multiple-objective optimal designs for the logit model - Semantic Scholar

MULTIPLE-OBJECTIVE OPTIMAL DESIGNS FOR THE LOGIT MODEL Wei Zhu, Hongshik Ahn Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, NY 11794-3600 Weng Kee Wong Department of Biostatistics University of California, Los Angeles Los Angeles, CA 90095-1772 Key Words and Phrases: Bayesian Compound Optimal Design; Design Eciency; Dose-response Experiment.

ABSTRACT The toxicity and eciency of a drug must be determined before it is approved for general use. Quantal dose-response experiments are routinely conducted to examine the response rates at dose levels of interest. Multiple-objective optimal designs for such studies are found and compared with the popular equal allocation rules.

1. INTRODUCTION The study of drug toxicity and eciency is a crucial part of the drug development process. \In Phase I trials, the problem is to identify one of a given set of doses that is closest to an acceptable level of toxicity", Richards, et al. (1997). Quantal dose-response experiments are routinely conducted to examine the response rates at several dose levels of interest. A common design strategy in these experiments is the simple uniform equal allocation rule. For example, Loose et al: (1993) conducted a double-blind, placebo-controlled, dose response study of Tenidap in Rheumatoid 1

Arthritis where equal number of subjects were allocated to 40mg, 80mg, 120mg/day or placebo in an eight-week trial; see also example 2 in Chaloner and Verdinelli (1995). In this paper, we compare the performance of such rules with Bayesian multiple-objective optimal designs for estimating a set of percentiles. This set of percentiles either comprises very low percentiles as in toxicity studies or very high percentiles as in ecacy studies. Since the response of interest in dose-response study is often binary, for example, toxic or otherwise, a usual model is the simple logit model log 1 ? (x()x) = (x ? ) ;

(1)

where  (x) = 1= [1 + exp (? (x ? ))] is the probability of a response at dosage x. The dosage levels of interest are usually con ned in an interval  selected by the researcher. The parameter is the slope in the logit scale. The location parameter  is the \median lethal dose" and denoted by LD50. In general, we use LD100 to denote the dose level x0 at which the probability of a response is  , 0    1. From (1), we have LD100 = x0 =  + = ; (2) where = log = (1 ?  ). Therefore the LD100 is a function of the model parameters T = (; ) and the response probability  . In this paper, we will consider designing a dose-response experiment to determine the dosage levels for some pre-speci ed toxicity/ecacy levels of interest. As an illustration, we assume the toxicity levels are the LD1, the LD5 and the LD10, and the ecacy levels are the LD99, the LD95 and the LD90. These dose levels may also be dierent in importance to the researchers; for example, the estimation of the LD1 may be most important, followed by that of the LD5, and the LD10. Consequently, dierent precision requirement might be imposed on the estimation of each percentile. A design which satis es several objectives of various importance simultaneously is called a multiple-objective optimal design (Cook and Wong, 1994). A brief overview of the traditional single-objective optimal designs for the dose2

response experiment is given in the next section. Bayesian multiple-objective optimal designs for estimating the three toxicity percentiles are developed in section 3, and section 4 enumerates the design eciencies of the uniform equal allocation rules for estimating these percentiles. In section 5, we deduce the corresponding results for ecacy studies from section 4.

2. BACKGROUND Suppose at the onset, a total of N subjects is allocated for the experiment. The design questions are the selection of the dosage levels and an allocation scheme for these subjects to the selected dosage levels in an optimal way. Most frequentists' design criteria for the logit model are based on the Fisher information matrix. Let the response data be denoted by y T = (y1;


; yN ), where yi = 1 or 0 depending on whether the ith subject responds or not, and suppose the contribution to the likelihood of an observation at the design point xi is  (yi j xi ; ), where  (1 j xi ; ) =  (xi ) and  (0 j xi ; ) = 1 ?  (xi). The (i; j ) th element of the normalized observed Fisher information matrix M (;  ) for the design  supported at x1;


; xN is given by Z 2 (3) [M (;  )]ij = ? @@@ log ( (y j x; ))  (dx) : i j

Notice that we have followed Kiefer's approximate design approach by replacing the summation sign with an integral. This means designs are treated as probability measures, see Kiefer (1959, 1985) for details and motivation behind this approach. In quantal dose-response experiment, interest is often on obtaining accurate estimates of percentiles (Wu, 1988; Chaloner and Larntz, 1989). Since the LD100 is a function of T = (; ), Wu (1985) developed some c ? optimal designs for estimating a function of the parameters , c (), by minimizing the asymptotic variance (rc ())T M ?1 (;  ) (rc ()) ; of the estimate. Here rc () is the vector of partial derivatives of c (). However, 3

the resulting design depends on the unknown parameters  and . One solution is to guess the values of the unknown parameters and construct locally optimal designs (Cherno, 1953). Another way is to use a prior distribution on the unknown parameters rather than a single best guess. Chaloner and Larntz (1989) proposed the Bayesian c ? optimality criterion given by h

i

 ( ) = E (rc ())T M ?1 (;  ) (rc ()) ; where the expectation is taken over the prior distribution of . The Bayesian c ? optimal design   is the one which minimizes  ( ). The design eciency of an arbitrary design  relative to   is de ned as

E () =  (  ) = () : It can be viewed roughly as the ratio of the variances for the estimated c () given by the two designs. Clearly, 0 < E ( )  1 and designs with high eciencies are desirable because they utilize less resources. For example, for a design with eciency 0.5, we need two replicates of it in order to do as well as the optimal design.

3. COMPOUND OPTIMAL DESIGNS The compound optimal design approach is one of the two approaches for nding a multiple-objective optimal design (Cook and Wong, 1994; Clyde and Chaloner, 1996). Suppose there are m distinct objectives as implemented by the convex func?
tions i , i = 1;


; m. Let i ( ) = i ( ) =i  i be a convex function of the eciency of the design  relative to  i , where  i is the i ? optimal design, i = 1;


; m. The compound optimal design  is the design which minimizes the weighted average of the m functions, m X  ( j ) = i i ( ) ; (4) i=1

P where 0  i  1 are user-selected constants and mi=1 i = 1.

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The optimal design for estimating the LD100 is a Bayesian c ? optimal design ?
with c () = LD100 , T = (; ), (rc ())T = 1; ? ?2 and = ln [= (1 ?  )]. Suppose the design  takes observations at xi (dosage level) with design weight pi (proportion of subjects allocated to xi ), where 0 < pi < 1, i = 1;


; K , and PK i=1 pi = 1. Then the inverse of its Fisher information matrix is 2

M (; )?1 = ?2 64

3

1=t + (x ? )2 =s

(x ? ) =s 7 5; 2=s

(x ? ) =s

P

P

where wi =  (xi ) (1 ?  (xi )), i = 1;


; K , t = Ki=1 pi wi , x = t?1 Ki=1 pi wi xi , P and s = Ki=1 pi wi (xi ? x)2. The Bayesian c ? optimality criterion for estimating the LD100 simpli es to n h io  ( ) = E ?2 t?1 + ( ? (x ? ))2 ?2 s?1 :

Substituting = 0 gives us the c ? optimality criterion for estimating  alone. Likewise, setting = 1 = ln (1=99), = 2 = ln (5=95) and = 3 = ln (1=9) lead to the c ? optimality criteria for estimating the 1st, 5th and 10th percentiles respectively. Therefore the criterion for estimating these three percentiles is  ( j ) =

3 X i=1

 

i () =  i : i

i

(5)

The compound optimal design can also be viewed as a Bayesian A-optimal design (Zhu, 1996) and can be found using a modi ed version of the Logit Design program developed by Chaloner and Larntz (1989). For each model parameter, two independent uniform priors are considered. The priors on  are U [?0:1; 0:1] and U [?1; 1], and the priors on are U [6:9; 7:1] and U [6; 8]. These priors range from very informative to vague and were used by Brown (1966) and, Chaloner and Larntz (1989). It appears that the optimal designs found here are more sensitive to the prior on  than on . This nding is in agreement with the situations considered in Chaloner and Larntz (1989). For the prior   5

Table 1: Compound Optimal Designs:   U [?0:1; 0:1],  U [6:9; 7:1]

T

1,0,0 0,1,0 0,0,1 0,.2,.8 0,.4,.6 0,.6,.4 0,.8,.2 .2,0,.8 .2,.2,.6 .2,.4,.4 .2,.6,.2 .2,.8,0 .4,0,.6 .4,.2,.4 .4,.4,.2 .4,.6,0 .6,0,.4 .6,.2,.2 .6,.4,0 .8,0,.2 .8,.2,0

x1 ; x2

-.346,.319 -.341,.265 -.311,.145 -.313,.184 -.317,.211 -.323,.231 -.331,.248 -.307,.224 -.312,.239 -.319,.253 -.327,.266 -.336,.281 -.311,.254 -.318,.265 -.326,.278 -.336,.291 -.319,.275 -.327,.287 -.338,.300 -.330,.296 -.341,.310

p1

.756 .881 .916 .903 .893 .885 .881 .863 .855 .849 .845 .844 .827 .821 .818 .817 .798 .795 .795 .775 .775

e1

1 .883 .662 .742 .796 .836 .865 .847 .879 .905 .926 .941 .923 .943 .959 .971 .966 .979 .988 .991 .997

e2

.921 1 .904 .947 .973 .988 .997 .975 .985 .992 .994 .993 .984 .986 .985 .980 .975 .971 .964 .955 .944

e3

.794 .923 1 .995 .984 .969 .949 .974 .961 .946 .927 .903 .938 .922 .903 .879 .897 .877 .852 .851 .824

U [?0:1; 0:1], the optimal design has two support points; when   U [?1; 1], the

optimal design has six support points. Some of these optimal designs and their eciencies for estimating each of the percentiles are shown in Tables 1 and 2. In these tables, e1 ; e2; e3 denote the eciencies of the compound optimal design for estimating the LD1, the LD5 and the LD10 respectively. Table 1 lists the two design support points x1 and x2 , and the design weight p1 associated with x1 . The other design weight p2 is simply (1 ? p1). Many compound optimal designs under the informative prior   U [?0:1; 0:1] are more than 90% ecient, while most of the six-point compound optimal designs under the non-informative prior are more than 95% ecient for estimating the three low percentiles.

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Table 2: Compound Optimal Designs (1):   U [?1; 1],  U [6; 8] T 1,0,0

0,1,0

0,0,1

0,.2,.8

0,.4,.6

0,.6,.4

0,.8,.2

.2,0,.8

.2,.2,.6

.2,.4,.4

.2,.6,.2

.2,.8,0

.4,0,.6

.4,.2,.4

.4,.4,.2

.4,.6,0

.6,0,.4

.6,.2,.2

.6,.4,0

.8,0,.2

.8,.2,0

?

:

1 05

:207 ?1:04 :231 ?1:03 :243 ?1:03 :240 ?1:03 :238 ?1:03 :236 ?1:04 :234 ?1:03 :236 ?1:03 :233 ?1:04 :231 ?1:04 :229 ?1:04 :226 ?1:04 :229 ?1:04 :226 ?1:04 :224 ?1:04 :222 ?1:04 :221 ?1:04 :219 ?1:04 :217 ?1:05 :215 ?1:05 :212

?:

567

:181 ?:575 :184 ?:573 :189 ?:573 :188 ?:572 :188 ?:572 :188 ?:571 :187 ?:568 :190 ?:568 :190 ?:568 :189 ?:567 :189 ?:566 :188 ?:567 :188 ?:567 :188 ?:567 :188 ?:565 :187 ?:567 :187 ?:566 :186 ?:566 :186 ?:567 :184 ?:566 :183

a Design

?:



165

:211 ?:196 :192 ?:194 :184 ?:195 :185 ?:191 :189 ?:188 :193 ?:187 :193 ?:176 :194 ?:177 :195 ?:173 :199 ?:170 :203 ?:167 :208 ?:172 :198 ?:169 :202 ?:165 :207 ?:164 :209 ?:164 :205 ?:164 :207 ?:162 :211 ?:165 :208 ?:163 :211

:214 :223 :150 :204 :154 :176 :158 :185 :167 :192 :172 :199 :169 :203 :194 :198 :193 :203 :200 :211 :205 :218 :208 :225 :208 :213 :210 :214 :216 :220 :215 :225 :219 :216 :217 :220 :218 :224 :218 :220 :216 :223

support points;

7

:575 :126 :451 :146 :465 :141 :477 :145 :489 :144 :493 :143 :487 :144 :531 :142 :532 :143 :541 :139 :547 :135 :549 :129 :560 :137 :561 :136 :567 :131 :566 :127 :575 :132 :573 :129 :573 :126 :577 :129 :574 :126

: a

1 09

:051b :751 :043 :776 :067 :792 :057 :805 :049 :811 :043 :806 :039 :879 :040 :895 :037 :919 :032 :941 :028 :959 :023 :965 :036 :987 :034 1:01 :032 1:03 :030 1:03 :039 1:04 :038 1:06 :037 1:06 :044 1:08 :044

b design

e1

e2

e3

1

.977

.961

.943

1

.975

.938

.990

1

.946

.993

.999

.951

.996

.999

.954

.999

.996

.954

1.00

.992

.968

.994

.997

.972

.996

.996

.976

.997

.994

.979

.998

.991

.979

.998

.986

.985

.993

.990

.988

.993

.988

.990

.993

.985

.992

.993

.981

.994

.989

.981

.996

.989

.978

.997

.988

.974

.999

.984

.971

.999

.982

.968

weights.

4. EFFICIENCY OF UNIFORM EQUAL ALLOCATION RULES IN DRUG TOXICITY STUDIES One simple dosage allocation scheme is to divide the dosage range evenly and allocate equal number of subjects to each dosage level. We consider three types of uniform equal allocation rules. Without loss of generality, we will assume  = [?1; 1]. The Type 1 scheme is to divide the entire interval  evenly and assign equal numbers of subjects to each dosage levels including the two end points -1 and 1. For example, a ve-point Type 1 design is equally supported at f-1, -.5, 0, .5, 1g. Type 2 design is the same as Type 1 except that the two end points are excluded. For instance, a three-point Type 2 design is supported evenly at f-.5, 0, .5g. This scheme may be reasonable if it is felt that the lowest dosage level corresponds to `placebo' and the highest dosage level may be toxic. The Type 3 scheme is more motivated by ethical concerns and avoids assigning any subjects to dose levels higher than the expected LD50 (0 in this case). The interval [?1; 0] is divided into even subintervals and subjects are assigned evenly to each division points including -1 and 0. A three-point Type 3 design would be equally supported at f-1, -0.5, 0g. It is unlikely that researchers will adopt an equal allocation rule with more than nine support points for a two-parameter model. Therefore, eight designs of each type with two to nine support points are considered. The eciencies of these designs for estimating each of the three percentiles are displayed in Table 3. Under the noninformative prior   U [?1; 1], the Type 1 six-point design is the most ecient uniform equal allocation rule. It is between 85% to 90% ecient for estimating the three percentiles. However, it is not as ecient as most of the compound optimal designs which are generally more than 98% ecient for estimating each percentiles. For the informative prior   U [?0:1; 0:1], we found that none of the uniform equal allocation rules is ecient. This echoes the observation of Abdelbasit and Plackett (1983) in that a blind procedure without utilizing the abundant information and the goals of the experiment is unlikely to be ecient. The implication of these results is that the popular simple equal allocation rules can be inecient when we have informative priors and our interest lies in the estimation of a set of low or high 8

Table 3: Eciencies of uniform equal allocation rules under dierent prior assumptions:   U [?0:1; 0:1],  U [6:9; 7:1] (*) and   U [?1; 1],  U [6; 8] (**) Type 1 Designs 2-point 3-point 4-point 5-point 6-point 7-point 8-point 9-point Type 2 Designs 2-point 3-point 4-point 5-point 6-point 7-point 8-point 9-point Type 3 Designs 2-point 3-point 4-point 5-point 6-point 7-point 8-point 9-point

e1 * .030 .075 .449 .357 .427 .418 .433 .438

e2 * .016 .074 .356 .314 .365 .360 .372 .377

e3 * .010 .077 .295 .287 .322 .322 .331 .336

e1 ** e2 ** e3 ** .003 .330 .772 .900 .915 .907 .898 .892

.003 .319 .733 .855 .868 .863 .856 .850

.003 .321 .727 .843 .858 .855 .849 .844

.827 .535 .598 .551 .549 .538 .532 .527

.664 .478 .517 .479 .475 .464 .460 .455

.560 .441 .464 .432 .426 .414 .413 .408

.023 .123 .271 .401 .496 .559 .602 .633

.016 .096 .231 .358 .453 .517 .561 .592

.012 .082 .208 .335 .432 .499 .545 .578

.053 .315 .395 .412 .412 .417 .417 .417

.055 .330 .432 .464 .478 .485 .491 .497

.060 .349 .464 .508 .521 .533 .544 .552

.001 .043 .051 .049 .046 .043 .041 .039

.001 .007 .009 .009 .009 .008 .008 .007

.000 .003 .004 .004 .004 .004 .004 .004

9

percentiles in the logit model.

5. EFFICIENCY OF UNIFORM EQUAL ALLOCATION RULES IN DRUG EFFICACY STUDIES The drug ecacy studies are very similar to the drug toxicity studies. The same model can be assumed, and optimal designs can be found along the same line. The only dierence is that we are more concerned with the highly eective dose levels, for example the LD90, the LD95 and the LD99. The uniform equal allocation rules of interest are the Type 1, the Type 2 and the re ection of the Type 3 designs with respect to the center 0, i.e: the uniform allocation over the interval [0; 1] rather than [?1; 0]. We will denote these new Type 3 designs as the Type 3+ designs. Assuming a symmetrical prior on  as in the previous section, and the LD100 optimal design is 8 9 < x  = > 1 : p1 >

x2


xK p2


pK

> = > ;

;

it can be shown that the optimal design for estimating the LD100(1 ?  ) is approximately 8 9 = ?xK ?xK?1


?x1 > : : pK ; pK?1


p1 > >


Furthermore, the eciencies of any symmetric design for estimating the LD100 and the LD100(1 ?  ) are approximately equal. This implies, for instance, that if a Type 1 or 2 design is not ecient for estimating the LD1, it is not ecient for estimating the LD99 either. For any Type 3 design  and its corresponding Type 3+ design  + , (  and  + are symmetrical to each other with respect to the center 0), the design  has the same eciency for estimating the LD100 as the design  + for estimating the LD100(1 ? ). Consequently, if none of the Type 3 designs is ecient for estimating, say, the LD5, then none of the Type 3+ designs is ecient for estimating the LD95. It is straightforward to show that the above statements also apply to the com10

pound optimal designs for estimating a linear combination of several percentiles. Thus, our results in the drug toxicity studies can be generalized to the drug ecacy studies as well. We omit the justi cations for these claims because of space consideration but details can be found in Zhu (1996).

ACKNOWLEDGMENT We thank Drs: Chaloner and Larntz for kindly sending us their Logit-Design program which was very helpful. The research of Wong is partially supported by a NIH award research grant R29 AR44177-01A1.

REFERENCES Abdelbasit, K. M. and Plackett, R. L. (1983). \Experimental Design for Binary Data," Journal of the American Statistical Association, 78, 90-98. Brown, B. W., Jr. (1966). \Planning a Quantal Assay of Potency," Biometrics, 22, 322-329. Chaloner, K. and Larntz, K. (1989). \Optimal Bayesian Design Applied to Logistic Regression Experiments," Journal of Statistical Planning and Inference, 21, 191-208. Chaloner, K. and Verdinelli, I. (1995). \Bayesian Experimental Design: A Review," Statistical Science, 10, 273-304. Cherno, H. (1953). \Locally Optimal Designs for Estimating Parameters," Annals of Mathematical Statistics, 24, 586-602. Clyde, M. and Chaloner, K. (1996). \The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems," Journal of the American Statistical Association, 91, 1236-1244. 11

Cook, R. D. and Wong, W. K. (1994). \On the equivalence of Constrained and Compound Optimal Designs," Journal of the American Statistical Association, 89, 687-692. Kiefer, J. (1959). \Optimum Experimental Designs," J. Roy. Statistical Soc., Ser. B, 21, 272-319. Kiefer, J. (1985). \Jack Carl Kiefer Collected Papers III, Design of Experiments," Springer-Verlag, New York. Loose, L. D., et al. (1993). \Double-blind, Placebo-controlled, Dose Response Study of Tenidap in Rheumatoid Arthritis," Arthritis Rheum, 36, S166. Abstract. Richards, W., Suresh, R. and Belanger, B. (1997). \A Comparison of CRM and Polya Urn Methods in Clinical Trials," JSM'97 Abstracts, 273. Wu, C. F. J. (1985). \Asymptotic Inference from Sequential Design in a Nonlinear Situation," Biometrika, 72, 553-8. Wu, C. F. J. (1988). \Optimal Design for Percentile Estimation of a Quantal Response Curve," In Optimal Design and Analysis of Experiments eds: Y. Dodge, V.V. Fedorov and H.P. Wynn. Elsevier Science Publication, North Holland. Zhu, W. (1996). \On the Optimal Designs of Multiple-objective Clinical Trials and Quantal Dose-response Experiments," Ph. D. Dissertation, Department of Biostatistics, School of Public Health, UCLA.

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