Two-group time-to-event continual reassessment method using likelihood estimation

    Two-group time-to-event continual reassessment method using likelihood estimation Amber Salter, John O’Quigley, Gary R. Cutter, Inmaculada B. Aban PII: DOI: Reference:

S1551-7144(15)30091-4 doi: 10.1016/j.cct.2015.09.016 CONCLI 1282

To appear in:

Contemporary Clinical Trials

Received date: Revised date: Accepted date:

16 June 2015 18 September 2015 21 September 2015

Please cite this article as: Salter Amber, O’Quigley John, Cutter Gary R., Aban Inmaculada B., Two-group time-to-event continual reassessment method using likelihood estimation, Contemporary Clinical Trials (2015), doi: 10.1016/j.cct.2015.09.016

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

PT

Two-Group Time-To-Event Continual Reassessment Method Using Likelihood Estimation

RI

Amber Salter, PhD1 , John O’Quigley, PhD2 , Gary R. Cutter, PhD1 Inmaculada B. Aban, PhD1 Department of Biostatistics, University of Alabama at Birmingham, Birmingham, AL USA

2

Universit´e Pierre et Marrie Curie - Paris VI, 75005 Paris, France

NU

SC

1

Amber Salter

School of Public Health

AC CE P

Birmingham, AL 35294-0022

TE

1665 University Blvd. Room 327

D

University of Alabama at Birmingham

MA

Corresponding Author:

Phone: 205 934-6725 [email protected]

Keywords:dose finding, phase I trial, patient heterogeneity, continual reassessment method, timeto-event, maximum likelihood

ACCEPTED MANUSCRIPT Abstract The presence of patient heterogeneity in dose finding studies is inherent (i.e. groups with

PT

different maximum tolerated doses). When this type of heterogeneity is not accounted for in the trial design, subjects may be exposed to toxic or suboptimal doses. Options to handle pa-

RI

tient heterogeneity include conducting separate trials or splitting the trial into arms. However,

SC

cost and/or lack of resources may limit the feasibility of these options. If information is shared between the groups, then both of these options do not benefit from using the shared informa-

NU

tion. Extending current dose finding designs to handle patient heterogeneity maximizes the utility of existing methods within a single trial. We propose a modification to the time-to-event

MA

continual reassessment method to accommodate two groups using a two-parameter model and maximum likelihood estimation. The operating characteristics of the design are investigated

D

through simulations under different scenarios including the scenario where one conducts two

AC CE P

method.

TE

separate trials, one for each group, using the one-sample time-to-event continual reassessment

1

ACCEPTED MANUSCRIPT 1

INTRODUCTION Dose finding trials investigate the relationship between dose and safety of a drug. The goal

PT

of these studies is to establish the optimal dose for a drug that is considered safe, usually based

RI

on the toxicity responses in those being tested. This dose is usually called the maximum tolerated

SC

dose (MTD) defined as the highest dose with the expected probability that is closest to the target rate of toxicity (TRT). The TRT is the level of risk researchers are willing to accept for how likely

NU

a patient is to experience a toxic event and is defined before the study begins. The TRT in cancer studies is usually between 0.2 to 0.3 while stroke studies use a TRT of 0.1 or less [1]. The MTD

MA

resulting from dose finding trials often provide the dose used in future trials. The impact of these studies is significant for the future of a drug.

D

Inclusion and exclusion criteria are used in an effort to obtain a homogeneous group of

TE

subjects. However, heterogeneity (i.e. groups with different MTDs) may still arise even with such criteria. Examples of sources of heterogeneity include being naive versus non-naive to treatment or

AC CE P

the presence or absence of a biomarker. We consider having 2 groups and assume the relationship between the dose and the probability of toxicity to be different but having some common information as will be defined later. An option available to the researcher when this heterogeneity is present is to conduct separate trials, one per group, or split the trial into two arms. However, these options do not utilize the information shared between the groups. Another option is to simply use the onesample CRM and ignore any heterogeneity which may be present. Using the one-sample CRM in this situation [2, 3] results in a single MTD being estimated that is an average of the two groups. Consequently, subjects are treated at overly toxic or suboptimal doses as shown by O’Quigley [4]. An alternative option is to use a design that accounts for the heterogeneity. O’Quigley [4, 5] proposed an extension of the CRM, the two-group CRM, to estimate the MTD for both groups in the same trial using a model with two parameters. The objective of this paper is to address the issue of patient heterogeneity based on the two-group two-parameter models used in the CRM, applied to the time-to-event CRM (TITE2

ACCEPTED MANUSCRIPT CRM) for 2 groups. Instead of estimating the next recommended dose at the end of follow-up, TITE-CRM estimates a recommended dose upon enrollment of a new subject utilizing any partial

PT

information available on subjects. In this paper, we derive the estimating equations to obtain maximum likelihood estimates and use simulations to investigate the operating characteristics of

RI

this design. In section 2, we define notations, assumptions and the model. Section 3 derives the

SC

estimating equations for the maximum likelihood estimator (MLE) and describes the estimation process. Section 4 discusses the simulation design and the results of our evaluation of the operating

Notation and Assumptions

TE

2.1

METHODS

D

2

MA

NU

characteristics. Finally, the last section provides some concluding remarks.

AC CE P

Group membership is denoted using superscripts while subscripts are used to index the subjects. We define n and m be the total number of subjects in each group so that the total sample (1)

(2)

size for the trial is N = n + m. Let Yj , j = 1, ..., n and Yl , l = 1, ..., m be the binary outcomes (1)

(1=toxicity, 0=no toxicity) for group 1 and group 2, respectively. Yj

and Yl

(2)

are independent

for all j and l. The clinical doses, d ∈ {d1 , ..., dr }, chosen for the trial (e.g. 10, 50, 100, 150, 300, 400 milligrams) are associated with a probability of toxicity. A dose-toxicity function, f (x, β), is used to describe the relationship between them. The function should be strictly increasing and although the dose-toxicity function is continuous, the estimation is for a set of discrete doses. The common choices for f (x, β) which satisfy the necessary conditions include the empiric function, β  tanh x + 1 eβ , and logistic function, f (x, β) = x , hyperbolic tangent function, f (x, β) = 2 exp(a + βx) f (x, β) = . To ensure the probabilities of toxicity range between 0 and 1, we 1 + exp(a + βx) solve for the x in f (x, β) where x is the standardized dose units representing the clinical dose levels in the dose-toxicity function. For example, given a dose, d3 of say 100 milligrams with an β

associated probability of toxicity of 0.2 and f (x, β) = xe , then the corresponding x3 is x3 = 3

ACCEPTED MANUSCRIPT }. To accommodate two groups, let the dose-toxicity function for the ith group be exp{ ln(0.2) eβ denoted by f (i) (x, β (i) ), i = 1, 2. We consider the case where the two groups have the same

PT

parametric form of the dose-toxicity function but different parameters denoted by f (x, β (i) ), i = 1, 2. The target rate of toxicity, θ, is assumed to be the same for both groups.

RI

If no information is shared between the two groups, conducting two separate trials is rea-

SC

sonable. However, the dose-toxicity relationship of one group may be related to the other group. For instance, the parametric function of the dose-toxicity relationship is the same for both groups

NU

and differ only on the parameter value or the probabilities of toxicity for a given set of doses are higher for the second group. In such cases, one can take advantage of that shared information and

MA

conduct a single trial. How can this information be incorporated in the model? One way to handle this is to introduce two parameters in the model as done in the two-group CRM by O’Quigley

D

[4]. This two-parameter model uses one parameter to describe the information they share and an-

TE

other parameter measures the difference between the two groups. As each sequential estimation is

AC CE P

conducted, both parameters are estimated simultaneously which then determine the recommended dose for the next subject enrolled for either group. Another approach is to use a shift model as described by O’Quigley [6, 7]. With this model, the differences between the groups are limited to a small finite set of differences. The second group is a shift (in either direction) of the dose-toxicity function specified for the first group. This results in the MTD being located at a higher, lower or the same dose level for the second group compared to the first. Specifying each shift allows for only one parameter to be estimated but requires estimation of the candidate models for each shift. The shift model with the highest estimated likelihood value is chosen and used to find the recommended dose for each group. For this paper, we will be focusing on the two-parameter model. There are multiple specifications of the parameters that allow for information to be shared between the two groups. We chose to use the additive relationship and define the two parameters as β (1) = β, β (2) = β + τ where the ranges of β and τ depend on the choice of f (x, β). Consequently, the dose-toxicity

4

ACCEPTED MANUSCRIPT function for group 1 is f (x, β) and the dose-toxicity function for group 2 is f (x, β + τ ).

Likelihood Estimation

PT

2.2

RI

The difference between CRM and TITE-CRM is the incorporation of partial information

SC

in the estimation. TITE-CRM estimates a recommended dose upon enrollment of a new subject by using data from subjects who have completed the trial (i.e. subjects who have experienced

NU

toxicity or been observed for the entire observation period without a toxicity) as well as subjects partially observed (i.e. subjects still in the observation period and have not experienced a toxicity

MA

or dropped out). TITE-CRM incorporates information from the partially observed subjects using a weight with a value between 0 and 1 in the likelihood. The weight is a function of the time-

D

to-toxicity for this subject and has value that is allowed to change as the trial progresses. At a

TE

given time point, a subject in the study has data of the form (y, x, w) where y denotes the binary outcome (1=toxicity, 0=non-toxicity), x is the dose given and w is the weight. In the likelihood,

AC CE P

the dose-toxicity function is adjusted by the weight, w, and is denoted by G(x, w, β) ≡ wf (x, β) which must be monotone increasing in w satisfying G(x, 0, β) = 0 and G(x, 1, β) = f (x, β) for all x, β [8]. A subject that experienced a toxicity or has completed the study at the time of estimation is fully weighted (i.e. weight=1) in the likelihood. We now extend the TITE-CRM to the two-group case. As with the one-sample likelihoodbased TITE-CRM, the two-group likelihood-based TITE-CRM needs at least one toxic outcome and non-toxic outcome in each group to begin using the full likelihood estimation. Define the initial dose escalation (IDE) stage as the first stage employed until both outcomes (toxicity and no toxicity) have been observed at least once in each group. There are multiple options for the IDE in the two-group setting [4]. These options involve choosing the dose level to start the trial followed by escalating each group independently or jointly to be able to move out of the IDE stage. During the IDE stage, a group may use the one-sample likelihood TITE-CRM once both outcomes have been observed in that group while waiting for the other group to observe both 5

ACCEPTED MANUSCRIPT outcomes. The second stage begins once both outcomes have been observed in both groups. At that time, the full two-group likelihood TITE-CRM can be employed.

PT

Recall that we assumed the same parametric form for the dose-toxicity function for two groups so that the adjusted dose-toxicity function for group i given data (y (i) , x(i) , w(i) ) is denoted

(1)

(1)

RI

by G(x(i) , w(i) , β (i) ) , with β (1) = β and β (2) = β + τ . Thus, at the enrollment of the (k + (1)

(2)

(2)

(2)

SC

1)th subject, the available data are: [(yj , xj , wj ), j = 1, . . . , k (1) ] and [(yl , xl , wl ), l = 1, . . . , k (2) ] where k (i) , i = 1, 2 denotes the number of subjects from the ith group and k = k (1) +

NU

k (2) . For ease of notation in the likelihood and estimating equations, we drop the superscript (1)

(2)

(1)

(2)

(1)

indicating group in y, x, and w, i.e. yj ≡ yj , yl ≡ yl , xj ≡ xj , xl ≡ xl , wj ≡ wj , and wl ≡ (2)

MA

wl . The MLE for β and τ based on the first k observations are solutions to the equations (1)

(1)

D

∂ ∂ k k X −wj ∂β f (xj , β) X f (xj , β) ∂ℓk ∂β yj (1 − yj ) = + ∂β f (xj , β) 1 − wj f (xj , β) j=1 j=1 (2)

∂ℓk = ∂τ

k(2) X l=1

(2)

+

k X

+ τ) yl + f (xl , β + τ )

k(2) X

yl

+ τ)

f (xl , β + τ )

AC CE P

l=1

∂ f (xl , β ∂β

TE

+

k X

∂ f (xl , β ∂τ

(1 − yl )

∂ −wl ∂β f (xl , β + τ )

=0

(1)

∂ −wl ∂τ f (xl , β + τ ) (1 − yl ) =0 1 − wl f (xl , β + τ ) l=1

(2)

l=1

1 − wl f (xl , β + τ )

∂f ∂f and are the partial derivatives of f (· , · ) with respect to β and τ , respectively. Since ∂β ∂τ the likelihood was based on the first k subjects, we denote the MLE at this stage as βˆk . The

where

solutions to these equations typically have no closed analytical form even for the commonly chosen function. The revised dose-toxicity function is obtained using βˆk and τˆk with the same target rate of toxicity, θ. The next dose for the k + 1 subject is determined by

(1) xk+1 = arg min |f (x, βˆk ) − θ| x

(dose for Group 1)

or (2) xk+1 = arg min |f (x, βˆk + τˆk ) − θ| x

6

(dose for Group 2)

ACCEPTED MANUSCRIPT where arg minx is the argument of the minimum corresponding to the dose level for which |f (x, βˆk − θ| or |f (x, βˆk + τˆk ) − θ| attain their respective minima. The next dose recommended depends on

PT

the group membership of the k + 1 subject. This procedure continues in this manner to obtain (1) (2) (1) (2) (1) (2) (βˆk+1 , τˆk+1 , xk+2 , xk+2 ), (βˆk+2 , τˆk+2 , xk+3 , xk+3 ), . . . , (βˆN , τˆN , xN +1 , xN +1 ), where βˆN and τˆN (1)

(2)

RI

are the MLEs of β and τ using data from all subjects. The doses, (xN +1 , xN +1 ), defined by

x

NU

and

SC

(1) xN +1 = arg min |f (x, βˆN ) − θ|

xN +1 = arg min |f (x, βˆN + τˆN ) − θ| (2)

x

D

MA

are the trial MTDs for each group. The trial ends when N subjects are enrolled.

TE

3 OPERATING CHARACTERISTICS

AC CE P

Conventionally when working with MLEs the interest for finite samples is to assess the properties of the estimators. For dose finding methods, the primary goal is to find the MTD, and in the CRM framework, the dose-toxicity function assumed and the parameter(s) associated with this function are conduits to assist in finding the MTD. What is relevant are the operating characteristics of the dose finding method. The operating characteristics describe how well the method recommends the correct dose as the MTD, how well the method allocates the doses, and how well the method minimizes the proportion of persons who experience toxicity [9]. We use simulations to investigate the performance of the two-group TITE-CRM from its operating characteristics.

3.1

Simulation Method Sample size, N , was fixed at 32 subjects with group size ratios of 16/16 and 20/12. The

one-sample case was also simulated. The target rate of toxicity was set at 0.2 as is within the range

7

ACCEPTED MANUSCRIPT used in cancer studies as previously mentioned. The dose-toxicity function chosen was the empiric β

function, f (x, β) = xe for group 1 and f (x, β) = xe

β+τ

for group 2, where −∞ < β, τ < ∞

PT

and x is the standardized units representing the 6 dose levels in the dose-toxicity function. Three different scenarios of dose-toxicity probabilities were used (see Table 1; true MTD in bold) denoted

RI

by DTR 1, DTR 2 and DTR 3. The initial working model used for each strategy and scenario is

SC

detailed in Table 2. The initial working models are the starting probabilities of toxicity associated with each dose level.

NU

Subject recruitment is in many ways unpredictable. In practice, when subjects present for enrollment, the timing and number of subjects for each group is difficult to predict. In keeping

MA

with this, for the simulations the size of each group was fixed but the group membership of a subject arriving was randomly assigned using a Bernoulli random variable. Also, the interval of time

D

between the arrival of each subject for trial enrollment followed a Weibull (5,4) distribution. This

TE

generated arrival times which ranged from almost simultaneous enrollment of subjects to enroll-

AC CE P

ments with no subjects undergoing follow-up (i.e. no partial information used in the estimation). It should be noted that the arrival times and group for each subject were generated in such a way that they were the same for all strategies but different for each dose-toxicity probability scenarios. The time to toxicity were generated from two distributions, a Beta (2,2) distribution to approximate the toxicities occurring over the entire observation window and a Beta (4,1) distribution to approximate the toxicities occurring late in the observation window. Conditional on experiencing a toxicity, the time to toxicity was generated for a subject in the observation window interval (0, T ) where T is the length of follow-up for the trial. The uniform weight, w(u, T ) =

u , T

was used for

the TITE-CRM where u is the amount of time the subject has been followed [8]. The IDE stage of the simulations used the MTD of the working model for each group as the starting dose. Escalation proceeds separately for each group, and if there were 3 toxicities at a dose level within a group, the dose would de-escalate. During the second stage, dose escalation and de-escalation were not restricted.

8

ACCEPTED MANUSCRIPT Simulations were ran on SAS version 9.4. The random number generator used the rand function to determine the outcome, group membership, time to toxicity and arrival time variables.

PT

The Newton-Raphson with ridging optimization method was used to obtain maximum likelihood estimators. One thousand simulations were performed for 24 scenarios tested looking at 4 different

RI

group allocations for 3 dose-toxicity relationships and 2 different time to toxicities resulting in a

Results

NU

3.2

SC

total of 24,000 trials simulated [10].

MA

Of the 24,000 trials simulated, DTR 2 was the only scenario where failed trials were observed and this occurred in 1 trial for the 20/12 group size. The one-sample allocations represent

D

separate trials conducted for each group using all N subjects and provide a comparison of perfor-

Recommended MTD Proportion

AC CE P

3.2.1

TE

mance.

Recall that for these simulations we started the trial at the MTD for each group based on the initial working DTR which was one dose level below the true MTD. We evaluate the proportion of times the correct MTD is recommended in the simulated trials. The corresponding one-sample proportions for each group is used as a benchmark of performance. Figures 1 and 2 show the proportion at each dose level that is recommended as the MTD in the simulations by group size for each DTR. For the one-sample case (n = 32), the recommended MTD proportions range from 51% to 79% for the different DTRs. The late onset toxicities have recommended MTD proportions which are consistent with the typical onset toxicities. The higher the true MTD dose level the higher proportion of time the true MTD is selected. The variation in the recommended MTD proportions within a group is due in part to the probabilities of toxicity associated with the dose levels adjacent to the true MTD dose level for that group. Table 1 shows

9

ACCEPTED MANUSCRIPT that for DTR 1 the dose level above the MTD for group 1 has a probability of toxicity of 0.35 (versus 0.20 at the MTD) while group 2 has a probability of toxicity of 0.40 (versus 0.18 at the

PT

MTD). Hence it is easier to distinguish the MTD for group 2 than group 1 in this case. As expected, there is a drop in the proportion the true MTD is recommended for both

RI

groups with decreasing sample size. Compared to the one-sample, the balanced group sizes (16-

SC

16) show a 14% to 25% decrease in proportion recommended. For the imbalanced group size, the larger group recommends the true MTD with only a slight reduction compared to the one-

NU

sample, but the loss for the other group compared to the one-sample is larger. This loss is expected compared to the one-sample as neither group use all 32 subjects to find the MTD. In the 16-

MA

16 group size, the recommended proportion in the one-sample uses double the subjects of the two-sample estimation and yet, the recommended MTD proportion does not suffer as much as it

D

could with the added advantage of getting information about another group. Additionally, there

TE

is little additional decrease in the recommended MTD proportion at the MTD when there are late

toxicities.

3.2.2

AC CE P

onset toxicities. In some cases, the proportions recommended is slightly higher for the late onset

In-trial Dose Allocation of Subjects Another characteristic of interest is the distribution of doses the subjects treated at each

dose. The aim is to treat as many subjects as possible at the MTD. While some subjects may be exposed to doses one level above the MTD, treating beyond that level should be minimized. Tables 3 and 4 show the proportion of subjects allocated to each dose across all simulated trials for typical and late onset toxicities, respectively. There is a general decrease in the in trial allocation from the one-sample TITE-CRM, yet most of the change is concentrated at the dose levels adjacent to the MTD. The DTRs for both typical and late onset toxicities have 7% or less of the trials allocated subjects to doses two or more dose levels above the MTD. Group 1 in DTR 2 and Group 2 in DTR 3 treat 10

ACCEPTED MANUSCRIPT more subjects at 2 or more dose levels higher than the MTD. However, this is consistent with the results for the one-sample case. The highest proportion of subjects are treated at the MTD for each

RI

Two-Group TITE-CRM in the Case of a Homogeneous Sample

SC

3.3

PT

DTR and group which is reassuring.

Recall that the design assumes the sample has two groups with different MTDs from the

NU

onset. However, this assumption can be incorrect and the sample is actually homogeneous (no heterogeneity exists). While the truth is usually unknown in practice, it is helpful to know the

MA

effect of this misspecification of group relationship in the event this design is used when the onesample TITE-CRM may be more applicable. We investigate the impact of this scenario (DTR 3)

D

on the recommended MTD proportions and in-trial dose allocations.

TE

The recommended MTD proportion (Figure 3) shows the same decrease in the recommended MTD proportions as the group size decreases. The in trial allocation of subjects still has a

AC CE P

high proportion in each group being treated at the MTD (Tables 3 and 4). Finally, the two types of toxicity onset do not appear to effect the results in this case.

4

DISCUSSION

In this paper, we address the issue of patient heterogeneity in a single trial for dose finding studies using TITE-CRM. We showed how the TITE-CRM can accommodate two groups with different MTDs by extending the two-group CRM design of O’Quigley [4] using likelihood-based methods. The general form of the estimating equations provides a flexible framework which can be used to customize the design to a specific trial. We examined the performance of the two-group TITE-CRM design in the presence of patient heterogeneity, including imbalances in group size reflective of different recruitment rates of the two groups and different magnitude and difference in the MTD between groups. The decision to conduct the design incorporating both groups into 11

ACCEPTED MANUSCRIPT a single trial instead of conducting 2 separate trials has the benefit of maximizing the human and financial resources with an impact on the recommended MTD dependent on the sample size

PT

allocation between the two groups. As a result of using likelihood estimation, additional challenges are presented in the two-

RI

group setting. Estimation requires the observation of at least one toxic and a non-toxic event in

SC

both groups before valid parameter estimates can be obtained. An optimal strategy to accomplish this is unclear. Of course, the Bayesian method could be used and eliminate the need for the two

NU

stages. Only one IDE algorithm was considered in the simulations where the IDE started at the true MTD of each group. Other IDE options could impact how subjects are used early in the

MA

trial and are expected to affect the operating characteristics of the design. Another issue resulting from using likelihood estimation stems from the need to use numerical methods to find solutions.

D

The method has the potential to be sensitive to the choice of optimization technique or starting

TE

parameter value [11]. Choosing a weight which needs to estimate another parameter or specifying

AC CE P

a multiplicative group relationship may cause more challenges in obtaining solutions. Potential issues that may result with other design specifications warrant further exploration in the planning phase of a trial.

Even in the presence of late onset toxicities in the limited number of scenarios evaluated, there does not appear to be much decrease in the performance of the method. Also reassuring is the fact that the in trial allocation of subjects does not seem to result in over-escalating the dose during the trial on average. It is possible that with a different and more rapid enrollment of subjects could have a more harmful effect on the in trial allocation.

12

ACCEPTED MANUSCRIPT References [1] M. Fisher, K. Cheung, G. Howard, and S. Warach, “New pathways for evaluating potential

PT

acute stroke therapies,” International Journal of Stroke, vol. 1, pp. 52–58, 2006.

RI

[2] J. O’Quigley, M. Pepe, and L. Fisher, “Continual reassessment method: A practical design

SC

for phase 1 clinical trials in cancer,” Biometrics, vol. 46, no. 1, pp. 33–48, 1990. [3] J. O’Quigley and L. Shen, “Continual reassessment method: A likelihood approach,” Bio-

NU

metrics, vol. 52, no. 2, pp. 673–684, 1996.

MA

[4] J. O’Quigley, L. Shen, and A. Gamst, “Two-sample continual reassessment method,” Journal of Biopharmaceutical Statistics, vol. 9, no. 1, pp. 17–44, 1999.

D

[5] J. O’Quigley and X. Paoletti, “Continual reassessment method for ordered groups,” Biomet-

TE

rics, vol. 59, pp. 430–440, 2003.

AC CE P

[6] J. O’Quigley, “Theoretical study of the continual reassessment method,” Journal of Statistical Planning and Inference, vol. 136, pp. 1765–1780, 2006. [7] J. O’Quigley and A. Iasonos, “Bridging solutions in dose-finding problems,” Statistics in Biopharmaceutical Research, vol. 6, no. 2, pp. 185–197, 2014. [8] Y. K. Cheung and R. Chappell, “Sequential designs for phase I clinical trials with late-onset toxicities,” Biometrics, vol. 56, pp. 1177–1182, 2000. [9] X. Paoletti and A. Kramar, “A comparison of model choices for the continual reassessment method in phase I cancer trials,” Statistics in Medicine, vol. 28, no. 24, pp. 3012–3028, 2009. [10] A. Salter, C. Morgan, and I. B. Aban, “Implementation of a two-group likelihood timeto-event continual reassessment method using sas,” Computer Methods and Programs in Biomedicine, p. Accepted, 2015. [11] Sas Institute Inc, SAS/STAT 9.3 User’s Guide. Cary, NC: SAS Institute Inc, 2011. 13

ACCEPTED MANUSCRIPT

6 0.80 0.70

PT

Dose Level 1 2 3 4 5 Dose-Toxicity Relationship 1 (DTR 1) Group 1 0.08 0.20 0.35 0.50 0.70 Group 2 0.01 0.05 0.18 0.40 0.55 Dose-Toxicity Relationship 2 (DTR 2) Group 1 0.02 0.19 0.31 0.45 0.51 Group 2 0.03 0.05 0.11 0.21 0.39 Dose-Toxicity Relationship 3 (DTR 3) Group 1 0.07 0.23 0.31 0.35 0.45 Group 2 0.07 0.23 0.31 0.35 0.45

SC

RI

0.63 0.50

0.57 0.57

AC CE P

TE

D

MA

NU

Table 1: True Dose-Toxicity Relationships (MTD in bold)

14

ACCEPTED MANUSCRIPT

1 Group 1 0.20 Group 2 0.10

Dose Level 2 3 4 5 0.30 0.50 0.70 0.80 0.20 0.30 0.50 0.70

6 0.90 0.80

AC CE P

TE

D

MA

NU

SC

RI

PT

Table 2: Working Model Dose-Toxicity Relationships

15

ACCEPTED MANUSCRIPT

Group Size

1

Group 1 Dose Level 2 3 4 5 6

1

Group 2 Dose Level 2 3 4 5

6

38 29 6 1 39 32 6 1 49 28 4 0

16 40 20 36 One-Sample 31

38 41 46

MA

16 27 20 23 One-Sample 19

RI

22 3 0 25 3 0 22 2 0

SC

40 42 48

One-Sample 2 22 0 16 2 32 0 12 4 37 0 DTR 2 One-Sample 2 15 0 16 2 25 0 12 4 33 0 DTR 3 One-Sample 25 49 0 16 23 46 0 12 28 46 0

NU

16 36 20 30 One-Sample 27

PT

DTR 1

D

18 4 1 19 4 1 18 4 0

59 16 46 17 43 15

2 2 2

35 36 37 26 36 21

12 1 9 1 7 0

19 25 21

1 1 1

5 5 4

0 0 0

0 0 0

AC CE P

TE

Table 3: In-Trial Dose Allocation of Subjects by Group Size for Each Dose-Toxicity Relationship (DTR): Proportion of Subjects at Each Dose Level (True MTD)

16

ACCEPTED MANUSCRIPT

Group Size

1

Group 1 Dose Level 2 3 4 5

6

1

Group 2 Dose Level 2 3 4 5

6

RI

SC

D

16 39 38 19 4 20 34 40 21 4 One-Sample 31 46 18 4

NU

16 26 37 29 6 20 21 37 35 6 One-Sample 19 49 28 4

One-Sample 2 22 59 0 0 16 2 31 45 0 0 12 4 38 41 0 0 DTR 2 One-Sample 2 15 35 1 0 16 2 25 37 1 0 12 3 32 37 0 0 DTR 3 One-Sample 25 49 19 1 0 16 22 46 26 1 0 12 24 47 23 0 0

MA

16 35 40 23 3 20 29 41 27 3 One-Sample 27 48 22 2

PT

DTR 1 16 18 15

2 3 2

0 0 0

36 12 26 9 21 7

1 1 0

5 5 5

0 0 0

1 1 1

AC CE P

TE

Table 4: In-Trial Dose Allocation of Subjects by Group Size for Each Dose-Toxicity Relationship (DTR) and Late Onset Toxicities: Proportion of Subjects at Each Dose Level (True MTD)

17

SC

RI

PT

ACCEPTED MANUSCRIPT

(b) DTR1: Group 2

D

MA

NU

(a) DTR1: Group 1

(d) DTR2: Group 2

TE

(c) DTR2: Group 1

AC CE P

Figure 1: Proportion Each Dose Level is Recommended as the Maximum Tolerated Dose (MTD) for Different Group Sizes by Dose-Toxicity Relationship (DTR) Group Size:

18

D

MA

NU

SC

RI

PT

ACCEPTED MANUSCRIPT

(b) DTR1: Group 2

AC CE P

TE

(a) DTR1: Group 1

(c) DTR2: Group 1

(d) DTR2: Group 2

Figure 2: Proportion Each Dose Level is Recommended as the Maximum Tolerated Dose (MTD) for Different Group Sizes and Late Onset Toxicities by Dose-Toxicity Relationship (DTR) Group Size:

19

D

MA

NU

SC

RI

PT

ACCEPTED MANUSCRIPT

(b) DTR3: Group 2 (Typical)

AC CE P

TE

(a) DTR3: Group 1 (Typical)

(c) DTR3: Group 1 (Late)

(d) DTR3: Group 2 (Late)

Figure 3: Proportion Each Dose Level is Recommended as the Maximum Tolerated Dose (MTD) for Different Group Sizes by Dose-Toxicity Relationship (DTR) Group Size:

20