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Experimental Design for Clonogenic Assays inChemotherapySalomon Minkin a ba Division of Epidemiology and Statistics, Ontario Cancer Institute, Toronto, M4X 1K9,Canadab Department of Medical Biophysics, University of TorontoPublished online: 27 Feb 2012.

To cite this article: Salomon Minkin (1993) Experimental Design for Clonogenic Assays in Chemotherapy, Journal of theAmerican Statistical Association, 88:422, 410-420

To link to this article: http://dx.doi.org/10.1080/01621459.1993.10476290

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Experimental Design for Clonogenic Assaysin Chemotherapy

SALOMON MINKIN*

One approach to estimating the number of cells with high growth potential in a cancer patient is the assessment of tumor cell colonyformation in semisolid medium. A potential indicator of the clinical response to a specificdrug is the capacity of the drug to reducethe formation of cell colonies. Often, a Poisson log-linear model provides an adequate representation of the dose-response curve.This model is then summarized by the dose required to reduce the number of colonies to a predetermined percentage of the maximumgrowth. But more general models are needed to account for the overdispersion and the resistant subpopulations often present inthese assays. This article explores alternative methods for selecting the dose levels to obtain precise estimates of the parameters ofinterest. Optimal dose allocations for a single patient or a group of patients are derived. Methods to incorporate the informationfrom pilot studies are discussed. A study of drug sensitivity with leukemia patients provides the framework and motivation for theproblem.

KEY WORDS: Dose-response curves; Optimal design; Overdispersion; Poisson regression.

1. INTRODUCTION

An important area in leukemia research aims to explainthe heterogeneity in the response to chemotherapy regimensby characterizing patients' relevant biological attributes. Aspecific example is establishing the association between sen­sitivity to a particular drug as observed in the laboratory andthe clinical outcome of a therapy based principally on thatdrug. Drug sensitivity is assessed by studying the effect of arange of concentrations in cultures generated from a patient'spretreatment leukemic cells. A dose-response curve is de­termined from the decrease in colony formation over allconcentrations. A numerical summary of the dose-responsecurve is then considered as a potentially useful predictor ofthe clinical outcome (McCulloch, Buick, Curtis, Messner,and Senn 1981).

Often, the dose-response curve can be represented by aPoisson log-linear model. This model is appealing becauseof its numerical and biological simplicity. It would be ex­pected to hold when there is a simple chemical interactionbetween the molecules of the drug and a molecular target inthe cell, as long as the target does not have a repair mecha­nism. But empirical evidence, rather than cell kinetics, isoften the main justification for using this model. Naturally,for several types ofdrugs the Poisson log-linear model doesnot hold. For example, departures from the log-linear modelare not uncommon for alkylating agents, for which the dose­response curves tend to exhibit an initial shoulder at lowdosages resembling the curves commonly seen when cellsare exposed to radiation. Other anticancer drugs often pro­duce curves showing a plateau for high doses, which indicatesa resistant subpopulation (Tannock 1987). Another depar­ture from the Poisson log-linear model-extra-Poisson vari-

* Salomon Minkin is Biostatistician, Division of Epidemiology and Sta­tistics, Ontario Cancer Institute, Toronto, M4X IK9, Canada, and AssociateProfessor, Department of Medical Biophysics, University of Toronto. Thiswork was partially supported by the Natural Sciences and Engineering Re­search Council of Canada and by the National Cancer Institute of Canada.The author thanks K. Sykora, T. Stukel, and the referees for many helpfulcomments and suggestions, and E. A. McCulloch for providing the dataused in the example.

ation-can be introduced by small variations in the exper­imental conditions.

These drug sensitivity studies typically consist of twostages. First, the assay technique is developed and a pilotstudy involving a small number of patients is conducted toestablish the potential usefulness of a given dose-responsesummary. Second, a larger study is conducted in which allrelevant clinical, demographic, and biological variables (in­cluding the dose-response curve summary) are determinedfollowing a standard protocol and are considered potentialpredictors of a response variable, typically either survivaltime or a binary variable such as success in achieving re­mission (see, for example, Curtis, Messner, Minden, Minkin,and McCulloch 1987). The pilot study provides prior infor­mation, which is employed for the selection of concentrationlevels to be used in the larger study. Such prior informationis particularly useful if the selection of concentration levelsis to be based on statistical arguments, because (as will beseen below) the models used for the observed colony countswill lead to parameter-dependent choices.

In this presentation the selection of concentration levelswill be discussed under different empirical models that mayprovide adequate summarizations of the dose-responsecurve. To begin with, the optimal choice is derived for thePoisson log-linear model when the parameter of interest isDp , the concentration at which the number of colonies percells plated is 100 X p% of the maximum (0 < p < 1). Therobustness in the efficiencyof such a design against departuresin the assumptions, such as extra-Poisson variation or thepresence of a resistant subpopulation, is assessed. Becausethe optimal choice for the Poisson log-linear model dependson the value of the slope, which is subject to considerablepatient-to-patient variation, I explore the implications ofus­ing the information obtained in the first stage for specifyinga distribution for the slope in the patient population underconsideration. The choice of concentration levels to mini­mize the expected value of the criterion is derived for the

© 1993 American Statistical AssociationJournal of the American Statistical Association

June 1993, Vol. 88, No. 422, Applications and Case Studies

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Minkin: Clonogenic Assay Experimental Design 411

m ni

2. OPTIMAL CHOICE OF CONCENTRATIONS

I( 0:, 1/(3) = L L [YihOi - CiexP( Oi )],i~] h~]

Denote by Yih the number of cell colonies observed whenc, cells of the patient have been plated with a concentrationXi of the drug, with i = 1, ... , m indexing concentrationsand h = I, ... , n, indexing replicates. Assume that Yih followsa Poisson distribution with mean CiexP(Oi). The model tobe used specifies that

(3)2q exp( -0) = (l - q)(o - 2)

tration Xi. Note that a choice of concentrations that mini­mizes the asymptotic variance of 11ti also minimizes theasymptotic variance of ti, because var(l 1ti) = var(ti)1 (34.The Appendix shows that the optimal choice is a two-pointdesign where n(l - q) cells are plated at X = 0 and theremaining nq cellsare plated at the concentration that reducesthe number of colonies to a fraction of the control given byexp( -0), where 0 = -(3x is the solution of the equation

(cf. Eq. A.3 with q = q3 and 1 - q = q, ). In these experimentsthe number of cells plated c, is typically very large, so thereis no loss of generality in ignoring the restriction that nqshould be an integer. Without this restriction, the optimalvalues of 0 and q are 2.557 and .782. That is, the optimaldesign is one with 22% of the cells plated at the con­trol (i.e., X = 0) and 78% of the cells plated at the dose thatreduces the number of colonies to 8% of control (i.e., X

= -2.5571(3).Clearly, the optimal choice of concentrations depends on

the actual value of (3. This is a common situation when deal­ing with nonnormal linear models (Silvey 1980). Neverthe­less, knowing the optimal choice is valuable, as it permitscalculation of the efficiency of a particular choice of con­centrations given by the ratio ofvar(l 1ti) under the optimalchoice relative to var(l 1(3) under the alternative choice. Inparticular, it is important to establish the loss of efficiencyresulting from attempting to produce the optimal allocationbased on an inaccurate guess of the slope parameter (3. Asindicated in Table 1, at least 80%efficiencyis preserved whenthe magnitude ofthe slope is overestimated by no more than43% or when it is underestimated by no more than 34%.

Having only two concentrations is a very serious short­coming of the optimal design from a practical perspective,since it makes it impossible to detect departures from thelog-linear assumption (I). Designs with four to six equallyspaced concentrations are practical alternatives which pro­vide a better indication of the nature of the dose-responserelation. If the same proportion of cells are plated at each ofthe concentrations, the maximum efficiencyis 79.6%, 76.0%,and 73.1% for the four-, five-, and six-concentration designs.If instead the proportion of cells at each concentration ischosen in an optimal way subject to no less than 10% of thecells being plated at each concentration, then it is possibleto increase these efficiencies by at least 10.8%, as indicatedin Table 1. Having a positive proportion of cells plated at

(1)0i = 0: + (3Xi, with (3 < O.

it is easy to find the inverse of the Fisher information matrixto obtain the asymptotic variance of I1ti,

(l1{3) = 2:7:, qiexp(O;) (2)var n(34{2:7:, 2:~] qiqjxfexP(Oi + OJ)'

- [2:7:] qi xi exP(0;)]2}

where n = 2: n,c, is the total number of cells plated and q,= n,c,1n is the proportion of cells exposed to the concen-

cases when the prior distribution ofthe negative of the slopebelongs to the Gaussian family or to the gamma family. Thecriterion adopted is the ratio of the parameter estimate vari­ance under a given set ofconcentration levels, relative to theminimum possible variance for that patient using the samenumber of observations. A procedure to incorporate priorinformation for the resistant subpopulation model is alsosuggested. The application of these results is illustrated withdata from an assay for the assessment of the cytotoxic effectsof the drug cytosine arabinoside in patients with acute my­eloblastic leukemia, which provides the motivation for thisstudy. Throughout the presentation, consideration is givento the implication of using the more practically appealingequally spaced designs with several concentrations.

Because (3 has been assumed to be negative, the maximumexpected number of colonies per cells plated occurs at thecontrol; that is, when Xi = O. Thus for that patient, the con­centration that reduces the expected number of colonies percells plated to 100 X p% of the maximum is Dp = 10g(p)1(3. Therefore, the parameter of interest is 1/(3, and in thiscase 0: is a nuisance parameter. For simplicity of notation,the fact that all the parameters are patient-dependent (i.e.,a c1onogenic assay is performed on each patient's pretreat­ment c1onogenic cells) has not been made explicit.

From the Poisson log-likelihood,

Table 1. Optimum Equally Spaced Designs With at Least 10% of the Observations at Each Design Point for the Poisson Log-Linear Model

Efficiency imervel"

Efficiency 80% 60%

2456

0,2.560, 1.27, 2.55, 3.82

0, .88, 1.77,2.65,3.540, .82, 1.65,2.47,3.29,4.12

.22, .78.21, .1, .59, .1

.21, .1, .1, .49, .1.20, .1, .1, .40, .1, .1

100.092.287.783.9

(.66, 1.43)(.70,1.37)(.75,1.31)(.80,1.23)

(.52,1.69)(.52, 1.72)(.54,1.73)(.54,1.74)

• Ratios between the guess and the true value of {3for which the design efficiency is at least as stated.

• m ~ number of concentrations.'01 = -{3xl'd ql ~ proportion of observations at concentration XI .

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412

each concentration is necessary ifmore than two concentra­tions are to be used. The choice of 10% as the minimumproportion at each concentration, although arbitrary, ensuresthat when there are 20 plates with the same number ofcells,at least two replicates are available at each concentration.Note that the proportion ofcells plated at the control remainsfairly constant in the designs listed in Table I. Also, thevalue of OJ with the largest proportion of cells is quite closeto 2.557.

Increasing the number of concentrations brings a deteri­oration in efficiency. But although the intervals with 80%efficiency become narrower when more concentrations areused, there is little difference between the 60% efficiency in­tervals for the different number of concentrations, withslightly wider intervals as the number of concentrations in­creases. This means that when the slope can be accuratelyspecified in advance, the two-concentration design is con­siderably more efficient; but when the prior estimate of theslope is poor, there is no efficiency advantage in using onlytwo concentrations.

3. DEPARTURES FROM THE ASSUMPTIONS

3.1 Overdispersion

As in many applications involving counts, these assaysoften exhibit more variation than would be expected underthe Poisson assumption. To be able to assess the presenceof such overdispersion, it is necessary to have replicates.Replicates can be incorporated in the design specified in theprevious section, because var( 1/&) is affected only by thetotal number ofcells plated at each concentration. Naturally,it is good practice to divide those cells, so several dishes areused for each of the two concentrations.

A common way of accommodating overdispersion is bypostulating a multiplicative random effect; that is, for a fixedvalue of the random variable II with mean I and variance T,

the mean and variance of the response Yih are both equal toIIlli. Then, unconditionally, the response Yih has mean Ili andvariance Ili + TilT. The equations to estimate the parametersspecifying Ili are based on quasi-likelihood methods (Me­Cullagh and Neider 1989), whereas the additional equationto estimate T is based on the method of moments (Breslow1984). If for simplicity we assume that all the plates receivethe same number of cells (i.e., c, = c), and if Ili = C exp(8i)

Journal of the American Statistical Association. June 1993

with 8i satisfying (I), then the asymptotic variance of 1/ t3 isof the same form as (2), except that the terms exp(8;) arereplaced by exp (e, )/[1 + T exp(8i)] (see Lawless 1987), thatis,

var(l / &)L~l qiexp(8;)/[1 + T exp(8;)]

To establish to what extent the efficiency of the designoptimal for T = 0, given in the previous section, is affectedwhen the true value of T is positive, we restrict attention todesigns with only two concentrations to simplify the analysis.(Note that as long as there are replicates, it is possible toestimate T.) Although one of the concentrations in the op­timal design should be the control where n(l - q) cells areplated, the equivalent of Equation (3) for determining 0= - Bx and the corresponding equation for q are

2(q + Tea)eXp( -0) = (l - q)(o - 2)

and

q2 - Tea(l - 2q) _ (0)(l - q)2 - exp .

Note that in this case the optimal design depends on thevalue of Tea. The solution to these equations for selectedvalues of Teacan be found in Table 2 (the values with 100%efficiency). Table 2 also shows the efficiency ofthese solutionsunder alternative values of Tea. In particular, the columncorresponding to 0 = 2.56 and q = .782 shows the lack ofrobustness of the design optimal for T = 0. That is, when e"(the expected count at the control) is large, even a smallamount ofoverdispersion may drastically affect the efficiencyof the design that is optimal for T = 0. For example, if T

= .05, then the efficiency of the design optimal for T = °drops to 91.6% when e" = 40, to 69.4% when e" = 160, andto 43.4% when e" = 640. Among the allocations displayedin Table 2, the one that retains highest minimum efficiencyover the range 0 ~ Tea ~ 256 is the one that is optimal forTea = 16; but even that one can only assure 60.7% efficiencywhen Tea is in the interval [0, 256].

Table 2. Efficiencies for the Log-Linear Model in the Presence of Overdispersion

s 2.56 3.01 3.25 3.57 3.94 4.36 4.82 5.31 5.82q .782 .731 .710 .687 .666 .646 .630 .615 .603

rea

0 100.0 92.9 85.5 74.9 62.4 49.5 37.6 27.4 19.42 91.6 100.0 98.0 91.3 80.0 65.9 51.3 38.2 27.34 82.2 97.9 100.0 97.2 88.6 75.4 60.1 45.4 32.88 69.4 90.6 97.1 100.0 96.6 86.5 71.8 55.8 41.1

16 55.7 78.5 88.2 96.5 100.0 96.1 84.9 69.3 52.732 43.4 64.6 75.3 86.8 96.2 100.0 95.8 83.9 67.664 33.6 51.6 61.6 73.7 86.1 96.0 100.0 95.6 83.3

128 26.2 40.9 49.5 60.6 73.3 86.0 96.0 100.0 95.5256 20.7 32.6 39.7 49.2 60.7 73.6 86.2 96.1 100.0

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Minkin: Clonogenic Assay Experimental Design 413

where OJ = - (3x;. The Appendix shows that the optimalallocation puts the same number of cells at each ofthe threeconcentrations and that the optimal choice of concentrationswhen °< 7r < I is 01 = 0, 03 as large as feasible, and 02 thesolution to the equation

Solutions to (6) are given in Table 3 for selected values of7r. Table 3 indicates that the contours of relative efficiencycorresponding to different values of 7r are remarkably similarto each other when viewed as functions of relative changein {3. This means that the effect of misspecifying {3 (whichchanges the value of (2), is practically independent of thevalue of 7r. Here efficiency is computed as the cube root ofthe result of dividing the minimum of (5) by the value of (5)at some 02. There are two reasons for taking cube roots tocompute efficiencies: (I) It makes it possible to viewefficiency

(5)

(6)exp( -(2)(2 - (2) = _-_27r_

(I - (2) I - 7r

Rather than trying to use a single point to summarizethe model, it is more reasonable to focus on the wholedose-response curve. A criterion that reflects the idea ofprecise estimation of all three parameters in model (4) isD-optimality, which maximizes the determinant of theinformation matrix. The information matrix F has en­tries Fjk = L:~1 neaqiAijAik/vi, where (Ail> Ai2, A i3)= «(1 - 7r )e- O; + 7r, -(1 - 7r )Oie-o;/ (3, I - e- O;) and Vi = Ai 1

or Vi = Ail + reaATI depending on whether extra­Poisson variation is absent or present. Here, again for sim­plicity in the case of overdispersion, we assumed that c, = C

and that this constant c is absorbed as part of ea. Note thatthe D-optimal design also minimizes the volume of theasymptotic confidence region for the parameters (a, (3, 7r),because such volume is proportional to the cube root of thereciprocal of the information matrix determinant.

At least three concentrations are required to obtain a pos­itive determinant. With three concentrations, the determi­nant of the information matrix when overdispersion is notpresent is

n3qtq2q3e3a(1 - 7r)2[e- O\-02(02 - od- e-01-03(03 - od + e-Or 03(03 - (2)]2

(32[e-01(1 - 7r) + 7r][e 02(1 - 7r) + 7r][e- 03(1 - 7r) + 7r]

3.2 Resistant Subpopulation

In assessing drug sensitivity by the decrease in colony for­mation, one pattern which emerges with certain frequencyin the dose-response curves shows an exponential decreasefor low dosages which gradually levels off to a plateau forhigh dosages (Tannock 1987). A simple model to accountfor such a plateau, suggested by Minkin (1991), is

J1.i = Ci[(1 - 7r)exp(a + (3x;) + 7r exp(a)]. (4)

This model is based on assuming the presence of two pop­ulations of clonogenic cells among the plated cells: one pop­ulation whose drug tolerance follows an exponential distri­bution with mean -1 / (3 and one population that is resistant.The parameter 7r represents the proportion of resistant clon­ogenic cells,and e" denotes the rate of clonogenic cellsplated.But, as with most models in the field, its main justificationis an empirical one. Minkin (1991) demonstrated that thismodel provides both a good fit and information of prognosticvalue for the type of experiments discussed here in Section5. If such a model is in fact plausible, then the criterion forselecting a design needs to be reevaluated. For the log-linearmodel, D; is a one-to-one function of the slope, and thus itcaptures the most relevant aspect of the dose-response curveirrespective of p, But for model (4), Dp focuses on just oneaspect of the dose-response curve, which may not be themost relevant one. In fact, Dp is not even defined when p< n . Although 7r is likely to be small, it may exceed p­particularly if p is taken as .1, which is not an uncommonchoice.

To assessthe effectof overdispersion on designs with morethan two concentrations, we searched for designs with fourand with six equally spaced concentrations having high ef­ficiency for rea = 16, with structure similar to the ones re­ported in Table I; that is, at least 10%ofcells plated at eachconcentration. The design with (01, ... , (4) = (0, 1.93,3.86,5.79) and (ql' ... , q4) = (.31, .1, .49, .1) has efficiencies of66.0%, 87.2%, and 59.6% when rea is equal to 0, 16, and256, while the design with (01) ... , (6) = (0, 1.22,2.45,3.67,4.90,6.12) and (ql,"" q6) = (.27, .1, .1, .33, .1,.1) hasefficiencies of 66.8%, 77.2%, and 57.5% for the same valuesof rea. Thus the loss of efficiency resulting from increasingthe number of concentrations gradually disappears as theprior guess of rea becomes less precise.

Table 3. D-Optimal Designs for the Resistant Subpopulation Model Without Overdispersion

Three-point designs with q, = q2 = q3 = 1/3 and 01 = 0

Interva/ b with efficiency of Efficiency of 02 = 1.501 Efficiency ofOptimum" a six-point

7r 02 90% 80% 03 = co 03 = 402 03 = 302 design C

.01 1.88 (.55,1.60) (.40,1.91) 97.8 89.2 70.1 88.7

.05 1.65 (.56,1.56) (.41,1.84) 99.7 95.7 85.4 92.2

.1 1.50 (.57,1.56) (.41, 1.85) 100 96.8 88.4 92.2

.15 1.41 (.57, 1.57) (.41,1.87) 99.8 96.9 89.4 91.7

.2 1.34 (.56, 1.58) (.41,1.88) 99.5 96.7 89.6 91.1

.3 1.25 (.56,1.59) (.41,1.91) 98.6 96.0 89.3 89.8

.4 1.19 (.56, 1.61) (.41, 1.94) 97.7 95.2 88.7 88.7

• When 03 = 00.

b Ratios between the guess and the true value of (3 for which the efficiency is at least as stated when using the design specified in the second column.'Design with 0,~ 1.611(; - 1) for; ~ 1, ... , 6 and (q, ... ,q,) ~ (.30, .27, .10, .10, .10, .13).

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414 Journal of the American Statistical Association, June 1993

as equivalent sample size, since (5) is a function of n3 ; and(2) As indicated before, the volume ofthe confidence region(ex, fl, 11") is inversely proportional to the cube root of (5).Although the optimal value for 02 increases with 11", the fifthcolumn of Table 3 demonstrates that a value for 02 near themiddle of the range of these optimal values (in this case1.501, which is optimal for 11" = .1) is almost fully efficient(efficiency ~ 97.7%) for all the values of 11" under consider­ation.

From a practical viewpoint, it is necessary to establish theeffect of using a finite value for 03. In general terms, 03 shouldbe large enough to provide information about the tail of themodel. Specifically, the sixth column in Table 3 indicatesthat the design with the same number of cells plated at (0(,02,03) = (0, 1.501,6.003) is more than 95% efficient if .05::::; 11" ::::; .40, whereas for very small values of 11", its efficiencydrops to 89%. Using a value of 03 as small as 4.502 bringsdown the efficiency to a still acceptable 85% as long as 11"~ .05. The low efficiency for 11" = .01 and 03 = 4.502 is theresult of having all three concentrations in the log-linear partof the model, so there is no information about 11".

Finally, the last column in Table 3 shows the efficiencyof using a six-point equally spaced design. The gap of 1.611between the design points and the relative number of cellsplated at each concentration were chosen to maximize theminimum efficiency when .01 ::::; 11" ::::; .40, subject to havingnot less than 10% of the cells at each concentration.

If overdispersion is present then the determinant of theinformation matrix given above is divided by the productof the terms 1+Tea[1I"+(1-1I")exp(-Oj)]. The only changein the optimal allocation is a more complicated equation fordetermining 02 that depends not only on 11" but also on Tea,namely

{exp( -02)(2 - 02)(1 - 11")(1 + 2Tea 1l" )+ 211"(1 - 02)(1 + Tea 1l" ) }

exp( - 202)( 1 - 11")2(7)

These optimal values for 02 for a wide range of values for 11"and Tea are given in Table 4. From the third column ofTable 4, it is apparent that the effect of Tea on the optimal02 is stronger when 11" is closer to 0, and, conversely, the effectof 11" on the optimal 02 is stronger for large Tea. But 11" hasalmost no effecton the contours of relative efficiency,whereasthese contours become narrower as re" increases. The sixthcolumn of Table 4 shows that a compromise value for 02 canproduce designs with reasonable efficiency (at least 85%) forthe set of values of 11" and re" considered in the table, but itis not possible to retain full efficiency with a single choice of02, as was the case when no overdispersion was present. Theseventh column indicates that using 03 = 6.633 provides al­most as high efficiency as 03 = 00 when 02 = 2.211, exceptwhen 11" is small and Tea is high. The last column in the tableshows the efficiencies of an equally spaced six-point designwith at least 10% of the cells plated at each concentration,chosen to maximize the minimum efficiency for the pairs ofvalues of( 11", Tea) considered in the table. Note that this six­point design, although having efficiencies lower and moreuniform than those of the three-point design reported in thesixth column, has efficiencies of at least 85.1%-not signif­icantly lower than the minimum efficiencyof85.6% reportedin the sixth column.

4. USING PRIOR INFORMAnON

4.1 Poisson Log-linear Model

Based on the results of a pilot study and perhaps additionalexperience with similar studies, it is often possible to estimatesome of the characteristics of the distribution of the param­eter of interest in the particular patient population underconsideration-specifically the location, scale, and to someextent the shape. When such information is available, itseems reasonable to try to identify a choice of concentrationswith good average properties. But there are some subjectivechoices that have strong influence on the selection of a design.

Table 4. D-Optimal Designs for the Resistant Subpopulation Model With Overdispersion

Three-point designs with q, = q2 = q3 = 1/3 and D, = 0

Efficiency of EfficiencyInterva/ b with efficiency of D2 = 2.211 ofa

Optimum" six-point7C rea D2 90% 80% D3 = OCJ D3 = 3D2 design C

.01 .3 1.95 (.56,1.58) (.41,1.87) 99.3 94.6 91.63 2.30 (.61, 1.47) (.46,1.70) 99.9 95.0 91.7

30 3.05 (.67,1.34) (.53,1.50) 92.9 87.6 87.8300 3.52 (.69, 1.29) (.55,1.43) 85.6 79.0 85.1

.10 .3 1.55 (.57,1.54) (.42,1.82) 93.4 92.3 90.33 1.77 (.61, 1.47) (.46, 1.72) 96.8 95.6 92.3

30 2.03 (.63,1.42) (.49,1.63) 99.5 98.0 93.2300 2.09 (.64,1.41) (.49,1.62) 99.8 98.2 93.2

.30 .3 1.28 (.57, 1.58) (.42,1.89) 85.6 84.9 85.13 1.41 (.58,1.54) (.44,1.82) 88.9 88.0 87.6

30 1.50 (.59, 1.51) (.45,1.78) 91.2 90.3 89.2300 1.51 (.60, 1.51) (.45, 1.78) 91.6 90.6 89.4

a When 03= 00.

b Endpoints given as multiples of 11 when trying to use the design specified in the third column.'Design with h,~ 1.938(; - 1) for i ~ 1, ...• 6 and (q" ... ,q.) ~ (.31, .23, .10, .10•. 10, .16).

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Minkin: Clonogenic Assay Experimental Design

One is the actual choice of the criterion to be averaged. It isequally reasonable to consider average variance, average rel­ative efficiency, or average relative variance (where relativevariance is the reciprocal of the efficiency). Clearly, the lackof invariance of the expectation operation under nonlineartransformations results in different choices, depending onthe particular criterion. Another subjective choice involvesthe choice ofa distribution family for the parameter, althoughthe pilot study might provide some guidance.

If the pilot study data is consistent with the Poisson log­linear model, then one approach to find the optimal designwith m equally spaced concentrations, one of them at thecontrol, is to determine the values x, ql, ... , qm-I that min­imize the expression

(8)

with qm = 1 - LJ!,1 1 qj' Note that to obtain the asymptoticvariance of 1/ri, the term in curly braces must be dividedby (34n exp( a). But when comparing allocations by the ratioof their variances, this additional term cancels out. Thus weare adopting a criterion that avoids dealing with fourth mo­ments and with the variability in the population of patientsassociated with the parameter a. This criterion is the mini­mum expected relative variance; that is,

. E{ var(1/ril{3,x,q) }mill (3 • A •

x,q mlllx({3),q({3) var[ 1/{31 (3, x({3), q({3)]

In most cases it is not possible to evaluate this expectationin closed form, so one must rely on a one-dimensional nu­merical integration, followed by an m-dimensional numericaloptimization. Such a procedure can be numerically unstable.But the case of m = 2 is relatively easy to deal with, as willbe demonstrated in the remainder of this section. Neverthe­less, keep in mind that the optimal design for the assumeddistribution of {3 might involve more than two concentra­tions.

If m = 2, then the criterion (8) reduces to

min E{3{exp( -(3x)/(X2q2) + 1/[x2(1 - q2)]}. (9)X,Q2

To find the optimum value ofx in (9), it is necessary to havean expression for mgL{3( x), the moment-generating functionof - {3, for x in an interval [0, A]. In some situations it maybe reasonable to approximate the distribution of - {3 withthe Gaussian distribution with mean Il and variance (52. Byusing the Gaussian moment-generating function to findE{3{ exp( - (3x)} in (9), and equating to 0 the partial deriva­tives with respect to x and q2, it can be shown that x is theonly positive solution of the equation

415

Equation (10) can be easily solved numerically. Its solu­tion is always in the interval (XL, Xu), where XL

= {[1l 2 + 8(1.27846)u 2] 1/ 2 - 1l}/(2u2

) and Xu = {[1l 2

+ 4(1.27846)u2] 1/2 - Il} / u

2, because 1.27846 is the solutionto exp( - y) - y = -1. The optimal value of q: is a simplefunction of the optimal x,

The fact that - {3 is restricted to be positive suggests thatrather than a Gaussian distribution, a skewed distributionsuch as the gamma with scale parameter II > 0 and shape orindex parameter K > 0 may be appropriate to describe thevariability in the values of - (3 associated with different pa­tients. In this case the optimal value ofx is the only positivesolution to the equation

XII(K + 2) - 2 = 2(1 - XII)"/2+1. (12)

From a second-order Taylor expansion of (12), the optimalx can be approximated by x ~ (4 - Y32/[K + 2] )/KII. Theoptimal q2 results from evaluating at the optimal X

(13)

In both the Gaussian and the gamma cases, the equationto find the optimum x depends on the parameters of theassumed distribution for - {3, so the role of the prior infor­mation will be in providing both an empirical assessment ofthe suitability of the assumed distribution for - (3 and esti­mates for the unknown parameters.

An alternative to parametric specification of mgL{3( x),viable only when the information available is in the form ofa large pilot study, is to use the empirical moment-generatingfunction. When M patients have been studied and their val­ues {3k have been estimated, it appears reasonable to treatthese M estimates as if they were observed values from thepopulation of possible slopes-although by doing so, one isdisregarding the error in estimation. Then the empirical mo­ment-generating function of - {3 is given by emgL (3( x)= Lf/=lexp(-{3kX)/M. Using this empirical moment-gen­erating function to estimate E[exp(-{3x)] in (9), yields theoptimal concentration X as the solution to

M

L ({3kx + 2)exp(-{3kX)/M= -2YemgL{3(x), (14)k~1

and q2 is determined by evaluating at the optimal x

q2 = YemgL{3(x)/[1 + YemgL{3(x)]. (15)

4.2 Resistant Subpopulation

As in Section 4.1, a reasonable criterion aims to maximizethe expected relative precision of the parameter estimates.In this case, this is implemented by minimizing the expectedratio of the volume of the joint confidence region for theparameters relative to the optimal volume for each value ofthe parameters; that is,

. E { det[F(O',{3,1l",Tlx,qr l 13}

min a {3 1r T • 113 .(x,q) '" mlll x(a,{3,1r,T),q(a,{3,1r,T)det[F(O', {3, 1l", Tlx(a, (3, 1l", T), q(a, {3, 1l", -u

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416 Journal of the American Statistical Association, June 1993

Table 5. Sensitivity in Methylcellulose Data

Ara-Cdose

Pat 0.0 0.5 1.0 2.0 3.0 5.0

13 7967 (-99, 261, 436) 3768 (-80, 436, 664) 3051 (-123, 193, 501) 1121 (-25, 88, 207) 350 (-110, 30, 51)37 7284(-636,284,304) 6588(-140,264,344) 5532 (-120,172,296) 4602 (-174, 478, 666) 4210 (-502,154,244) 1760 (-256,104.164)44 4230 (-722. 270. 730) 980 (-44.40.120) 590 (-18. 18.82) 484 (-52. 16,32) 56 (-12, 52, 64) 26 (-2. 10, 14)

2 4014 (-62,250,330) 3876 (-272, 46, 172) 3264 (-58, 100, 150) 2724 (-102.156.636) 1754 (-60.110.138) 86 (-18,14.46)16 3848 (-124. 92.1284) 116(124) 56 (29) 23 (23) 22 (8) 8(8)38 3780(-60,1672.1824) 476 (-116. 56. 136) 236 (-92. 4, 56) 76 (0, 32, 44) 28 (-4. 28, 32) 20 (0. 16. 16)17 3692 (-860,992.1248) 2516 (-256,512,596) 1868(-396,236,244) 556 (-316,64,120) 140 (-44, 220. 352) 110 (-42, 62. 158)65 3596(-132.208.580) 2380(-340,80,468) 1728(-226,116,208) 1040 (-72.171.440) 498 (-42. 37.122) 180 (-44,22.60)41 2764 (-44, 68, 348) 120 (-16,184.272) 112 (-12. 28,164) 56 (-8,12.44) 4(4) 0(0)34 2736(-540,620.1240) 928(-128.828,1072) 588 (-40.16.140) 432 (-32,160,172) 316 (-8,52,148) 276 (-152. 32. 44)15 2728 (-216.188,392) 656 (-76, 88,104) 260 (-80. 28. 68) 70 (-40, 22, 50) 0(0.4,8) 0(2)43 2600(-1080,376,564) 224 (-40,28,36) 132 (-12, 44, 48) 128 (-32. 2, 52) 80 (-24.12,80) 60 (-10.30,100)62 2468 (-532,252,472) 1700 (-56,120,680) 1388 (-128, 264, 744) 1180 (-400, 4, 608) 728 (-100,144.296) 524 (-80. 76, 284)57 2244 (-364,8,1556) 604 (-140,112.236) 436(-188,132,168) 92 (-8. 28, 56) 0(0,8,8) 0(0)11 2176 (-112. 796, 872) 456 (-48,130,156) 276 (-68.136,172) 180 (-64,100.132) 100 (-40. 36, 80) 52 (-12,8.64)28 1996 (-788,300,1728) 504 (-112.8,136) 80 (-26, 20, 40) 64 (-8,11,12) 0(0.4.8) 0(8)45 1844 (-200,334.388) 364 (-32.18.36) 198 (-18. 34. 66) 10 (-2,14.22) 0(0) 0(0)26 1842 (-306, 66. 286) 1004 (-80. 46. 134) 296 (-56,24.64) 106 (-6.14.16) 60 (-4,4,18) 6 (2)

1 1660 (400) 296 (65) 212 (108) 156 (8) 128 (36) 20(4)59 1588(-788,368,764) 64(4) 20(20) 12 (4) 0(8) 0(0)22 1342 (-94, 78, 278) 592 (16) 272 (120) 292 (18) 140(4) 76(54)39 1336 (-248. 320, 636) 430 (-50, 178,378) 200 (-8,58.192) 100 (0,24,192) 40 (-24. 4. 10) 14(-6,18.30)54 1328 (-336,168,472) 832 (-100, 36, 96) 664 (-110. 40. 120) 412 (-84.12,44) 296(-61.8,100) 242 (-34. 82, 90)31 1276 (-80.240,280) 668 (136) 436 (76) 153 (21) 92 (28) 28 (1)35 1268 (216) 848 (92) 436 (316) 108 (304) 36(32) 0(0)42 1240 (-72.388. 992) 224 (-8, 16, 136) 40 (-4, 12,44) 32 (-16,4,24) 24 (0. 4, 24) 8 (-8. 4, 24)12 1232 (-152. 76, 92) 676 (-52, 36, 224) 432 (-112. 296. 404) 320 (-40. 200, 272) 160 (-32, O. 32) 80 (-4,36,55)7 1208 (-240, 340. 632) 148 (-16. 92. 100) 64 (-8. 8. 8) 48 (-12, O.0) 24 (-8, 4, 8) 8 (0)

21 1192(114) 762 (130) 558 (106) 464 (44) 40(58) 28(0)14 1160 (-128, 24. 316) 616 (-64,36,88) 448 (-56, 44, 60) 186 (-12. 30. 42) 52 (-4. 28, 60) 2 (-2. 4. 6)6 1132 (-60.100,220) 652 (-332. 312) 428 (-20.168.192) 304 (-32. 28. 84) 4 (-4.0,4) 4(4)9 1112 (-132.180.256) 592 (-192, 88, 508) 176 (-16.60,224) 80 (-24, 8, 36) 40 (-20. 20, 22) 12 (-4, 4, 8)

18 1088 (-248. 80, 496) 76 (-20.12,44) 48 (-8, 0,12) 12 (-4.12,24) 10 (-4. 6. 18) 4(0)56 1084 (-118,68,484) 212(-4,56,194) 40 (-24, 7, 16) 4 (-4, 0, 0) 0(4) 0(4)19 1076 (-520,1044.1464) 36 (-32. 8,12) 8 (-4. O.12) 0(0,4.8) 0(0.4,4) 0(0)20 1072 (462) 156 (171) 100 (170) 76 (102) 38 (90) 8 (12)5 1068 (-184,244,408) 376 (-16, 88, 272) 212 (-8, 40,156) 112 (-8, 92. 100) 16 (0, 32, 44) 0(0)

58 1052 (732) 521 (411) 422 (154) 24 (106) 16 (12) 0(8)52 1024 (-68, 28, 32) 904 (-80, 8. 168) 280(-24.209,220) 92 (-36, 44, 64) 20 (-12. O. 12) 0(4)32 916 (-76. 44. 160) 140 (-4.28,36) 44 (-4,8,16) 20 (-8, 8,12) 16 (4) 0(0)55 904 (-280, 140,408) 12 (-12. 4, 20) 0(0.4,16) 0(0,4.4)8 876 (-68. 200, 304) 726 (-122.10.74) 632 (-200, 36. 56) 308 (-56,188.192) 128 (-24.164,204) 36 (-28,4.32)

61 812 (52) 404 (84) 360 (12) 316 (24) 128 (-44, 8. 36) 20 (2)63 688 (176) 140(52) 130(20) 48 (48) 4 (16) 0(10)24 680 (-52,6.20) 330 (34) 110(-10.116) 42 (4)40 532 (-72,56,68) 40 (-20.16.32) 28 (-16,12.16) 4(0) 0(0) 0(0)27 489 (-13,35.47) 60 (-14,10,58) 24 (-4. 4,18) 10 (-2,4,6) 4(2) 0(0)29 460 (-140, 40.116) 180 (4) 80 (28) 16 (8) 8 (4) 0(0)25 416(116) 240 (22) 120 (24) 55 (27) 42 (4) 22 (6)33 368 (-144.136.220) 60 (-12, 4. 20) 28 (-4, 4.12) 20 (-8, 4, 24) 16 (-8. 4, 8) 8 (-4. 4. 4)10 312 (-27.12.88) 64 (-4,0,8) 52 (-4,8,8) 16 (0. 4. 8) 0(0) 0(0)50 288 (-48. 44, 72) 84 (-20. 4, 24) 48 (-20.4. 16) 8 (-4, 0.12) 8(0.2,2)46 269 (-62, 41, 78) 76 (-16. 53. 77) 46(-22,1.35) 39 (-14,4,19) 18 (-8. 3. 9) 11 (-3,4.13)30 264 (-64. 24, 36) 104 (-24.36,44) 60 (-28. 4,12) 8 (0.12.18) 0(4) 0(0)47 200 (-90.188,188) 40 (-12. 4, 28) 12 (-8. 0, 8) 0(4) 0(0) 0(0)

3 160 (-32,84,124) 32 (0. 4. 24) 24 (0,4.8) 12 (-4,0,8) 8 (0) 0(0)23 158 (-18. 22. 53) 96 (16) 72(-2,50) 52(0) 36 (4)48 120 (0.136) 64 (-14,32.52) 4 (12) 2 (6) 0(4)51 92 (-12, 48, 60) 0(0,8.12) 4(0, O.0) 0(0) 0(0)49 73 (-6,3,8) 31 (-6.6.8) 21 (-3,3,6) 0(0,0,0)60 48 (40) 8(0) 4(4) 0(0)64 46(22) 16 (24) 16 (8) 8(8) 0(0) 0(0)4 28 (-8,0,20) 16 (0. 8.24) 4 (-4, 4,12) 0(0.12.16) 0(0) 0(0)

53 12 (-4. 4,12) 0(4) 0(4) 0(0) 0(0) 0(0)36 12 (-12, 4. 4) 0(0,0,4) 0(0, O.4) 0(0, O. 0) 0(0. O.0) 0(0,0,0)

NOTE: First figure is largest observation s median. In parentheses, additional observations as deviations from first figure.

If overdispersion is not present, then this expression does can be closely approximated as C7r 2l(n 3e3a) , where the con-not depend on a, because the only dependence on a of the stant Cdoes not involve any parameter. Thus if only threedeterminant of the information matrix is through the mul- concentrations XI = 0, X2, X3 = OC! with the same number oftiplicative factor e3a . This is one advantage of considering cells are used, then the intermediate concentration X2 is cho-volumes relative to the minimum. A drawback of using rel- sen byative volumes is that the minimum volume is a function of

. { exp(l3x2) (1 - 7r) + 7r r37rthat cannot be given explicitly (cf. [6]). But the minimum min E' ". (16)of the reciprocal of the determinant of the information matrix X2 13, 7r(l - 7r)2x~exp(2I3x2)

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Minkin: Clonogenic Assay Experimental Design

Note that the criterion depends on the joint distribution of(j and 7r in the population, and it seems unlikely that a usefulparametric specification of such a distribution can be pos­tulated. Therefore, when a large pilot study of M patients isavailable, expectation in (16) can be replaced by the averageover the M estimated values of ({jk, 7rd, k = 1, ... , M.More complicated expressions can be derived for more prac­tical designs, such as when X3 is a finite multiple of X2, andfor equally spaced designs with more than three concentra­tions.

If overdispersion is present, then the minimum of the re­ciprocal of the determinant of the information matrix is afunction of 7r and Tea that can be closely approximated byc( 1 + Tea7r )( 1 + Tea)27r2 I( n3e3a), where, as earlier, the con­stant c does not involve any parameter. Using this approx­imation, the intermediate concentration in the optimal three­point design is found by

. { if;({j, 7r, x2)[1 + Teaif;W, 7r, X2)] }lt3mIn 1S , (17)

X2 7r(1 + Tea)(l - 7r)2x~exp(2/h2)

where if;({j, 7r, X2) = {exp((jx2)(1 - 7r) + 7r}, and the ex­pectation is now over the joint distribution in the populationof the values for ({j, tr, T, a). Clearly, despite the increasingcomplexity of the criterion, the strategy of replacing expec­tation by average over the patients is still feasible. But patientswith a log-linear dose-response present a difficulty, becausewhen 7r = 0, the minimum of the reciprocal information is0. A simple approach is to ignore those patients, becauseany allocation with only finite concentrations would havezero efficiency.

5. AN EXAMPLE

Peripheral blood samples were obtained prior to treatmentfrom 67 patients with acute myeloblastic leukemia referredto the Princess Margaret Hospital in Toronto. To determinepatient sensitivity in culture to cytosine arabinoside (Ara­C), mononuclear cells were plated in methylcellulose con­taining growth factors plus different concentrations of Ara­C. Colonies of more than 20 cells were counted after five toseven days in culture. The experimental plan established theuse ofsix concentrations ofAra-C: 0, .5, 1,2,3, and 5 X 10-6

mol/L, with four plates for each concentration. In somecases, however, not enough cells were available; in aboutone-third of the cases, only two plates were used for a par­ticular concentration, and in one-eighth of the cases fewerconcentrations were used. In fact, for two patients only twoconcentrations were used. These two patients are excludedin this illustration, because it is not feasible to use a three­parameter model. The data set is shown in Table 5.

As an illustration of the techniques introduced in Section4.1, consider first the log-linear model (1) under the as­sumption of Poisson variation. Using maximum likelihood,the 65 estimated slopes were negatively skewed with a meanof -1.80 and a standard deviation of 1.34. Q-Q plots (notshown here) indicated that a gamma distribution provides abetter fit than the Gaussian distribution for the negative ofthe estimated slopes. Figure 1provides a comparison betweenthe average values ofexp( -fix) and the moment-generating

417

0.0 0.2 0.4 0.6 0.8 1.0

x

Figure 1. Average of exp(-{3x) Based on the 65 Estimated Slopes. Themoment generating functions of the fitted Gamma (higher curve) andGaussian distribution (lower curve) are also shown.

function of both the gamma and Gaussian distribution withparameters estimated from the data. Both distributions pro­vide good approximations when x is small, whereas when xincreases an underestimation is observed for the Gaussianapproximation and an overestimation is observed for thegamma. Clearly, the approximation provided by the gammadistribution remains accurate over a wider range than doesthe Gaussian approximation.

Under the gamma assumption and using the maximumlikelihood estimates K= 2.15 and;' = .84, the optimal two­point design obtained from (13) and (12) is to take 71% ofthe observations at the Ara-C concentration of .68 X 10-6

mol/L, and the remaining observations at the control. Usingnumerical integration and optimization, the optimal four­point design with concentrations (0, x, 2x, 3x) was foundto be x = .54 with (ql> ... , q4) = (.24, .44, .24, .07). Thevalue of the objective function for this four-point design is95% of the corresponding value for the optimal two-pointdesign. These results and the ones reported later are sum­marized in Table 6, which also shows for comparison theoptimal design for the Poisson log-linear model reported inSection 2.

If the fitted Gaussian distribution is used instead to providean expression for 1S[exp(-(jx»), then the optimal two-pointdesign obtained from (11) and (10) takes fewer observationsat the control and uses a higher concentration than does theoptimal two-point design under the gamma prior. The reasonfor this can be seen in Figure 1, because in this case theGaussian moment-generating function does not produce aslarge values for large concentrations as does the gamma.Therefore, high concentrations are not penalized as heavilyunder the Gaussian prior. The optimal design with concen­trations (0, x, 2x, 3x) is very similar to the two-point design.

When the 65 estimated slopes are used to determine theempirical moment-generating function emgLIJ( x), solving(15) and (14) yields an optimal allocation that is remarkablysimilar to the allocation provided by the gamma distribution.

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Mode/b Prior

PII nonePII gammaPII GaussianPII empiricalDrs empirical

Journal of the American Statistical Association, June 1993

Table 6. Suggested Designs for the Example

Design with minimum m· More practical alternativeRelative

Doses (qt • . . . • qm) m gape (qt •. . . • qm) efficiencyd

(0, -2.56/fJ) (.22, .78) 4 -1.27/fJ (.21, .1, .59, .1) 92(0, .68) (.29, .71) 4 .54 (.24, .44, .24, .07) 105(0, .82) (.26, .74) 4 .80 (.25, .73, 0, .03) 102(0, .67) (.28, .72) 4 .57 (.24, .60, 0, .16) 105

(0, .81, CXJ) (.33, .33, .33) 12 .62 (1/12, ... , 1/12) 98

• m ~ number of concentrations.• PII ~ Poisson log linear; Drs ~ Overdispersed, resistant subpopolanon,C Designs have concentrations, 0, x, 2x, ... ,(m - 1)x. Here x is the gap between concentrations.d Efficiency of the alternative relative to the design with minimum number of doses.

The best four-point equally spaced allocation also uses con­centrations very similar to those provided by the gammaprior, although it differs substantially in the proportions al­located to each concentration.

Taking the 65 estimated slopes as the population ofslopes,a comparison of the different allocations considered herecan be made by forming the ratio of the criterion evaluatedat the optimal four-point allocation for this population (i.e.,q = (.24, .60, .0, .16), x = .57), relative to the value of thecriterion at alternative allocations. The four-point allocationderived under the Gaussian assumption yields a ratio of .87,whereas the four-point allocation derived under the gammaassumption is almost fully efficient, giving a ratio of .99.Interestingly, the experimental plan used to generate the es­timated slopes (i.e., equal allocation to the six concentrations:0, .5, 1, 2, 3, 5) yields a ratio of .60. Therefore, under theassumptions of this comparison, this experimental plan isonly 60% efficient for the Poisson log-linear model.

But an examination of the dose-response curves revealsthat the Poisson log-linear model is not an appropriate onefor these data. There is clear evidence of a resistant subpop­ulation in at least one-third ofthe curves, and overdispersionis a common occurrence. These facts are illustrated in Table7, where data from one of the 65 patients is given. By com­paring the second and third columns, it is obvious that thevariance exceeds the mean by one order of magnitude, in­dicating overdispersion. The fit provided by the resistantsubpopulation model (given in column 5) is significantlycloser to the means (reported in column 3) than are the fittedvalues provided by the log-linear model (given in column4). The last column shows that the overdispersion expression

Table 7. Ara-C Sensitivity in Culture: Dose-Response Curvefor Patient 54

Resistantsubpopulation b

Ara-C Sample Sample Log-linear·iii + iii~dose variance mean III III

0 113,600 1,404 1,026 1,360 54,894.5 6,752 840 878 911 24,9041 9,158 676 751 646 12,7342 2,980 405 550 400 5,0223 4,411 308 403 314 3,1765 3,740 276 216 275 2,460

'Ii = 6.93, {j = -.31, f = .100.• Ii = 7.22, {j~ -1.06, .. = .198, f = .029.

for the mean-variance relationship captures the general trendof the sample variances (column 2).

A procedure to fit the model (4) allowing for overdispersionusing quasi-likelihood methods was described by Minkin(1991). Following that procedure to estimate the parametersresulted in 35 of the 65 patients having a positive value of1r. The estimated parameters for these 35 patients are listedin Table 8. Using the approximate criterion (17) with ex­pectation replaced by average over these 35 patients yieldsas the recommended design to allocate i of the cells at eachof'x, = 0,X2 = .846 X 10-6 mol/L, and x, as large as possible.As long as X3 > 6.193, the efficiency exceeds 90%. Amongthe equally spaced designs with the same number of cellsplated at each concentration, the best one uses the 12 con-

Table B. Parameter Estimates for Patients With n- > a

Pat ex -Ii n- T

1 6.96 .82 .00658 .3266 7.37 1.19 .00070 .5827 7.16 3.71 .02274 .1369 7.03 1.30 .01209 .138

11 7.52 1.62 .04944 .23312 7.05 .70 .05601 .09415 7.83 2.02 .00026 .36917 8.43 1.14 .02946 .15218 6.87 3.81 .01578 .30120 6.64 .88 .00744 .46322 7.13 1.14 .08679 .06723 5.09 .47 .14913 .04824 6.53 1.60 .02372 .05225 6.10 1.22 .06212 .01826 7.38 1.20 .00282 .08828 7.46 1.98 .00158 .58931 7.17 1.00 .01717 .02533 5.98 3.75 .04402 .19534 7.97 1.81 .11349 .09538 8.18 2.95 .01334 .23039 7.14 1.31 .01739 .26343 7.84 5.88 .04542 .10844 8.03 1.19 .00741 .19346 5.45 1.40 .07047 .15248 5.27 2.33 .01087 .34750 5.67 2.15 .02727 .07452 7.27 1.35 .00012 .14354 7.22 1.06 .19810 .02955 6.86 8.70 .00350 .88256 7.18 3.45 .00143 .14758 7.39 1.47 .00180 .27859 6.93 2.05 .00004 .99962 7.80 .52 .18104 .05363 6.36 1.28 .00660 .40765 8.17 .63 .01132 .027

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(A.2)

Minkin: Clonogenic Assay Experimental Design

centrations: 0, .65,2(.65), ... ,11(.65). If the exact mini­mum of the reciprocal of the information is used instead ofthe approximation c(l + Teu1l" )( 1 + Teu)211"2 f(n 3e 3u

) , thenthese concentrations are slightly reduced from .846 to .808,from 6.193 to 6.046, and from .65 to .625, with the 12­concentration design being also the optimal equally spaceddesign, with a 98.5% efficiency in comparison with the bestthree-point design with X3 = 00.

In this example the variability in the estimated parametersof the dose-response curves has been ignored. Although someapproaches can be developed to incorporate it in the esti­mation of the prior distributions, they are unlikely to havemuch effect in this type of application, where the estimationerror is very small in comparison with the variability amongdose-response curves. For example, in estimating the 65slopes in the Poisson log-linear model, only five of the stan­dard errors exceeded .134, which is one-tenth of the samplestandard deviation of the estimated slopes. If the samplevariance of the estimated slopes is corrected by subtractingthe average of the squared standard errors, then the standarddeviation of the Gaussian prior would be reduced by .005.This change is so small that, to the reported accuracy, theoptimal two- and four-point designs for the Gaussian priorare identical to those identified here. The fact that the prioris so broad also implies that there is little difference betweenthe posterior distribution and the likelihood function. Thuswhether maximum likelihood or Bayesian estimation is usedwould have practically no effect on the criteria consideredhere.

6. CONCLUDING REMARKS

This application illustrates some problems that must beaddressed when trying to apply the theoretical results of op­timal experimental design. These include (a) establishing theeffects of parameter misspecification, (b) identifying morepractical designs that are nearly optimal, (c) assessing therobustness against departures of both the systematic and thestochastic components of the model, and (d) incorporatingcriteria for judging the performance of a particular designto be used in a series of experiments. Although the presen­tation does not attempt to generalize the results to a widerfamily of models, it emphasizes an approach to dealing withthe practical aspects of designing an experiment.

In the pharmacokinetic literature there have been severalattempts to deal with these issues for models similar to theones considered here. Useful reviews of this work have beengiven by Endrenyi (1981) and by Landaw (1985). In partic­ular, several strategies that are alternatives to the minimi­zation of average relative variance introduced in Section 4have been used to deal with the large variability in the pop­ulation of dose-response curves (see D'Argenio 1990 andreferences therein). The strategy of using adaptive designswhich sequentially try to optimize each experiment has beenused in other areas, particularly in quantal-response problems(Minkin 1987, Wetherill 1963). But this strategy is not prac­tical in the present context, due mainly to the fact that thecolony formation assessment is made after the cells havebeen plated for one week.

419

An interesting suggestion put forward by one referee is toadopt a design that maximizes the ability of detecting theassociation between assay parameters and clinical outcome,because this is the eventual goal of the experiment. Minkin(1991) reported an association between clinical outcome andthe parameter 11" in the resistant subpopulation model (4),so this suggestion would lead to focus attention on estimatingthis parameter. But now a new series of experiments is to becarried out to confirm this association and to establish thecombined ability to predict clinical outcome by this clono­genic assay and another that assesses different biologicalcharacteristics of the leukemic cells. Thus at this point it isprobably wiser to consider the estimation of all the param­eters of the dose-response curve as equally important.

APPENDIX: DERIVATION OF THE OPTIMAL DESIGNS

A.1 Poisson log-linear Model

To find the minimum of var(l / (1) in Equation (2) with respectto the choice of concentrations and the proportion of cells platedat each concentration under the restriction that (3 < 0, assume with­out loss of generality that 0, 2 ... 2 Om. Because (3 and n areconstants, and X; - x, = (0; - OJ)/(3, an equivalent problem is tominimize

exp(O,) ~ q;exp(-.:l;)i=l

i»]

where .:l;= 01 - 0;. Clearly, 01 should be as large as possible, whichimplies that Xl, the lowest concentration, should be 0.

Now, after equating to °the partial derivatives of (A. I ) withrespect to each .:lk, it turns out that each .:lk, k = 2, ... , m has tosatisfy the same quadratic equation a2.:l~ + al.:lk + ao = 0, where,using the fact that .:lj = 0, the coefficients a2, a" and ao are a2

= L~I qjexp(-.:lj), al = -2 L~I qj(.:lj + l)exp(-.:lj), and ao= L~I L~l q;qj.:l;(2 + .:lj)exp(-.:l; - .:lNL~1 qjexp(-.:lj).The two solutions to this quadratic equation are .:lk = 0 - 2 and.:lk = 0, where

0-2 = j~1 qj.:ljexp(-.:lj) / j~1 qjexp(-.:lJ.

Thus the optimal design has at most three different dosages: XI = 0,X2 = -(0 - 2)/(3, and X3 = -0/(3. Moreover, the best strategy is toavoid taking observations at X2.

Set m = 3, .:l2 = 0 - 2, and .:l3 = O. From (A.2) it follows that

q3 = ql(o - 2)exp(0)!2, (A.3)

so the objective function (A. I) is equal to [exp(OI)ql(o - 2)0)r'.From this it immediately follows that in the optimal allocation q2

= 0, because ql + q2 + q3 = I and qk 2 °for k = 1, 2, 3. Then

ql = 2/[2 + (0 - 2)exp(0)],

where we have ignored the restriction that nq, should be an integer.Finally, because 0 2 0, the optimal 0 satisfies

(0 - 2)exp(0/2) = 2.

The solution to these last two equations is 0 = 2.55693 and ql= .21781.

A.2 Resistant Subpopulation

To minimize (5), note first that because q3 = 1 - ql - q2, itfollows by taking derivatives with respect to ql and q2 that the op-

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timal allocation has ql = q2 = Q3. Now set 0 ::s 01 < 02 < 03 andrecall that the restrictions on the parameters are /3 < 0, ex > 0, ando ::s 1r ::S I. The optimal choice includes 03 = 00, because as afunction of 03, (5) is increasing. This follows from the fact that thedenominator is clearly a decreasing function of 03 and the terminside the square brackets in the numerator is 0 at 03 = 02 andmonotonic for 03 > 02> 01 ~ O.Then it is simple to show that with03 = 00, the expression (5) is decreasing for 01 in the interval[0,02), and thus 01 = 0 is in the optimal choice. Finally, the expres­sion (6) for determining 02 ensures that the derivative of (5) withrespect to 02 is 0 when 03 = 00, 01 = 0, and 0 < 1r < I. Note thatwhen 1r = I, the information matrix is singular, whereas for 1r = 0the determinant of the information goes to infinity when 0 = 01< 02 < 03 = 00.

[Received October 1991. Revised September 1992.]

REFERENCES

Breslow, N. E. (1984), "Extra-Poisson Variation in Log-Linear Models,"Applied Statistics, 33, 38-44.

Curtis, J. E., Messner, H. A., Minden, M. D., Minkin, S., and McCulloch,E. A. (1987), "High-Dose Cytosine Arabinoside in the Treatment of AcuteMyelogenous Leukemia: Contributions to Outcome ofClinical and Lab­oratory Attributes," Journal ofClinical Oncology, 5,532-543.

Journal of the American Statistical Association, June 1993

D'Argenio, D. Z. (1990), "Incorporating Prior Parameter Uncertainty inthe Design of Sampling Schedules for Pharmacokinetic Parameter Esti­mation Experiments," Mathematical Biosciences, 99, 105-118.

Endrenyi, L. (1981), "Design of Experiments for Estimating Enzyme andPharmacokinetic Experiments," in Kinetic Data Analysis, Design andAnalysis ofEnzyme and Pharmacokinetic Experiments, ed. L. Endrenyi,New York: Plenum Press, pp. 137-167.

Landaw, E. M. (1985), "Optimal Design for Individual Parameter Estimationin Pharmacokinetics," in Variability in Drug Therapy: Description, Es­timation and Control, eds. M. Rowland, L. M. Sheiner, and J. L. Steimer,New York: Raven Press, pp. 187-197.

Lawless, J. F. (1987), "Negative Binomial and Mixed Poisson Regression,"Canadian Journal ofStatistics, 15,209-225.

McCullagh, P., and Neider, J. A. (1989), Generalized Linear Models (2nded.), London: Chapman and Hall.

McCulloch, E. A., Buick, R. N., Curtis, J. E., Messner, H. A., and Senn,J. S. (1981), "The Heritable Nature of Clonal Characteristics in AcuteMyeloblastic Leukemia," Blood, 58, 105-108.

Minkin, S. (1987), "Optimal Designs for Binary Data," Journal of theAmerican Statistical Association, 82, 1098-1103.

--- (1991), "A Statistical Model for In-Vitro Assessment of PatientSensitivity to Cytotoxic Drugs," Biometrics, 47,1581-1591.

Silvey, S. D. (1980), Optimal Design, London: Chapman and Hall.Tannock, I. F. (1987), "Biological Properties of Anticancer Drugs," in The

Basic Science ofOncology, eds. I. F. Tannock and R. P. Hill, New York:Pergamon Press, pp. 278-291.

Wetherill, G. B. (1963), "Sequential Estimation of Quantal ResponseCurves," Journal ofthe Royal Statistical Society, Ser. B, 25, 1-48.

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