Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

High dimensional multivariate mixed models for binary questionnaire data

Summary.  Questionnaires that are used to measure the effect of an intervention often consist of different sets of items, each set possibly measuring another concept. Mixed models with set-specific random effects are a flexible tool to model the different sets of items jointly. However, computational problems typically arise as the number of sets increases. This is especially true when the random-effects distribution cannot be integrated out analytically, as with mixed models for binary data. A pairwise modelling strategy, in which all possible bivariate mixed models are fitted and where inference follows from pseudolikelihood theory, has been proposed as a solution. This approach has been applied to assess the effect of physical activity on psychocognitive functioning, the latter measured by a battery of questionnaires.     

Allowing for non-ignorable non-response in the analysis of voting intention data

We apply some log-linear modelling methods, which have been proposed for treating non-ignorable non-response, to some data on voting intention from the British General Election Survey. We find that, although some non-ignorable non-response models fit the data very well, they may generate implausible point estimates and predictions. Some explanation is provided for the extreme behaviour of the maximum likelihood estimates for the most parsimonious model. We conclude that point estimates for such models must be treated with great caution. To allow for the uncertainty about the non-response mechanism we explore the use of profile likelihood inference and find the likelihood surfaces to be very flat and the interval estimates to be very wide. To reduce the width of these intervals we propose constraining confidence regions to values where the parameters governing the non-response mechanism are plausible and study the effect of such constraints on inference. We find that the widths of these intervals are reduced but remain wide.     

Modelling spatially correlated data via mixtures: a Bayesian approach

Summary. The paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixture that vary from one location to another. Our specific focus is on Poisson-distributed data, and applications in disease mapping. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and an unknown number of components. We propose two alternative models for spatially dependent weights, based on transformations of autoregressive Gaussian processes: in one (the logistic normal model), the mixture component labels are exchangeable; in the other (the grouped continuous model), they are ordered. Reversible jump Markov chain Monte Carlo algorithms for posterior inference are developed. Finally, the performances of both of these formulations are examined on synthetic data and real data on mortality from a rare disease.     

Bayesian diffusion process models with time-varying parameters

The diffusion process is a widely used statistical model for many natural dynamic phenomena but its inference is very complicated because complete data describing the diffusion sample path is not necessarily available. In addition, data is often collected with substantial uncertainty and it is not uncommon to have missing observations. Thus, the observed process will be discrete over a finite time period and the marginal likelihood given by this discrete data is not always available. In this paper, we consider a class of nonstationary diffusion process models with not only the measurement error but also discretely time-varying parameters which are modeled via a state space model. Hierarchical Bayesian inference for such a diffusion process model with time-varying parameters is applied to financial data.     

Simultaneous modelling of multiple traffic safety performance indicators by using a multivariate generalized linear mixed model

Summary.  Traffic safety in the UK is one of the increasing number of areas where central government sets targets based on 'outcome-focused' performance indicators (PIs). Judgments about such PIs are often based solely on rankings of raw indicators and simple league tables dominate centrally published analyses. There is a considerable statistical literature examining health and education issues which has tended to use the generalized linear mixed model (GLMM) to address variability in the data when drawing inferences about relative performance from headline PIs. This methodology could obviously be applied in contexts such as traffic safety. However, when such models are applied to the fairly crude data sets that are currently available, the interval estimates generated, e.g. in respect of rankings, are often too broad to allow much real differentiation between the traffic safety performance of the units that are being considered. Such results sit uncomfortably with the ethos of 'performance management' and raise the question of whether the inference from such data sets about relative performance can be improved in some way. Motivated by consideration of a set of nine road safety performance indicators measured on English local authorities in the year 2000, the paper considers methods to strengthen the weak inference that is obtained from GLMMs of individual indicators by simultaneous, multivariate modelling of a range of related indicators. The correlation structure between indicators is used to reduce the uncertainty that is associated with rankings of any one of the individual indicators. The results demonstrate that credible intervals can be substantially narrowed by the use of the multivariate GLMM approach and that multivariate modelling of multiple PIs may therefore have considerable potential for introducing more robust and realistic assessments of differential performance in some contexts.     

Forecasting with non-homogeneous hidden Markov models

We present a Bayesian forecasting methodology of discrete-time finite state-space hidden Markov models with non-constant transition matrix that depends on a set of exogenous covariates. We describe an MCMC reversible jump algorithm for predictive inference, allowing for model uncertainty regarding the set of covariates that affect the transition matrix. We apply our models to interest rates and we show that our general model formulation improves the predictive ability of standard homogeneous hidden Markov models.     

Data-driven sensitivity analysis to detect missing data mechanism with applications to structural equation modelling

Missing data are a common problem in almost all areas of empirical research. Ignoring the missing data mechanism, especially when data are missing not at random (MNAR), can result in biased and/or inefficient inference. Because MNAR mechanism is not verifiable based on the observed data, sensitivity analysis is often used to assess it. Current sensitivity analysis methods primarily assume a model for the response mechanism in conjunction with a measurement model and examine sensitivity to missing data mechanism via the parameters of the response model. Recently, Jamshidian and Mata (Post-modelling sensitivity analysis to detect the effect of missing data mechanism, Multivariate Behav. Res. 43 (2008), pp. 432–452) introduced a new method of sensitivity analysis that does not require the difficult task of modelling the missing data mechanism. In this method, a single measurement model is fitted to all of the data and to a sub-sample of the data. Discrepancy in the parameter estimates obtained from the the two data sets is used as a measure of sensitivity to missing data mechanism. Jamshidian and Mata describe their method mainly in the context of detecting data that are missing completely at random (MCAR). They used a bootstrap type method, that relies on heuristic input from the researcher, to test for the discrepancy of the parameter estimates. Instead of using bootstrap, the current article obtains confidence interval for parameter differences on two samples based on an asymptotic approximation. Because it does not use bootstrap, the developed procedure avoids likely convergence problems with the bootstrap methods. It does not require heuristic input from the researcher and can be readily implemented in statistical software. The article also discusses methods of obtaining sub-samples that may be used to test missing at random in addition to MCAR. An application of the developed procedure to a real data set, from the first wave of an ongoing longitudinal study on aging, is presented. Simulation studies are performed as well, using two methods of missing data generation, which show promise for the proposed sensitivity method. One method of missing data generation is also new and interesting in its own right.     

Theoretic and empirical data-inclusive process characterization

Summary.  In process characterization the quality of information that is obtained depends directly on the quality of process model. The current quality revolution is now providing a strong stimulus for rethinking and re-evaluating many statistical ideas. Among these are the role of theoretic knowledge and data in statistical inference and some issues in theoretic–empirical modelling. With this concern the paper takes a broad, pragmatic view of statistical inference to include all aspects of model formulation. The estimation of model parameters traditionally assumes that a model has a prespecified known form and takes no account of possible uncertainty regarding model structure. But in practice model structural uncertainty is a fact of life and is likely to be more serious than other sources of uncertainty which have received far more attention. This is true whether the model is specified on subject-matter grounds or when a model is formulated, fitted and checked on the same data set in an iterative interactive way. For that reason novel modelling techniques have been fashioned for reducing model uncertainty. Using available knowledge for theoretic model elaboration the techniques that have been created approximate the exact unknown process model concurrently by accessible theoretic and polynomial empirical functions. The paper examines the effects of uncertainty for hybrid theoretic–empirical models and, for reducing uncertainty, additive and multiplicative methods of model formulation are fashioned. Such modelling techniques have been successfully applied to perfect a steady flow model for an air gauge sensor. Validation of the models elaborated has revealed that the multiplicative modelling approach allows us to attain a satisfactory model with small discrepancy from empirical evidence.     

A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials

The objective of this research was to demonstrate a framework for drawing inference from sensitivity analyses of incomplete longitudinal clinical trial data via a re‐analysis of data from a confirmatory clinical trial in depression. A likelihood‐based approach that assumed missing at random (MAR) was the primary analysis. Robustness to departure from MAR was assessed by comparing the primary result to those from a series of analyses that employed varying missing not at random (MNAR) assumptions (selection models, pattern mixture models and shared parameter models) and to MAR methods that used inclusive models. The key sensitivity analysis used multiple imputation assuming that after dropout the trajectory of drug‐treated patients was that of placebo treated patients with a similar outcome history (placebo multiple imputation). This result was used as the worst reasonable case to define the lower limit of plausible values for the treatment contrast. The endpoint contrast from the primary analysis was ? 2.79 (p = .013). In placebo multiple imputation, the result was ? 2.17. Results from the other sensitivity analyses ranged from ? 2.21 to ? 3.87 and were symmetrically distributed around the primary result. Hence, no clear evidence of bias from missing not at random data was found. In the worst reasonable case scenario, the treatment effect was 80% of the magnitude of the primary result. Therefore, it was concluded that a treatment effect existed. The structured sensitivity framework of using a worst reasonable case result based on a controlled imputation approach with transparent and debatable assumptions supplemented a series of plausible alternative models under varying assumptions was useful in this specific situation and holds promise as a generally useful framework. Copyright © 2012 John Wiley & Sons, Ltd.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Bayesian methods for dealing with missing data problems

Missing data, a common but challenging issue in most studies, may lead to biased and inefficient inferences if handled inappropriately. As a natural and powerful way for dealing with missing data, Bayesian approach has received much attention in the literature. This paper reviews the recent developments and applications of Bayesian methods for dealing with ignorable and non-ignorable missing data. We firstly introduce missing data mechanisms and Bayesian framework for dealing with missing data, and then introduce missing data models under ignorable and non-ignorable missing data circumstances based on the literature. After that, important issues of Bayesian inference, including prior construction, posterior computation, model comparison and sensitivity analysis, are discussed. Finally, several future issues that deserve further research are summarized and concluded.     

Weighted composite quantile regression analysis for nonignorable missing data using nonresponse instrument

Efficient statistical inference on nonignorable missing data is a challenging problem. This paper proposes a new estimation procedure based on composite quantile regression (CQR) for linear regression models with nonignorable missing data, that is applicable even with high-dimensional covariates. A parametric model is assumed for modelling response probability, which is estimated by the empirical likelihood approach. Local identifiability of the proposed strategy is guaranteed on the basis of an instrumental variable approach. A set of data-based adaptive weights constructed via an empirical likelihood method is used to weight CQR functions. The proposed method is resistant to heavy-tailed errors or outliers in the response. An adaptive penalisation method for variable selection is proposed to achieve sparsity with high-dimensional covariates. Limiting distributions of the proposed estimators are derived. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies. An application to the ACTG 175 data is analysed.     

Bayesian analysis of the Box-Cox transformation model based on left-truncated and right-censored data

In this paper, we discuss the inference problem about the Box-Cox transformation model when one faces left-truncated and right-censored data, which often occur in studies, for example, involving the cross-sectional sampling scheme. It is well-known that the Box-Cox transformation model includes many commonly used models as special cases such as the proportional hazards model and the additive hazards model. For inference, a Bayesian estimation approach is proposed and in the method, the piecewise function is used to approximate the baseline hazards function. Also the conditional marginal prior, whose marginal part is free of any constraints, is employed to deal with many computational challenges caused by the constraints on the parameters, and a MCMC sampling procedure is developed. A simulation study is conducted to assess the finite sample performance of the proposed method and indicates that it works well for practical situations. We apply the approach to a set of data arising from a retirement center.     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.     

Local sensitivity approximations for selectivity bias

Observational data analysis is often based on tacit assumptions of ignorability or randomness. The paper develops a general approach to local sensitivity analysis for selectivity bias, which aims to study the sensitivity of inference to small departures from such assumptions. If M is a model assuming ignorability, we surround M by a small neighbourhood N defined in the sense of Kullback–Leibler divergence and then compare the inference for models in N with that for M . Interpretable bounds for such differences are developed. Applications to missing data and to observational comparisons are discussed. Local approximations to sensitivity analysis are model robust and can be applied to a wide range of statistical problems.     

Maximum-Entropy Prior Uncertainty and Correlation of Statistical Economic Data

Empirical estimates of source statistical economic data such as trade flows, greenhouse gas emissions, or employment figures are always subject to uncertainty (stemming from measurement errors or confidentiality) but information concerning that uncertainty is often missing. This article uses concepts from Bayesian inference and the maximum entropy principle to estimate the prior probability distribution, uncertainty, and correlations of source data when such information is not explicitly provided. In the absence of additional information, an isolated datum is described by a truncated Gaussian distribution, and if an uncertainty estimate is missing, its prior equals the best guess. When the sum of a set of disaggregate data is constrained to match an aggregate datum, it is possible to determine the prior correlations among disaggregate data. If aggregate uncertainty is missing, all prior correlations are positive. If aggregate uncertainty is available, prior correlations can be either all positive, all negative, or a mix of both. An empirical example is presented, which reports relative uncertainties and correlation priors for the County Business Patterns database. In this example, relative uncertainties range from 1% to 80% and 20% of data pairs exhibit correlations below ?0.9 or above 0.9. Supplementary materials for this article are available online.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact uniformly most powerful postselection confidence distributions

A conditioning on the event of having selected one model from a set of possibly misspecified normal linear regression models leads to the construction of uniformly optimal conditional confidence distributions. They can be used for valid postselection inference. The constructed conditional confidence distributions are finite sample exact and encompass all information regarding the focus parameter in the selected model. This includes the construction of optimal postselection confidence intervals at all significance levels and uniformly most powerful hypothesis tests.