Adaptive Optimal Designs for DoseFinding Studies with TimetoEvent Outcomes
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Abstract
We consider optimal design problems for dosefinding studies with censored Weibull timetoevent outcomes. Locally Doptimal designs are investigated for a quadratic dose–response model for logtransformed data subject to right censoring. Twostage adaptive Doptimal designs using maximum likelihood estimation (MLE) model updating are explored through simulation for a range of different dose–response scenarios and different amounts of censoring in the model. The adaptive optimal designs are found to be nearly as efficient as the locally Doptimal designs. A popular equal allocation design can be highly inefficient when the amount of censored data is high and when the Weibull model hazard is increasing. The issues of sample size planning/early stopping for an adaptive trial are investigated as well. The adaptive Doptimal design with early stopping can potentially reduce study size while achieving similar estimation precision as the fixed allocation design.
KEY WORDS
adaptive design censoring Doptimal design dose finding Weibull distributionINTRODUCTION
Dose–response studies play an important role in clinical drug development. Such studies are typically randomized, multiarmed, placebocontrolled parallel group designs involving several dose levels of an investigational drug. The study goals may be to estimate the drug’s dose–response profile with respect to some primary outcome measure and to identify a dose or doses to be tested in subsequent confirmatory phase III trials. Optimization of a trial design can allow an experimenter to achieve study objectives most efficiently with a given sample size.
Many clinical trials use timetoevent outcomes as primary study endpoints. The outcome could be, for example, progressionfree survival in oncology, duration of viral shredding in virology, time from treatment administration until pain symptoms disappear in studies of migraine, time to onset/duration of anesthesia in dentistry, or time to first relapse in multiple sclerosis. Optimal experimental designs for multiarm timetoevent outcome trials are warranted, but finding and implementing such designs in practice may be challenging due to uncertainty about the model for event times, delayed and potentially censored outcomes, and dependence of optimal designs for survival models on model parameters that are unknown at the trial outset (the socalled locally optimal designs).
Recently, there has been an increasing interest in research and application of optimal designs for experiments with timetoevent outcomes. For problems where the design space is discrete (e.g., treatment is a classification factor), the design optimization involves finding optimal allocation proportions to the given treatment groups to maximize efficiency of treatment comparisons for selected study objectives (1). Optimal allocation designs for survival trials with two or more treatment arms were studied in (2, 3, 4, 5, 6) to name a few. These optimal allocation designs can be implemented in practice by means of response–adaptive randomization with established statistical properties (7).
Another class of problems involve dose–response studies where the design space is an interval and therefore the dose level is measured on a continuous scale. In this case, for a given dose–response regression model, an optimal design problem is to determine a set of optimal doses and the probability mass distribution at these doses to maximize some convex criterion of the model Fisher information matrix. Optimal designs for twoparameter exponential regression models with different censoring mechanisms were investigated in (8, 9, 10). Optimal designs for most efficient estimation of specific quantiles of censored Weibull or lognormal observations were developed in (11). Optimal design problems for accelerated failure time (AFT) models with loglogistic distribution for event times and type I and random censoring were considered in (12). Optimal designs for partial likelihood (Cox’s proportional hazard) regression models were investigated in (13) and (14). One common and important observation from these papers is that censoring impacts the structure of optimal designs. Also, most of the optimal designs developed in these papers are locally optimal and cannot be implemented unless reliable guesstimates of the model parameters are available to an experimenter. Solutions to this problem include: (i) maximin designs that maximize the worst efficiency with respect to the locally optimal designs over the range of potential model parameter values, (ii) Bayesian optimal designs that maximize average efficiency with respect to the locally optimal designs for a given prior distribution on the parameters, or (iii) adaptive optimal designs which sequentially or periodically update the estimates of the model parameters and direct future dose assignments to the targeted optimal design.
In this paper, we overcome the limitation of locally optimal designs by developing adaptive optimal designs for dosefinding trials with Weibull timetoevent outcomes. We focus on the Weibull family of distributions because this family is widely used in the analysis of timetoevent data and its utility is well documented (15). The “Materials and Methods” section provides necessary statistical background. We study locally Doptimal designs for the most precise estimation of the dose–response relationship, both in situations with and without censoring. In the same section, we propose multistage adaptive optimal designs using maximum likelihood (MLE) updating. In the “Results” section, we investigate the locally Doptimal design structure and its efficiency compared to the popular equal allocation design. A striking result is that equal allocation designs can be highly inefficient in the presence of heavy censoring. We also report results of a simulation study to investigate operating characteristics of two nonadaptive designs (uniform and locally Doptimal) and a twostage adaptive Doptimal design using MLE updating under 24 different dose–response scenarios and different amount of censoring in the model. We find that the proposed adaptive Doptimal designs can significantly improve efficiency of a dosefinding trial; in fact, adaptive optimal designs can be nearly as efficient as the locally optimal designs. Furthermore, we present simulation results in a more complex setting, assuming the trial has prespecified criteria which can potentially enable stopping of the trial once model parameters have been estimated with due precision, thereby potentially reducing the total sample size in the study. The “Discussion” section provides some concluding remarks and outlines future work.
The R code used to generate all results in this paper is fully documented and is available for download from the journal website.
MATERIALS AND METHODS
Statistical Background
In timetoevent trials, observations are likely to be censored. Standard methods of survival analysis are based on the assumption of noninformative right censoring. The event time T is rightcensored by a fixed or random time C > 0, if T and C are independent and one observes (t, δ), where t = min(T, C) and δ_{ i } = I{T_{ i } ≤ C_{ i }}, where I{⋅} is an indicator function. The random variables t and δ are, in general, not independent; in fact, the joint distribution of (t, δ) can be quite complex as it depends on the censoring mechanism in the trial.
It should be noted that our considered structure of the likelihood function in Eq. (2) using standardized logtransformed times as observations is in line with (16), and this enables development of optimal designs for a broad class of parametric models. Specifically, the Fisher information matrix in Eq. (3) has the same general structure for the Weibull, loglogistic, lognormal, and some other distributions, and the key step is derivation of the quantity in Eq. (4) which is determined by the distribution of the model error term for the standardized logtransformed event times.
Alternatively, one can consider the likelihood function using timetoevent data on the original (untransformed) scale as was done, for example in (17). In this case, the elemental Fisher information matrix is readily available for the common timetoevent models, including censored cases (17,18). Applying the generalized regression approach developed in (18) (cf. Chapters 1.6 and 5.4 from (19)), one can derive similar results and facilitate construction of optimal designs in a streamlined manner. This approach merits further investigation but it is beyond the scope of our current work.
Weibull Model
For such a model, the hazard function of T conditional on x is h(t x) = b^{−1} exp(−(β_{0} + β_{1}x + β_{2}x^{2})/b)t^{1/b − 1}. Therefore, for a given x, the hazard is monotone increasing if 0<b < 1, it is constant if b = 1, and it is monotone decreasing if b > 1. Also, Median(T x) = exp(β_{0} + β_{1}x + β_{2}x^{2}){log(2)}^{ b }. Our motivation for choosing the model in Eq. (5) is that such a model is very flexible and it covers various (non)linear dose–response shapes. This model was considered, for instance, in (11) in the context of optimal estimation of specific quantiles of the Weibull distribution, in application to reliability studies.
A direct calculation shows that for the model in Eq. (5), Eq. (4) is simplified to \( {\lambda}_i={e}^{w_i} \).
Type I Censoring
As τ → ∞, there is no censoring in the model, in which case A_{ x } = E(δ_{ x }) = 1, B_{ x } = 1 − γ, and D_{ x } = π^{2}/6 − 1 + (1 − γ)^{2}, where γ = 0.577215… (Euler’s constant). Therefore, without censoring, the Fisher information matrix in Eq. (6) does not depend on θ.
OPTIMAL DESIGNS
Locally Doptimal Design
 a)
The design ξ^{∗} minimizes − log ∣ M(ξ, θ)∣.
 b)
The design ξ^{∗} minimizes \( \underset{x\in \mathcal{X}}{\max }\ \mathrm{trace}\left\{{\boldsymbol{M}}^{1}\left(\xi, \boldsymbol{\theta} \right){\boldsymbol{M}}_x\left(\boldsymbol{\theta} \right)\right\}4 \).
 c)
For all\( x\in \mathcal{X} \), the derivative function d(x, ξ, θ) = trace{M^{−1}(ξ, θ)M_{ x }(θ)} − 4 ≤ 0, with the equality holding at each support point of ξ^{∗}.
With censoring, the Doptimal design is more complex as it depends on θ and the censoring mechanism in the trial. In this work, it is found using a firstorder (exchange) algorithm (21) implemented using the R software; the code is fully documented and available in the supplementary online materials.
Adaptive Doptimal Designs
A major limitation of the locally Doptimal design is its dependence on the true values of model parameters and the amount of censoring in the experiment—these aspects are frequently unknown at the study planning stage. An adaptive design is a natural approach to handle such uncertainty. In practice, many dose–response trials are performed in a staged manner. At each stage, a cohort of eligible patients is enrolled and allocated to the study treatments. Patient outcome data can be monitored sequentially or periodically throughout the trial to update knowledge on the underlying dose–response relationship and facilitate informed decisions (e.g., to change allocation of subsequent cohorts to the “most informative” dose levels).
The idea of constructing an adaptive strategy for updating the design under model uncertainty can be traced to the work of Box and Hunter (22) where the authors proposed choosing design points sequentially to maximize an incremental increase of information at each step. See also Chapter 5.3 of (19) and references therein for further background on adaptive optimal designs.

Stage 1: n^{(1)} patients are allocated to doses according to some initial design ξ^{(1)}.

Interim updating: For k = 2, …, ν, fit model in Eq. (5) using accrued data from stages 1, …, (k − 1) to obtain an updated estimate\( {\widehat{\boldsymbol{\theta}}}^{\left(k1\right)} \) of θ. Compute the optimal design for the kth stage, ξ^{(k)}, based on \( {\widehat{\boldsymbol{\theta}}}^{\left(k1\right)} \) and the information accumulated up to this point, \( {\overset{\sim }{\xi}}^{\left(1,\dots, k1\right)} \).

Stage k = 2, … , ν: n^{(k)} patients are allocated to doses according to ξ^{(k)}.
 1.
At the interim analysis k = 2, …, ν, fit the model in Eq. (5) to the cumulative outcome data from cohorts 1, …, (k − 1) to obtain \( {\widehat{\boldsymbol{\theta}}}_{\mathrm{MLE}}^{\left(k1\right)} \), the MLE of θ.
 2.
Amend the cohort design as
In practice, an adaptive design with one interim analysis (i.e., ν = 2, a twostage design) is a reasonable choice both from statistical and operational perspectives (e.g., (24,25)). Each interim analysis requires database lock, data cleaning, analysis, review, and interpretation of results—if multiple interim looks are planned then study timelines can be delayed which may be undesirable from a business perspective.
RESULTS
Locally Doptimal Design
Specifically, we consider six choices for the parameter b in Eq. (1) which defines the hazard pattern of a Weibull distribution: four cases of 0 < b < 1 (monotone increasing hazard): b = 0.4, b = γ = 0.57721…(Euler^{′}s gamma), b = 0.65, and b = 0.8; the case of an exponential distribution (constant hazard): b = 1; and one case of b > 1 (monotone decreasing hazard): b = 1.5. In addition, we consider four choices for β = (β_{0}, β_{1}, β_{2}) that determines the shape of a dose–response, namely monotone increasing (Shape I (β_{0} = 1.9, β_{1} = 0.6, β_{2} = 2.8); Ushape with a minimum in [0,1] (Shape II (β_{0} = 3.4, β_{1} = − 7.6, β_{2} = 9.4); unimodal with a maximum in [0, 1] (Shape III (β_{0} = 3.5, β_{1} = 4.7, β_{2} = − 3.1); and Sshape (Shape IV (β_{0} = 3.1, β_{1} = 4.2, β_{2} = − 2.1).
Clearly, 0 < D_{eff}(θ) ≤ 1 for any value of θ. D_{eff}(θ) = 1 indicates that ξ_{ U } is as efficient as ξ^{∗}. A value of D_{eff}(θ) = 0.90 implies that ξ_{ U } is 90% as efficient as ξ^{∗}; in other words, an experiment using ξ_{ U } would require 10% more subjects than an experiment using ξ^{∗} to achieve the same level of estimation precision of θ.
Simulation Study to Compare Fixed and Adaptive Designs
Here, we present results of several simulation studies to compare operating characteristics of various experimental designs. In the subsection “Fixed Total Sample Size”, we investigate two nonadaptive designs (uniform and locally Doptimal) and a twostage adaptive Doptimal design using MLE updating, if the total sample size for the experiment is fixed and predetermined (e.g. according to budgetary and/or logistical considerations). In the subsection “Designs with Early Stopping Criteria”, we present results in a more complex setting, assuming the trial can potentially be stopped early (before the total planned sample size has been reached), provided that predefined requirements of estimation accuracy have been achieved.
Fixed Total Sample Size
Operating characteristics of three designs (singlestage uniform, singlestage locally Doptimal, and twostage adaptive Doptimal) were evaluated under various experimental scenarios, using 1000 simulation runs for each design/scenario combination. The scenarios included 24 choices of dose–response relationships (Fig. 1), different choices of the total sample size (n =150; 300; 450), different amounts of censoring in the study and, for the adaptive design, different initial cohort sizes. For censoring, we considered type I censoring schemes with parameter τ selected in such a way that the total average probability of event in the experiment for the uniform design in Eq. (7) is one of 25%, 50%, or 75%. Note that the singlestage locally Doptimal design should be viewed here as a theoretical benchmark; in practice, it cannot be implemented because the true model parameter values are unknown at the beginning of the study.
We also examined design estimation accuracy for other values of event probabilities. Outputs similar to those in Fig. 4 and Fig. S1, S2, and S3 in the Supplemental Appendix have been generated for average event probabilities of 25% and 75% (results not shown here). In the case of 75% event probability, the designs have, overall, better estimation accuracy than in the case of 50% probability. However, when only 25% of observations, on average, are events, estimation of dose–response is challenging. Our simulations showed that all three designs (uniform, locally Doptimal, and twostage adaptive Doptimal) in the 25% event probability case had lower estimation accuracy (higher bias and higher variance of estimated mean/median dose–response profiles) than in the 50% event probability case. Therefore, in situations when a high amount of censoring is expected, increasing the study size is necessary.
Designs with Early Stopping Criteria
So far, we have assumed that the study sample size is fixed and predetermined. However, an investigator may want to have a more flexible adaptive design which allows for the possibility to stop the trial early, before the target sample size is reached. Here, we propose an adaptive design with early stopping based on the accuracy of estimation precision. Similar ideas have been explored in the context of extensions of a continual reassessment method (27), by several authors (28, 29, 30).
To use the stopping rule Eq. (9) in an adaptive design, the first cohort is randomized to dose groups according to the uniform (nonadaptive) design in Eq. (7). Based on observed data, the stopping criterion in Eq. (9) is checked, and if it is met then the study is stopped; otherwise, the next cohort is randomized to dose groups according to the updated Doptimal design in Eq. (8). This procedure is repeated until either the stopping criterion is met or the maximum sample size is reached.
We performed simulations to compare three designs using the stopping criterion in Eq. (9). The experimental scenarios included four choices of dose–response relationships (cf. Fig. 1) (four shapes and parameter b = γ) and three choices for the total average probability of event (25%, 50%, or 75%). The three designs were: (i) Uniform design: for each cohort of patients, each subject is randomized to one of the doses 0, 0.5, or 1 with equal probability; (ii) Locally Doptimal design: assuming that the true model parameters are known, each cohort of patients is randomized among the optimal doses according to the true Doptimal design; and (iii) Adaptive Doptimal design with a stopping rule: described above. For each design, interim analyses were made after every 90 subjects (i.e., the number of subjects in each cohort is 90) and the trial was stopped when the stopping criteria was met or when the maximum sample size in the study was reached (set to n_{max} = 2,500). The randomization probabilities were applied individually to every subject in the given cohort. All results were obtained based on 1000 simulation runs for each design/scenario combination.
We ran additional simulations with other choices of the parameter η (results not shown here). In particular, we found that for η = 0.25, the final sample size upon termination was overall lower than in the case of η=0.20, and for some scenarios it was equal to the size of the initial cohort. This makes good sense as the targeted quality of estimation with η = 0.25 is a lower bar than in the case of η=0.20, and therefore, a smaller sample size can fulfill the experimental objectives. For η=0.15, we observed that the final sample size was higher and required more than one adaptation.
Difference in Median Sample Size Upon Termination for a Scenario with Early Stopping Rule (b = γ, average probability of event =50%), with Various Initial Cohort Sizes
(b = γ) average probability of event = 50%  

Initial cohort size  Shape I  Shape II  
Adaptive vs. Doptimal  Adaptive vs. uniform  Adaptive vs. Doptimal  Adaptive vs. uniform  
30  50%  − 675%  50%  − 675% 
60  0%  − 200%  50%  − 350% 
90  0%  − 200%  50%  − 200% 
120  0%  − 150%  50%  − 150% 
150  33%  − 100%  50%  − 100% 
Difference in Median Sample Size Upon Termination for a Scenario with Early Stopping Rule (b = γ, initial cohort size = 90), with Various Values of Pr(Event)
(b = γ) initial cohort size = 90  

Pr(Event)  Shape I  Shape II  
Adaptive vs. Doptimal  Adaptive vs. uniform  Adaptive vs. Doptimal  Adaptive vs. uniform  
25%  0%  − 283%  50%  − 650% 
50%  0%  − 200%  50%  − 200% 
75%  50%  0%  50%  0% 
DISCUSSION
The methodology proposed in this paper is applicable to dose–response studies with timetoevent outcomes, where events are assumed to follow a Weibull distribution and are subject to rightcensoring. We focused on the Weibull family of distributions because this family is widely used in survival analysis and its utility is well documented (15). A quadratic regression for the logtransformed event times provides flexibility and covers various (non)linear dose–response shapes for the median timetoevent dose–response relationship. In our study, the experimental design settings corresponded to threearm trials. Such designs are very common in randomized phase II clinical studies where the three arms correspond, for instance, to placebo, low dose of the drug, and high dose of the drug (e.g., the maximum tolerated dose).
In general, the Doptimal design can be quite different from the popular uniform (equal allocation) design due to the dependence on both model parameters and the amount of censoring in the model. However, the Doptimal design cannot be directly implemented unless reliable guesstimates of the model parameters are available. To overcome the limitation of local optimality, we proposed a twostage adaptive Doptimal design which performs dose assignments adaptively, according to updated knowledge on the dose–response curve at an interim analysis. Simulations under various experimental scenarios show that the proposed twostage adaptive design provides a very good approximation and it is nearly as efficient as the true Doptimal design. A particular advantage of the adaptive Doptimal design compared to the uniform design has been observed in scenarios when the Weibull model hazard is increasing. Since the hazard pattern is frequently unknown at the trial outset, a twostage adaptive Doptimal design provides a scientifically sound approach to dose finding in timetoevent settings. Higher statistical efficiency can potentially translate into reduction in study sample size. We showed that by adding a stopping criterion prescribing that the experiment should stop once model parameters have been estimated with due precision, one can add even more flexibility to adaptive Doptimal designs. In this paper, we explored one simple and practical stopping criterion (cf. Eq. (9)). Other criteria can be considered as well. In particular, we investigated a stopping criterion which prescribes stopping the study once the maximum value of the coefficient of variation for estimating each component of the model parameter vector is less than or equal to a predetermined constant. The results and conclusions were generally similar to the case of the early stopping based on the Dcriterion. The detailed results are available from the first author upon request.
In addition to potential benefits of adaptive Doptimal designs, locally Doptimal designs themselves provide useful tools in real dosefinding experiments. For instance, they provide theoretical measures of statistical estimation precision against which other experimental designs (e.g., with different number of treatment arms and/or different allocation ratios) can be compared.
As noted above, the optimal designs considered in this paper are based on the Weibull family of distributions. If the Weibull model is misspecified, then loss in efficiency is possible. To handle model uncertainty at the design stage, one could consider several parametric candidate models (e.g., Weibull, loglogistic, lognormal), and consider a twostage design for which the firststage data are used to estimate each model from the candidate set, and then select the “best” one for the second stage. This idea is similar in the spirit to the MCPMod methodology (31), but its further development is beyond the scope of the current paper.
Another important consideration is the censoring scheme. In this paper, we focused on type I censoring. On the other hand, in many clinical trials, patient enrollment times are random (e.g., follow a Poisson process), and therefore, event times are censored by random followup times. This calls for using more complex censoring schemes in the study. Optimal designs for timetoevent trials with censoring driven by random enrollment were obtained recently in (17) and (32). These two papers provide general theoretical results applicable for a broad class of timetoevent models. An interesting and important future research topic is construction of adaptive optimal designs for studies with censoring driven by random enrollment.
Twostage designs with Bayesian updating were recently investigated in the context of adaptive MCPMod procedures (23), and they were found to outperform adaptive designs with MLE updating. Implementation of Bayesian updating and its comparison with MLE updating in timetoevent dosefinding trials is an important future work. Some preliminary results were obtained in (34).
In any experiment involving multiple treatment arms, it is important that treatment allocation involves randomization—this allows mitigation of various experimental biases (35). For adaptive Doptimal designs, both dose levels and target allocation proportions are calibrated through the course of the experiment. Treatment allocation ratio for a given cohort of subjects is frequently different from equal allocation. To implement an unequal allocation in practice, one can use randomization procedures with established statistical properties such as brick tunnel randomization (36) or wide brick tunnel randomization (37). These procedures preserve the allocation ratio at each step and lead to valid statistical inference at the end of the trial.
Finally, we would like to highlight that successful implementation of any methodology relies on validated statistical software. The R code used to generate results in this paper is fully documented and can be used to generate additional results under userdefined experimental scenarios.
CONCLUSION
The current paper developed adaptive Doptimal designs for dosefinding experiments with censored timetoevent outcomes. The proposed designs overcome a limitation of local Doptimal designs by performing response–adaptive allocation to most informative dose levels according to predefined statistical criteria. These designs are flexible and maintain a high level of statistical estimation efficiency, which can potentially translate into reduction in study sample size. All results presented in this paper are fully reproducible with the R code which can be downloaded from the journal website.
Notes
Acknowledgements
The research leading to these results has received support from the Innovative Medicines Initiative Joint Undertaking under grant agreement No. 115156 (the DDMoRe project), resources of which were composed of financial contributions from the European Union’s Seventh Framework Programme (FP7/2007–2013) and EFPIA companies’ in kind contribution. The DDMoRe project was also supported by financial contributions from academic and SME partners. The authors would like to acknowledge three anonymous reviewers whose comments led to an improved version of the paper.
Supplementary material
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