Analyzing complex capture-recapture data in the presence of individual and temporal covariates and model uncertainty

SUMMARY: We consider the issue of analyzing complex ecological data in the presence of covariate information and model uncertainty. Several issues can arise when analyzing such data, not least the need to take into account where there are missing covariate values. This is most acutely observed in the presence of time-varying covariates. We consider mark-recapture-recovery data, where the corresponding recapture probabilities are less than unity, so that individuals are not always observed at each capture event. This often leads to a large amount of missing time-varying individual covariate information, because the covariate cannot usually be recorded if an individual is not observed. In addition, we address the problem of model selection over these covariates with missing data. We consider a Bayesian approach, where we are able to deal with large amounts of missing data, by essentially treating the missing values as auxiliary variables. This approach also allows a quantitative comparison of different models via posterior model probabilities, obtained via the reversible jump Markov chain Monte Carlo algorithm. To demonstrate this approach we analyze data relating to Soay sheep, which pose several statistical challenges in fully describing the intricacies of the system.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

When mechanism matters: Bayesian forecasting using models of ecological diffusion

Ecological diffusion is a theory that can be used to understand and forecast spatio‐temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white‐tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression‐based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.     

Bayesian semiparametric nonlinear mixed-effects joint models for data with skewness, missing responses, and measurement errors in covariates

Summary It is a common practice to analyze complex longitudinal data using semiparametric nonlinear mixed-effects (SNLME) models with a normal distribution. Normality assumption of model errors may unrealistically obscure important features of subject variations. To partially explain between- and within-subject variations, covariates are usually introduced in such models, but some covariates may often be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. Inferential procedures can be complicated dramatically when data with skewness, missing values, and measurement error are observed. In the literature, there has been considerable interest in accommodating either skewness, incompleteness or covariate measurement error in such models, but there has been relatively little study concerning all three features simultaneously. In this article, our objective is to address the simultaneous impact of skewness, missingness, and covariate measurement error by jointly modeling the response and covariate processes based on a flexible Bayesian SNLME model. The method is illustrated using a real AIDS data set to compare potential models with various scenarios and different distribution specifications.     

New Insights into Handling Missing Values in Environmental Epidemiological Studies

Missing data are unavoidable in environmental epidemiologic surveys. The aim of this study was to compare methods for handling large amounts of missing values: omission of missing values, single and multiple imputations (through linear regression or partial least squares regression), and a fully Bayesian approach. These methods were applied to the PARIS birth cohort, where indoor domestic pollutant measurements were performed in a random sample of babies'' dwellings. A simulation study was conducted to assess performances of different approaches with a high proportion of missing values (from 50% to 95%). Different simulation scenarios were carried out, controlling the true value of the association (odds ratio of 1.0, 1.2, and 1.4), and varying the health outcome prevalence. When a large amount of data is missing, omitting these missing data reduced statistical power and inflated standard errors, which affected the significance of the association. Single imputation underestimated the variability, and considerably increased risk of type I error. All approaches were conservative, except the Bayesian joint model. In the case of a common health outcome, the fully Bayesian approach is the most efficient approach (low root mean square error, reasonable type I error, and high statistical power). Nevertheless for a less prevalent event, the type I error is increased and the statistical power is reduced. The estimated posterior distribution of the OR is useful to refine the conclusion. Among the methods handling missing values, no approach is absolutely the best but when usual approaches (e.g. single imputation) are not sufficient, joint modelling approach of missing process and health association is more efficient when large amounts of data are missing.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Meta-analysis of studies with missing data

Summary .  Consider a meta-analysis of studies with varying proportions of patient-level missing data, and assume that each primary study has made certain missing data adjustments so that the reported estimates of treatment effect size and variance are valid. These estimates of treatment effects can be combined across studies by standard meta-analytic methods, employing a random-effects model to account for heterogeneity across studies. However, we note that a meta-analysis based on the standard random-effects model will lead to biased estimates when the attrition rates of primary studies depend on the size of the underlying study-level treatment effect. Perhaps ignorable within each study, these types of missing data are in fact not ignorable in a meta-analysis. We propose three methods to correct the bias resulting from such missing data in a meta-analysis: reweighting the DerSimonian–Laird estimate by the completion rate; incorporating the completion rate into a Bayesian random-effects model; and inference based on a Bayesian shared-parameter model that includes the completion rate. We illustrate these methods through a meta-analysis of 16 published randomized trials that examined combined pharmacotherapy and psychological treatment for depression.     

Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records

Large amounts of longitudinal health records are now available for dynamic monitoring of the underlying processes governing the observations. However, the health status progression across time is not typically observed directly: records are observed only when a subject interacts with the system, yielding irregular and often sparse observations. This suggests that the observed trajectories should be modeled via a latent continuous‐time process potentially as a function of time‐varying covariates. We develop a continuous‐time hidden Markov model to analyze longitudinal data accounting for irregular visits and different types of observations. By employing a specific missing data likelihood formulation, we can construct an efficient computational algorithm. We focus on Bayesian inference for the model: this is facilitated by an expectation‐maximization algorithm and Markov chain Monte Carlo methods. Simulation studies demonstrate that these approaches can be implemented efficiently for large data sets in a fully Bayesian setting. We apply this model to a real cohort where patients suffer from chronic obstructive pulmonary disease with the outcome being the number of drugs taken, using health care utilization indicators and patient characteristics as covariates.     

缺失数据和贝叶斯系统发育分析精确度之探索

The effect of missing data on phylogenetic methods is a potentially important issue in our attempts to reconstruct the Tree of Life. If missing data are truly problematic, then it may be unwise to include species in an analysis that lack data for some characters (incomplete taxa) or to include characters that lack data for some species. Given the difficulty of obtaining data from all characters for all taxa (e.g., fossils), missing data might seriously impede efforts to reconstruct a comprehensive phylogeny that includes all species. Fortunately, recent simulations and empirical analyses suggest that missing data cells are not themselves problematic, and that incomplete taxa can be accurately placed as long as the overall number of characters in the analysis is large. However, these studies have so far only been conducted on parsimony, likelihood, and neighbor-joining methods. Although Bayesian phylogenetic methods have become widely used in recent years, the effects of missing data on Bayesian analysis have not been adequately studied. Here, we conduct simulations to test whether Bayesian analyses can accurately place incomplete taxa despite extensive missing data. In agreement with previous studies of other methods, we find that Bayesian analyses can accurately reconstruct the position of highly incomplete taxa (i.e., 95% missing data), as long as the overall number of characters in the analysis is large. These results suggest that highly incomplete taxa can be safely included in many Bayesian phylogenetic analyses.     

Two Bayesian methods for estimating parameters of the von Bertalanffy growth equation

The von Bertalanffy growth equation (VBGE) is commonly used in ecology and fisheries management to model individual growth of an organism. Generally, a nonlinear regression is used with length-at-age data to recover key life history parameters: L _∞ (asymptotic size), k (the growth coefficient), and t ₀ (a time used to calculate size at age 0). However, age data are often unavailable for many species of interest, which makes the regression impossible. To confront this problem, we have developed a Bayesian model to find L _∞ using only length data. We use length-at-age data for female blue shark, Prionace glauca, to test our hypothesis. Preliminary comparisons of the model output and the results of a nonlinear regression using the VBGE show similar estimates of L _∞ . We also developed a full Bayesian model that fits the VBGE to the same data used in the classical regression and the length-based Bayesian model. Classical regression methods are highly sensitive to missing data points, and our analysis shows that fitting the VBGE in a Bayesian framework is more robust. We investigate the assumptions made with the traditional curve fitting methods, and argue that either the full Bayesian or the length-based Bayesian models are preferable to classical nonlinear regressions. These methods clarify and address assumptions␣made in classical regressions using von Bertalanffy growth and facilitate more detailed stock assessments of species for which data are sparse.     

Bayesian sensitivity analysis methods to evaluate bias due to misclassification and missing data using informative priors and external validation data

Background: Recent research suggests that the Bayesian paradigm may be useful for modeling biases in epidemiological studies, such as those due to misclassification and missing data. We used Bayesian methods to perform sensitivity analyses for assessing the robustness of study findings to the potential effect of these two important sources of bias. Methods: We used data from a study of the joint associations of radiotherapy and smoking with primary lung cancer among breast cancer survivors. We used Bayesian methods to provide an operational way to combine both validation data and expert opinion to account for misclassification of the two risk factors and missing data. For comparative purposes we considered a “full model” that allowed for both misclassification and missing data, along with alternative models that considered only misclassification or missing data, and the naïve model that ignored both sources of bias. Results: We identified noticeable differences between the four models with respect to the posterior distributions of the odds ratios that described the joint associations of radiotherapy and smoking with primary lung cancer. Despite those differences we found that the general conclusions regarding the pattern of associations were the same regardless of the model used. Overall our results indicate a nonsignificantly decreased lung cancer risk due to radiotherapy among nonsmokers, and a mildly increased risk among smokers. Conclusions: We described easy to implement Bayesian methods to perform sensitivity analyses for assessing the robustness of study findings to misclassification and missing data.     

Bayesian analysis of survival data with missing censoring indicators

In some large clinical studies, it may be impractical to perform the physical examination to every subject at his/her last monitoring time in order to diagnose the occurrence of the event of interest. This gives rise to survival data with missing censoring indicators where the probability of missing may depend on time of last monitoring and some covariates. We present a fully Bayesian semi‐parametric method for such survival data to estimate regression parameters of the proportional hazards model of Cox. Theoretical investigation and simulation studies show that our method performs better than competing methods. We apply the proposed method to analyze the survival data with missing censoring indicators from the Orofacial Pain: Prospective Evaluation and Risk Assessment study.     

An exploration of fixed and random effects selection for longitudinal binary outcomes in the presence of nonignorable dropout

We explore a Bayesian approach to selection of variables that represent fixed and random effects in modeling of longitudinal binary outcomes with missing data caused by dropouts. We show via analytic results for a simple example that nonignorable missing data lead to biased parameter estimates. This bias results in selection of wrong effects asymptotically, which we can confirm via simulations for more complex settings. By jointly modeling the longitudinal binary data with the dropout process that possibly leads to nonignorable missing data, we are able to correct the bias in estimation and selection. Mixture priors with a point mass at zero are used to facilitate variable selection. We illustrate the proposed approach using a clinical trial for acute ischemic stroke.     

Infinite hidden Markov models for multiple multivariate time series with missing data

Exposure to air pollution is associated with increased morbidity and mortality. Recent technological advancements permit the collection of time-resolved personal exposure data. Such data are often incomplete with missing observations and exposures below the limit of detection, which limit their use in health effects studies. In this paper, we develop an infinite hidden Markov model for multiple asynchronous multivariate time series with missing data. Our model is designed to include covariates that can inform transitions among hidden states. We implement beam sampling, a combination of slice sampling and dynamic programming, to sample the hidden states, and a Bayesian multiple imputation algorithm to impute missing data. In simulation studies, our model excels in estimating hidden states and state-specific means and imputing observations that are missing at random or below the limit of detection. We validate our imputation approach on data from the Fort Collins Commuter Study. We show that the estimated hidden states improve imputations for data that are missing at random compared to existing approaches. In a case study of the Fort Collins Commuter Study, we describe the inferential gains obtained from our model including improved imputation of missing data and the ability to identify shared patterns in activity and exposure among repeated sampling days for individuals and among distinct individuals.     

Assessing agreement between multiple raters with missing rating information, applied to breast cancer tumour grading

Background

We consider the problem of assessing inter-rater agreement when there are missing data and a large number of raters. Previous studies have shown only ‘moderate’ agreement between pathologists in grading breast cancer tumour specimens. We analyse a large but incomplete data-set consisting of 24177 grades, on a discrete 1–3 scale, provided by 732 pathologists for 52 samples.

Methodology/Principal Findings

We review existing methods for analysing inter-rater agreement for multiple raters and demonstrate two further methods. Firstly, we examine a simple non-chance-corrected agreement score based on the observed proportion of agreements with the consensus for each sample, which makes no allowance for missing data. Secondly, treating grades as lying on a continuous scale representing tumour severity, we use a Bayesian latent trait method to model cumulative probabilities of assigning grade values as functions of the severity and clarity of the tumour and of rater-specific parameters representing boundaries between grades 1–2 and 2–3. We simulate from the fitted model to estimate, for each rater, the probability of agreement with the majority. Both methods suggest that there are differences between raters in terms of rating behaviour, most often caused by consistent over- or under-estimation of the grade boundaries, and also considerable variability in the distribution of grades assigned to many individual samples. The Bayesian model addresses the tendency of the agreement score to be biased upwards for raters who, by chance, see a relatively ‘easy’ set of samples.

Conclusions/Significance

Latent trait models can be adapted to provide novel information about the nature of inter-rater agreement when the number of raters is large and there are missing data. In this large study there is substantial variability between pathologists and uncertainty in the identity of the ‘true’ grade of many of the breast cancer tumours, a fact often ignored in clinical studies.     

A Bayesian network approach incorporating imputation of missing data enables exploratory analysis of complex causal biological relationships

Bayesian networks can be used to identify possible causal relationships between variables based on their conditional dependencies and independencies, which can be particularly useful in complex biological scenarios with many measured variables. Here we propose two improvements to an existing method for Bayesian network analysis, designed to increase the power to detect potential causal relationships between variables (including potentially a mixture of both discrete and continuous variables). Our first improvement relates to the treatment of missing data. When there is missing data, the standard approach is to remove every individual with any missing data before performing analysis. This can be wasteful and undesirable when there are many individuals with missing data, perhaps with only one or a few variables missing. This motivates the use of imputation. We present a new imputation method that uses a version of nearest neighbour imputation, whereby missing data from one individual is replaced with data from another individual, their nearest neighbour. For each individual with missing data, the subsets of variables to be used to select the nearest neighbour are chosen by sampling without replacement the complete data and estimating a best fit Bayesian network. We show that this approach leads to marked improvements in the recall and precision of directed edges in the final network identified, and we illustrate the approach through application to data from a recent study investigating the causal relationship between methylation and gene expression in early inflammatory arthritis patients. We also describe a second improvement in the form of a pseudo-Bayesian approach for upweighting certain network edges, which can be useful when there is prior evidence concerning their directions.     

Bayesian estimation of movement and survival probabilities from capture-recapture data

The Arnason–Schwarz model is usually used for estimatingsurvival and movement probabilities of animal populations fromcapture-recapture data. The missing data structure of this capture-recapturemodel is exhibited and summarised via a directed graph representation.Taking advantage of this structure we implement a Gibbs samplingalgorithm from which Bayesian estimates and credible intervalsfor survival and movement probabilities are derived. Convergenceof the algorithm is proved using a duality principle. We illustrateour approach through a real example.     

Sufficient dimension reduction via bayesian mixture modeling

Dimension reduction is central to an analysis of data with many predictors. Sufficient dimension reduction aims to identify the smallest possible number of linear combinations of the predictors, called the sufficient predictors, that retain all of the information in the predictors about the response distribution. In this article, we propose a Bayesian solution for sufficient dimension reduction. We directly model the response density in terms of the sufficient predictors using a finite mixture model. This approach is computationally efficient and offers a unified framework to handle categorical predictors, missing predictors, and Bayesian variable selection. We illustrate the method using both a simulation study and an analysis of an HIV data set.     

Bayesian back-calculation and nowcasting for line list data during the COVID-19 pandemic

Surveillance is critical to mounting an appropriate and effective response to pandemics. However, aggregated case report data suffers from reporting delays and can lead to misleading inferences. Different from aggregated case report data, line list data is a table contains individual features such as dates of symptom onset and reporting for each reported case and a good source for modeling delays. Current methods for modeling reporting delays are not particularly appropriate for line list data, which typically has missing symptom onset dates that are non-ignorable for modeling reporting delays. In this paper, we develop a Bayesian approach that dynamically integrates imputation and estimation for line list data. Specifically, this Bayesian approach can accurately estimate the epidemic curve and instantaneous reproduction numbers, even with most symptom onset dates missing. The Bayesian approach is also robust to deviations from model assumptions, such as changes in the reporting delay distribution or incorrect specification of the maximum reporting delay. We apply the Bayesian approach to COVID-19 line list data in Massachusetts and find the reproduction number estimates correspond more closely to the control measures than the estimates based on the reported curve.     

Estimating haplotype frequencies and standard errors for multiple single nucleotide polymorphisms

Estimating haplotype frequencies becomes increasingly important in the mapping of complex disease genes, as millions of single nucleotide polymorphisms (SNPs) are being identified and genotyped. When genotypes at multiple SNP loci are gathered from unrelated individuals, haplotype frequencies can be accurately estimated using expectation-maximization (EM) algorithms (Excoffier and Slatkin, 1995; Hawley and Kidd, 1995; Long et al., 1995), with standard errors estimated using bootstraps. However, because the number of possible haplotypes increases exponentially with the number of SNPs, handling data with a large number of SNPs poses a computational challenge for the EM methods and for other haplotype inference methods. To solve this problem, Niu and colleagues, in their Bayesian haplotype inference paper (Niu et al., 2002), introduced a computational algorithm called progressive ligation (PL). But their Bayesian method has a limitation on the number of subjects (no more than 100 subjects in the current implementation of the method). In this paper, we propose a new method in which we use the same likelihood formulation as in Excoffier and Slatkin's EM algorithm and apply the estimating equation idea and the PL computational algorithm with some modifications. Our proposed method can handle data sets with large number of SNPs as well as large numbers of subjects. Simultaneously, our method estimates standard errors efficiently, using the sandwich-estimate from the estimating equation, rather than the bootstrap method. Additionally, our method admits missing data and produces valid estimates of parameters and their standard errors under the assumption that the missing genotypes are missing at random in the sense defined by Rubin (1976).