Multilevel models for longitudinal variables prognostic for survival

The issue of modelling the joint distribution of survival time and of prognostic variables measured periodically has recently become of interest in the AIDS literature but is of relevance in other applications. The focus of this paper is on clinical trials where follow-up measurements of potentially prognostic variables are often collected but not routinely used. These measurements can be used to study the biological evolution of the disease of interest; in particular the effect of an active treatment can be examined by comparing the time profiles of patients in the active and placebo group. It is proposed to use multilevel regression analysis to model the individual repeated observations as function of time and, possibly, treatment. To address the problem of informative drop-out—which may arise if deaths (or any other censoring events) are related to the unobserved values of the prognostic variables—we analyse sequentially overlapping portions of the follow-up information. An example arising from a randomized clinical trial for the treatment of primary biliary cirrhosis is examined in detail.     

Stability of Bayesian decisions

Stability of a Bayes decision is analysed with respect to small changes of the probability measure on the space of the states of nature. The problem leads to the study of continuity aspects of the infimum of the Bayes functional. These are approached through the epigraphical convergence of integral functionals as the minimal setting for convergence of infima.     

A novel equivalence probability weighted power prior for using historical control data in an adaptive clinical trial design: A comparison to standard methods

A standard two-arm randomised controlled trial usually compares an intervention to a control treatment with equal numbers of patients randomised to each treatment arm and only data from within the current trial are used to assess the treatment effect. Historical data are used when designing new trials and have recently been considered for use in the analysis when the required number of patients under a standard trial design cannot be achieved. Incorporating historical control data could lead to more efficient trials, reducing the number of controls required in the current study when the historical and current control data agree. However, when the data are inconsistent, there is potential for biased treatment effect estimates, inflated type I error and reduced power. We introduce two novel approaches for binary data which discount historical data based on the agreement with the current trial controls, an equivalence approach and an approach based on tail area probabilities. An adaptive design is used where the allocation ratio is adapted at the interim analysis, randomising fewer patients to control when there is agreement. The historical data are down-weighted in the analysis using the power prior approach with a fixed power. We compare operating characteristics of the proposed design to historical data methods in the literature: the modified power prior; commensurate prior; and robust mixture prior. The equivalence probability weight approach is intuitive and the operating characteristics can be calculated exactly. Furthermore, the equivalence bounds can be chosen to control the maximum possible inflation in type I error.     

Semiparametric analysis of longitudinal data with informative observation times and censoring times

We focus on regression analysis of irregularly observed longitudinal data which often occur in medical follow-up studies and observational investigations. The model for such data involves two processes: a longitudinal response process of interest and an observation process controlling observation times. Restrictive models and questionable assumptions, such as Poisson assumption and independent censoring time assumption, were posed in previous works for analysing longitudinal data. In this paper, we propose a more general model together with a robust estimation approach for longitudinal data with informative observation times and censoring times, and the asymptotic normalities of the proposed estimators are established. Both simulation studies and real data application indicate that the proposed method is promising.     

A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967 Jeffreys , H. ( 1967 ). Theory of Probability , 3rd rev. ed. . London : Oxford University Press . [Google Scholar]), Lindley (1956 Lindley , D. V. ( 1956 ). On a measure of the information provided by an experiment . Ann. Mathemat. Statist. 27 : 986 – 1005 .[Crossref] , [Google Scholar]); (1961 Lindley , D. V. ( 1961 ). The use of prior probability distributions in statistical inference and decisions . In: Neyman , J. , ed. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability . Vol. 1. Berkeley : University of California Press , pp. 453 – 468 . [Google Scholar]), Hartigan (1964 Hartigan , J. ( 1964 ). Invariant priors distributions . Ann. Mathemat. Statist. 35 : 836 – 845 .[Crossref] , [Google Scholar]), Bernardo (1979 Bernardo , J. M. ( 1979 ). Reference posterior distributions for Bayesian inference . J. Roy. Statist. Soc. 41 ( 2 ): 113 – 147 . [Google Scholar]), Zellner (1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani (1989 Tibshirani , R. ( 1989 ). Noninformative priors for one parameters of many . Biometrika 76 : 604 – 608 .[Crossref], [Web of Science ®] , [Google Scholar]), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.     

Analysis of Frechet Distribution Using Reference Priors

This article develops the Bayesian estimators in the context of reference priors for the two-parameter Frechet distribution. The general forms of the second-order matching priors are also derived in case of any parameter of interest and concluded that the reference prior is also a second order matching prior. Since the Bayesian estimators cannot be obtained in closed form, they are obtained using Monte Carlo simulation and Laplace approximation. The Bayesian and maximum likelihood estimates are compared via simulation study. Two real-life data sets are analyzed for illustration and comparison purpose.     

An examination of distributed lag model coefficients estimated with smoothness priors

The estimator for the coefficients of a distributed lag model with smoothness priors proposed by Shi Her (1973) is examined and an alternative ridge-type estimator is obtained utilizing restricted least squares approach, A Monte Carlo experiment is carried out to compare the behavior of these estimators.