fitting emax models to clinical trial dose–response data

Fitting Emax models to clinical trialdose–response dataSimon Kirby,� Phil Brain, and Byron Jones

We consider fitting Emax models to the primary endpoint for a parallel group dose–response clinical trial. Such models can bedifficult to fit using Maximum Likelihood if the data give little information about the maximum possible response.Consequently, we consider alternative models that can be derived as limiting cases, which can usually be fitted. Furthermorewe propose two model selection procedures for choosing between the different models. These model selection proceduresare compared with two model selection procedures which have previously been used. In a simulation study we find that themodel selection procedure that performs best depends on the underlying true situation. One of the new model selectionprocedures gives what may be regarded as the most robust of the procedures. Copyright r 2010 John Wiley & Sons, Ltd.

Keywords: clinical trial; dose–response; Emax model; model selection; non-linear model

1. INTRODUCTION

The Emax models

Y ¼ E01Emax � D

ED501D1e e� Nð0; s2Þ ð1Þ

and

Y ¼ E01Emax � Dl

EDl

501Dl1e e� Nð0; s2Þ ð2Þ

where Y denotes a response of interest, D represents dose,E0, Emax, ED50 and l are unknown parameters and e is a randomerror assumed to be Normally and independently distributedwith constant variance are popular choices for modelling thedose–response relationship in a parallel group clinical trial.E0 represents the placebo response, E01Emax represents themaximum possible response as dose approaches infinity, ED50 isthe dose that produces half of the Emax effect and l representsthe slope and shape of the curve. Model (1) is a special case ofmodel (2) obtained by setting l= 1.

These models are popular choices for modelling the dose–response relationship in a parallel group clinical trial because theexpected response is monotonically related to dose, and there isa lower and upper asymptote for the expected response. Theseproperties are basic desirable features for modelling manyclinical trial dose–response curves. In practice, these models havebeen found to fit well to many datasets.

As well as being used to estimate mean response at differentdoses, models (1) and (2) can be used to estimate the doserequired to give a minimum difference of interest from placebo.Taking the expectation of Y, setting the difference of interestfrom placebo equal to D, and re-expressing in terms of the doserequired we have for model (2)

EDDð2Þ ¼ED50

Emax

D� 1

� �1=l

where EDDð2Þ is the effective dose to produce a difference of Dfrom placebo for model (2), provided that Emax4D. Setting l= 1gives the corresponding result for model (1).

Maximum Likelihood estimation of models (1) and (2) fails,however, when there is insufficient information in the data toestimate all of the parameters. In particular, estimates of theED50 and Emax parameters will fail to converge when there isinsufficient information about the maximum possible responseas dose increases. This is more likely to happen if data are notobtained for sufficiently large doses. This can be the case for theperfectly valid reason that larger doses are not used because oftolerability and safety concerns. It should be noted that if modelestimation fails because of lack of information about theparameters in the data then methods used for improving theconvergence of non-linear models – see, e.g. Bates and Watts[1], Ratkowsky [2] and Ross [3] – are unlikely to help.

An illustration of a dataset for which Maximum Likelihoodfitting of models (1) and (2) fails using the function nls in the Rpackage [4] is shown in Figure 1. The options used within R forthis fit were: the Gauss-Newton method; the default tolerancelimit for convergence; a maximum of 100 allowed iterations;convergence limits for the parameter values such that thelog(ED50) had to be greater than or equal to �10 and less thanor equal to log(250) and l had to lie between 0.1 and 10(inclusive). The data are simulated means for 80 subjects perdose obtained using parameter values and variance set equal to

14

3

MAIN PAPER

(wileyonlinelibrary.com) DOI: 10.1002/pst.432 Published online 28 April 2010 in Wiley Online Library

Pharmaceut. Statist. 2011, 10 143–149 Copyright r 2010 John Wiley & Sons, Ltd.

Statistics, Pfizer Limited, Sandwich, Kent, UK

*Correspondence to: Simon Kirby, Statistics, Pfizer Limited, Sandwich, Kent, UK.E-mail: [email protected]

the estimated parameter values and variance from an actualclinical trial. Out of 1000 simulations using the same parametervalues 288 produced datasets for which the fit of model (2)failed to converge and 37 produced datasets for which the fit ofmodel (1) failed to converge. This example is unusual in thatthere is a very large signal-to-noise ratio (the largest treatmentdifference is 9.05 and the standard deviation is 7.1) and largesample size so it represents a particularly favourable situation forfitting models (1) and (2). Even so it is evident that lack ofconvergence of the model fits can be a problem. In morecommon, less favourable situations one can expect lack ofconvergence to be a bigger issue.

In Section 2 of this article we consider alternatives to models(1) and (2) which can be obtained as limiting cases when theED50 and Emax parameters are allowed to approach infinity. Thenin Section 3 we propose two model selection procedures forchoosing between models (1) and (2) and the models obtainedas limiting cases. Two other model selection procedures,which have previously been used, are described in Section 4. InSection 5 we apply the model selection procedures to a particulardataset. A simulation study to compare the performance of allfour model selection procedures is described in Section 6. Theresults of the simulation study are presented in Section 7. Finallyin Section 8 there is a discussion of the results obtained.

2. ALTERNATIVES TO THE Emax MODELS THATCAN BE OBTAINED AS LIMITING CASES

For the expected value of Y, dividing the numerator anddenominator of the second term on the right hand side ofmodel (2) through by EDl

50 and letting the value of the ED50 andEmax parameters approach infinity but with the ratio Emax=EDl

50

held constant (see Appendix, available online as supportinginformation) model (2) approaches

EðYÞ ¼ E01Slope � Dl ðPÞ

where

Slope ¼Emax

EDl50

We use the label (P) for this model to stand for Power model asthe expected value of Y has a power relationship with dose. Thismodel can be expected to fit well to data when the trueunderlying model is model (2) and ED50 is large relative to themaximum value of D.

It can be seen that it is reasonable to consider that the ratioEmax=EDl

50 is held constant because this ratio has the interpretationof being the tangent to the curve for model (2) when E(Y) is

plotted against Dl and D is zero. It should generally still be possibleto estimate the value of this tangent even when the individualparameters cannot be estimated because there is insufficientinformation about the maximum possible response. When l= 1model (P) becomes the equation for a simple linear regressiongiven by

EðYÞ ¼ E01Slope � D ðLindoseÞ

where

Slope ¼Emax

ED50

Model (Lindose) can be expected to fit well to data when thetrue underlying model is model (1) and ED50 is large relative tothe maximum value of D.

For model (P) the dose required to give a minimum differenceof interest from placebo is, provided that Slope40,

DSlope

� �1=l

Setting l= 1 gives the corresponding result for model (Lindose).Schoemaker et al. [5] have previously considered the idea of

dividing the right hand side of the expected response for model(1) by ED50 to give a new parameter S0 = Emax/ED50 in the contextof estimating concentration–effect relationships. Their aim in doingthis was to give a more stable parameter that can be estimatedand interpreted as the initial sensitivity to drug at low concentra-tions. This differs from our aim of considering limiting cases toobtain models, which can be fitted to dose–response data fromparallel group clinical trials. Our interest is in estimating responsesfor different doses and estimating doses required to obtain at leasta given response rather than interpreting parameter values. Duttaet al. [6] also noted for model (2) with dose replaced byconcentration that the estimated curve can be used forinterpolation even when the parameters are poorly estimated.

3. TWO MODEL SELECTION PROCEDURESFOR THE Emax MODELS AND THEALTERNATIVE MODELS

The relationships between the models (1), (2), (P) and (Lindose)can be displayed as shown in Figure 2. Model (2) is the mostgeneral of the models and consequently appears at the top ofthe figure. One can move from model (2) to model (1) if l= 1.Alternatively one can move from model (2) to model (P) if l6¼1but the limiting form of model (2) as ED50 and Emax approachinfinity applies. From both model (1) and model (P) one canmove to model Lindose. In the first case model Lindose isobtained as the limiting form of model (1) as ED50 and Emax

approach infinity. In the second case setting l= 1 for the Powermodel gives model Lindose. Clearly it would be helpful to have amodel selection procedure to decide between the models whenfitting them to data. It should be stated, however, that ourstarting point is that either model (1) or model (2) is theunderlying truth and our aim in choosing between models is toestimate responses for given doses or a dose for a givendifference in response from placebo as well as possible. Wepresume that at least five doses are used so that at least1 degree of freedom for error is available when the models are1

44

Figure 1. Plot of an example dataset for which the fits of models (1) and (2) do notconverge using the nls function within the R Statistics package.

S. Kirby, P. Brain and B. Jones

Copyright r 2010 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2011, 10 143–149

fitted to mean data. In this section we propose two modelselection procedures although there are many others that couldbe considered. We assume throughout that model fitting isdone using Maximum Likelihood (which is equivalent toOrdinary Least Squares for the models considered in this article).

The first model selection procedure is as follows:

1. Try to fit model (2). If this model can be successfully fittedthen it is the selected model.

2. If model (2) cannot be fitted then, if possible, fit both models(1) and (P) and select the one with the smallest residual sumof squares.

3. If model (2) cannot be fitted and only one of models (1) and(P) can be fitted then choose the model that can be fitted.

4. If none of the models (2), (1) and (P) can be fitted then selectmodel Lindose.

We use the shorthand RSS to refer to this model selectionprocedure. This is because if a choice has to be made betweenmodels (1) and (P) it is made by choosing the model with thesmallest residual sum of squares. A fit is regarded as successful ifthe fitting process converges.

The second model selection procedure is:

1. Calculate the value of Akaike’s Information Criterion (AIC)[7] given by

�2 ln L12p

where L is the maximized likelihood and p is the number offitted parameters, for each of models (2), (1) and (P), if theycan be fitted. Select the model that has the smallest AICvalue and can be fitted. AIC seeks to balance model fit, givenby the maximized likelihood and the number of parametersin the model. It is used here to enable comparisons to bemade between non-nested models (comparisons where onemodel is not a special case of the other).

2. If none of the models (2), (1) and (P) can be fitted thenselect model Lindose.

We use the shorthand AIC to refer to the second modelselection procedure. Again this shorthand is adoptedbecause of the way that a choice between models is made.

4. TWO OTHER MODEL SELECTIONPROCEDURES, WHICH DO NOT USE THEALTERNATIVE MODELS

In this section we describe two other model selectionprocedures which we have known to be used which allow forthe failure of Maximum Likelihood estimation for models (1) and(2). Again we assume that all model fitting is done usingMaximum Likelihood.

One procedure that has been used (in the R DoseResponsepackage) is as follows

1. Select model (2) if it can be fitted.2. If model (2) cannot be fitted but model (1) can be then select

model (1).3. If neither model (2) nor model (1) can be fitted then fit the

model from the following set of three which has the smallestresidual sum of squares:

ðiÞ model Lindose

ðiiÞ Y ¼ a1b � logðD1kÞ1e e� Nð0; s2Þ

ðiiiÞ Y ¼ c1d � expðDÞ1e e� Nð0; s2Þ

where a, b, c and d are parameters to be estimated and k is asmall pre-specified constant value. We subsequently refer tomodels (ii) and (iii) as models Linlogdose and Linexpdose,respectively. Models Lindose, Linlogdose and Linexpdose havepreviously been suggested along with model (1) and a logisticmodel using dose as a set of candidate models for a monotonicdose–response curve by Dette et al. [8]. A logistic model usingdose typically gives a similar fit to model (2). A logistic modelusing log dose is equivalent to model (2).

We label this model selection procedure by the shorthandLinRSS for the choice between the three linear regressionmodels if neither model (2) nor model (1) can be fitted.

A second procedure that we have known to be used byClinical Pharmacologists is

1. Select model (2) if it can be fitted.2. If model (2) cannot be fitted but model (1) can be then select

model (1).3. If neither model (2) nor model (1) can be fitted then fit model

Lindose.

We refer to this procedure by the shorthand Emaxlin torepresent the fit of an Emax model if it is possible to fit one andthe fit of a simple linear regression on dose otherwise.

5. APPLICATION OF THE FOUR MODELSELECTION PROCEDURES TO A DATA SET

To illustrate the use of the four model selection procedures weapply them to the analysis of a real set of data. The data areillustrated in Figure 3 and are those that underlie the simulationresults presented in the Introduction. The data are meanchanges from baseline in a score for five doses of a drug. Eachmean was based on approximately the same number of subjects(the number of subjects ranged from 77 to 81).

Application of model selection procedures LinRSS and Emaxlinusing the R package (with the options described in Section 6)results in model (1) being chosen. Model selection procedures 1

45

Figure 2. Diagrammatic representation of hierarchy of models (1), (2), (P) and(Lindose) described in the text.



RSS and AIC, however, choose model (P) which has a smallerresidual sum of squares than model (1).

6. A SIMULATION STUDY TO COMPARE THEFOUR MODEL SELECTION PROCEDURES

To compare the four model selection procedures we haveused scenarios based on some of those previously adoptedby the PhRMA Working Group on Adaptive Dose-RangingStudies [9].

The scenarios adopted by the PhRMA Working Group werebased on a clinical trial for neuropathic pain. The primaryendpoint was the change from baseline to 6 weeks in a VASscore, which was considered to be Normally distributed with avariance of 4.5. We consider the improvement in pain scorefrom baseline rather than the change so that a positive responserepresents an improvement in a patient’s condition ratherthan a deterioration. Like the PhRMA Working group we take, forsimplicity, the true placebo effect, represented by E0 inthe Emax models, to be zero. We set Emax to be equal to 1.8and then consider three possible values for the ED50. Theseare 20, 50 or 80. These three values correspond to low,medium or high values for the ED50 relative to the dose rangewhich is taken to be 0–80. The value for l is set equal to 1,corresponding to model (1) or set equal to 2. The value of2 was chosen because a study of observed concentration–response relationships has suggested that such a value can be

considered reasonably typical [6]. Three sets of doses are used.These are:

1. 0, 10, 20, 30, 40, 50, 60, 70 and 802. 0, 20, 30, 40, 50, 60 and 803. 0, 20, 40, 60 and 80

As in the PhRMA Working Group study, the total sample size isset to be approximately either 150 or 250. The total sample sizeis divided equally between the dose groups such that to thenearest integer it is 150 or 250. Finally, unlike the PhRMAWorking Group study we allow the variance to vary, being 4.5 asstated above, half this value (2.25) or double this value (9.0).

The total number of scenarios given by all of the possiblecombinations of parameter values, sets of doses, total samplesize and values for the variance is 108.

The six basic Emax curves are shown in Figure 4 with possibledose sets offset above the curves. The top row of graphs showthe expected response when l= 1 for ED50 = 20,50 or 80 (theED50 value increasing from left to right). The bottom row ofgraphs show the expected response when l= 2 for the samevalues of the ED50. It can be seen that as the ED50 increases lessinformation is obtained about the maximum possible response.Thus we expect Maximum Likelihood estimation to fail morefrequently as the ED50 increases.

To assess the different model selection procedures thefollowing performance measures were calculated for thedifferences in predicted response between each dose andplacebo:

1. the mean and maximum absolute percentage bias,2. the mean and maximum root mean square error expressed as

a percentage of the true mean response,3. the mean and minimum coverage probability for the nominal

95% confidence interval.

Also calculated were the percentage bias, root mean squareerror (expressed as a percentage of the true dose) and coverageprobability for a nominal 95% confidence interval for the doseestimated to give an increase in response over placebo of

14

6

Figure 3. Plot of an example dataset for which the four model selectionprocedures are applied.

Figure 4. Plot of dose–response relationship for 3 and 4 -parameter Emax models used for simulations (l= 1 in the top row and l= 2 in the bottom row, ED50 increasesfrom 20 to 50 to 80 across the columns and the three possible sets of doses are marked above each curve).



0.8 units (which may be regarded as a conservatively large valuefor an important difference in this setting [10]).

The standard errors required for the calculation of theconfidence intervals for the difference between the responsefor each dose and placebo and for the dose required to give anincrease in response over placebo of 0.8 were obtained by thedelta method [7].

The simulations were carried out by simulating mean datarather than by simulating individual data values. One thousandsimulations were carried out for each scenario. The number ofsimulations was restricted to 1000 because of the large numberof scenarios examined. This number of simulations was sufficientto draw qualitative conclusions (see Sections 7 and 8).

The confidence intervals were calculated using t-distributionsand the residual mean square from the model fits to themeans. The possibility of using a within dose group estimate ofvariance together with a critical value from the Normaldistribution was explored by simulating a common withingroup variance but the resulting coverage probabilities werefound to be very liberal.

The simulations and model fitting were done using the Rpackage. Specifically the nls function was used with the Gauss-Newton fitting method. Bounds of �10 to 10 (inclusive) were setfor allowable values of the log of the ED50 (the parameterizationused in the model fit for models (1) and (2)) and 0.1 to 10(inclusive) for l. The default tolerance was used with up to 100iterations allowed for each model fit. A dose was recorded as notestimable if EmaxpD in models (1) or (2) or the estimated dosewas negative or greater than 100. The value of k for modelLinlogdose was set equal to 0.01. It should be noted that

although the R package was used to carry out the simulationssimilar results are to be expected using any Statistics packagefor non-linear modelling. This is because the primary reason forthe non-linear models failing to fit is lack of information aboutthe model parameters in the data. For the same reasonchanging the model fit options within the R Statistics packagewould be expected to produce similar results.

7. RESULTS OF THE SIMULATION STUDY

Results are presented in Tables I and II for model selectionprocedures RSS, LinRSS and Emaxlin for the scenarios for fivedoses, total sample size approximately equal to 250 andvariance equal to 4.5. The results for five doses for a totalsample size approximately equal to 150 and the results for sevenand nine doses all with variance equal to 4.5 demonstrate asimilar pattern to the results presented in this article and arereferred to below. The results for the other values of thevariance were also similar and are also referred to below. Resultsfor model selection procedure AIC are omitted because they areeither similar to or worse than those for the RSS model selectionprocedure.

The proportion of times each of the six models was selectedby each of the model selection procedures is shown in Table I.This table displays a number of key features about the modelselections. First, it is apparent that the frequency with whichmodels (1) and (2) can be fitted declines as the ED50 increases.For a given value of the ED50 the frequency with which thesemodels can be fitted is lower when l= 1. Although not shown inthis article the frequency with which the models can be fitted is

14

7

Table I. Probability a model is selected classified by model selection procedure for simulation scenarios for five doses andapproximate total sample size of 250 (variance = 4.5).

Model selected

Model selection procedure (1) (2) P Lindose Linlogdose Linexpdose

l= 1, ED50 = 20RSS 0.238 0.287 0.355 0.120 — —LinRSS 0.238 0.603 — 0.019 0.103 0.037Emaxlin 0.238 0.603 — 0.159 — —








14

8

Ta

ble

II.

Sum

mar

ies

of

sim

ula

tio

nre

sult

sfo

rm

od

el

sele

ctio

np

roce

du

res

for

five

do

ses

and

app

roxi

mat

eto

tal

sam

ple

size

of

25

0(v

aria

nce

=4

.5).

Dif

fere

nce

inre

spo

nse

be

twe

en

eac

hd

ose

and

pla

ceb

oD

ose

req

uir

ed

toin

cre

ase

resp

on

seb

y0

.8u

nit

sco

mp

are

dw

ith

pla

ceb

o

Mo

de

lse

lect

ion

pro

ced

ure

Me

an,

max

imu

mab

solu

tep

erc

en

tag

eb

ias

Me

an,m

axim

um

ffiffiffiffiffiffi MSE

p EðYÞ�

10

0M

ean

,min

imu

mco

vera

ge

pro

bab

ility

%(9

5%

c.i.)

Pe

rce

nta

ge

bia

s

ffiffiffiffiffiffi MSE

p D�

10

0

Co

vera

ge

pro

bab

il-it

y%

(95

%c.

i.)P

rop

ort

ion

of

tim

es

do

seco

uld

be

est

imat

ed

l=

1,

ED5

0=

20

RSS

6.9

,1

2.3

34

.5,

46

.09

3.0

,8

7.0

57

.51

41

.99

3.1

0.9

7Li

nR

SS5

.0,

8.4

34

.6,

42

.28

7.3

,8

6.9

38

.01

13

.89

5.0

0.8

9E m

axlin

7.3

,1

3.6

34

.8,

46

.79

2.0

,8

4.4

62

.81

41

.99

4.1

0.9

6l

=1

,ED

50

=5

0R

SS1

.8,

2.8

45

.3,

63

.59

3.6

,8

8.9

0.8

59

.29

4.0

0.8

7Li

nR

SS1

0.5

,1

6.7

47

.8,

62

.89

0.4

,8

9.4

�1

7.0

52

.39

4.7

0.8

1E m

axlin

0.8

,1

.94

4.1

,6

1.3

92

.9,

86

.0�

0.7

57

.49

4.7

0.8

6l

=1

,ED

50

=8

0R

SS2

.7,

5.5

56

.5,

79

.09

4.1

,8

9.8

�1

9.3

43

.39

2.7

0.7

1Li

nR

SS1

9.6

,3

6.2

65

.5,

88

.49

1.3

,8

9.7

�3

4.9

50

.28

9.6

0.6

9E m

axlin

1.6

,4

.05

4.4

,7

4.2

94

.1,

91

.3�

22

.24

3.3

92

.50

.69

l=

2,

ED5

0=

20

RSS

2.5

,3

.22

9.7

,4

3.9

95

.3,

93

.77

.57

5.9

93

.91

.00

Lin

RSS

2.6

,4

.72

9.4

,4

2.6

94

.3,

93

.51

.36

1.5

94

.60

.97

E ma

xlin

2.3

,3

.02

9.4

,4

3.5

94

.8,

92

.28

.27

4.2

94

.01

.00

l=

2,

ED5

0=

50

RSS

10

.6,

34

.55

4.4

,1

10

.79

4.3

,9

1.1

0.8

39

.49

4.8

0.9

7Li

nR

SS4

7.7

,1

17

.98

7.5

,1

94

.18

7.9

,8

0.4

�1

5.5

42

.69

4.0

0.8

5E m

axlin

17

.3,

49

.95

2.9

,1

08

.99

3.3

,9

0.3

�3

.93

7.5

95

.10

.97

l=

2,

ED5

0=

80

RSS

33

.0,

10

4.1

99

.2,

22

6.3

95

.0,

92

.0�

13

.83

0.0

94

.90

.79

Lin

RSS

13

3.4

,3

56

.81

90

.7,

47

0.0

86

.7,

80

.0�

36

.74

8.8

89

.00

.77

E ma

xlin

52

.7,

15

2.4

10

2.2

,2

35

.79

3.9

,9

0.3

�1

8.2

32

.69

1.1

0.7

6



also lower for the smaller sample size. Other noticeable featuresare that model selection procedure LinRSS chooses the modelLinlogdose more frequently or about as frequently as the modelLinexpdose when the ED50 = 20 but less frequently when theED50 equals 50 or 80. Model selection procedure RSS choosesmodel P with reasonably constant probability for l= 1 but withincreasing probability for l= 2 as the ED50 increases.

The results in Table II for l= 1 show that model selectionprocedure LinRSS is similar to or better than the other modelselection procedures when the ED50 = 20 apart from the meancoverage probability for differences between the doses andplacebo for which it is worse. There is also a lower probability forthis procedure of being able to estimate the dose giving adifference from placebo of 0.8. However, when the ED50

increases to 50 LinRSS shows a worse performance fordifferences from placebo for absolute percentage bias and rootmean square error. This deterioration in performance comparedwith the other model selection methods becomes much moremarked when ED50 = 80. The performance of the Emaxlin and RSSprocedures is similar for all three values of the ED50.

For l= 2 a similar pattern of results is observed as the value ofthe ED50 increases but this time the differences between LinRSSand the other model selection procedures are greater. Again theresults for the Emaxlin and RSS model selection procedures aresimilar apart from for ED50 = 80 when there is some evidencethat the RSS procedure performs better.

The results for the smaller total sample size of approximately150 for 5 doses which are not reproduced in this article show asimilar pattern with the performance of the LinRSS procedurebecoming worse than that of the other two procedures as theED50 increases. However there is a tendency for the absolutepercentage bias and root mean square error for the differencesbetween the doses and placebo to be smaller for Emaxlin thanfor RSS for l= 1 as the ED50 increases but for the reverse toapply for l= 2.

The results for seven and nine doses, not reproduced in thisarticle, tend to follow the same pattern described above for fivedoses for l= 1 and 2 and approximate total sample sizes of 150and 250.

The results for the other values of the variance are similar tothose for when the variance is set equal to 4.5 although theadvantage for LinRSS when ED50 = 20 is more pronounced whenthe variance is large, the sample size equal to 150 and l= 1.

8. CONCLUSIONS AND DISCUSSION

The results of the simulation study show that the modelselection procedure LinRSS mostly performs best when theED50 = 20, a relatively low value in the dose range. However theperformance of this procedure deteriorates markedly for someof the measures as the ED50 increases to become high relative tothe studied dose range. The results for model selectionprocedures Emaxlin and RSS are mostly fairly similar with someevidence in favour of the RSS procedure when l= 2 and thereverse applying when l= 1. Although the simulation study isbased on a specific situation it is expected that these results willapply qualitatively more generally.

Based on the results presented in this article, model selectionprocedure LinRSS is likely to perform relatively well when theED50 is low in the dose range. Otherwise if it is felt likely thatl= 1 model selection Emaxlin might be preferred and if l= 2model selection procedure RSS is preferred. If a choice of model

selection procedure robust to the underlying truth is requiredthen the model selection procedure RSS could be chosen foruse. This is because it is not too much worse than either LinRSSwhen the ED50 is relatively low in the dose range or procedureEmaxlin when l= 1 and can be noticeably better than bothprocedures when l= 2. Dette et al. [8] have considered therelated topic of deriving optimal designs robust to model mis-specification.

A different way of dealing with lack of information about themaximum possible response to that considered in this article isto use a Bayesian approach. Such an approach is described byThomas [11].

In this article we have not discussed nonparametric approachesto the estimation of minimum effective doses. This is because ourinterest is in estimating mean responses and effective doseswhen a three- or four-parameter Emax model is assumed to be thecorrect model.

REFERENCES

[1] Bates DM, Watts DG. Nonlinear regression analysis and itsapplications. Wiley: New York, 1988.

[2] Ratkowsky DA. Nonlinear regression modelling: a unified practicalapproach. Marcel Dekker: New York, 1983.

[3] Ross GJS. Nonlinear estimation. Springer: New York, 1990.[4] R Development Core Team. R: a language and environment for

statistical computing. R Foundation for Statistical Computing:Vienna, Austria, 2007.

[5] Schoemaker RC, van Gerven JMA, Cohen AF. Estimating potencyfor the Emax-model without attaining maximal effects. Journal ofPharmacokinetics and Biopharmaceutics 1998; 26(5):581–593.

[6] Dutta S, Matsumoto Y, Ebing WF. Is it possible to estimate theparameters of the sigmoid Emax model with truncated data typicalof clinical studies? Journal of Pharmaceutical Sciences 1996;85(2):232–239.

[7] Morgan BJT. Applied stochastic modelling. Arnold: London, 2000.[8] Dette H, Bretz F, Pepelyshev A, Pinheiro J. Optimal designs for

dose-finding studies. Journal of the American Statistical Association2008; 103(483):1225–1237. DOI: 10.1198/016214508000000427.

[9] Bornkamp B, Bretz F, Dmitrienko A, Enas G, Gaydos B, Hsu C, Konig F,Krams M, Liu Q, Neuenschwander B, Parke T, Pinheiro J, Roy A,Sax R, Shen F. Innovative approaches for designing and analyzingadaptive dose-ranging trials. Journal of Biopharmaceutical Statistics2007; 17:965–995. DOI: 10.1080/10543400701643848.

[10] Kirby S, Chuang-Stein C, Morris M. Determining a minimumclinically important difference between treatments for a patientreported outcome. Journal of Biopharmaceutical Statistics;2010; 20:5.

[11] Thomas N. Hypothesis testing and Bayesian estimation using asigmoid Emax model applied to sparse dose-response designs.Journal of Biopharmaceutical Statistics 2006; 16(5):657–677. DOI:10.1080/10543400600860469.

APPENDIX

Dividing the second term on the right-hand side of model (2) forE(Y) by EDl

20 gives

EðYÞ ¼ E01

Emax

EDl50

!Dl

11Dl

EDl50

As Emax and ED50 approach infinity with the ratio Slope ¼ Emax=EDl50

held constant the model approaches

EðYÞ ¼ E01Slope � Dl

1 14

9



fitting emax models to clinical trial dose–response data

Documents