Bayesian Inference for a Class of Start-Up Demonstration Tests with Rejection of Units upon the Observation of d Failures

This article considers Bayesian inference, posterior and predictive, in the context of a start-up demonstration test procedure in which rejection of a unit occurs when a pre-specified number of failures is observed prior to obtaining the number of consecutive successes required for acceptance. The method developed for implementing Bayesian inference in this article is a Markov chain Monte Carlo (MCMC) method incorporating data augmentation. This method permits the analysis to go forth, even when the results of the start-up test procedure are not completely recorded or reported. An illustrative example is included.     

Statistical inference for start-up demonstration tests with rejection of units upon observing d failures

In this paper, we consider the statistical inference for the success probability in the case of start-up demonstration tests in which rejection of units is possible when a pre-fixed number of failures is observed before the required number of consecutive successes are achieved for acceptance of the unit. Since the expected value of the stopping time is not a monotone function of the unknown parameter, the method of moments is not useful in this situation. Therefore, we discuss two estimation methods for the success probability: (1) the maximum likelihood estimation (MLE) via the expectation-maximization (EM) algorithm and (2) Bayesian estimation with a beta prior. We examine the small-sample properties of the MLE and Bayesian estimator. Finally, we present an example to illustrate the method of inference discussed here.     

Waiting time distributions for a run with additional constraints

This paper proposes a method for obtaining the exact probability of occurrence of the first success run of specified length with the additional constraint that at every trial until the occurrence of the first success run the number of successes up to the trial exceeds that of failures. For the sake of the additional constraint, the problem cannot be solved by the usual method of conditional probability generating functions. An idea of a kind of truncation is introduced and studied in order to solve the problem. Concrete methods for obtaining the probability in the cases of Bernoulli trials and time-homogeneous {0,1}$\{0,1\}$-valued Markov dependent trials are given. As an application of the results, a modification of the start-up demonstration test is studied. Numerical examples which illustrate the feasibility of the results are also given.     

Hypothesis testing for Markov chain Monte Carlo

Testing between hypotheses, when independent sampling is possible, is a well developed subject. In this paper, we propose hypothesis tests that are applicable when the samples are obtained using Markov chain Monte Carlo. These tests are useful when one is interested in deciding whether the expected value of a certain quantity is above or below a given threshold. We show non-asymptotic error bounds and bounds on the expected number of samples for three types of tests, a fixed sample size test, a sequential test with indifference region, and a sequential test without indifference region. Our tests can lead to significant savings in sample size. We illustrate our results on an example of Bayesian parameter inference involving an ODE model of a biochemical pathway.     

Structural change and unit roots

Recent research (though sometimes disputed) has demonstrated that structural breaks can affect inference about unit roots. The present paper presents a selective survey of the literature on structural breaks, on tests for unit roots and on tests for unit roots under structural breaks. Both the classical and Bayesian literature are reviewed and some suggestions for further research are made.     

A bayesian group sequential analysis procedure

In this work a method is developed for determining the expected sample size, 2nN, required by a group sequential test using a Bayesian approach. This method is proved to be superior to some recently developed methods. It gives a specific technique to determine the maximum number of groups, N, as well as the group, size, 2n. The proposed method allows a very early termination of the experiment when the alternative hypothesis is true, i.e. when i there is real difference between the treatments under consideration.     

Bayesian analysis of dynamic panel data by penalized quantile regression

Existing literature on quantile regression for panel data models with individual effects advocates the application of penalization to reduce the dynamic panel bias and increase the efficiency of the estimators. In this paper, we consider penalized quantile regression for dynamic panel data with random effects from a Bayesian perspective, where the penalty involves an adaptive Lasso shrinkage of the random effects. We also address the role of initial conditions in dynamic panel data models, emphasizing joint modeling of start-up and subsequent responses. For posterior inference, an efficient Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of a real data set, we assess the performance of the proposed Bayesian method.     

Inference and Decision Making for 21st-Century Drug Development and Approval

ABSTRACT

The cost and time of pharmaceutical drug development continue to grow at rates that many say are unsustainable. These trends have enormous impact on what treatments get to patients, when they get them and how they are used. The statistical framework for supporting decisions in regulated clinical development of new medicines has followed a traditional path of frequentist methodology. Trials using hypothesis tests of “no treatment effect” are done routinely, and the p-value < 0.05 is often the determinant of what constitutes a “successful” trial. Many drugs fail in clinical development, adding to the cost of new medicines, and some evidence points blame at the deficiencies of the frequentist paradigm. An unknown number effective medicines may have been abandoned because trials were declared “unsuccessful” due to a p-value exceeding 0.05. Recently, the Bayesian paradigm has shown utility in the clinical drug development process for its probability-based inference. We argue for a Bayesian approach that employs data from other trials as a “prior” for Phase 3 trials so that synthesized evidence across trials can be utilized to compute probability statements that are valuable for understanding the magnitude of treatment effect. Such a Bayesian paradigm provides a promising framework for improving statistical inference and regulatory decision making.     

Linear Bayesian Inference for Accelerated Weibull Model

In this paper, we present a Bayesian approach for inference from accelerated life tests when the underlying life model is Weibull. Our approach is based on the General Linear Models framework of West, Harrison and Migon (1985). We discuss inference for the model and show that computable results can be obtained using linear Bayesian methods. We illustrate the usefulness of our approach by applying it to some actual data from accelerated life tests. This revised version was published online in July 2006 with corrections to the Cover Date.     

An efficiency Bayesian unit root test in Unobserved-ARCH models

This paper investigates the new prior distribution on the Unobserved-Autoregressive Conditional Heteroscedasticity (ARCH) unit root test. Monte Carlo simulations show that the sample size is seriously effective in efficiency of Bayesian test. To improve the performance of Bayesian test for unit root, we propose a new Bayesian test that is robust in the presence of stationary and nonstationary Unobserved-ARCH. The finite sample property of the proposed test statistic is evaluated using Monte Carlo studies. Applying the developed method, we test the policy of daily exchange rate of the German Marc with respect to the Greek Drachma.     

Bayesian prediction for two-stage sequential analysis in clinical trials

This article deals with a Bayesian predictive approach for two-stage sequential analyses in clinical trials, applied to both frequentist and Bayesian tests. We propose to make a predictive inference based on the notion of satisfaction index and the data accrued so far together with future data. The computations and the simulation results concern an inferential problem, related to the binomial model.     

Robust inference for group sequential trials

For ethical reasons, group sequential trials were introduced to allow trials to stop early in the event of extreme results. Endpoints in such trials are usually mortality or irreversible morbidity. For a given endpoint, the norm is to use a single test statistic and to use that same statistic for each analysis. This approach is risky because the test statistic has to be specified before the study is unblinded, and there is loss in power if the assumptions that ensure optimality for each analysis are not met. To minimize the risk of moderate to substantial loss in power due to a suboptimal choice of a statistic, a robust method was developed for nonsequential trials. The concept is analogous to diversification of financial investments to minimize risk. The method is based on combining P values from multiple test statistics for formal inference while controlling the type I error rate at its designated value.This article evaluates the performance of 2 P value combining methods for group sequential trials. The emphasis is on time to event trials although results from less complex trials are also included. The gain or loss in power with the combination method relative to a single statistic is asymmetric in its favor. Depending on the power of each individual test, the combination method can give more power than any single test or give power that is closer to the test with the most power. The versatility of the method is that it can combine P values from different test statistics for analysis at different times. The robustness of results suggests that inference from group sequential trials can be strengthened with the use of combined tests.     

Bayesian unit-root tests for Stochastic Volatility models

In this article, we consider Bayesian inference procedures to test for a unit root in Stochastic Volatility (SV) models. Unit-root tests for the persistence parameter of the SV models, based on the Bayes Factor (BF), have been recently introduced in the literature. In contrast, we propose a flexible class of priors that is non-informative over the entire support of the persistence parameter (including the non-stationarity region). In addition, we show that our model fitting procedure is computationally efficient (using the software WinBUGS). Finally, we show that our proposed test procedures have good frequentist properties in terms of achieving high statistical power, while maintaining low total error rates. We illustrate the above features of our method by extensive simulation studies, followed by an application to a real data set on exchange rates.     

A local unit root test in mean for financial time series

A common financial trading strategy involves exploiting mean-reverting behaviour of paired asset prices. Since a unit root test can be used to determine which pairs of assets appear to exhibit mean-reverting behaviour, we propose a new Bayesian unit root to detect the presence of a local unit root vs. mean-reverting nonlinear smooth transition heteroskedastic alternative hypotheses. This test procedure is based on the posterior odds. For simultaneous estimation and inference, we employ an adaptive Bayesian Markov chain Monte Carlo scheme, which utilizes a mixture prior specification to solve the likelihood identification problem of the smoothing parameter and the autoregressive coefficient with a unit root. The size and power properties of the proposed method are examined via a simulation study. An empirical study examines the mean-reverting behaviour of price differential between stock and future.     

Bayesian computation for the superposition of nonhomogeneous poisson processes

Bayesian inference for the superposition of nonhomogeneous Poisson processes is studied. A Markov-chain Monte Carlo method with data augmentation is developed to compute the features of the posterior distribution. For each observed failure epoch, a latent variable is introduced that indicates which component of the superposition model gives rise to the failure. This data-augmentation approach facilitates specification of the transitional kernel in the Markov chain. Moreover, new Bayesian tests are developed for the full superposition model against simpler submodels. Model determination by a predictive likelihood approach is studied. A numerical example based on a real data set is given.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

Efficient Bayesian sampling plans for exponential distributions with type-I-censored samples

The aim of this paper is to introduce an efficient Bayesian sampling procedure for exponential distribution with type-I censoring. An online inspection method is suggested to reach a Bayes decision prior the termination time of life test. Bayesian sampling plans (BSPs) with quadratic loss function are established to illustrate the use of the proposed method. Some BSPs are tabulated, and the performance of the proposed BSPs is compared with two existing competitive methods. Numerical results indicate that a significant reduction in the experimental time over the conventional BSP can be achieved when the online inspection method is applied.     

Randomization as a basis for inference in noninferiority trials

Noninferiority testing in clinical trials is commonly understood in a Neyman-Pearson framework, and has been discussed in a Bayesian framework as well. In this paper, we discuss noninferiority testing in a Fisherian framework, in which the only assumption necessary for inference is the assumption of randomization of treatments to study subjects. Randomization plays an important role in not only the design but also the analysis of clinical trials, no matter the underlying inferential field. The ability to utilize permutation tests depends on assumptions around exchangeability, and we discuss the possible uses of permutation tests in active control noninferiority analyses. The other practical implications of this paper are admittedly minor but lead to better understanding of the historical and philosophical development of active control noninferiority testing. The conclusion may also frame discussion of other complicated issues in noninferiority testing, such as the role of an intention to treat analysis.     

Unit Roots: Bayesian Significance Test

The unit root problem plays a central role in empirical applications in the time series econometric literature. However, significance tests developed under the frequentist tradition present various conceptual problems that jeopardize the power of these tests, especially for small samples. Bayesian alternatives, although having interesting interpretations and being precisely defined, experience problems due to the fact that that the hypothesis of interest in this case is sharp or precise. The Bayesian significance test used in this article, for the unit root hypothesis, is based solely on the posterior density function, without the need of imposing positive probabilities to sets of zero Lebesgue measure. Furthermore, it is conducted under strict observance of the likelihood principle. It was designed mainly for testing sharp null hypotheses and it is called FBST for Full Bayesian Significance Test.     

On semiparametric pivotal bayesian inference for quantiles

In semiparametric inference we distinguish between the parameter of interest which may be a location parameter, and a nuisance parameter that determines the remaining shape of the sampling distribution. As was pointed out by Diaconis and Freedman the main problem in semiparametric Bayesian inference is to obtain a consistent posterior distribution for the parameter of interest. The present paper considers a semiparametric Bayesian method based on a pivotal likelihood function. It is shown that when the parameter of interest is the median, this method produces a consistent posterior distribution and is easily implemented, Numerical comparisons with classical methods and with Bayesian methods based on a Dirichlet prior are provided. It is also shown that in the case of symmetric intervals, the classical confidence coefficients have a Bayesian interpretation as the limiting posterior probability of the interval based on the Dirichlet prior with a parameter that converges to zero.