



?:218 :1864 ?:172 :1775 ?:132 :1702

:095 :1821 :151 :1833 :182 :1950

:434 :1468 :491 :1756 :515 :2096

:801 :0847 :867 :1479 :901 :2074

e1 1 .912

e2

e3

.877 .685 1

.686 .877

.912 1

Table 69: Symmetrical Designs (1);  2 (?1; 1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3)

?:871 :1454 ? :871 (:1; :8; :1) :1473 ? :876 (:15; :7; :15) :1438 :877 (:2; :6; :2) ?:1440 :878 (:25; :5; :25) ?:1437 :880 (:3; :4; :3) ?:1431 (:05; :9; :05)

?:500 :1705 ?:489 :1776 ?:503 :1706 ?:500 :1723 ?:499 :1735 ?:500 :1730



?:171 :1772 ?:143 :1896 ?:166 :1852 ?:162 :1856 ?:159 :1871 ?:160 :1866

73

:152 :1852 :190 :1810 :164 :1847 :168 :1844 :171 :1835 :168 :1839

:4941 :1752 :515 :1626 :501 :1714 :503 :1703 :505 :1698 :503 :1708

:870 :1464 :876 :1418 :875 :1443 :877 :1434 :880 :1424 :881 :1427

e1

e2

e3

.913 1.00 .913 .913 1.00 .913 .913 1.00 .913 .913 1.00 .913 .913 1.00 .913 .913 .999 .913

Table 70: Symmetrical Designs (2);  2 (?1; 1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (1=3; 1=3; 1=3) (:35; :3; :35) (:4; :2; :4) (:45; :1; :45) (:5; 0; :5)

?:882 :1423 ?:882 :1421 ?:884 :1419 ?:885 :1417 ?:887 :1411

?:502 :1723 ?:502 :1723 ?:503 :1720 ?:503 :1724 ?:504 :1720



?:163 :1866 ?:163 :1866 ?:163 :1876 ?:162 :1882 ?:163 :1888

:166 :1853 :167 :1857 :167 :1860 :169 :1869 :169 :1875

:503 :1712 :504 :1711 :504 :1706 :507 :1699 :507 :1699

e1

:882 :1424 :882 :1422 :884 :1419 :886 :1409 :887 :1408

e2

e3

.913 .999 .913 .913 .999 .913 .914 .999 .914 .914 .999 .914 .914 .999 .914

Table 71: Non-symmetrical Designs (1);  2 (?1; 1), 2 (6; 9);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :1; :9) (0; :2; :8) (0; :3; :7) (0; :4; :6) (0; :5; :5)

?:821 :0854 ?:830 :0921 ?:841 :0958 ?:847 :1026 ?:852 :1095

?:466 :1378 ?:466 :1440 ?:482 :1431 ?:482 :1489 ?:484 :1528



?:140 :1749 ?:140 :1724 ?:157 :1727 ?:151 :1771 ?:155 :1751

74

:183 :1975 :178 :1940 :166 :1956 :1731 :1914 :169 :1921

:518 :2042 :514 :2024 :509 :2020 :509 :1951 :509 :1922

:900 :2001 :898 :1951 :895 :1908 :893 :1848 :891 :1783

e1

e2

e3

.717 .904 .999 .746 .925 .995 .772 .944 .990 .796 .959 .983 .818 .971 .975

Table 72: Non-symmetrical Designs (2);  2 (?1; 1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (0; :6; :4) (0; :7; :3) (0; :8; :2) (0; :9; :1)

?:855 :1171 ?:858 :1254 ?:861 :1325 ?:864 :1407

?:483 :1595 ?:480 :1658 ?:486 :1667 ?:487 :1717



?:146 :1804 ?:138 :1822 ?:151 :1788 ?:149 :1805

:180 :1870 :183 :1811 :168 :1810 :171 :1790

:512 :1847 :510 :1800 :501 :1811 :501 :1750

:889 :1713 :885 :1655 :880 :1600 :875 :1531

e1

e2

e3

.839 .981 .966 .858 .989 .955 .877 .995 .943 .895 .999 .929

Table 73: Non-symmetrical Designs (3);  2 (?1; 1), 2 (6; 9);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; 0; :9) (:1; :1; :8) (:1; :2; :7) (:1; :3; :6)

?:840 :0921 ?:852 :0945 ?:856 :1014 ?:860 :1084

?:465 :1515 ?:495 :1393 ?:495 :1446 ?:492 :1512



?:1236 :1828 ?:171 :1763 ?:166 :1796 ?:159 :1818

75

:197 :1884 :161 :2007 :167 :1969 :174 :1948

:522 :1943 :509 :2015 :509 :1957 :513 :1889

:900 :1908 :897 :1877 :895 :1818 :894 :1748

e1

e2

e3

.757 .933 .993 .781 .949 .988 .803 .962 .981 .824 .973 .973

Table 74: Non-symmetrical Designs (4);  2 (?1; 1), 2 (6; 9);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:1; :4; :5) (:1; 5; :4) (:1; :6; :3) (:1; :7; :2)

?:862 :1163 ?:864 :1240 ?:867 :1316 ?:870 :1380

?:488 :1593 ?:486 :1647 ?:487 :1690 ?:492 :1729



?:150 :1819 ?:146 :1824 ?:144 :1859 ?:147 :1864

76

:178 :1891 :180 :1866 :186 :1852 :178 :1794

:512 :1843 :512 :1794 :515 :1724 :507 :1727

:891 :1690 :888 :1629 :885 :1559 :880 :1505

e1

e2

e3

.843 .983 .964 .862 .990 .954 .879 .995 .942 .897 .999 .928

Table 75: Non-symmetrical Designs (5);  2 (?1; 1), 2 (6; 9);   Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:2; 0; :8) (:2; :1; :7) (:2; :2; :6) (:2; :3; :5) (:2; :4; :4) (:2; :5; :3)

?:861 :1024 ?:866 :1081 ?:870 :1142 ?:872 :1211 ?:874 :1286 ?:874 :1376

?:491 :1478 ?:497 :1516 ?:502 :1526 ?:501 :1588 ?:497 :1665 ?:494 :1709



?:162 :1793 ?:164 :1805 ?:172 :1802 ?:165 :1836 ?:155 :1863 ?:152 :1867

:172 :2009 :166 :1950 :160 :1956 :167 :1910 :176 :1869 :176 :1828

:518 :1932 :509 :1908 :506 :1891 :507 :1830 :510 :1765 :508 :1730

:899 :1764 :894 :1740 :892 :1683 :889 :1625 :887 :1551 :883 :1490

e1

e2

e3

.809 .965 .979 .829 .975 .971 .847 .984 .962 .865 .990 .952 .881 .995 .941 .898 .999 .928

Table 76: Non-symmetrical Designs (6);  2 (?1; 1), 2 (6; 9);  

Beta (2; 2) ;  Beta (1; 2). (1; 2; 3) (:3; 0; :7) (:3; :1; :6) (:3; :2; :5) (:3; :3; :4) (:4; 0; :6) (:4; :1; :5)

?:875 :1143 ?:876 :1217 ?:877 :1291 ?:879 :1357 ?:881 :1282 ?:884 :1343

?:503 :1538 ?:500 :1597 ?:499 :1643 ?:501 :1673 ?:502 :1642 ?:505 :1676



?:170 :1823 ?:163 :1848 ?:160 :1858 ?:165 :1840 ?:161 :1886 ?:166 :1866

77

:163 :1957 :169 :1915 :171 :1893 :165 :1886 :173 :1898 :167 :1898

:508 :1873 :509 :1820 :509 :1769 :505 :1755 :511 :1766 :507 :1744

:894 :1666 :892 :1604 :889 :1546 :885 :1489 :892 :1526 :888 :1474

e1

e2

e3

.851 .985 .960 .867 .991 .951 .883 .995 .940 .898 .998 .927 .885 .995 .938 .899 .998 .927

0.4

Efficiency

0.6

0.8

1.0

Appendix C: Figures for the LD50 and Slope Estimation Problem

0.0

0.2

e1 e2

0.0

0.2

0.4

0.6

0.8

1.0

lambda

Figure 6: Eciency plot of compound optimal designs for the estimation of the LD50 and the slope when   U [?0:1; 0:1] and  U [6:9; 7:1].

78

1.0 0.8 0.6 0.4

Efficiency 0.0

0.2

e1 e2

0.0

0.2

0.4

0.6

0.8

1.0

lambda

0.90

Efficiency

0.95

1.00

Figure 7: Eciency plot of compound optimal designs for the estimation of the LD50 and the slope when   U [?0:1; 0:1] and  U [6; 8].

0.80

0.85

e1 e2

0.0

0.2

0.4

0.6

0.8

1.0

lambda

Figure 8: Eciency plot of compound optimal designs for the estimation of the LD50 and the slope when   U [?1; 1] and  U [6:9; 7:1]. 79

1.00 0.95 0.90

Efficiency

0.80

0.85

e1 e2

0.0

0.2

0.4

0.6

0.8

1.0

lambda

Figure 9: Eciency plot of compound optimal designs for the estimation of the LD50 and the slope when   U [?1; 1] and  U [6; 8].

80

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