Bias modelling in evidence synthesis

Summary.  Policy decisions often require synthesis of evidence from multiple sources, and the source studies typically vary in rigour and in relevance to the target question. We present simple methods of allowing for differences in rigour (or lack of internal bias) and relevance (or lack of external bias) in evidence synthesis. The methods are developed in the context of reanalysing a UK National Institute for Clinical Excellence technology appraisal in antenatal care, which includes eight comparative studies. Many were historically controlled, only one was a randomized trial and doses, populations and outcomes varied between studies and differed from the target UK setting. Using elicited opinion, we construct prior distributions to represent the biases in each study and perform a bias-adjusted meta-analysis. Adjustment had the effect of shifting the combined estimate away from the null by approximately 10%, and the variance of the combined estimate was almost tripled. Our generic bias modelling approach allows decisions to be based on all available evidence, with less rigorous or less relevant studies downweighted by using computationally simple methods.     

Estimates of human immunodeficiency virus prevalence and proportion diagnosed based on Bayesian multiparameter synthesis of surveillance data

Summary.  Estimates of the number of prevalent human immunodeficiency virus infections are used in England and Wales to monitor development of the human immunodeficiency virus–acquired immune deficiency syndrome epidemic and for planning purposes. The population is split into risk groups, and estimates of risk group size and of risk group prevalence and diagnosis rates are combined to derive estimates of the number of undiagnosed infections and of the overall number of infected individuals. In traditional approaches, each risk group size, prevalence or diagnosis rate parameter must be informed by just one summary statistic. Yet a rich array of surveillance and other data is available, providing information on parameters and on functions of parameters, and raising the possibility of inconsistency between sources of evidence in some parts of the parameter space. We develop a Bayesian framework for synthesis of surveillance and other information, implemented through Markov chain Monte Carlo methods. The sources of data are found to be inconsistent under their accepted interpretation, but the inconsistencies can be resolved by introducing additional 'bias adjustment' parameters. The best-fitting model incorporates a hierarchical structure to spread information more evenly over the parameter space. We suggest that multiparameter evidence synthesis opens new avenues in epidemiology based on the coherent summary of available data, assessment of consistency and bias modelling.     

A model of toxoplasmosis incidence in the UK: evidence synthesis and consistency of evidence

Summary.  We present a Bayesian evidence synthesis model combining data on seroprevalence, seroconversion and tests of recent infection, to produce estimates of current incidence of toxoplasmosis in the UK. The motivation for the study was the need for an estimate of current average incidence in the UK, with a realistic assessment of its uncertainty, to inform a decision model for a national screening programme to prevent congenital toxoplasmosis. The model has a hierarchical structure over geographic region, a random-walk model for temporal effects and a fixed age effect, with one or more types of data informing the regional estimates of incidence. Inference is obtained by using Markov chain Monte Carlo simulations. A key issue in the synthesis of evidence from multiple sources is model selection and the consistency of different types of evidence. Alternative models of incidence are compared by using the deviance information criterion, and we find that temporal effects are region specific. We assess the consistency of the various forms of evidence by using cross-validation where practical, and posterior and mixed prediction otherwise, and we discuss how these measures can be used to assess different aspects of consistency in a complex evidence structure. We discuss the contribution of the various forms of evidence to estimated current average incidence.     

Bias Reduction of Likelihood Estimators in Semiparametric Frailty Models

Abstract. Frailty models with a non‐parametric baseline hazard are widely used for the analysis of survival data. However, their maximum likelihood estimators can be substantially biased in finite samples, because the number of nuisance parameters associated with the baseline hazard increases with the sample size. The penalized partial likelihood based on a first‐order Laplace approximation still has non‐negligible bias. However, the second‐order Laplace approximation to a modified marginal likelihood for a bias reduction is infeasible because of the presence of too many complicated terms. In this article, we find adequate modifications of these likelihood‐based methods by using the hierarchical likelihood.     

Missing time-dependent covariates in human immunodeficiency virus dynamic models

Summary. The study of human immunodeficiency virus dynamics is one of the most important areas in research into acquired immune deficiency syndrome in recent years. Non-linear mixed effects models have been proposed for modelling viral dynamic processes. A challenging problem in the modelling is to identify repeatedly measured (time-dependent), but possibly missing, immunologic or virologic markers (covariates) for viral dynamic parameters. For missing time-dependent covariates in non-linear mixed effects models, the commonly used complete-case, mean imputation and last value carried forward methods may give misleading results. We propose a three-step hierarchical multiple-imputation method, implemented by Gibbs sampling, which imputes the missing data at the individual level but can pool information across individuals. We compare various methods by Monte Carlo simulations and find that the multiple-imputation method proposed performs the best in terms of bias and mean-squared errors in the estimates of covariate coefficients. By applying the favoured multiple-imputation method to clinical data, we conclude that there is a negative correlation between the viral decay rate (a virological response parameter) and CD4 or CD8 cell counts during the treatment; this is counter-intuitive, but biologically interpretable on the basis of findings from other clinical studies. These results may have an important influence on decisions about treatment for acquired immune deficiency syndrome patients.     

Statistical correction for functional metagenomic profiling of a microbial community with short NGS reads

By sequence homology search, the list of all the functions found and the counts of reads being aligned to them present the functional profile of a metagenomic sample. However, a significant obstacle has been observed in this approach due to the short read length associated with many next-generation sequencing technologies. This includes artificial families, cross-annotations, length bias and conservation bias. The widely applied cut-off methods, such as BLAST E-value, are not able to solve the problems. Following the published successful procedures on the artificial families and the cross-annotation issue, we propose in this paper to use zero-truncated Poisson and Binomial (ZTP-Bin) hierarchical modelling to correct the length bias and the conservation bias. Goodness of fit of the modelling and cross-validation for the prediction using a bioinformatic simulated sample show the validity of this approach. Evaluated on an in vitro-simulated data set, the proposed modelling method outperforms other traditional methods. All three steps were then sequentially applied on real-life metagenomic samples to show that the proposed framework will lead to a more accurate functional profile of a short-read metagenomic sample.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

Order-free co-regionalized areal data models with application to multiple-disease mapping

With the ready availability of spatial databases and geographical information system software, statisticians are increasingly encountering multivariate modelling settings featuring associations of more than one type: spatial associations between data locations and associations between the variables within the locations. Although flexible modelling of multivariate point-referenced data has recently been addressed by using a linear model of co-regionalization, existing methods for multivariate areal data typically suffer from unnecessary restrictions on the covariance structure or undesirable dependence on the conditioning order of the variables. We propose a class of Bayesian hierarchical models for multivariate areal data that avoids these restrictions, permitting flexible and order-free modelling of correlations both between variables and across areal units. Our framework encompasses a rich class of multivariate conditionally autoregressive models that are computationally feasible via modern Markov chain Monte Carlo methods. We illustrate the strengths of our approach over existing models by using simulation studies and also offer a real data application involving annual lung, larynx and oesophageal cancer death-rates in Minnesota counties between 1990 and 2000.     

Models for potentially biased evidence in meta-analysis using empirically based priors

Summary.  We present models for the combined analysis of evidence from randomized controlled trials categorized as being at either low or high risk of bias due to a flaw in their conduct. We formulate a bias model that incorporates between-study and between-meta-analysis heterogeneity in bias, and uncertainty in overall mean bias. We obtain algebraic expressions for the posterior distribution of the bias-adjusted treatment effect, which provide limiting values for the information that can be obtained from studies at high risk of bias. The parameters of the bias model can be estimated from collections of previously published meta-analyses. We explore alternative models for such data, and alternative methods for introducing prior information on the bias parameters into a new meta-analysis. Results from an illustrative example show that the bias-adjusted treatment effect estimates are sensitive to the way in which the meta-epidemiological data are modelled, but that using point estimates for bias parameters provides an adequate approximation to using a full joint prior distribution. A sensitivity analysis shows that the gain in precision from including studies at high risk of bias is likely to be low, however numerous or large their size, and that little is gained by incorporating such studies, unless the information from studies at low risk of bias is limited. We discuss approaches that might increase the value of including studies at high risk of bias, and the acceptability of the methods in the evaluation of health care interventions.     

Sparse Markov Chains for Sequence Data

Finite memory sources and variable‐length Markov chains have recently gained popularity in data compression and mining, in particular, for applications in bioinformatics and language modelling. Here, we consider denser data compression and prediction with a family of sparse Bayesian predictive models for Markov chains in finite state spaces. Our approach lumps transition probabilities into classes composed of invariant probabilities, such that the resulting models need not have a hierarchical structure as in context tree‐based approaches. This can lead to a substantially higher rate of data compression, and such non‐hierarchical sparse models can be motivated for instance by data dependence structures existing in the bioinformatics context. We describe a Bayesian inference algorithm for learning sparse Markov models through clustering of transition probabilities. Experiments with DNA sequence and protein data show that our approach is competitive in both prediction and classification when compared with several alternative methods on the basis of variable memory length.     

A sign based loss approach to model selection in nonparametric regression

In parametric regression models the sign of a coefficient often plays an important role in its interpretation. One possible approach to model selection in these situations is to consider a loss function that formulates prediction of the sign of a coefficient as a decision problem. Taking a Bayesian approach, we extend this idea of a sign based loss for selection to more complex situations. In generalized additive models we consider prediction of the sign of the derivative of an additive term at a set of predictors. Being able to predict the sign of the derivative at some point (that is, whether a term is increasing or decreasing) is one approach to selection of terms in additive modelling when interpretation is the main goal. For models with interactions, prediction of the sign of a higher order derivative can be used similarly. There are many advantages to our sign-based strategy for selection: one can work in a full or encompassing model without the need to specify priors on a model space and without needing to specify priors on parameters in submodels. Also, avoiding a search over a large model space can simplify computation. We consider shrinkage prior specifications on smoothing parameters that allow for good predictive performance in models with large numbers of terms without the need for selection, and a frequentist calibration of the parameter in our sign-based loss function when it is desired to control a false selection rate for interpretation.     

Semi parametric estimation of employment duration models

Semi parametric methods provide estimates of finite parameter vectors without requiring that the complete data generation process be assumed in a finite-dimensional family. By avoiding bias from incorrect specification, such estimators gain robustness, although usually at the cost of decreased precision. The most familiar semi parametric method in econometrics is ordi¬nary least squares, which estimates the parameters of a linear regression model without requiring that the distribution of the disturbances be in a finite-parameter family. The recent literature in econometric theory has extended semi parametric methods to a variety of non-linear models, including models appropriate for analysis of censored duration data. Horowitz and Newman make perhaps the first empirical application of these methods, to data on employment duration. Their analysis provides insights into the practical problems of implementing these methods, and limited information on performance. Their data set, containing 226 male controls from the Denver income maintenance experiment in 1971-74, does not show any significant covariates (except race), even when a fully parametric model is assumed. Consequently, the authors are unable to reject the fully parametric model in a test against the alternative semi parametric estimators. This provides some negative, but tenuous, evidence that in practical applications the reduction in bias using semi parametric estimators is insufficient to offset loss in precision. Larger samples, and data sets with strongly significant covariates, will need to be interval, and if the observation period is long enough will eventually be more loyal on average for those starting employment spells near the end of the observation period.     

An Examination of the Conceptual Issues Involved in Developing Credit-Scoring Models

Multiple discriminant analysis (MDA) is frequently used to develop statistical credit-scoring models for loan evaluation purposes. Current legislative efforts to insure that credit is being granted in a nondiscriminatory manner have focused considerable attention on the reliability of such models. This article examines the theoretical requirements of the MDA model in the context of a realistic lending situation and illustrates the extent of bias when these theoretical assumptions are not fully met. The article concludes that failure to rigorously meet all the theoretical assumptions of the statistical model may not be as critical as insuring that credit managers fully understand the limitations of these types of decision tools. Furthermore, the evidence indicates that statistical models other than multiple discriminant analysis are possibly more relevant to the credit-granting decision.     

On Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

Modelling and simulation to improve decision‐making in clinical development

Modelling and simulation are buzz words in clinical drug development. But is clinical trial simulation (CTS) really a revolutionary technique? There is not much more to CTS than applying standard methods of modelling, statistics and decision theory. However, doing this in a systematic way can mean a significant improvement in pharmaceutical research. This paper describes in simple examples how modelling could be used in clinical development. Four steps are identified: gathering relevant information about a drug and the disease; building a mathematical model; predicting the results of potential future trials; and optimizing clinical trials and the entire clinical programme. We discuss these steps and give a number of examples of model components, demonstrating that relatively unsophisticated models may also prove useful. We stress that modelling and simulation are decision tools and point out the benefits of integrating them with decision analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluations of Bayesian and maximum likelihood methods in PK models with below‐quantification‐limit data

Pharmacokinetic (PK) data often contain concentration measurements below the quantification limit (BQL). While specific values cannot be assigned to these observations, nevertheless these observed BQL data are informative and generally known to be lower than the lower limit of quantification (LLQ). Setting BQLs as missing data violates the usual missing at random (MAR) assumption applied to the statistical methods, and therefore leads to biased or less precise parameter estimation. By definition, these data lie within the interval [0, LLQ], and can be considered as censored observations. Statistical methods that handle censored data, such as maximum likelihood and Bayesian methods, are thus useful in the modelling of such data sets. The main aim of this work was to investigate the impact of the amount of BQL observations on the bias and precision of parameter estimates in population PK models (non‐linear mixed effects models in general) under maximum likelihood method as implemented in SAS and NONMEM, and a Bayesian approach using Markov chain Monte Carlo (MCMC) as applied in WinBUGS. A second aim was to compare these different methods in dealing with BQL or censored data in a practical situation. The evaluation was illustrated by simulation based on a simple PK model, where a number of data sets were simulated from a one‐compartment first‐order elimination PK model. Several quantification limits were applied to each of the simulated data to generate data sets with certain amounts of BQL data. The average percentage of BQL ranged from 25% to 75%. Their influence on the bias and precision of all population PK model parameters such as clearance and volume distribution under each estimation approach was explored and compared. Copyright © 2009 John Wiley & Sons, Ltd.     

Likelihood analysis for a class of beta mixed models

Beta regression is a suitable choice for modelling continuous response variables taking values on the unit interval. Data structures such as hierarchical, repeated measures and longitudinal typically induce extra variability and/or dependence and can be accounted for by the inclusion of random effects. In this sense, Statistical inference typically requires numerical methods, possibly combined with sampling algorithms. A class of Beta mixed models is adopted for the analysis of two real problems with grouped data structures. We focus on likelihood inference and describe the implemented algorithms. The first is a study on the life quality index of industry workers with data collected according to an hierarchical sampling scheme. The second is a study assessing the impact of hydroelectric power plants upon measures of water quality indexes up, downstream and at the reservoirs of the dammed rivers, with a nested and longitudinal data structure. Results from different algorithms are reported for comparison including from data-cloning, an alternative to numerical approximations which also allows assessing identifiability. Confidence intervals based on profiled likelihoods are compared with those obtained by asymptotic quadratic approximations, showing relevant differences for parameters related to the random effects. In both cases, the scientific hypothesis of interest was investigated by comparing alternative models, leading to relevant interpretations of the results within each context.     

Conservative prior distributions for variance parameters in hierarchical models

Bayesian hierarchical models typically involve specifying prior distributions for one or more variance components. This is rather removed from the observed data, so specification based on expert knowledge can be difficult. While there are suggestions for “default” priors in the literature, often a conditionally conjugate inverse‐gamma specification is used, despite documented drawbacks of this choice. The authors suggest “conservative” prior distributions for variance components, which deliberately give more weight to smaller values. These are appropriate for investigators who are skeptical about the presence of variability in the second‐stage parameters (random effects) and want to particularly guard against inferring more structure than is really present. The suggested priors readily adapt to various hierarchical modelling settings, such as fitting smooth curves, modelling spatial variation and combining data from multiple sites.     

Semiparametric hierarchical selection models for bayesian meta analysis

Meta-analysis refers to quantitative methods for combining results from independent studies in order to draw overall conclusions. Hierarchical models including selection models are introduced and shown to be useful in such Bayesian meta-analysis. Semiparametric hierarchical models are proposed using the Dirichlet process prior. These rich class of models combine the information of independent studies, allowing investigation of variability both between and within studies, and weight function. Here we investigate sensitivity of results to unobserved studies by considering a hierarchical selection model with including unknown weight function and use Markov chain Monte Carlo methods to develop inference for the parameters of interest. Using Bayesian method, this model is used on a meta-analysis of twelve studies comparing the effectiveness of two different types of flouride, in preventing cavities. Clinical informative prior is assumed. Summaries and plots of model parameters are analyzed to address questions of interest.