A Bayesian piecewise exponential phase II design for monitoring a time-to-event endpoint

A robust Bayesian design is presented for a single-arm phase II trial with an early stopping rule to monitor a time to event endpoint. The assumed model is a piecewise exponential distribution with non-informative gamma priors on the hazard parameters in subintervals of a fixed follow up interval. As an additional comparator, we also define and evaluate a version of the design based on an assumed Weibull distribution. Except for the assumed models, the piecewise exponential and Weibull model based designs are identical to an established design that assumes an exponential event time distribution with an inverse gamma prior on the mean event time. The three designs are compared by simulation under several log-logistic and Weibull distributions having different shape parameters, and for different monitoring schedules. The simulations show that, compared to the exponential inverse gamma model based design, the piecewise exponential design has substantially better performance, with much higher probabilities of correctly stopping the trial early, and shorter and less variable trial duration, when the assumed median event time is unacceptably low. Compared to the Weibull model based design, the piecewise exponential design does a much better job of maintaining small incorrect stopping probabilities in cases where the true median survival time is desirably large.     

Approximating distributions of random functionals of Ferguson-Dirichlet priors

We explore the possibility of approximating the Ferguson-Dirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in conjunction with the use of a random stopping rule. This allows us to set in advance the closeness to the distributions of interest. The distribution of the stopping rule is derived, and the practicability of the simulating procedure is discussed. Sufficient conditions for convergence of random functionals are provided. The numerical applications provided just sketch the idea of the variety of nonparametric procedures that can be easily and safely implemented in a Bayesian setting.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

Pivotal Quantities Based on Sequential Data: A Bootstrap Approach

A bootstrap algorithm is provided for obtaining a confidence interval for the mean of a probability distribution when sequential data are considered. For this kind of data the empirical distribution can be biased but its bias is bounded by the coefficient of variation of the stopping rule associated with the sequential procedure. When using this distribution for resampling the validity of the bootstrap approach is established by means of a series expansion of the corresponding pivotal quantity. A simulation study is carried out using Wang and Tsiatis type tests and considering the normal and exponential distributions to generate the data. This study confirms that for moderate coefficients of variation of the stopping rule, the bootstrap method allows adequate confidence intervals for the parameters to be obtained, whichever is the distribution of data.     

Optimal,nearly optimal and robust procedures for the exponential distribution with relative linear exponential loss

In this paper, the problem of sequentially estimating the mean of the exponential distribution with relative linear exponential loss and fixed cost for each observation is considered within the Bayesian framework. An optimal procedure with a deterministic stopping rule is derived. Since the corresponding value of the optimal deterministic stopping rule cannot be obtained directly, an approximate optimal deterministic stopping rule and an asymptotically pointwise optimal rule are proposed. In addition, we propose a robust procedure with a deterministic stopping rule, which does not depend on the parameters of the prior distribution. All of the proposed procedures are shown to be asymptotically optimal. Some numerical studies are conducted to investigate the performances of the proposed procedures. A real data set is provided to illustrate the use of the proposed procedures.     

A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution

A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution is considered. A reasonable sequential stopping rule, based on a one-step look-ahead procedure is derived. The procedure can be applied to estimate the infestation level of insects in a given field crop area.     

Sequential estimation for covariate-adjusted response-adaptive designs

In clinical trials, a covariate-adjusted response-adaptive (CARA) design allows a subject newly entering a trial a better chance of being allocated to a superior treatment regimen based on cumulative information from previous subjects, and adjusts the allocation according to individual covariate information. Since this design allocates subjects sequentially, it is natural to apply a sequential method for estimating the treatment effect in order to make the data analysis more efficient. In this paper, we study the sequential estimation of treatment effect for a general CARA design. A stopping criterion is proposed such that the estimates satisfy a prescribed precision when the sampling is stopped. The properties of estimates and stopping time are obtained under the proposed stopping rule. In addition, we show that the asymptotic properties of the allocation function, under the proposed stopping rule, are the same as those obtained in the non-sequential/fixed sample size counterpart. We then illustrate the performance of the proposed procedure with some simulation results using logistic models. The properties, such as the coverage probability of treatment effect, correct allocation proportion and average sample size, for diverse combinations of initial sample sizes and tuning parameters in the utility function are discussed.     

Revisit of a randomized response model for estimating a rare sensitive attribute under probability proportional to size sampling using Poisson probability distribution

The present article deals with the estimation of mean number of individuals possess a rare sensitive attribute using Poisson probability distribution, when the population consists of clusters. Unbiased estimation procedures for the mean number of individuals have been suggested and their properties are discussed when the parameter of a rare non-sensitive unrelated attribute is assumed to be known as well as unknown. The suggested estimation procedure is further discussed for situation of stratified cluster population. Empirical studies are carried out to show the dominance of proposed method and resultant estimators over a well-known contemporary estimator.     

Some sequential Bernoulli selection procedures with modified Bechhofer-Kulkarni stopping rules

Sequential methods for choosing the better of two Bernoulli populations are discussed using a Bayesian framework and when the maximum number of observations is fixed. Performance characteristics of the designs are obtained by using Monte Carlo simulation. Several sampling rules are considered, together with a stopping rule due to Bechhofer and Kulkarni (1982) and some modifications which use posterior estimates of the unknown probabilities.     

Informing the selection of futility stopping thresholds: case study from a late-phase clinical trial

In an environment where (i) potential risks to subjects participating in clinical studies need to be managed carefully, (ii) trial costs are increasing, and (iii) there are limited research resources available, it is necessary to prioritize research projects and sometimes re-prioritize if early indications suggest that a trial has low probability of success. Futility designs allow this re-prioritization to take place. This paper reviews a number of possible futility methods available and presents a case study from a late-phase study of an HIV therapeutic, which utilized conditional power-based stopping thresholds. The two most challenging aspects of incorporating a futility interim analysis into a trial design are the selection of optimal stopping thresholds and the timing of the analysis, both of which require the balancing of various risks. The paper outlines a number of graphical aids that proved useful in explaining the statistical risks involved to the study team. Further, the paper outlines a decision analysis undertaken which combined expectations of drug performance with conditional power calculations in order to produce probabilities of different interim and final outcomes, and which ultimately led to the selection of the final stopping thresholds.     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

Optimal group sequential designs constrained on both overall and stage one error rates

Abstract

Optimized group sequential designs proposed in the literature have designs minimizing average sample size with respect to a prior distribution of treatment effect with overall type I and type II error rates well-controlled (i.e., at final stage). The optimized asymmetric group sequential designs that we present here additionally consider constrains on stopping probabilities at stage one: probability of stopping for futility at stage one when no drug effect exists as well as the probability of rejection when the maximum effect size is true at stage one so that accountability of group sequential design is ensured from the first stage throughout.     

The construction of optimal histograms

In this note, a stopping rule is derived that gives the optimal histogram bin width to approximate a probability density function, The procedure minimizes the mean integrated square error between the density function and the histogram, and requires no prior knowledge of the density function.     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

Effect of the sample on the posterior probability in bayesian analysis

This paper examines Bayesian posterior probabilities as a function of selected elements within the set of data, x, when the prior distribution is assumed fixed. The posterior probabilities considered here are those of the parameter vector lying in a subset of the total parameter space. The theorems of this paper provide insight into the effect of elements within x on this posterior probability. These results have applications, for example, in the study of the impact of outliers within the data and in the isolation of misspecified parameters in a model.     

An application of lomax distributions in receiver operating characteristic(roc)curve analysis

Receiver operating characteristic(ROC)curves are useful for studying the performance of diagnostic tests. ROC curves occur in many fields of applications including psychophysics, quality control and medical diagnostics. In practical situations, often the responses to a diagnostic test are classified into a number of ordered categories. Such data are referred to as ratings data. It is typically assumed that the underlying model is based on a continuous probability distribution. The ROC curve is then constructed from such data using this probability model. Properties of the ROC curve are inherited from the model. Therefore, understanding the role of different probability distributions in ROC modeling is an interesting and important area of research. In this paper the Lomax distribution is considered as a model for ratings data and the corresponding ROC curve is derived. The maximum likelihood estimation procedure for the related parameters is discussed. This procedure is then illustrated in the analysis of a neurological data example.     

Shortest and longest length of success runs in binary sequences

The shortest and the longest length of success runs statistics in binary sequences are considered. The sequences are arranged on a line or on a circle. Exact probabilities of these statistics are derived, both in closed formulae via combinatorial analysis, as well as recursively. Furthermore, their joint probability distribution function and cumulative distribution function are obtained. The results are developed first for Bernoulli trials (i.i.d. binary sequences), and then they are generalized to the Polya–Eggenberger sampling scheme. For the latter case, the length of the longest success run is related to other success runs statistics and to reliability of consecutive systems.     

Probability of success for phase III after exploratory biomarker analysis in phase II

The probability of success or average power describes the potential of a future trial by weighting the power with a probability distribution of the treatment effect. The treatment effect estimate from a previous trial can be used to define such a distribution. During the development of targeted therapies, it is common practice to look for predictive biomarkers. The consequence is that the trial population for phase III is often selected on the basis of the most extreme result from phase II biomarker subgroup analyses. In such a case, there is a tendency to overestimate the treatment effect. We investigate whether the overestimation of the treatment effect estimate from phase II is transformed into a positive bias for the probability of success for phase III. We simulate a phase II/III development program for targeted therapies. This simulation allows to investigate selection probabilities and allows to compare the estimated with the true probability of success. We consider the estimated probability of success with and without subgroup selection. Depending on the true treatment effects, there is a negative bias without selection because of the weighting by the phase II distribution. In comparison, selection increases the estimated probability of success. Thus, selection does not lead to a bias in probability of success if underestimation due to the phase II distribution and overestimation due to selection cancel each other out. We recommend to perform similar simulations in practice to get the necessary information about the risk and chances associated with such subgroup selection designs.     

Optimal on-line detection of outside observations

Suppose that a Markov process X₁,X₂,… appears consecutively. There are two random moments of time θ₁ and θ₂, independent of (X_n). The distribution of the process (X_n) changes for the first time at instant θ₁ and for the second time at instant θ₂. Our objective is to find a stopping rule based only upon the former values of (X_n) which maximizes the probability that the moment of stopping is between θ₁ and θ₂. A sufficient condition is given under which an optimal stopping time is finite. The maximal probability corresponding to the optimal rule is found.     

Optimal sequential estimation procedures of a function of a probability of success under LINEX loss

In this paper, we investigate the problem of estimating a function g(p), where p is the probability of success in a sequential sample of independent identically Bernoulli distributed random variables. As a loss associated with estimation we introduce a generalized LINEX loss function. We construct a sequential procedure possessing some asymptotically optimal properties in the case when p tends to zero. In this approach to the problem, the conditions are given, under which the stopping time is asymptotically efficient and normal, and the corresponding sequential estimator is asymptotically normal. The procedure constructed guarantees that its sequential risk is asymptotically equal to a prescribed constant.