Monotone response surface of multi-factor condition: estimation and Bayes classifiers

We formulate the estimation of monotone response surface of multiple factors as the inverse of an iteration of partially ordered classifier ensembles. Each ensemble (called PIPE-classifiers) is a projection of Bayes classifiers on the constrained space. We prove the inverse of PIPE-classifiers (iPIPE) exists, and propose algorithms to efficiently compute iPIPE by reducing the space over which optimisation is conducted. The methods are applied in analysis and simulation settings where the surface dimension is higher than what the isotonic regression literature typically considers. Simulation shows iPIPE-based credible intervals achieve nominal coverage probability and are more precise compared to unconstrained estimation.

Practical implementation of the partial ordering continual reassessment method in a Phase I combination-schedule dose-finding trial

There is a growing medical interest in combining several agents and optimizing their dosing schedules in a single trial in order to optimize the treatment for patients. Evaluating at doses of several drugs and their scheduling in a single Phase I trial simultaneously possess a number of statistical challenges, and specialized methods to tackle these have been proposed in the literature. However, the uptake of these methods is slow and implementation examples of such advanced methods are still sparse to date. In this work, we share our experience of proposing a model-based partial ordering continual reassessment method (POCRM) design for three-dimensional dose-finding in an oncology trial. In the trial, doses of two agents and the dosing schedule of one of them can be escalated/de-escalated. We provide a step-by-step summary on how the POCRM design was implemented and communicated to the trial team. We proposed an approach to specify toxicity orderings and their a-priori probabilities, and developed a number of visualization tools to communicate the statistical properties of the design. The design evaluation included both a comprehensive simulation study and considerations of the individual trial behavior. The study is now enrolling patients. We hope that sharing our experience of the successful implementation of an advanced design in practice that went through evaluations of several health authorities will facilitate a better uptake of more efficient methods in practice.

A nonparametric Bayesian method for dose finding in drug combinations cancer trials

We propose an adaptive design for early-phase drug-combination cancer trials with the goal of estimating the maximum tolerated dose (MTD). A nonparametric Bayesian model, using beta priors truncated to the set of partially ordered dose combinations, is used to describe the probability of dose limiting toxicity (DLT). Dose allocation between successive cohorts of patients is estimated using a modified continual reassessment scheme. The updated probabilities of DLT are calculated with a Gibbs sampler that employs a weighting mechanism to calibrate the influence of data vs the prior. At the end of the trial, we recommend one or more dose combinations as the MTD based on our proposed algorithm. We apply our method to a Phase I clinical trial of CB-839 and Gemcitabine that motivated this nonparametric design. The design operating characteristics indicate that our method is comparable with existing methods.

Two-part models for repeatedly measured ordinal data with "don't know" category

Ordinal data (eg, "low," "medium," "high"; graded response on a Likert scale) with an additional "don't know" category are frequently encountered in the medical, social, and behavioral science literature. The handling of a "don't know" option presents unique challenges as it often "destroys" the ordinal nature of the data. Commonly, nominal models are employed which ignore the partial ordering and have a complicated interpretation, especially in situations with repeatedly measured outcomes. We propose two-part models that easily accommodate longitudinal partially ordered (semiordinal) data. The most easily interpretable formulation consists of a random effect logistic submodel for "don't know" vs all the other categories combined, and a random effect ordinal submodel for the ordered categories. Correlated random effects account for statistical dependence within individual. An extension allowing for nonproportionality of odds for the predictor effects in the ordinal submodel is also considered. Maximum likelihood estimation is performed using adaptive Gaussian quadrature in SAS PROC NLMIXED. A simulation study is performed to evaluate the performance of the estimation algorithm in terms of bias and efficiency, and to compare the results of joint and separate models of the two parts, and of proportional and nonproportional model formulations. The methods are motivated and illustrated on a dataset from a study of adolescents' perceptions of nicotine strength of JUUL e-cigarettes. Using the proposed approach we show that adolescents perceive 5% nicotine content as relatively low, a misconception more pronounced among past month nonusers than among past month users of JUUL e-cigarettes.

A comparison of phase I dose-finding designs in clinical trials with monotonicity assumption violation

BACKGROUND/AIMS

In oncology, new combined treatments make it difficult to order dose levels according to monotonically increasing toxicity. New flexible dose-finding designs that take into account uncertainty in dose levels ordering were compared with classical designs through simulations in the setting of the monotonicity assumption violation. We give recommendations for the choice of dose-finding design.

METHODS

Motivated by a clinical trial for patients with high-risk neuroblastoma, we considered designs that require a monotonicity assumption, the Bayesian Continual Reassessment Method, the modified Toxicity Probability Interval, the Bayesian Optimal Interval design, and designs that relax monotonicity assumption, the Bayesian Partial Ordering Continual Reassessment Method and the No Monotonicity Assumption design. We considered 15 scenarios including monotonic and non-monotonic dose-toxicity relationships among six dose levels.

RESULTS

The No Monotonicity Assumption and Partial Ordering Continual Reassessment Method designs were robust to the violation of the monotonicity assumption. Under non-monotonic scenarios, the No Monotonicity Assumption design selected the correct dose level more often than alternative methods on average. Under the majority of monotonic scenarios, the Partial Ordering Continual Reassessment Method selected the correct dose level more often than the No Monotonicity Assumption design. Other designs were impacted by the violation of the monotonicity assumption with a proportion of correct selections below 20% in most scenarios. Under monotonic scenarios, the highest proportions of correct selections were achieved using the Continual Reassessment Method and the Bayesian Optimal Interval design (between 52.8% and 73.1%). The costs of relaxing the monotonicity assumption by the No Monotonicity Assumption design and Partial Ordering Continual Reassessment Method were decreases in the proportions of correct selections under monotonic scenarios ranging from 5.3% to 20.7% and from 1.4% to 16.1%, respectively, compared with the best performing design and were higher proportions of patients allocated to toxic dose levels during the trial.

CONCLUSIONS

Innovative oncology treatments may no longer follow monotonic dose levels ordering which makes standard phase I methods fail. In such a setting, appropriate designs, as the No Monotonicity Assumption or Partial Ordering Continual Reassessment Method designs, should be used to safely determine recommended for phase II dose.

Assessing and Grouping Chemicals Applying Partial Ordering Alkyl Anilines as an Illustrative Example

AIM AND OBJECTIVE

In chemistry, there is a long tradition in classification. Usually, methods are adopted from the wide field of cluster analysis. The present study focusses on the application of partial ordering methodology for the classification of 21 alkyl substituted anilines.

MATERIALS AND METHODS

The analyses are based on the concepts from partial order methodology and cluster analyses. Here, with the example of 21 alkyl anilines, we show that concepts taken out from the mathematical discipline of partially ordered sets may be applied for classification. The chemical compounds are described by a multi-indicator system. For the present study four indicators, mainly taken from the field of environmental chemistry were applied and a graph of the ordering (Hasse diagram) was constructed.

RESULTS

A Hasse diagram is an acyclic, transitively reduced, triangle-free graph that may have several graph-theoretical components. The Hasse diagram has been directed from a structural chemical point of view. Two cluster analysis methods are applied (K-means and a hierarchical cluster method) and compared with the results from the Hasse diagram. In both cases, the partitioning of the set of 21 compounds by the component structure of the Hasse diagram appears to be better interpretable.

CONCLUSION

It is shown that the partial ordering approach indeed can be used for classification in the present case. However, it must be clearly stated that a guarantee for meaningful results, in general, cannot be given. For that, further theoretical work is needed.

Resolve conflicting rankings of outcomes in network meta-analysis: Partial ordering of treatments

Network meta-analysis has evolved into a core method for evidence synthesis in health care. In network meta-analysis, 3 or more treatments for a given medical condition are compared, based on a number of clinical studies, usually randomized controlled trials. Often, many different endpoints are investigated, related to different aspects of the patient's outcome, such as efficacy, safety, acceptability, or costs of a treatment. Different outcomes may lead to different rankings of the treatments. We use the existing theory of partially ordered sets and show how the relations between the treatments in a network meta-analysis can be illustrated by Hasse diagrams, that is, directed graphs showing the partial order relations, and by structured scatter plots and biplots.

Practical designs for Phase I combination studies in oncology

Phase I trials evaluating the safety of multidrug combinations are becoming more common in oncology. Despite the emergence of novel methodology in the area, it is rare that innovative approaches are used in practice. In this article, we review three methods for Phase I combination studies that are easy to understand and straightforward to implement. We demonstrate the operating characteristics of the designs through illustration in a single trial, as well as through extensive simulation studies, with the aim of increasing the use of novel approaches in Phase I combination studies. Design specifications and software capabilities are also discussed.

Phase I design for completely or partially ordered treatment schedules

The majority of methods for the design of phase I trials in oncology are based upon a single course of therapy, yet in actual practice, it may be the case that there is more than one treatment schedule for any given dose. Therefore, the probability of observing a dose-limiting toxicity may depend upon both the total amount of the dose given, as well as the frequency with which it is administered. The objective of the study then becomes to find an acceptable combination of both dose and schedule. Past literature on designing these trials has entailed the assumption that toxicity increases monotonically with both dose and schedule. In this article, we relax this assumption for schedules and present a dose-schedule finding design that can be generalized to situations in which we know the ordering between all schedules and those in which we do not. We present simulation results that compare our method with other suggested dose-schedule finding methodology.

Specifications of a continual reassessment method design for phase I trials of combined drugs

In studies of combinations of agents in phase I oncology trials, the dose-toxicity relationship may not be monotone for all combinations, in which case the toxicity probabilities follow a partial order. The continual reassessment method for partial orders (PO-CRM) is a design for phase I trials of combinations that leans upon identifying possible complete orders associated with the partial order. This article addresses some practical design considerations not previously undertaken when describing the PO-CRM. We describe an approach in choosing a proper subset of possible orderings, formulated according to the known toxicity relationships within a matrix of combination therapies. Other design issues, such as working model selection and stopping rules, are also discussed. We demonstrate the practical ability of PO-CRM as a phase I design for combinations through its use in a recent trial designed at the University of Virginia Cancer Center.