A Bayesian approach to dose-finding studies for cancer therapies: incorporating later cycles of therapy.

Research Article Received 24 January 2013,

Accepted 12 February 2014

Published online 4 March 2014 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/sim.6132

A Bayesian approach to dose-finding studies for cancer therapies: incorporating later cycles of therapy Karen Sinclair* † and Anne Whitehead We consider phase I dose-finding studies for cytotoxic drugs in cancer, where the objective is identification of a target dose .TD100ı / associated with the probability ı of a dose-limiting toxicity (DLT). Previous authors have presented a design utilising a Bayesian decision procedure based on a logistic regression model to describe the relationship between dose and the risk of a DLT (LRDP). A cautious prior, chosen to ensure that the first cohort of patients are given the lowest dose, is combined with binary observations of DLTs to update model parameters and choose a safe dose for the next cohort. This process continues with each new cohort of patients. Typically, only DLTs occurring in the first treatment cycle are included. To incorporate data from later cycles, a new Bayesian decision procedure based on an interval-censored survival model (ICSDP) has been developed. This models the probability that the first DLT occurs in each specific cycle via the probability of a DLT during a specific cycle, conditional on having no DLT in any previous cycle. The second cohort of patients start after responses have been obtained from the first cycle of the first cohort, and subsequently, dose selection for each new cohort is based on DLTs observed across all completed cycles for all patients. A simulation study comparing the ICSDP and LRDP showed that the ICSDP induces faster updating of the current estimate of the target dose, leading to shorter trials and fewer patients, whilst keeping the same level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd. Keywords:

cancer trial; decision procedure; dose finding; interval censored; survival

1. Introduction The aim of a phase I dose-finding study of a cytotoxic drug in cancer is usually to investigate the safety profile of the new drug. A target toxicity level (TTL) of ı is pre-specified, where ı is the probability of observing a dose-limiting toxicity (DLT). The objective is to identify a target dose TD100ı that produces this TTL to take forward to later trials. Dose-finding studies can be divided into two main categories: rule-based and model-based. Rule-based designs (such as the 3C3 [1] or the Rolling 6 [2]) start with the lowest dose and escalate one dose level at a time for each new cohort. The decision to escalate or stop is based on a set of rules. Model-based designs are often implemented within a Bayesian framework (such as the Continual Reassessment Method (CRM) [3] or the Escalation With Overdose Control (EWOC) [1]). They start with the specification of prior distributions for model parameters. Data on the occurrence of DLTs from the first cohort are combined with the prior to obtain a posterior distribution. An appropriate dose to administer to the next cohort is selected depending on a suitable gain function. The procedure is repeated until a specified accuracy criterion has been achieved, a safety rule is violated or the maximum number of patients has been observed. It is usually assumed that the relationship between the probability of a DLT and dose is monotonically increasing. Cytotoxic treatments are generally given in cycles, with treatment given during the first part of each cycle. However, most existing designs only consider the occurrence of DLTs during the first cycle of therapy. This results in data of the form of one binary observation per patient. Thus, observations of DLTs that occur in later cycles during the treatment regimen are ignored. Ignoring data from later cycles

Copyright © 2014 John Wiley & Sons, Ltd.

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Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, U.K. *Correspondence to: Karen Sinclair, Novartis Pharma AG, CH-4002 Basel, Switzerland. † E-mail: [email protected]

K. SINCLAIR AND A. WHITEHEAD

could lead to the selection of an inappropriate dose for later studies. For example, drug tolerance may develop over time requiring a higher dose to be administered, or an accumulation of the drug throughout the cycles may indicate that a smaller dose should be administered later on. Therefore, a dose may be associated with the required probability ı for the first cycle, but there is no information about how the drug works in later cycles. In order to address this concern, we consider following up patients through all their treatment cycles and focus on the time to the first occurrence of a DLT. The time-to-event (TITE)-CRM [4] uses this approach to consider whether a DLT has occurred by a specific time point in the study. This method considers patients to be recruited one at a time throughout the study where analyses of the data are conducted at specified time points during the study. At any particular analysis time, patients contribute an event if a DLT is observed by the specified time point, a non-event if they survive the treatment period without an event and a partial non-event if they are part way through treatment and have not yet experienced an event. The partial non-events are weighted in a way that changes depending on how far through the trial they are (i.e. the risk of having an event later is less than earlier, so the longer the treatment period that a patient has survived without a DLT, the larger the weight given to the partial non-event). Cheung and Chappell [4] consider simple linear weights and adaptive schemes, but they consider the former to be oversimplistic and the latter appear to be complicated to implement. The specification of the weights, which is required before the start of the study so that partial non-events can be included effectively from the beginning, requires knowledge of the dose–response relationship during treatment cycles and how it changes from one cycle to the next. This information is not usually available before starting an early phase trial. Furthermore, the TITE-CRM requires the calculation of complex Bayes estimates. An alternative would be to use a maximum likelihood approach, but this requires the observation of both DLTs and no DLTs, and until this occurs, a sequential-rule-based design is followed, which is quite inefficient. Bekele et al. [5] present a Bayesian approach similar to the TITE-CRM, which incorporates a safety stopping rule in case the predicted risk of toxicity is unacceptably high. We propose the use of an interval-censored survival (ICS) model [6], which models the probability that the first occurrence of a DLT is in a particular treatment cycle by considering a complementary log– log link transformation of the risk of a DLT during a treatment cycle conditional on having no DLT in any previous cycle. It therefore includes information on whether or not a patient has experienced a DLT in every treatment cycle they received. The ICS model allows different intercepts for the dose–response relationship for each cycle but has a common overall effect of dose, and it deals with non-informative censoring and dropout intuitively. All treatment cycles completed by a patient are included in the analysis up to and including the one in which the first DLT occurred or they withdrew from treatment. The ICS model is considered within a Bayesian decision procedure based on modal estimates, which can be implemented using standard statistical software for generalised linear models, which calculates maximum likelihood estimates. The inclusion of later cycles of therapy requires consideration to be given to the TTL. If interest lies in the occurrence of DLTs during the first s cycles, then the TTL might be specified in relation to the cumulative probability of a DLT occurring within the first s cycles. Cumulative probabilities can be readily obtained from the ICS model. In this paper, we compare a Bayesian decision procedure based on an ICS model (ICSDP) with an existing Bayesian decision procedure based on a logistic regression model (LRDP) [7], in the cases where DLTs from the first cycle of therapy are included (LRDP1) and where DLTs from the first three cycles of therapy are included (LRDP3). The LRDP and ICSDP are described in Sections 2 and 3, respectively, and Section 4 focuses on the simulation study comparing the ICSDP, LRDP1 and LRDP3. Finally, in Section 5, we present our conclusions and recommendations.

2. Existing Bayesian decision procedures for dose finding

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Model-based Bayesian decision procedures for dose finding [8] generally consist of five major components. These are the choice of a parametric model to depict the relationship between dose and the probability of toxicity, the choice of prior distributions for the model parameters, the set of possible doses, the gain function and the calculation of the posterior distributions. In this section, we discuss these in the context of the LRDP used for comparison with the proposed ICSDP. Copyright © 2014 John Wiley & Sons, Ltd.

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The LRDP uses the logistic regression model log

pj 1 pj

D ˛ C ˇ log dj ;

(1)

where pj is the probability of a DLT at dose dj ; j D 1; : : : ; k. The term ‘log’ refers to the natural logarithm throughout this paper. The likelihood function based on the DLT data is given by k Y

n r r pj j 1 pj j j ;

(2)

j D1

where of the nj subjects receiving dose dj ; rj experiences a DLT. Prior distributions are usually chosen so that for safety reasons, the lowest dose is given to the first cohort of patients (typically of size three). Whitehead and Williamson [7] consider the choice of independent beta distributions for the probability of a DLT at two different doses. We consider the minimum and maximum doses, d1 and dk . The beta prior for pj ; j D 1; k, with parameters rj0 and qj0 can be thought of as prior pseudo-data comprising n0j D rj0 C qj0 observations at dose dj , where rj0 is the number of patients with a DLT and qj0 is the number of patients who do not experience a DLT. Therefore, the posterior distribution also takes a Beta form because it is conjugate with the Binomial data. Both prior and posterior modal estimates can be obtained by fitting a logistic regression model using maximum likelihood methods in standard statistical software. The expected value of the beta distri. bution with parameters rj0 and qj0 is pj0 D r 0j n0j . To ensure that the lowest dose is selected for the first cohort of subjects, p10 is set equal to the TTL ı. The value chosen for pk0 is one that would be deemed too high if observed in the actual escalation procedure. This produces a prior dose–response curve that shows high doses to be unsafe. After fixing p10 and pk0 , the number of observations can be chosen to reflect the strength of the prior and, for each of the two dose levels, is typically chosen to be equal to the number of patients in each cohort. The number of DLTs rj0 can then be calculated as 0

n0j p j . Relative to the data collected during the study, the amount of pseudo-data is small. Table I shows the prior pseudo-data to be used with the LRDP1 and LRDP3 in the simulation study. Here, a cohort 0 size of 3 was used, giving n01 D nk D 3. For the LRDP1, p10 D 0:2 and pk0 D 0:5. For the LRDP3, p10 was increased to 0.316, and the assumption of a common slope for the LRDP1 and LRDP3 would give pk0 D 0:649. The gain function can be selected to optimise some specific criteria. For example, the patient gain function [7] is used to choose the dose that has probability of a DLT closest to ı, and the variance gain function chooses the dose that minimises the variance of the estimated target dose. In the setting of dose finding for cancer therapies, the patient gain function [7] is often used to select the dose for the next cohort of patients. This gain function gj is given by 1 gj D 2 ; ı pOj

(3)

where pOj is the current estimate of the probability of a DLT for dose dj . The selected dose is the one that has probability of a DLT closest to the TTL ı.

Table I. Pseudo-data used in the simulations for the LRDP1 and LRDP3. Procedure, TTL

Dose dj

pj0

n0j

rj0 D n0j pj0

d1

0.2

3

0.6

ı1 D 0:2

dk

0.5

3

1.5

LRDP3

d1

0.316

3

0.948

ı3 D 0:316

dk

0.649

3

1.947


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LRDP1

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After allocating the lowest dose to the first cohort, the cohort is observed for occurrences of DLTs. Posterior modal estimates of the parameters are used to calculate estimates of the probability of toxicity at different doses, which are then used within the gain function to select one of the dose levels for the next cohort. The escalation procedure itself continues until certain stopping criteria are met or the pre-specified maximum number of cohorts have completed. We consider the following forms of stopping rule. The safety rule states that if the dose with closest modal posterior probability of DLT to the TTL exceeds some pre-defined probability of DLT, then that dose will not be allocated to any patient and the procedure will be stopped. The accuracy rule is based on the limits of the asymptotic 95% credible interval (CI) for the TD100ı . The CI is created for the log.TD100ı /, and the upper and lower limits are exponentiated to produce the CI for the TD100ı . The estimate and 95% CI of the log.TD100ı / are given by ı ˛O log 1ı c 100ı D (4) log TD ˇO and

p p c 100ı C 1:96 var ; c 100ı 1:96 var; log TD log TD

where T c 100ı ; c 100ı I 1 .˛; ˇ/ r log TD var D r log TD c 100ı is the first derivative vector for log .TD100ı /, and I 1 .˛; ˇ/ is the inverted observed r log TD information matrix for the parameters [7]. Once the ratio of the upper limit of the CI to the lower limit of the CI for the TD100ı falls below a certain value (R), the estimate of the TD100ı is considered sufficiently accurate. If there are no toxicities at the highest dose level, the procedure would either stop when the maximum number of patients had been recruited or due to the accuracy rule. If the highest dose were repeatedly administered with no toxic events observed, the precision of the estimated TD will converge quite quickly, and the accuracy rule will force the trial to stop. Further investigation of even higher doses may then be considered. In practice, this is unlikely to occur, because the relationship between dose and P(DLT) is usually assumed to be monotonically increasing and the dose levels are usually selected to cover the extreme range of doses that are deemed suitable for investigation. The LRDP is a useful and efficient approach to dose finding when there is a fixed period of observation on each subject. For cancer therapies, generally only one cycle of treatment is used for the analysis of the occurrence of DLTs, and the TTL ı is set in respect of the risk of observing a DLT during the first cycle of therapy. Patients will usually be monitored for occurrences of DLTs whilst they remain in the study, but data from later cycles of treatment are not included in the analysis or escalation procedure. One way of incorporating data from later cycles would be to extend the fixed period of inclusion of DLTs in the analysis to the total number of cycles. However, this would increase the length of time between the starting dates of successive cohorts, as each cohort has to complete all cycles before the next one starts, leading to an unfeasibly long trial. The easiest compromise might be to consider the first few cycles (e.g. three cycles), which from previous knowledge, may be assumed to contain most of the DLTs. Another possibility is to extend the model to include data from later cycles but recruit new cohorts after each cycle of therapy. Thus, as a patient completes an additional cycle of therapy, they contribute further data to the analysis. A natural extension of the logistic regression model for one cycle would be the proportional odds model. Suppose that interest lies in the first s treatment cycles and that treatment cycle l starts at time tl1 and finishes at time tl ; l D 1; : : : s, where t0 D 0. Let pj l be the probability that the first occurrence of a DLT for a patient on dose dj is in cycle l, and pj;sC1 be the probability that no DLT occurs during the first s cycles for a patient on dose dj . The probability that a DLT occurs within the first l treatment cycles for a patient on dose dj is given by pj .tl /, where

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pj .tl / D

l X

pj m :

(5)

mD1


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As the pj m ; m D 1; : : :; s C 1 are probabilities relating to mutually exclusive events, the sum of pj m ; m D 1; : : :; s C 1 is equal to 1. The proportional odds model is given by pj .tl / D ˛l C ˇ log dj ; l D 1; : : : ; s: (6) log 1 pj .tl / Note that for the LRDP based on one treatment cycle (LRDP1), pj D p j .t1 /, and ˛ D ˛1 , and for the LRDP based on three treatment cycles (LRDP3), pj D p j .t3 /, and ˛ D ˛3 : The likelihood function for the proportional odds model is given by k sC1 Y Y

r

pjjl l D

j D1 lD1

k sC1 Y Y

r pj .tl / pj .tl1 / j l ;

(7)

j D1 lD1

where pj .t0 / D 0; rj l ; l D 1; : : : s, is the number of patients on dose dj who have their first DLT during cycle l, and rj;sC1 is the number of patients who do not have a DLT during the first s cycles. One problem with the proportional odds model is that it does not easily account for patients who have not completed all s cycles of treatment at the time of the analysis or who drop out of the study early. This would require additional terms in the likelihood described in Equation (7), which would lead to a likelihood function that is intractable when combined with the familiar beta pseudo-data prior. We do not develop the proportional odds model further here, but it will be used for data generation in the simulation study in Section 4.

3. Interval-censored survival decision procedure 3.1. The interval-censored survival model A similar concept to the proportional odds model is an ICS model, which is based on the probability of having an event (DLT) in a given period (cycle) conditional on the fact that an event has not occurred in a previous cycle. Because of the use of conditional probabilities rather than cumulative probabilities, non-informative censoring is easily accounted for. Now 8 l D1 < j1 l D 2; : : : ; s pj l D 1 j1 1 j 2 : : : 1 j;l1 j l (8) : l DsC1 1 j1 1 j 2 : : : 1 j;l1 .1 j l / where j l is the conditional probability of a DLT in cycle l for patient on dose j , given that they did not have a DLT in previous cycles. The likelihood function is given by [6] k sC1 Y Y j D1 lD1

r

pjjl l D

k Y s Y

q r j jl l 1 j l j l ;

(9)

j D1 lD1

where qj l is equal to the number of patients who have completed l cycles of therapy without experiencing a DLT. Under a proportional hazards model 1 j l

j Sj .tl / S0 .tl / e D ; D Sj .tl1 / S0 .tl1 /

(10)

where l is an intercept term for cycle l and dependent solely on the cycle of interest. Copyright © 2014 John Wiley & Sons, Ltd.

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where Sj .tl / is the probability of not having a DLT in cycle l or earlier for a patient on dose dj , S0 .tl / is the corresponding baseline survivor function and j D log dj . This leads to the following linear model for the complementary log–log transformation of j l S0 .tl / C j D l C log dj ; log log 1 j l D log log (11) S0 .tl1 /


The probability that a DLT occurs within the first l treatment cycles, l D 1; : : : s, for a patient on dose dj can be calculated from the ICS model using Equations (5) and (8). TTLs may be defined in relation to the probability that a DLT occurs within the first l treatment cycles, for l D 1; : : : s. When based on these cumulative probabilities, the TTL will increase with the number of cycles. Let ıl denote the TTL based on l treatment cycles. Our focus here will be on ıs . 3.2. The prior and posterior distribution Because the likelihood based on the conditional probabilities of a DLT is of a Binomial form, independent beta distributions may be used as prior distributions for the j l . This is carried out to produce a conjugate beta posterior distribution for the j l . In a similar way to that implemented for the LRDP, both prior and posterior modal estimates can be obtained by fitting a binary regression model with a complementary log–log link function using maximum likelihood methods in standard statistical software. We consider providing independent beta priors for the j l for j D 1; k and l D 1; : : :; s. The beta prior for j l has parameters rj0l and qj0l , which can be thought of as prior pseudo-data comprising n0jl D rj0l C qj0l observations at dose dj in cycle l, where rj0l is the number of patients with a DLT and qj0l is the number of patients who do not experience a DLT. Our choice of values for the TTLs and beta parameters has been motivated by the data on DLTs from molecularly targeted agents analysed by Postel-Vinay et al. [9]. These indicated that the probability of a DLT investigated during the first treatment cycle was typically about 0.2 and that the prevalence of the first occurrence of a DLT halved for each successive treatment cycle. Furthermore, it was found that over 90% of DLTs had occurred by cycle 3 while less than 50% occur during cycle 1. The simulation study in Section 4 was therefore based on three cycles, to maintain simplicity whilst increasing information. The pseudo-data for the ICSDP used in the simulation study are presented in Table II. Because there are equivalent amounts of pseudo-data included in cohort 1, the pseudo-data for each procedure is equally informative. Expected values for the conditional probabilities were set at 0 0 0 11 D 0:2; 12 D 0:1; 13 D 0:05, which is equivalent to p10 .t1 / D 0:2; p10 .t2 / D 0:28; p10 .t3 / D 0:316. If the TTLs are chosen to be ı1 D 0:2; ı2 D 0:28; ı3 D 0:316, then this prior ensures that the lowest 0 dose is selected for the first cohort of subjects. To be consistent with the LRDP1, k1 D 0:5. Assum0 0 ing a common slope parameter gives k2 D 0:2791; k3 D 0:1473. These values are equivalent to pk0 .t1 / D 0:5; pk0 .t2 / D 0:640; pk0 .t3 / D 0:693, and produces a prior dose–response curve that shows high doses to be unsafe. As for the LRDP, the number of subjects for each dose level can be chosen to be equal to the number of patients in each cohort. In Table II, n0j1 D 3. The values of n0jl for l > 1 are calculated by subtracting the expected number of DLTs that have occurred in earlier cycles from n0j1 . The number of DLTs rj0l can 0

then be calculated as n0jl . jl After the first cohort has been observed for one cycle, the observations are combined with the pseudodata and analysed again using the ICS model. Posterior modal estimates of the parameters in the model will be obtained and fed into the link function. This will then give updated estimates for the j l ; pj l and pj .tl /, which can be used within a gain function to choose which dose to allocate to the next cohort. Once each new cohort has been allocated a dose and observed, all of the data, including the pseudo-data and those from later cycles for earlier cohorts, are included in the calculation of the posterior modal estimates.

Table II. Pseudo-data used in the simulations for the ICSDP. Dose dj , cycle

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d1 , cycle 1 d1 , cycle 2 d1 , cycle 3 dk , cycle 1 dk , cycle 2 dk , cycle 3


j0l

n0jl

rj0l D n0jl j0l

0.2 0.1 0.05 0.5 0.2791 0.1473

3 2.4 2.16 3 1.5 1.08135

0.6 0.24 0.108 1.5 0.41865 0.1593

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3.3. The gain function The gain function to be used in this procedure is the patient gain [7], given by 1 gj D 2 ; ıs pOj .ts / where pOj .ts / is the current estimate of the probability of a DLT in the first s cycles for dose dj . The dose to be allocated is selected on the basis of the dose that has the modal posterior probability of a DLT closest to the TTL after s cycles of treatment. With this function, the procedure simply picks the dose believed to be closest in the probability of a DLT to the TTL. The incorporation of pseudo-data should ensure that the escalation proceeds slowly. However, when the data outweighs the prior information, doses can either escalate or de-escalate depending on which dose appears to be closest to the TTL. 3.4. Conducting the escalation procedure This procedure continues until stopping criteria are met. Similar safety and accuracy rules can be used with the ICSDP as with the LRDP. We consider the same safety rule as for the LRDP, but now based on the probability of a DLT within the first s cycles. For example, suppose the TTL for the LRDP is 0.2 and the safety stopping rule for the LRDP is invoked when the closest dose to achieving a TTL of 0.2 has an estimated probability of a DLT greater than 0.30. Assuming that the prevalence of the first occurrence of a DLT halves for each successive treatment cycle, the conditional probabilities of a DLT for cycles 2 and 3 based on a cycle 1 value of 0.30 are 0.15 and 0.075, respectively. The safety stopping rule for a procedure with three cycles would then be invoked when the closest dose to achieving a TTL of 0.316 has an estimated probability of a DLT greater than 0.44. The accuracy rule is based on the ratio of the limits of the exponentiated 95 % CI for the log TD100ıs . The estimate and 95% CI of log TD100ıs are given by # " . log .1 ıs / c 100ıs D log O log TD e O1 e O2 : : : e Os and p p c 100ı 1:96 var; log TD c 100ı C 1:96 var ; log TD where T c 100ıs I 1 .1 ; 2 ; : : : ; s ; / r log TD c 100ıs ; var D r log TD c 100ıs is the first I 1 .1 ; 2 ; : : : ; s ; / is the inverted observed information matrix and r log TD c 100ıs with respect to each of the parameters. Further details are provided derivative vector of log TD in the Appendix for the case of s D 3. Once the ratio of the upper limit of the CI to the lower limit of the CI for the TD100ı falls below a certain value (R), the estimate of the TD100ı is considered sufficiently accurate. For the simulation study, R has been set at 4, which generally provides good estimates of the TD after a reasonable number of cohorts. We note that the asymptotic normality assumption used in the construction of the credibility interval may not be ideal for phase I studies with small numbers of patients, but we have used this approach to be consistent with LRDP [10].

4. Simulation study 4.1. Simulation method


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In order to investigate the properties of the ICSDP as a new dose-finding procedure, it was compared with the LRDP, an existing design to which it is most similar. As the data from the paper of Postel-Vinay et al. [9] indicated that the vast majority of DLTs occur by the end of cycle 3, we selected the case of three cycles. The LRDP1 and LRDP3, based on one and three cycles, respectively, were used in the comparison.


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The models selected for the data simulation were based on scenario 1 from [10], which used the LRDP1 with a TTL of 0.2. In this scenario, nine different dose levels 60; 120; 200; 300; 420; 630; 945; 1400; 1700 mg/m2 were available, and the TD20 and TD50 were set at 366 and 799 mg/m 2 , respectively. We extended this scenario to the ICSDP based on three cycles and assumed that the conditional probabilities of a DLT would halve in each successive cycle. Thus, for the target dose of 366 mg/m 2 , the conditional probabilities were 366;1 D 0:2; 366;2 D 0:1; 366;3 D 0:05, and the TTLs were ı1 D 0:2 for the LRPD1 and ı3 D 0:316 for the LRDP3 and ICSDP. This also meant that p366;1 D 0:2; p366;2 D 0:08; p366;3 D 0:036, and p366 .t1 / D 0:2; p366 .t2 / D 0:28; p366 .t3 / D 0:316. The cohort size was 3, and the maximum number of cohorts was set at 20. The accuracy stopping rule was based on achieving R < 4. For the LRDP1, the safety stopping rule was implemented when the closest dose to achieving a TTL ı1 D 0:2 had an estimated probability of toxicity greater than 0.30. For the LRDP3 and ICSDP, the safety stopping rule was implemented when the closest dose to achieving a TTL ı3 D 0:316 had an estimated probability of toxicity greater than 0.44, the value calculated assuming the same slope as for the LRDP1. The maximum number of cohorts was set at 20. Results are based on 1000 simulations of the procedures. In reality, the actual dose that would be recommended for further investigation would be rounded to the nearest possible dose. However, because the actual information about the exact TD is known for simulation, the estimated TD on the continuous can be compared to the true TD to assess the precision of the different procedure’s estimates. Data were simulated under three different dose–response models. The first dose–response model was a proportional odds model, using Equation (6) with ˛1 D 11:8733; ˛2 D 11:4317; ˛3 D 11:2594; ˇ D 1:7767. The intercept terms were obtained by setting p366;1 D 0:2; p366;2 D 0:08; p366;3 D 0:036; p799;1 D 0:5. Four categories were created: (1) first DLT occurs in cycle 1; (2) first DLT occurs in cycle 2; (3) first DLT occurs in cycle 3; and (4) no DLT occurs during the first three cycles. The outcome category for a subject on dose dj was simulated from a uniform distribution in the proportions pj1 ; pj 2 ; pj 3 ; .1pj .t3 //, for categories 1 and 4, respectively. This model is similar to the logistic regression model, but additionally imposes a common odds ratio for the cumulative probabilities. The ICS model was used for the second dose–response model. This modelled the j l using Equation (11), with 1 D 10:0694; 2 D 10:8198; 3 D 11:5396; D 1:4518. These values were obtained by setting 366;1 D 0:2; 366;2 D 0:1; 366;3 D 0:05; 799;1 D 0:5. The occurrence or not of a DLT at dose dj in the first cycle was simulated as a binary observation with parameter j1 . If a DLT did not occur in the first cycle, the occurrence or not of a DLT in the second cycle was simulated as a binary observation with parameter j 2 . If a DLT did not occur in the first or second cycle, the occurrence or not of a DLT in the third cycle was simulated as a binary observation with parameter j 3 . The third dose–response model was a proportional odds model with dose as a covariate in the model as opposed to log (dose). The same four outcome categories were created as before using intercepts ˛1 D 2:5575, ˛2 D 2:1157 and ˛3 D 1:9434 and slope ˇ D 0:0032. The intercept terms were obtained by setting p366;1 D 0:2; p366;2 D 0:08; p366;3 D 0:036; p799;1 D 0:5. The different simulation models were selected to demonstrate the robustness of the methods to the violation of the model assumptions. The logistic regression model and the proportional odds model both assume a constant log odds ratio across doses. The ICS model assumes a constant log hazard ratio across doses. Figure 1a–e illustrates the differences between the simulation and analysis models for the three scenarios. In the simulations, it was assumed that once a patient had experienced a DLT, they would not contribute any data from future cycles on the occurrence of a DLT. For most of the simulations, it was assumed that there was no censoring. However, for the ICS dose–response model, two additional simulation studies were conducted to explore the effect of censoring on the results. In the first of these, non-informative censoring, which might occur because of withdrawal from the study for progressive disease or other unforeseeable reason unrelated to the occurrence of a DLT, was considered. We use a 10% rate of non-informative censoring. In the second, we considered a 10% rate of informative censoring. Informative censoring will occur when patients who are more likely to have a DLT withdraw from the study for a reason associated with an increased risk of a DLT. For example, the risk of a DLT may be correlated with the occurrence of lower grade toxicities, and patients might withdraw early after experiencing low grade toxicities. The 10% rate of informative censoring was introduced by censoring patients who would have experienced a DLT if they had not withdrawn. The TTL investigated here is 31.6% over three cycles. Therefore, it can be expected that on average, approximately 30% of Copyright © 2014 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2665–2680


Figure 1. Models used for data simulation and data analysis. (a) Data simulated by proportional odds model with log(dose) (dashed line), analysis using interval-censored survival model (solid line), TTL D ı3 D TD31:6 , TTD D 366 mg/m2 ; (b) data simulated by interval-censored survival model with log(dose) (solid line), analysis using binary logistic regression on first three cycles (dashed line), TTL D ı3 D TD31:6 , TTD D 366 mg/m2 ; (c) data simulated by interval-censored survival model with log(dose) (solid line), analysis using binary logistic regression on first cycle (dashed line), TTL D ı1 D TD20 , TTD D 366 mg/m2 ; (d) data simulated by proportional odds model with dose (dotted line), analysis using interval-censored survival model (solid line) or binary logistic regression on first three cycles (dashed line) with log(dose), TTL D ı3 D TD31:6 , TTD D 366 mg/m2 ; (e) data simulated by proportional odds model with dose (dotted line), analysis using binary logistic regression on first cycle with log(dose) (dashed line), TTL D ı1 D TD20 , TTD D 366 mg/m2 .

patients will be observed to have a DLT. Approximately one third of the patients experiencing DLTs were censored. That is, of the subjects for whom a DLT is simulated, a censoring indicator is simulated from a Bernoulli random variable with probability equal to 0.33. 4.2. Simulation results


Statist. Med. 2014, 33 2665–2680

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Table III shows the results for data generated under the proportional odds model with log dose. Thus, for the LRDP1 and LRDP3, the same model is being used for data generation and analysis. Figure 1a shows that fitting an ICS model to these data results in a similar curve to the LRDP3 but is likely to lead to a slight overestimate of the TD31:6 . The mean estimates of the TD20 for the LRDP1 and of the TD31:6 for the LRDP3 are close to the true target dose (TTD) of 366 mg/m2 . The LRDP1 stops less frequently for accuracy than the LRDP3 (73% versus 90%). For the LRDP1, approximately 27% of trials stop due to having recruited the maximum number of cohorts, and of the 73% stopping for accuracy, 7% only stop for accuracy during the 20th cohort, so on average, 34% of trials require the maximum number of cohorts. No trials stop for safety reasons. The difference between the LRDP1 and LRDP3 in terms of the frequency of stopping due to the accuracy rule may be explained by the difference between the two TTLs. Dose estimation tends to be more precise the closer the TTL is to 50%. As expected, the mean estimate for the TD31:6 from the ICSDP is


Table III. Results from 1000 simulations with data generated from the proportional odds model with log dose and true target dose .TTD/ D 366 mg/m2 .

Design LRDP1 ı1 D TD20

Variable TD20 No. of cohorts

LRDP3 ı3 D TD31:6

TD31:6 No. of cohorts

ICSDP ı3 D TD31:6


Stopping reason % Mean estimate Accuracy Safety Max no. 371.9

73.5

0.0

26.5

(2.5, 97.5) percentiles of estimates and ratio 97.5/2.5

8 89.6

0.1

10.3

14.87 381.5 14.67

(232.2, 538.8) 2.32

0.0

6.5

85.3

20

(247.7, 575.3) 2.32 178.4 742.2 8

82.2

20

1.3 714.1 1

93.5

% in Max .TTD ˙ 30%/

(227.9, 554.1) 2.43 169.4 750.3

16.92 360.0

Min

83.0

20

2674

slightly higher than the TTD of 366 mg/m 2 . Relative to the LRDP1, the ICSDP stops more frequently due to the accuracy rule (93% versus 73%) and, on average, requires fewer cohorts (14.7 versus 16.9). Of the 93%, only 3% stopped due to the accuracy rule during cohort 20, so approximately 10% of trials required the maximum number of patients. No trials stopped for safety reasons. The LRDP3 gave a mean estimate of the TD31:6 close to the TTD of 366 mg/m 2 , stopped frequently due to the accuracy rule (90%) and required about the same number of cohorts on average as the ICSDP. Of the three designs, the final estimate of the target dose is within a 30% limit of the TTD of 366 mg/m2 most often for the LRDP3, although there is little difference between the three. Under this scenario, the LRDP3 would be expected to be the best performer. However, although the expected number of cohorts for the LRDP3 is similar to that for the ICSDP, the actual length of the study would be much longer because new cohorts are only started after every three cycles rather than after every cycle with the ICSDP. Table IV shows the results for data generated under the ICS model with log dose. Thus, for the ICSDP, the same model is being used for data generation and analysis. Figure 1b,c indicates that the LRDP1 should produce a reasonably accurate estimate of the TD20 and that the LRDP3 is likely to lead to a slight underestimate of the TD31:6 . Under this scenario, the ICSDP would be expected to be the best performer. For the ICSDP, the mean estimate for the TD31:6 is close to the TTD, and the final estimate is within a 30% limit of the TTD of 366 mg/m 2 most often out of the three procedures. The reason for stopping with the ICSDP is generally due to the accuracy rule with approximately 10% of the simulated trials reaching the maximum number of cohorts required. The safety rule is never used. For the LRDP1, the accuracy rule is used approximately the same number of times with fewer trials (approximately 6%) stopping due to the maximum number of patients, and the safety rule is used approximately 4% of the time. However, about 30% stopped due to the accuracy rule during the 20th cohort, so approximately 35% of trials required the maximum number of patients. For the ICSDP, only an extra 3% stopped for the accuracy rule in the final cohort, so approximately 13% of trials required the maximum number of patients. The LRDP3 produced similar results to those from the first scenario, although as expected, the mean estimate of the TD31:6 was a slight underestimate. The number of trials stopping due to the accuracy rule is approximately 89%, with approximately 10% stopping due to recruiting the maximum number of patients and 0.6% stopping for safety reasons. Only an extra 3% stopped due to the accuracy rule in the final cohort showing that 13% of trials lasted the maximum duration. Table V shows the results for data generated under the proportional odds model with dose. Consequently, none of the procedures match the data generation model. Figure 1d,e shows that all three procedures will lead to an underestimation of the target dose, with the LRDP1 and LRDP3 being worse. In this scenario, the ICSDP performed slightly better than the other two, although the mean number of cohorts was not much smaller than for the LRDP1. For all procedures, the frequency of stopping for accuracy has reduced relative to the first two scenarios. For the LRDP3, 34% of the trials stopped due Copyright © 2014 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2665–2680


Table IV. Results from 1000 simulations with data generated from the ICS model with log dose and true target dose .TTD/ D 366mg/m2 .



LRDP3 ı3 D TD31:6


ICSDP ı3 D TD31:6



90.6

3.6

5.8


8 89.3

0.6

10.1

(226.3, 523.6) 2.31

14.80 371.2

0.0

9.6

7

83.4

20

(243.2, 561.0) 2.31 186.9 690.4

15.02

78.5

20

1.3 683.7 1

90.4


(217.7, 589.6) 2.71 144.3 994.6

17.13 354.1

Min

85.5

20

Table V. Results from 1000 trials simulated with data generated from the proportional odds model with dose and true target dose .TTD/ D 366mg/m2 .



LRDP3 ı3 D TD31:6


ICSDP ı3 D TD31:6



64.1

17.3

18.7


(152.5, 553.2) 3.63 52.6

17.61 321.1

3 60.3

6.0

33.7

(1.3, 562.0) 419.4

16.00 347.1 17.01

Min

0.1

37.6

(154.9, 576.6) 3.72

750.4

61.7

20

6.6 1062.7 3

66.0

20

1.3 1250.4 1

62.4


68.0

20


Statist. Med. 2014, 33 2665–2680

2675

to recruiting the maximum number of patients and 6% stopped due to safety reasons. An additional 4% of the trials actually stopped for accuracy in the final cohort, so 38% of the trials lasted the maximum duration. For the LRDP1, 19% of trials stopped due to recruiting the maximum number of patients, but over 17% of trials stopped for safety reasons. Approximately 45% of trials lasted the maximum duration because 27% achieved accuracy in the last possible cohort. For the ICSDP, 38% of trials stopped due to recruiting the maximum number of patients with only an extra 5% stopping for accuracy in the last possible cohort. Only 0.1% of the trials were stopped for safety reasons. The ICSDP produced the most estimates of the target dose within a 30% limit of the TTD. Because comparison of the procedures is complex due to the different sample sizes used by the different procedures, we compared the procedures in the case where they all continued to the maximum number of patients allowed, 20 cohorts of three patients (60 patients). Despite the safety and accuracy stopping rule being removed, we did leave in the safety rule associated with the first cohort. If after the first cohort, where all patients were treated at the lowest dose, none of the doses were deemed safe to administer, the trial would stop. The results of the simulation, using the same simulation setting as in Table IV, are shown in Table VI. All of the mean estimates of the TD are closer to the true TD of 366 than in Table IV. Compared with the LRDP1, the ICSDP has greater precision; there are slightly


Table VI. Results from 1000 simulations with data generated from the ICS model with log dose and true target dose .TTD/ D 366mg/m2 for a fixed number of patients (20 cohorts of size 3).

Design LRDP1 ı1 D TD20 LRDP3 ı3 D TD31:6

Variable

Mean

(2.5, 97.5) percentiles of estimate and ratio 97.5/2.5

TD20

367.3

(216.7,558.6) 2.58

No. of cohorts

20

TD31:6

360.0

No. of cohorts ICSDP ı3 D TD31:6

(238.3,506.6) 2.13

19.94

TD31:6

362.1

No. of cohorts

20

(218.8,536.8) 2.45

Min

Max

% in .TTD ˙ 30%/

144.3

994.5

82.3

20

20

1.34

628.1

1

20

144.9

777.1

20

20

89.7

80.0

Table VII. Results from 1000 simulations with data generated from the ICS model with log dose and true target dose .TTD/ D 366mg/m2 with 10% non-informative censoring.



LRDP3 ı3 D TD31:6


ICSDP ı3 D TD31:6



67.7

1.0

31.3

(2.5, 97.5) percentiles of estimates and ratio 97.5/2.5 (203.9,592.6) 2.91

17.39 357.0

85.3

0.3

14.4

16.81

(224.9,536.4) 2.39

0.0

22.1

83.4

20

(241.8,556.6) 2.30 190.0 752.3 8

75.8

20

1.3 836.1 1

77.9


3.6 1199.1 1

15.45 369.7

Min

85.7

20

2676

fewer estimates within a 30% limit of the true TD, but the range of the estimates is not as extreme. The LRDP3 still has one occurrence where the trial stopped for safety after observing one cohort. The ICSDP performs slightly worse than in Table IV, which is due to the asymptotic variance for the ICS model. This variance relies on information from the number of patients administered a dose as well as the number of toxicities observed at that dose. After precision of the TD estimate is claimed, if the escalation continues, more doses may be administered. When these doses are ones that have been slightly underrepresented previously, the inclusion of these additional data can cause the asymptotic variance to increase once again and in many cases, the ratio of the credible interval creeps back above the threshold for which accuracy is claimed. In this case, the final estimate of the TD is no longer deemed as precise, and the variability across the trials is increased. The results from incorporating non-informative censoring are shown in Table VII. As expected, the estimates of the TD are similar to those in Table IV. Because of the reduced amount of information, the length of the trials are slightly longer, and there is slightly less difference between the LRDP1 and the ICSDP. The proportion of trials with an estimated TD within a 30% limit of the TTD is slightly worse for the LRDP1 but remains the same for the other two procedures. The occurrence of stopping for safety is reduced for both of the LRDPs but is still non-existent for the ICSDP, and the proportion of trials stopping for accuracy is reduced for all procedures. This is because more trials are stopping because the maximum number of patients have been recruited, due to the reduced information obtained throughout the trial. Copyright © 2014 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2665–2680


Table VIII. Results from 1000 simulations with data generated from the ICS model with log dose and true target dose .TTD/ D 366mg/m2 with 10% informative censoring.

Design

Variable

LRDP1 ı1 D TD20

TD20 No. of cohorts

LRDP3 ı3 D TD31:6


ICSDP ı3 D TD31:6



75.7

0.2

24.1

(2.5, 97.5) percentiles of estimates and ratio 97.5/2.5 (254.9,730.1) 2.86

16.76 431.8

87.9

0.3

11.8

14.73

(267.8,627.6) 2.34

0

11.3

67.2

10

(292.3,675.4) 2.31 213.4 1032.9 7

58.0

20

1.34 946.0 1

88.7


3.63 1225.9 1

15.12 457.3

Min

58.3

20

The results of incorporating informative censoring are shown in Table VIII. The estimate TDs are further away from the TTD than those in Table IV. This is to be expected, as one third of the events that occur are not observed. Thus, it would appear that the drug is not causing as many DLTs as expected and suggests that a higher dose is tolerated. The trials tend to stop more frequently because the maximum number of patients have been recruited, but some stopping for safety is still observed with the LRDPs. The proportion of trials in which the estimated TD is within a 30% limit of the TTD is now much lower, again to be expected because the trials are producing much higher estimates of the TD. A comparison of the three procedures still shows that the ICSDP and LRDP1 produce comparable results, but the ICSDP still does this in a reduced time with greater precision. The ICSDP also stops for accuracy more frequently and produces slightly more estimates within the target region of 30%.

5. Discussion


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This paper has presented the ICSDP as a practical extension to the existing LRDP1 for phase 1 dosefinding studies for cytotoxic drugs in cancer, in order to include data from later cycles of therapy. A simulation study comparing the ICSDP with the LRDP1 and the LRDP3, an extended version of the LRDP, which includes data from three cycles, has shown that under the scenarios considered, the ICSDP induces faster updating of the current estimates of the target dose, leading to shorter trials and fewer patients, whilst keeping the same level of accuracy. When comparing the designs under the same scenario, the ICSDP produces more precise results with estimates that are comparable to the other designs. This is still evident when the number of cohorts is fixed to be the same for all procedures. The ICS model is a natural choice to use for phase 1 dose-finding studies for cytotoxic drugs in cancer as interest lies in the time to occurrence of the first DLT. It allows for censoring and dropout naturally whilst still including as much information as soon as possible from all participating patients. This is evident in the case of non-informative censoring. None of the procedures could handle the informative censoring, which led to a reduced risk of DLTs for the drug, but all three procedures investigated behaved similarly. The ICSDP allows a varying dose–response relationship over time, which is reasonable to assume based on data from Postel-Vinay et al. [9], and leads to the straightforward calculation of the cumulative probability of a DLT within a first fixed number of cycles. The ICSDP has advantages over the TITE-CRM [4], in that it can be implemented in most standard statistical software for fitting generalized linear models for binary data with a complementary log–log link and that non-informative censoring is easily accommodated. Furthermore, a weight function needs


2678

to be developed and incorporated in order to utilise the TITE-CRM efficiently. A limitation of the ICSDP is that it assumes a model for the data, and this may not be appropriate in some situations, for example, if the effect of log-dose in the complementary log–log transformation is not linear or if the linear coefficient of log-dose differs from one treatment cycle to the next. To test the adequacy of the ICS model, a more complicated model could be fitted, for example, by including additional polynomial terms for log-dose or a cycle by log-dose interaction term, and a comparison with the ICS model undertaken using a likelihood ratio test. A further assumption that has been made is that there is no cumulative dose effect, that is, the probabilities of having a toxicity in each cycle are conditionally independent. A cumulative dose effect may manifest itself as an increasing effect of log-dose with increasing cycle number. In this case, it may be more appropriate to consider drug concentrations or cumulative doses instead of the dose administered in the particular cycle. The ICSDP described assumes that patients who enter the trial at a certain dose level remain on that dose for all cycles of therapy. Simulations have been undertaken to investigate whether allowing intrapatient dose adjustments (changing dose between cycles) would be beneficial in either speeding up the ICSDP or creating more accurate results. A comparison was made between the ICSDP developed in this paper and an ICSDP which allowed patients to change doses between cycles depending on what the current estimate of the target dose was believed to be. In the first case where intrapatient adjustments were allowed for all patients throughout the trial, the simulations showed that estimates of the target dose were more variable, due to having a larger number of patients on one particular dose at any one time. When intrapatient adjustments were allowed for the first three cohorts only for the purpose of reducing the amount of underdosing, the estimates of the target dose were even worse. Because there were very few observations for the low doses, the overall dose–response relationship was not well estimated, and there was little consistency in the target dose estimates across the simulations. For the third case where the first three cohorts stayed on the dose they started on until the end of therapy, so that observations were observed for low doses, and subsequent patients were allowed intrapatient adjustments, the target dose estimates improved relative to the second case but were worse relative to the first case and were still much worse that in the original ICSDP. Shen and O’Quigley [11] demonstrate that when allocated doses are concentrated around specific doses, the recommended target dose then tends to converge to those doses. Particularly, if dosing is concentrated around the TTD, the recommended TD will be close to the true TD. This is supportive of the results seen here. The ICSDP was developed using the patient gain function. This gain function selects the dose that current belief suggests is the dose with closest probability of toxicity to the TTL. An alternative would be the variance gain, which selects the dose to maximise the information about the dose–response model so that the final estimate of the TD can be found with maximum precision. In order to do this, doses from the entire range need to be administered. Although choosing low doses is not a safety issue (although it may be an ethical one), choosing high doses with a large probability of toxicity to administer to patients is not safe. A safety rule needs to be implemented to ensure that overly toxic doses are not administered. Simulations have been undertaken to investigate the ICSDP with the variance gain function, modified to only allow doses that have a probability of toxicity below some safety threshold. In comparison with the ICSDP with the patient gain function, the ICSDP with the modified variance gain function resulted in much shorter average trial durations, but with less accurate estimates of the TD. The use of the variance gain led to more observations on a few doses, which produced a greater contribution to the variance causing individual trials to stop for accuracy reasons. However, there was an increase in the variation of the TD estimates across the simulations as different doses were repeatedly administered in different trials because of the random nature of simulation. For the ICSDP, as with many existing dose-finding designs, the main response of interest is the first occurrence of a DLT. The focus here is to escalate doses to find a tolerable dose to take forward to the next phase of clinical development, so the time to the first occurrence of DLT seems a reasonable choice. However, there may be interest in repeat occurrences of toxicities, although these are more likely to be of a lower grade than those defined as a DLT. Repeated measures analyses could be conducted to incorporate later events in order to develop a greater understanding of the pattern of toxicities over time. The ICSDP described does not contain covariates, either at baseline or time varying. It should be relatively straightforward to incorporate baseline covariates; the target dose of interest would be in the form of a function of covariates (i.e. age, gender and biomarker level). The incorporation of timevarying covariates would be more complicated as the ICS model already has different relationships for different periods, so interactions between the time-varying covariate and the dose over time may be required. Copyright © 2014 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2665–2680


Appendix: Asymptotic variance of estimate of log .TD100ı / for the ICS model with three cycles The likelihood function for the ICS model given by Equation (7) leads to the following log-likelihood function for the ICS model with three cycles `D

3 n k X X

o rj l log j l C q j l log 1 j l

j D1 lD1

First-order derivatives are given for l D 1; : : :; 3 by 3 X

log 1 j l ˚ @` D r j l j l r j l C qj l @l j l j D1 k 3 X

X log 1 j l ˚ @` r j l j l r j l C qj l : log dj D @ j l j D1

lD1

Second-order derivatives are given for l; m D 1; : : :; 3; l ¤

m by

k X @2 ` D Rj l ; @l2 j D1

where Rj l D

˚ 2 ˚

C rj l j l log 1 j l 1 log 1 j l nj l j l 2 log 1 j l rj l log 1 j l j l 2

and nj l D rj l C qj l k

3

j D1

lD1

X 2 X @2 ` D log dj Rj l 2 @ @2 ` D0 @l @m k

X @2 ` log dj Rj l : D @l @ j D1

Fisher’s observed information matrix is given by I .1 ; 2 ; 3 ; / 2 k P Rj1 0 0 6 6 j D1 6 k P 6 6 0 Rj 2 0 6 j D1 6 D6 k P 6 0 0 Rj 3 6 6 j D1 6 k k k P P 4 P log dj Rj1 log dj Rj 2 log dj Rj 3 j D1

j D1

k P j D1 k P j D1 k P j D1 k P

j D1

j D1

log dj Rj1 log dj Rj 2 log dj Rj 3 3 2 P log dj Rj l

3 7 7 7 7 7 7 7 7 7 7 7 7 5

lD1

The estimate of log TD100ı3 is given by


h

log.1ıs / e O1 e O2 e O3

O

i

2679

log c 100ıs D log TD

:

Statist. Med. 2014, 33 2665–2680


c 100ıs / is obtained using the delta method and is given by The asymptotic variance of log.TD T c 100ıs I 1 .1 ; 2 ; 3 ; / r log TD c 100ıs ; r log TD c 100ıs is the first derivative vector of log TD c 100ıs with respect to each of the where r log TD parameters. Now, @ log TD100ıs el ; D @l .e1 C e2 C e3 / @ log TD100ıs 1 D log TD100ıs : @

Acknowledgements This work was supported by MRC strategic grant G0800792/87129. We wish to thank the referees for their helpful comments, which have led to an improved version of the paper.

References 1. Berry S, Carlin B, Lee J, Muller P. Bayesian Adaptive Methods for Clinical Trials, Vol. 38. Chapman & Hall/CRC Biostatistics Series: Boca Raton FL, 2010. 2. Skolnik J, Barrett J, Jayaraman B, Patel D, Adamson P. Shortening the timeline of pediatric phase I trials: the rolling six design. Journal of Clinical Oncology 2008; 26(2):190–195. 3. O’Quigley J, Pepe M, Fisher L. Continual reassessment method: a practical design for phase I clinical trials in cancer. Biometrics 1990; 46(1):33–48. 4. Cheung K, Chappell R. Sequential designs for phase I clinical trials with late-onset toxicities. Biometrics 2000; 56(4):1177–1182. 5. Bekele N, Shen JY, Thall PF. Monitoring late-onset toxicities in phase I trials using predicted risks. Biostatistics 2008; 9(3):442–457. 6. Collett D. Modelling Survival Data in Medical Research, Second Edition. Chapman and Hall/CRC: Boca Raton FL, 2003. 7. Whitehead J, Williamson D. Bayesian decision procedures based on logistic regression models for dose-finding studies. Journal of Biopharmaceutical Statistics 1998; 8(3):445–467. 8. Whitehead J, Brunier H. Bayesian decision procedures for dose determining experiments. Statistics in Medicine 1995; 14(9):885–893. 9. Postel-Vinay S, Gomez-Roca C, Molife L, Anghan B, Levy A, Judson I, De Bono J, Soria J, Kaye S, Paoletti X. Phase I trials of molecularly targeted agents: should we pay more attention to late toxicities? Journal of Clinical Oncology 2011; 29(13):1728–1735. 10. Zhou Y, Whitehead J. Practical implementation of Bayesian dose-escalation procedures. Drug Information Journal 2003; 37(1):45–59. 11. Shen L, O’Quigley J. Consistency of continual reassessment method under model misspecification. Biometrika 1996; 83(2):395–405.

2680 Copyright © 2014 John Wiley & Sons, Ltd.

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