Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

Objective Bayesian analysis of spatial data with uncertain nugget and range parameters

The authors develop default priors for the Gaussian random field model that includes a nugget parameter accounting for the effects of microscale variations and measurement errors. They present the independence Jeffreys prior, the Jeffreys‐rule prior and a reference prior and study posterior propriety of these and related priors. They show that the uniform prior for the correlation parameters yields an improper posterior. In case of known regression and variance parameters, they derive the Jeffreys prior for the correlation parameters. They prove posterior propriety and obtain that the predictive distributions at ungauged locations have finite variance. Moreover, they show that the proposed priors have good frequentist properties, except for those based on the marginal Jeffreys‐rule prior for the correlation parameters, and illustrate their approach by analyzing a dataset of zinc concentrations along the river Meuse. The Canadian Journal of Statistics 40: 304–327; 2012 © 2012 Statistical Society of Canada     

Frequency coverage properties of a uniform shrinkage prior distribution

A uniform shrinkage prior (USP) distribution on the unknown variance component of a random-effects model is known to produce good frequency properties. The USP has a parameter that determines the shape of its density function, but it has been neglected whether the USP can maintain such good frequency properties regardless of the choice for the shape parameter. We investigate which choice for the shape parameter of the USP produces Bayesian interval estimates of random effects that meet their nominal confidence levels better than several existent choices in the literature. Using univariate and multivariate Gaussian hierarchical models, we show that the USP can achieve its best frequency properties when its shape parameter makes the USP behave similarly to an improper flat prior distribution on the unknown variance component.     

Weighted total least-squares joint adjustment with weight correction factors

Abstract

A joint adjustment involves integrating different types of geodetic datasets, or multiple datasets of the same data type, into a single adjustment. This paper applies the weighted total least-squares (WTLS) principle to joint adjustment problems and proposes an iterative algorithm for WTLS joint (WTLS-J) adjustment with weight correction factors. Weight correction factors are used to rescale the weight matrix of each dataset while using the Helmert variance component estimation (VCE) method to estimate the variance components, since the variance components in the stochastic model are unknown. An affine transformation example is illustrated to verify the practical benefit and the relative computational efficiency of the proposed algorithm. It is shown that the proposed algorithm obtains the same parameter estimates as the Amiri-Simkooei algorithm in our example; however, the proposed algorithm has its own computational advantages, especially when the number of data points is large.     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

Addressing potential prior‐data conflict when using informative priors in proof‐of‐concept studies

Bayesian methods are increasingly used in proof‐of‐concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior‐data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior‐data conflict by comparing the observed data to the prior predictive distribution and resorting to a non‐informative prior if prior‐data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one‐component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior‐credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration. Copyright © 2015 John Wiley & Sons, Ltd.     

A BAYESIAN METHOD FOR THE CHOICE OF THE SAMPLE SIZE IN EQUIVALENCE TRIALS

In this paper we consider a Bayesian predictive approach to sample size determination in equivalence trials. Equivalence experiments are conducted to show that the unknown difference between two parameters is small. For instance, in clinical practice this kind of experiment aims to determine whether the effects of two medical interventions are therapeutically similar. We declare an experiment successful if an interval estimate of the effects‐difference is included in a set of values of the parameter of interest indicating a negligible difference between treatment effects (equivalence interval). We derive two alternative criteria for the selection of the optimal sample size, one based on the predictive expectation of the interval limits and the other based on the predictive probability that these limits fall in the equivalence interval. Moreover, for both criteria we derive a robust version with respect to the choice of the prior distribution. Numerical results are provided and an application is illustrated when the normal model with conjugate prior distributions is assumed.     

Bayesian variable selection for multioutcome models through shared shrinkage

Variable selection over a potentially large set of covariates in a linear model is quite popular. In the Bayesian context, common prior choices can lead to a posterior expectation of the regression coefficients that is a sparse (or nearly sparse) vector with a few nonzero components, those covariates that are most important. This article extends the “global‐local” shrinkage idea to a scenario where one wishes to model multiple response variables simultaneously. Here, we have developed a variable selection method for a K‐outcome model (multivariate regression) that identifies the most important covariates across all outcomes. The prior for all regression coefficients is a mean zero normal with coefficient‐specific variance term that consists of a predictor‐specific factor (shared local shrinkage parameter) and a model‐specific factor (global shrinkage term) that differs in each model. The performance of our modeling approach is evaluated through simulation studies and a data example.     

BAYESIAN HYPER‐LASSOS WITH NON‐CONVEX PENALIZATION

The Lasso has sparked interest in the use of penalization of the log‐likelihood for variable selection, as well as for shrinkage. We are particularly interested in the more‐variables‐than‐observations case of characteristic importance for modern data. The Bayesian interpretation of the Lasso as the maximum a posteriori estimate of the regression coefficients, which have been given independent, double exponential prior distributions, is adopted. Generalizing this prior provides a family of hyper‐Lasso penalty functions, which includes the quasi‐Cauchy distribution of Johnstone and Silverman as a special case. The properties of this approach, including the oracle property, are explored, and an EM algorithm for inference in regression problems is described. The posterior is multi‐modal, and we suggest a strategy of using a set of perfectly fitting random starting values to explore modes in different regions of the parameter space. Simulations show that our procedure provides significant improvements on a range of established procedures, and we provide an example from chemometrics.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Alternatives to post‐processing posterior predictive p values

The posterior predictive p value (ppp) was invented as a Bayesian counterpart to classical p values. The methodology can be applied to discrepancy measures involving both data and parameters and can, hence, be targeted to check for various modeling assumptions. The interpretation can, however, be difficult since the distribution of the ppp value under modeling assumptions varies substantially between cases. A calibration procedure has been suggested, treating the ppp value as a test statistic in a prior predictive test. In this paper, we suggest that a prior predictive test may instead be based on the expected posterior discrepancy, which is somewhat simpler, both conceptually and computationally. Since both these methods require the simulation of a large posterior parameter sample for each of an equally large prior predictive data sample, we furthermore suggest to look for ways to match the given discrepancy by a computation‐saving conflict measure. This approach is also based on simulations but only requires sampling from two different distributions representing two contrasting information sources about a model parameter. The conflict measure methodology is also more flexible in that it handles non‐informative priors without difficulty. We compare the different approaches theoretically in some simple models and in a more complex applied example.     

Measures of Variability for Model-Based Seasonal Adjustment Procedures

An algorithm is derived that develops measures of variability for the estimates of the nonseasonal component computed from a model-based seasonal adjustment procedure. The measures of variability are developed from signal extraction theory. Properties of components of the variance are developed, and the behavior of the variance is investigated for one popular time series model. The results are illustrated by using real data.     

Effect of assay measurement error on parameter estimation in concentration–QTc interval modeling

Linear mixed‐effects models (LMEMs) of concentration–double‐delta QTc intervals (QTc intervals corrected for placebo and baseline effects) assume that the concentration measurement error is negligible, which is an incorrect assumption. Previous studies have shown in linear models that independent variable error can attenuate the slope estimate with a corresponding increase in the intercept. Monte Carlo simulation was used to examine the impact of assay measurement error (AME) on the parameter estimates of an LMEM and nonlinear MEM (NMEM) concentration–ddQTc interval model from a ‘typical’ thorough QT study. For the LMEM, the type I error rate was unaffected by assay measurement error. Significant slope attenuation ( > 10%) occurred when the AME exceeded > 40% independent of the sample size. Increasing AME also decreased the between‐subject variance of the slope, increased the residual variance, and had no effect on the between‐subject variance of the intercept. For a typical analytical assay having an assay measurement error of less than 15%, the relative bias in the estimates of the model parameters and variance components was less than 15% in all cases. The NMEM appeared to be more robust to AME error as most parameters were unaffected by measurement error. Monte Carlo simulation was then used to determine whether the simulation–extrapolation method of parameter bias correction could be applied to cases of large AME in LMEMs. For analytical assays with large AME ( > 30%), the simulation–extrapolation method could correct biased model parameter estimates to near‐unbiased levels. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian sample size determination for phase IIA clinical trials using historical data and semi‐parametric prior's elicitation

The Simon's two‐stage design is the most commonly applied among multi‐stage designs in phase IIA clinical trials. It combines the sample sizes at the two stages in order to minimize either the expected or the maximum sample size. When the uncertainty about pre‐trial beliefs on the expected or desired response rate is high, a Bayesian alternative should be considered since it allows to deal with the entire distribution of the parameter of interest in a more natural way. In this setting, a crucial issue is how to construct a distribution from the available summaries to use as a clinical prior in a Bayesian design. In this work, we explore the Bayesian counterparts of the Simon's two‐stage design based on the predictive version of the single threshold design. This design requires specifying two prior distributions: the analysis prior, which is used to compute the posterior probabilities, and the design prior, which is employed to obtain the prior predictive distribution. While the usual approach is to build beta priors for carrying out a conjugate analysis, we derived both the analysis and the design distributions through linear combinations of B‐splines. The motivating example is the planning of the phase IIA two‐stage trial on anti‐HER2 DNA vaccine in breast cancer, where initial beliefs formed from elicited experts' opinions and historical data showed a high level of uncertainty. In a sample size determination problem, the impact of different priors is evaluated.     

Bayesian predictive distribution for a Poisson model with a parametric restriction

Abstract

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.     

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u‐shape very similar, but not equal, to a β‐distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian predictive densities based on latent information priors

Construction methods for prior densities are investigated from a predictive viewpoint. Predictive densities for future observables are constructed by using observed data. The simultaneous distribution of future observables and observed data is assumed to belong to a parametric submodel of a multinomial model. Future observables and data are possibly dependent. The discrepancy of a predictive density to the true conditional density of future observables given observed data is evaluated by the Kullback-Leibler divergence. It is proved that limits of Bayesian predictive densities form an essentially complete class. Latent information priors are defined as priors maximizing the conditional mutual information between the parameter and the future observables given the observed data. Minimax predictive densities are constructed as limits of Bayesian predictive densities based on prior sequences converging to the latent information priors.     

Control Charts for the Variance and Coefficient of Variation Based on Their Predictive Distribution

This article develops a control chart for the variance of a normal distribution and, equivalently, the coefficient of variation of a log-normal distribution. A Bayesian approach is used to incorporate parameter uncertainty, and the control limits are obtained from the predictive distribution for the variance. We evaluate this control chart by examining its performance for various values of the process variance.     

The Use of Permutation Tests for Variance Components in Linear Mixed Models

Standard asymptotic chi-square distribution of the likelihood ratio and score statistics under the null hypothesis does not hold when the parameter value is on the boundary of the parameter space. In mixed models it is of interest to test for a zero random effect variance component. Some available tests for the variance component are reviewed and a new test within the permutation framework is presented. The power and significance level of the different tests are investigated by means of a Monte Carlo simulation study. The proposed test has a significance level closer to the nominal one and it is more powerful.