Adaptive modelling of dose–response relationships using smoothing splines

PHARMACEUTICAL STATISTICS

Pharmaceut. Statist. 2009; 8: 346–355

Published online 9 February 2009 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/pst.363

Adaptive modelling of dose– response

relationships using smoothing splines

Simon Kirby�,y, Peter Colman and Mark MorrisStatistics, Pfizer Global Research and Development, Sandwich, Kent, UK

We consider the use of smoothing splines for the adaptive modelling of dose–response relationships.

A smoothing spline is a nonparametric estimator of a function that is a compromise between the fit to

the data and the degree of smoothness and thus provides a flexible way of modelling dose–response

data. In conjunction with decision rules for which doses to continue with after an interim analysis, it

can be used to give an adaptive way of modelling the relationship between dose and response. We fit

smoothing splines using the generalized cross-validation criterion for deciding on the degree of

smoothness and we use estimated bootstrap percentiles of the predicted values for each dose to decide

upon which doses to continue with after an interim analysis. We compare this approach with a

corresponding adaptive analysis of variance approach based upon new simulations of the scenarios

previously used by the PhRMA Working Group on Adaptive Dose-Ranging Studies. The results

obtained for the adaptive modelling of dose–response data using smoothing splines are mostly

comparable with those previously obtained by the PhRMA Working Group for the Bayesian Normal

Dynamic Linear model (GADA) procedure. These methods may be useful for carrying out

adaptations, detecting dose–response relationships and identifying clinically relevant doses. Copyright

r 2009 John Wiley & Sons, Ltd.

Keywords: dose response; smoothing spline; adaptive; generalized cross validation; bootstrap

1. INTRODUCTION

There is increasing interest in the PharmaceuticalIndustry in modelling the relationship between thedose of a drug administered and the main efficacy

response. This relationship is typically explored inPhase 2 clinical trials. The increased interest inmodelling the dose–response relationship hascome about at least partly because of the greaterpower that can be obtained from modelling thisrelationship as opposed to treating doses of a drugas unrelated treatments.

An extension to modelling the dose–responserelationship when all the data is available at theend of a trial is to change the doses being studiedyE-mail: [email protected]

*Correspondence to: Simon Kirby, Statistics, Pfizer GlobalResearch and Development, Ramsgate Road, Sandwich, Kent,CT13 9NJ, UK.

Copyright r 2009 John Wiley & Sons, Ltd.

as the clinical trial progresses in an attempt torefine information about doses likely to haveeffects of interest. We refer to such an approachas adaptive modelling of the dose–responserelationship.

Previous work that has looked at the adaptivemodelling of dose response has included para-metric approaches [1] and the use of a BayesianNormal Dynamic Linear model [1–3]. In this paperwe consider fitting smoothing splines for theadaptive modelling of dose–response relationships.Our motivation for doing this is to investigatewhether smoothing splines might be a usefulmethod for such adaptive modelling.

Section 2 outlines the smoothing spline ap-proach to the adaptive modelling of dose–responserelationships. In Section 3 we describe a simplecorresponding adaptive analysis of variance (AN-OVA) approach, which is used as a comparator forthe smoothing spline. The simulation scenariosand measures of performance used to assessthese two adaptive approaches are described inSection 4. Full details of the way the smoothingspline and ANOVA methods were implementedare given in Section 5 and the simulation resultsare presented in Section 6. Finally, conclusionsand a discussion of the results appear in Section 7.

2. THE USE OF SMOOTHINGSPLINES FOR THE ADAPTIVEMODELLING OF DOSE–RESPONSEDATA

Smoothing splines can be derived as the solutionof a minimization problem as follows [4]. We wishto find the function g() on some interval [a, b], outof all functions that are differentiable on thisinterval and that have absolutely continuous firstderivatives, that minimizes the penalized sum ofsquares

Xni¼1

fYi � gðxiÞg2 þ a

Z b

a

fg00ðtÞg2dt ð1Þ

where the Yi are observed responses, the xi arevalues of a predictor variable, a is a so-called

smoothing parameter and g00() is the secondderivative of the function. The first part of theexpression is the usual residual sum of squareswhile the second part imposes a penalty forfunctions that are rougher. The value of a controlsthe importance attached to the function beingsmooth. Larger values of a impose a greaterpenalty on lack of smoothness. Hence, largervalues of a give smoother fitting curves whilesmaller values of a give more jagged curves. In thecontext of modelling dose–response data for themain efficacy variable in a clinical trial, the Yi arethe efficacy responses for each subject and the xiare the doses of drug given to each subject.

The function gðÞ that minimizes (1) for a givenvalue of a is a natural cubic spline, i.e. for givenreal numbers x1; . . . ;xn on an interval [a, b] suchthat ao x1o x2o � � � o xno b, the function g:� is a cubic polynomial on each of the intervals

(a,x1),(x1,x2),(x2,x3),y,(xn,b);� is continuous;� has continuous first and second derivatives on

the interval and has second and third deriva-tives zero at the endpoints a and b.

The value of the smoothness parameter can bedetermined in a number of ways, including fixing avalue based on a priori considerations, graphingfitted curves or using cross-validation. In thispaper we use a variant of the last of these.

Cross-validation chooses the value of a thatminimizes

CVðaÞ ¼ n�1Xni¼1

fYi � gð�iÞðxi; aÞg2 ð2Þ

where gð�iÞ is the value of the function g() at xiwhen the ith observation is excluded from thederivation of the function. In this paper we use avariant of cross validation called generalized crossvalidation (GCV). The deletion residuals requiredfor the calculation of the cross-validation score in(2) can be obtained from the ordinary residuals bydividing by a factor, which depends on theobservation deleted. GCV replaces these factorsby their average value [4].

Copyright r 2009 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2009; 8: 346–355DOI: 10.1002/pst

Adaptive modelling of dose–response relationships 347

Smoothing splines can be simply used for theadaptive modelling of the relationship betweendose and response in a clinical trial as follows.Primary efficacy data are collected for someproportion (o1) of the total number of subjectsoriginally planned to be enrolled in the trial and asmoothing spline is fit to these data. On the basisof pre-specified criteria, the trial is stopped forfutility, stopped for success or continued with allor some of the doses. If appropriate, furtherinterim analyses can be conducted in the same wayor else the trial can be allowed to run tocompletion. Subsequent analyses can be carriedout by using all of the accumulated data to fit asmoothing spline.

3. AN ADAPTIVE ANOVAPROCEDURE FORDOSE–RESPONSE DATA

The usual one-way ANOVA method for responsesthat are Normally and independently distributedcan simply be implemented in an adaptive way asfollows. At one or more interim analyses criticalvalues are used to decide which doses to continuewith. Then at the final analysis each dose is testedagainst placebo using an adjusted critical value.The adjusted critical value is chosen to ensurecontrol of the Type I error for the test of the nullhypothesis of no dose–response relationship whenone or more significant tests of each dose againstplacebo are taken as evidence of a dose–responserelationship. In this paper we use such an adaptiveANOVA procedure. It should be noted that thisdiffers from the ANOVA procedure described byBornkamp et al. [1], which uses a fixed trial designbut allows an adaptive choice of analysis method.

4. SIMULATION SCENARIOS ANDPERFORMANCE MEASURES

We use the set of simulation scenarios previouslystudied by the PhRMA Working Group onAdaptive Dose-Ranging Studies [1]. Although

the same simulation scenarios are used new datasets were generated.

Total sample sizes of approximately 150 or250 were used for each of the following sets ofdoses:� 0, 1, 2, 3, 4, 5, 6, 7, 8� 0, 2, 3, 4, 5, 6, 8� 0, 2, 4, 6 and 8

The Available Doses – the doses available forsubsequent use – were considered to be those inthe first set above.

The number of subjects per dose was rounded tothe nearest integer to give a total sample size asclose as possible to 150 or 250.

The response profiles used for each of the sixcombinations of sample size and dose sets were:� Y ¼ e ðflatÞ� Y ¼ �ð1:65=8Þd þ e ðlinearÞ� Y ¼ 0:015� 1:73=f1þ exp½1:2ð4� dÞ�g þ eðlogisticÞ

� Y ¼ �ð1:65=3Þd þ ð1:65=36Þd2 þ e ðumbrellaÞ� Y ¼ �1:81d=ð0:79þ dÞ þ e ðEmaxÞ� Y ¼ �1:70d5=ð45 þ d5Þ þ e ðSigmoid EmaxÞ

where Y represents the main efficacy response, dthe dose and e is an error term which is assumed tobe Normally and independently distributed withconstant variance. These response profiles areshown in Figure 1.

The response was taken to be change frombaseline to 6 weeks in a VAS pain score and wasassumed to have a variance equal to 4.5. Theclinically relevant effect was set at a change of �1.3units from baseline and the maximum effectwithin the observed range was a change of �1.65units from baseline (except for the logistic responseprofile). It should be emphasized that thisdefinition of a clinically relevant effect is notfor a difference from placebo. The definition isthe one adopted by the PhRMA WorkingGroup and is used here to enable comparisonsto be made with the results obtained by theWorking Group.

The trial accrual period was taken to be 16weeks for all of the simulations. Further, weassumed that accrual was at a constant rate.


348 S. Kirby, P. Colman and M. Morris

The following performance measures used bythe PhRMA Working Group were estimated(fuller details are given in Section 5.2):(1) the probability of detecting a dose–response

relationship(2) the joint probability of detecting a dose–

response relationship and identifying a clini-cally relevant effect within the dose range

(3) the percentage bias in the selection of thetarget dose (defined below) given by

100Eðdt argÞ � dt arg

dt arg

!

where E denotes the expected value, dt arg isthe estimated target dose and dt arg is the truetarget dose

(4) the absolute percentage error in estimatingthe target dose given by

100Ejdt arg � dt argj

dt arg

(5) the joint probability of detecting a dose–response relationship and selecting a dosethat is within the target dose interval (definedbelow)

Figure 1. Dose–response profiles used in simulation.



(6) the average absolute prediction error multi-plied by 100 and standardized by the absolutevalue of the target efficacy of �1.3.

The target dose (true or estimated) is defined asthe smallest dose that produces an effect (true orestimated) greater than, or equal to, the clinicallyrelevant effect. If necessary this dose was roundedto the nearest integer. To allow for sensitivity todeviations around the true target dose, a targeteffect interval is defined. This is taken to be theinterval obtained by being within 710% of theclinically relevant effect. A corresponding targetdose interval is derived from each generatingequation (without error) and includes all dosesthat give an effect that is within the target effectinterval. Rounded values for the limits of thetarget dose intervals were obtained taking intoaccount clinical considerations such that the limitswere not necessarily the closest available dose [1].

5. IMPLEMENTATION OF THEADAPTIVE SMOOTHING SPLINEAND ANOVA PROCEDURES

The simulation results were obtained using Rversion 2.3.1 [5]. The function smooth.spline wasused to fit smoothing splines. The results for thesmoothing spline procedure were calculated from3500 simulations and those for the adaptiveANOVA procedure were calculated from 10 000simulations.

5.1. Interim analysis

For both the adaptive smoothing spline andANOVA approaches a single interim analysiswas carried out at approximately a quarter of theinitial total sample size to see whether some dosescould be dropped or the trial stopped for futility.The option of stopping a trial for success was notconsidered.

For the adaptive smoothing spline, doses weredropped at the interim analysis if the 5th percentileof the estimated parametric bootstrap distribution

for the predicted response for a dose lay above�1.3 (the clinically relevant effect). The parametricbootstrap results were obtained by taking onethousand samples (with treatment group sizesequal to the group sizes at the interim analysis)from Normal distributions with means equal tothe fitted values and a common variance estimatedfrom the smoothing spline fit.

For the adaptive ANOVA approach, doses weredropped if the lower limit of the 90% confidenceinterval for the mean of a dose lay above �1.3.

For both adaptive approaches the subjects whowould have been allocated to the doses droppedafter 10 weeks of the trial (the time of the interimanalysis) were then allocated equally to the dosesretained, rounding down the number allocated toeach dose if necessary. If all of the doses weredropped at an interim analysis then no furtherrecruitment was assumed to take place. Placebowas not dropped at the interim analysis unlessnone of the doses were continued.

5.2. Final analysis

5.2.1. Measure 1: probability of detecting doseresponse

For both adaptive procedures a dose–responserelationship was considered to have been found ifthe response for at least one of the doses in thetrial differed significantly from placebo using aone-sided test with critical values derived asdescribed below.

For the adaptive smoothing spline procedure acommon percentile was used to obtain a criticalvalue from each estimated bootstrap distributionof the difference between the predicted responsefor a dose and placebo. The estimated bootstrapdistributions were obtained in the same way as forthe interim analysis. The common percentile waschosen such that the Type I error for the test of thenull hypothesis of no dose–response relationshipwas unlikely to be greater than 0.05. The percentilewas estimated by simulation from 3500 flatdose–response curves – the most which wascomputationally feasible. To address the precision



obtainable from this number of simulations thepercentile to be used was chosen as the lowest thatgave an estimated probability of falsely detecting adose–response relationship less than or equal to0.042. This ensures that the upper limit of theusual 95% confidence interval calculated for theType I error rate lies below 0.05.

For the adaptive ANOVA procedure an esti-mated common critical value was used at the finalanalysis for a one-sided test of whether thedifference between each dose and placebo differedfrom zero. Ten thousand samples from flatdose–response curves were simulated to estimateeach critical value. To ensure that the Type I errorfor the test of the null hypothesis of no dose–response relationship for the adaptive ANOVAprocedure was likely to be greater than that for theadaptive smoothing spline procedure the commonfinal critical value was chosen by sequentiallysearching through quantiles from the standardNormal distribution in decreasing order of size insteps of 0.001. The critical value chosen was thehighest that gave an estimated Type I error rategreater than or equal to 0.05. The lower limit forthe usual 95% confidence interval for the differ-ence in Type I error rates between the ANOVAand spline procedures is then calculated to beapproximately zero.

5.2.2. Measure 2: joint probability of detectingdose–response and identifying a clinically relevanteffect within the dose range

For the adaptive smoothing spline a dose–response relationship was considered to have beenfound if there was a statistically significantdose–response relationship as defined above.A clinically relevant effect was considered to befound if there was a predicted response for anAvailable Dose, which was below the clinicallyrelevant effect of �1.3.

For the adaptive ANOVA procedure a dose–response relationship was similarly established ifthere was a statistically significant dose–responserelationship as defined above while a clinicallyrelevant effect was considered to be present if the

mean response for at least one dose used wasbelow �1.3.

5.2.3. Measures 3, 4, 5: target dose

For the adaptive smoothing spline procedure andthe adaptive ANOVA procedure the estimatedtarget dose was selected to be the smallest dosethat had a predicted/actual mean below �1.3 inthose simulations where a dose–response relation-ship was found. Because predictions for thesmoothing spline are on a continuous scale fordose the estimated dose giving a clinically relevanteffect was rounded to the nearest integer to giveone of the eight non-zero Available Doses.Measures 3 and 4 were only calculated forsimulations where both a dose–response relation-ship and a clinically relevant effect were found.

5.2.4. Measure 6: prediction error

The average absolute prediction error (multipliedby 100 and divided by the absolute value of thetarget effect) was calculated only if both astatistically significant dose–response relationshipand a dose giving a clinically relevant effect werefound. The average was calculated using all of theAvailable Doses for the smoothing spline proce-dure but only the doses actually used for theANOVA procedure.

6. RESULTS OF THE SIMULATIONSTUDY

The estimated probability of detecting a dose–response relationship for the adaptive smoothingspline and ANOVA procedures is given in Table I.It can be seen from the table that in every case theestimated probability of detecting a dose–responserelationship for the smoothing spline procedure isgreater than or equal to that for the ANOVAprocedure.

Table II shows the estimated joint probability ofdetecting a dose–response relationship and finding



a clinically relevant effect in the dose range. Againit can be seen that the estimated probability for thesmoothing spline procedure is greater than orequal to that for the ANOVA procedure.

The estimated percentage bias in the target dosefor the scenarios for each of the adaptiveprocedures is shown in Table III. The smoothingspline procedure performs better for the lineardose–response profile and when five doses areused. Otherwise the percentage biases are smallerfor the ANOVA procedure. The estimated percen-tage absolute error (Table IV) is lower for thesmoothing spline procedure for the linear, sigmoi-dal Emax and logistic dose–response profiles. Theresults for the remaining two dose–responseprofiles are mixed.

The estimated joint probability of detecting adose–response relationship and selecting a dose inthe target interval is given for each procedure inTable V. Apart from two cases for the Emaxscenario the estimated probability is always great-er for the adaptive smoothing spline procedure

than for the adaptive ANOVA procedure. Zeroesappear in the table for the ANOVA procedure fortwo of the response profiles because the target doseinterval is a single dose that was not used.

Finally the estimated average absolute predic-tion errors are given in Table VI. The averageabsolute prediction error is smaller for thesmoothing spline procedure when all AvailableDoses are used and in all cases except one when sixof the eight non-zero Available Doses are used.However, it should be noted that the averageabsolute prediction error for the ANOVA proce-dure has only been calculated for the dosesactually used. The average absolute predictionerror for the smoothing spline would be smaller ifit was also calculated only for the doses used. Thissuggests that interpolated values produced by thesmoothing spline may not be useful.

It is of interest to compare the results obtainedin this paper with those in the paper published bythe PhRMA Working Group on Adaptive Dose-Ranging Studies [1]. We focus on comparisons

Table I. Estimated probability of detecting dose response (smoothing spline result followed by ANOVA result).

Dose response profile

Maximum total sample size Number of doses Linear Umbrella Emax Sigmoidal Emax Logistic

250 5 0.99, 0.97 0.99, 0.99 1.00, 1.00 1.00, 0.98 1.00, 0.99250 7 0.98, 0.90 0.98, 0.97 0.98, 0.97 0.99, 0.93 1.00, 0.95250 9 0.98, 0.84 0.99, 0.93 0.96, 0.94 0.99, 0.89 1.00, 0.92150 5 0.95, 0.85 0.95, 0.92 0.95, 0.95 0.98, 0.89 0.98, 0.92150 7 0.88, 0.67 0.88, 0.81 0.88, 0.84 0.94, 0.73 0.96, 0.78150 9 0.90, 0.61 0.91, 0.74 0.83, 0.77 0.96, 0.68 0.97, 0.71

Five doses used were: 0, 2, 4, 6 and 8, seven doses used were: 0, 2, 3, 4, 5, 6 and 8, nine doses used were: 0, 1,y,8.

Table II. Estimated joint probability of detecting dose–response and identifying a clinically relevant dose (smoothingspline result followed by ANOVA result).


Maximum total sample size Number of doses Linear Umbrella Emax Sigmoidal Emax Logistic

250 5 0.94, 0.92 0.98, 0.98 0.99, 0.99 1.00, 0.96 1.00, 0.98250 7 0.92, 0.87 0.97, 0.97 0.98, 0.97 0.98, 0.92 0.98, 0.94250 9 0.94, 0.84 0.98, 0.93 0.96, 0.94 0.98, 0.89 0.99, 0.92150 5 0.88, 0.82 0.93, 0.92 0.95, 0.94 0.94, 0.86 0.95, 0.90150 7 0.81, 0.66 0.87, 0.81 0.87, 0.84 0.91, 0.73 0.94, 0.77150 9 0.84, 0.60 0.89, 0.74 0.83, 0.77 0.94, 0.68 0.95, 0.71




against the General Adaptive Dose Allocation(GADA) and Dopt methods described in thePhRMA paper because these are the only adaptivemethods in the sense used in this paper consideredby the PhRMA Working Group and because the

GADA method was considered the best of themethods studied by the Group. The GADAmethod uses a Bayesian Normal Dynamic LinearModel to allocate each new subject to a dose whichminimizes the variance of a parameter of interest.

Table III. Estimated percentage bias for target dose selection (smoothing spline result followed by ANOVA result).


Maximum totalsample size Number of doses Linear Umbrella Emax Sigmoidal Emax Logistic

250 5 3.7, 11.5 25.5, 41.0 45.9, 63.6 13.7, 23.8 10.5, 22.4250 7 2.2, �2.2 17.9, 14.9 38.6, 38.0 11.1, 9.0 8.3, 7.0250 9 2.6, �10.6 21.7, 9.2 33.9, 12.6 11.5, 5.7 8.4, 3.6150 5 0.7, 5.3 27.5, 37.5 55.0, 65.7 14.8, 20.3 11.1, 18.9150 7 �0.9, �12.3 18.8, 11.1 45.6, 37.4 11.3, 3.6 8.8, 2.9150 9 �0.7, �19.7 22.5, 1.6 41.3, 8.7 11.8, �0.8 8.8, �3.1


Table IV. Estimated percentage absolute error for target dose selection (smoothing spline result followed by ANOVAresult).



250 5 10.8, 19.9 32.3, 51.4 54.0, 63.6 17.1, 28.6 14.2, 27.2250 7 11.0, 22.4 27.3, 28.0 48.2, 38.0 15.0, 16.6 12.6, 15.3250 9 10.9, 22.3 30.2, 30.5 49.8, 41.6 15.2, 16.4 12.5, 15.8150 5 12.7, 21.7 38.7, 52.3 65.0, 65.7 19.2, 28.6 16.7, 27.6150 7 12.8, 26.2 30.9, 28.1 56.8, 37.4 16.8, 17.9 14.3, 17.4150 9 12.9, 28.4 34.3, 32.7 59.4, 41.8 17.4, 19.8 15.0, 19.8

Five doses used were: 0, 2, 4, 6 and 8, seven doses used were: 0, 2, 3, 4, 5, 6 and 8, nine doses used were: 0,1,y, 8.

Table V. Estimated joint probability of detecting dose–response and correctly selecting a dose in the target interval(smoothing spline result followed by ANOVA result).



250 5 0.72, 0.38 0.66, 0.59 0.64, 0.50 0.32, 0.00 0.40, 0.00250 7 0.68, 0.25 0.70, 0.63 0.67, 0.80 0.40, 0.37 0.46, 0.40250 9 0.70, 0.36 0.67, 0.56 0.59, 0.55 0.37, 0.34 0.45, 0.37150 5 0.59, 0.31 0.51, 0.48 0.52, 0.48 0.29, 0.00 0.34, 0.00150 7 0.52, 0.15 0.57, 0.49 0.52, 0.69 0.33, 0.27 0.41, 0.30150 9 0.56, 0.20 0.54, 0.39 0.44, 0.42 0.32, 0.23 0.38, 0.24

Five doses used were: 0, 2, 4, 6 and 8, seven doses used were: 0, 2, 3, 4, 5, 6 and 8, nine doses used were: 0, 1,y,8. Zeroes

are present for the ANOVA procedure for two of the response profiles because the target dose interval is a single dose

that was not used.



The PhRMA paper does not explicitly state whichparameter of interest was chosen but does say thattypically the variance of the response at the targetdose is used. The Dopt method chooses a doseallocation scheme that minimizes the D optimalitycriterion at each interim analysis. Although notexplicitly stated the results for the Dopt method inthe PhRMA paper are for alogistic model with a polynomial trend (Dmitrienko,personal communication).

For the probability of detecting dose responsethe performance of the adaptive smoothing splineprocedure is comparable with all of the otherprocedures described in the PhRMA paper withthe exception of the GADA (Bayesian NormalDynamic Linear model) procedure. The probabil-ity of detecting dose response using the smoothingspline procedure is similar to that using theGADA procedure for n5 250 but noticeably lowerfor n5 150 with the GADA method unaffected bythe reduced sample size.

The joint probability of detecting a dose–responserelationship and identifying a clinically relevant doseis broadly similar for the smoothing spline andGADA procedures and bigger than for all of theother methods considered in the PhRMA paper.

The percentage bias and percentage absoluteerror in target dose estimation for the smoothingspline procedure are broadly similar to those forthe GADA procedure although the bias and errorfor the smoothing spline do not become as extremeas for the GADA procedure. The performance ofboth of these procedures is as bad as or worse than

that of the other procedures studied by thePhRMA Working Group.

For the most part the joint probability ofdetecting a dose–response relationship andcorrectly identifying a dose in the target intervalis similar for the smoothing spline and GADAprocedures and is bigger than for the otherprocedures studied by the PhRMA WorkingGroup.

Finally, we note that the average absoluteprediction error obtained for the smoothing splineprocedure is similar to that for the Dopt procedurewhen all of the Available Doses are used but isotherwise bigger and is always bigger than thatobtained when the GADA procedure is used.

7. CONCLUSIONS AND DISCUSSION

In comparing the adaptive smoothing spline andadaptive ANOVA procedures described in Section5 we have shown that the smoothing splineprocedure performs better at determining that adose–response relationship exists. It also has abigger joint probability of detecting a dose–response relationship and a clinically relevanteffect in the dose range. The adaptive smoothingspline procedure has a lower percentage absoluteerror for the target dose for linear, sigmoidalEmax and logistic dose–response profiles andusually has a bigger probability of jointly detectinga dose–response relationship and correctly

Table VI. Estimated average absolute prediction error relative to target effect (smoothing spline result followed byANOVA result).



250 5 24.4, 22.5 28.1, 20.1 31.7, 19.6 30.4, 21.7 29.2, 21.2250 7 19.8, 27.9 21.6, 23.7 24.2, 23.3 23.0, 26.6 22.3, 26.0250 9 14.9, 32.9 17.0, 28.0 19.0, 27.1 17.9, 31.0 12.0, 30.3150 5 34.6, 32.5 37.3, 27.7 40.4, 26.7 37.6, 31.2 38.2, 29.7150 7 29.3, 47.1 30.4, 36.5 33.5, 35.1 30.1, 43.6 28.9, 40.5150 9 21.3, 57.7 23.1, 44.8 27.0, 42.3 22.5, 51.9 21.9, 49.5




selecting a dose in the target interval. However, theestimated percentage error for the target dose issmaller for the adaptive ANOVA procedure unlessthe dose–response profile is linear or five doses areused. The average absolute prediction error issmaller for the smoothing spline procedure whenall Available Doses are used and is usually betterwhen six of the non-zero Available Doses are used.This suggests that interpolated values produced bythe smoothing spline may not be useful.

Comparing the results obtained for the smooth-ing spline procedure with those previously ob-tained for the GADA procedure used by thePhRMA Working Group on Adaptive Dose-Ranging Trials we note that the probability ofdetecting dose response for the larger sample size,the joint probability of detecting dose responseand identifying a clinically relevant dose and thejoint probability of detecting dose response andcorrectly selecting a dose in the target interval areall similar. Both methods suffer from similarlylarge biases and errors for target dose estimation.Finally we note that the average absolute predic-tion error for the smoothing spline procedure isbigger than for the GADA procedure andparticularly big if not all Available Doses are used(when the error is averaged over the AvailableDoses).

In comparing the GADA and smoothing splineprocedures it should be noted that the adaptationused in this paper for the smoothing splineprocedure is much simpler than that used for theGADA procedure. A single interim analysis isundertaken, doses are dropped if they showinsufficient evidence of efficacy and the subjectsremaining to be allocated to treatment are simplyallocated equally to the doses retained. Thiscompares with computing an optimum treatmentallocation for each new subject for the GADAprocedure. In principle one could also use a morecomplex allocation method for the smoothingspline procedure.

The relatively poor results for the bias and errorin target dose estimation for the smoothing splineand GADA procedures and for the predictionerrors for the smoothing spline procedure suggestthat these procedures may be most useful forcarrying out adaptations, detecting dose–responserelationships and detecting doses with clinicallyrelevant effects. Estimation of the target dose andprediction of response for the Available Doses atthe end of a trial might then be best carried outusing parametric fits suggested by these nonpara-metric procedures.

ACKNOWLEDGEMENTS

We acknowledge the very helpful comments of thereferees, which have led to an improved version ofthis paper. We also wish to acknowledge helpfulinput on an earlier version of this paper fromtwo Pfizer colleagues, Jacqui Spanton andNeal Thomas.

REFERENCES

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2. Berry DA et al. Adaptive Bayesian designs for dose-ranging drug trials. In: Case studies in BayesianStatistics V, Gatsonis C, Kass RE, Carlin B,Carriquiry A, Gelman A, Verdinelli I, West M(eds). Springer: New York, 2002; 99–181.

3. Smith MK et al. Implementation of a Bayesianadaptive design in a proof of concept study. Pharma-ceutical Statistics 2006; 5:39–50. DOI: 10.1002/pst.198.

4. Green PJ, Silverman BW. Nonparametric regressionand generalized linear models: a roughness penaltyapproach. Chapman & Hall: London, 1994.

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Adaptive modelling of dose–response relationships using smoothing splines

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