When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robus...
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When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple dataaugmentation (DA) algorithm and a corresponding Haar PX-DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX-DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log-normal, inverted Gamma (with shape parameter larger than d/2) or Frechet (with shape parameter larger than d/2). The results also apply to certain subsets of the Gamma, F and Weibull families.
We consider the intractable posterior density that results when the one-way logistic analysis of variance model is combined with a flat prior. We analyze Polson, Scott and Windle's (2013) dataaugmentation (DA) al...
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We consider the intractable posterior density that results when the one-way logistic analysis of variance model is combined with a flat prior. We analyze Polson, Scott and Windle's (2013) dataaugmentation (DA) algorithm for exploring the posterior. The Markov operator associated with the DA algorithm is shown to be trace-class.
In previous phase I/II oncology trials for drug combinations, a number of methods have been studied to determine the dose combination for the next cohort. However, there is a risk that trial durations will be unfeasib...
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In previous phase I/II oncology trials for drug combinations, a number of methods have been studied to determine the dose combination for the next cohort. However, there is a risk that trial durations will be unfeasibly long if methods for evaluating safety and efficacy are based on the best overall response and toxicity during trial design. In this study, we propose an approach to shorten the duration of drug trials in oncology. In this method, the dose combination to be allocated to the next cohort is decided before all data for patients in the current cohort is known and best overall response is determined. The efficacy of drug combinations in patients for whom the best overall response has not been determined is treated as missing data. The missing data mechanism is modeled by nonparametric prior processes. The probabilities of efficacy and toxicity are estimated after applying dataaugmentation to missing data, and the dose combination to be allocated to the next cohort is decided using these probabilities. Simulation studies from the present study show that this proposed approach would shorten trial durations without the low-performing of the trial design in comparison to existing approaches. Shortening trial durations would enable patients with the targeted disease to receive effective therapy at an earlier stage. This also enables clinical trial sponsors to use fewer patients in drug trials, which would lead to a reduction in the costs associated with clinical development.
A practical impediment in adaptive clinical trials is that outcomes must be observed soon enough to apply decision rules to choose treatments for new patients. For example, if outcomes take up to six weeks to evaluate...
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A practical impediment in adaptive clinical trials is that outcomes must be observed soon enough to apply decision rules to choose treatments for new patients. For example, if outcomes take up to six weeks to evaluate and the accrual rate is one patient per week, on average three new patients will be accrued while waiting to evaluate the outcomes of the previous three patients. The question is how to treat the new patients. This logistical problem persists throughout the trial. Various ad hoc practical solutions are used, none entirely satisfactory. We focus on this problem in phase I-II clinical trials that use binary toxicity and efficacy, defined in terms of event times, to choose doses adaptively for successive cohorts. We propose a general approach to this problem that treats late-onset outcomes as missing data, uses dataaugmentation to impute missing outcomes from posterior predictive distributions computed from partial follow-up times and complete outcome data, and applies the design's decision rules using the completed data. We illustrate the method with two cancer trials conducted using a phase I-II design based on efficacy-toxicity trade-offs, including a computer stimulation study. Supplementary materials for this article are available online.
We study MCMC algorithms for Bayesian analysis of a linear regression model with generalized hyperbolic errors. The Markov operators associated with the standard data augmentation algorithm and a sandwich variant of t...
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We study MCMC algorithms for Bayesian analysis of a linear regression model with generalized hyperbolic errors. The Markov operators associated with the standard data augmentation algorithm and a sandwich variant of that algorithm are shown to be trace-class. (C) 2014 Elsevier B.V. All rights reserved.
One of the most widely used data augmentation algorithms is Albert and Chibs (1993) algorithm for Bayesian probit regression. Polson, Scott, and Windle (2013) recently introduced an analogous algorithm for Bayesian lo...
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One of the most widely used data augmentation algorithms is Albert and Chibs (1993) algorithm for Bayesian probit regression. Polson, Scott, and Windle (2013) recently introduced an analogous algorithm for Bayesian logistic regression. The main difference between the two is that Albert and Chibs (1993) truncated normals are replaced by so-called Polya-Gamma random variables. In this note, we establish that the Markov chain underlying Polson, Scott, and Windles (2013) algorithm is uniformly ergodic. This theoretical result has important practical benefits. In particular, it guarantees the existence of central limit theorems that can be used to make an informed decision about how long the simulation should be run.
Let pi denote the intractable posterior density that results when the standard default prior is placed on the parameters in a linear regression model with iid Laplace errors. We analyze the Markov chains underlying tw...
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Let pi denote the intractable posterior density that results when the standard default prior is placed on the parameters in a linear regression model with iid Laplace errors. We analyze the Markov chains underlying two different Markov chain Monte Carlo algorithms for exploring pi. In particular, it is shown that the Markov operators associated with the dataaugmentation (DA) algorithm and a sandwich variant are both trace-class. Consequently, both Markov chains are geometrically ergodic. It is also established that for each i is an element of (1, 2, 3, ...}, the ith largest eigenvalue of the sandwich operator is less than or equal to the corresponding eigenvalue of the DA operator. It follows that the sandwich algorithm converges at least as fast as the DA algorithm. (C) 2013 Elsevier Inc. All rights reserved.
The sandwich algorithm (SA) is an alternative to the dataaugmentation (DA) algorithm that uses an extra simulation step at each iteration. In this paper, we show that the sandwich algorithm always converges at least ...
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The sandwich algorithm (SA) is an alternative to the dataaugmentation (DA) algorithm that uses an extra simulation step at each iteration. In this paper, we show that the sandwich algorithm always converges at least as fast as the DA algorithm, in the Markov operator norm sense. We also establish conditions under which the spectrum of SA dominates that of DA. An example illustrates the results. (C) 2011 Elsevier B.V. All rights reserved.
Most common regression models for analyzing binary random variables are logistic and probit regression models. However it is well known that the estimates of regression coefficients for these models are not robust to ...
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Most common regression models for analyzing binary random variables are logistic and probit regression models. However it is well known that the estimates of regression coefficients for these models are not robust to outliers [26]. The robit regression model [1, 16] is a robust alternative to the probit and logistic models. The robit model is obtained by replacing the normal (logistic) distribution underlying the probit (logistic) regression model with the Student's t-distribution. We consider a Bayesian analysis of binary data with the robit link function. We construct a dataaugmentation (DA) algorithm that can be used to explore the corresponding posterior distribution. Following [10] we further improve the DA algorithm by adding a simple extra step to each iteration. Though the two algorithms are basically equivalent in terms of computational complexity, the second algorithm is theoretically more efficient than the DA algorithm. Moreover, we analyze the convergence rates of these Markov chain Monte Carlo (MCMC) algorithms. We prove that, under certain conditions, both algorithms converge at a geometric rate. The geometric convergence rate has important theoretical and practical ramifications. Indeed, the geometric ergodicity guarantees that the ergodic averages used to approximate posterior expectations satisfy central limit theorems, which in turn allows for the construction of asymptotically valid standard errors. These standard errors can be used to choose an appropriate (Markov chain) Monte Carlo sample size and allow one to use the MCMC algorithms developed in this paper with the same level of confidence that one would have using classical (iid) Monte Carlo. The results are illustrated using a simple numerical example.
We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which ar...
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We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let pi denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n >= d + k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from pi in the special case where n = d + k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore pi when n > d + k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu's (1996) [7] dataaugmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student's t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates. Published by Elsevier Inc.
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