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.     

Modelling under-reporting in epidemics

Under-reporting of infected cases is crucial for many diseases because of the bias it can introduce when making inference for the model parameters. The objective of this paper is to study the effect of under-reporting in epidemics by considering the stochastic Markovian SIR epidemic in which various reporting processes are incorporated. In particular, we first investigate the effect on the estimation process of ignoring under-reporting when it is present in an epidemic outbreak. We show that such an approach leads to under-estimation of the infection rate and the reproduction number. Secondly, by allowing for the fact that under-reporting is occurring, we develop suitable models for estimation of the epidemic parameters and explore how well the reporting rate and other model parameters can be estimated. We consider the case of a constant reporting probability and also more realistic assumptions which involve the reporting probability depending on time or the source of infection for each infected individual. Due to the incomplete nature of the data and reporting process, the Bayesian approach provides a natural modelling framework and we perform inference using data augmentation and reversible jump Markov chain Monte Carlo techniques.     

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

When responding to infectious disease outbreaks, rapid and accurate estimation of the epidemic trajectory is critical. However, two common data collection problems affect the reliability of the epidemiological data in real time: missing information on the time of first symptoms, and retrospective revision of historical information, including right censoring. Here, we propose an approach to construct epidemic curves in near real time that addresses these two challenges by 1) imputation of dates of symptom onset for reported cases using a dynamically-estimated “backward” reporting delay conditional distribution, and 2) adjustment for right censoring using the NobBS software package to nowcast cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number (R_t) in real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We evaluate how these real-time estimates compare with more complete epidemiological data that became available later. We explore the impact of the different assumptions on the estimates, and compare our estimates with those obtained from commonly used surveillance approaches. Our framework can help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health systems in other locations.     

Bayesian estimation of two-part joint models for a longitudinal semicontinuous biomarker and a terminal event with INLA: Interests for cancer clinical trial evaluation

Two-part joint models for a longitudinal semicontinuous biomarker and a terminal event have been recently introduced based on frequentist estimation. The biomarker distribution is decomposed into a probability of positive value and the expected value among positive values. Shared random effects can represent the association structure between the biomarker and the terminal event. The computational burden increases compared to standard joint models with a single regression model for the biomarker. In this context, the frequentist estimation implemented in the R package frailtypack can be challenging for complex models (i.e., a large number of parameters and dimension of the random effects). As an alternative, we propose a Bayesian estimation of two-part joint models based on the Integrated Nested Laplace Approximation (INLA) algorithm to alleviate the computational burden and fit more complex models. Our simulation studies confirm that INLA provides accurate approximation of posterior estimates and to reduced computation time and variability of estimates compared to frailtypack in the situations considered. We contrast the Bayesian and frequentist approaches in the analysis of two randomized cancer clinical trials (GERCOR and PRIME studies), where INLA has a reduced variability for the association between the biomarker and the risk of event. Moreover, the Bayesian approach was able to characterize subgroups of patients associated with different responses to treatment in the PRIME study. Our study suggests that the Bayesian approach using the INLA algorithm enables to fit complex joint models that might be of interest in a wide range of clinical applications.     

Back to basics for Bayesian model building in genomic selection

Numerous Bayesian methods of phenotype prediction and genomic breeding value estimation based on multilocus association models have been proposed. Computationally the methods have been based either on Markov chain Monte Carlo or on faster maximum a posteriori estimation. The demand for more accurate and more efficient estimation has led to the rapid emergence of workable methods, unfortunately at the expense of well-defined principles for Bayesian model building. In this article we go back to the basics and build a Bayesian multilocus association model for quantitative and binary traits with carefully defined hierarchical parameterization of Student's t and Laplace priors. In this treatment we consider alternative model structures, using indicator variables and polygenic terms. We make the most of the conjugate analysis, enabled by the hierarchical formulation of the prior densities, by deriving the fully conditional posterior densities of the parameters and using the acquired known distributions in building fast generalized expectation-maximization estimation algorithms.     

Bayesian analysis of severe acute respiratory syndrome: the 2003 Hong Kong epidemic

This paper analyzes data arising from a Severe Acute Respiratory Syndrome (SARS) epidemic in Hong Kong in 2003 involving 1755 cases. A discrete time stochastic model that uses a back-projection approach is proposed. Markov Chain Monte Carlo (MCMC) methods are developed for estimation of model parameters. The algorithm is further extended to integrate numerically over unobserved variables of the model. Applying the method to SARS data from Hong Kong, a value of 3.88 with a posterior standard deviation of 0.09 was estimated for the basic reproduction number. An estimate of the transmission parameter at the beginning of the epidemic was also obtained as 0.149 with a posterior standard deviation of 0.003. A reduction in the transmission parameter during the course of the epidemic forced the effective reproduction number to cross the threshold value of one, seven days after control interventions were introduced. At the end of the epidemic, the effective reproduction number was as low as 0.001 suggesting that the epidemic was brought under control by the intervention measures introduced.     

Bayesian Modeling of Haplotype Effects in Multiparent Populations

A general Bayesian model, Diploffect, is described for estimating the effects of founder haplotypes at quantitative trait loci (QTL) detected in multiparental genetic populations; such populations include the Collaborative Cross (CC), Heterogeneous Socks (HS), and many others for which local genetic variation is well described by an underlying, usually probabilistically inferred, haplotype mosaic. Our aim is to provide a framework for coherent estimation of haplotype and diplotype (haplotype pair) effects that takes into account the following: uncertainty in haplotype composition for each individual; uncertainty arising from small sample sizes and infrequently observed haplotype combinations; possible effects of dominance (for noninbred subjects); genetic background; and that provides a means to incorporate data that may be incomplete or has a hierarchical structure. Using the results of a probabilistic haplotype reconstruction as prior information, we obtain posterior distributions at the QTL for both haplotype effects and haplotype composition. Two alternative computational approaches are supplied: a Markov chain Monte Carlo sampler and a procedure based on importance sampling of integrated nested Laplace approximations. Using simulations of QTL in the incipient CC (pre-CC) and Northport HS populations, we compare the accuracy of Diploffect, approximations to it, and more commonly used approaches based on Haley–Knott regression, describing trade-offs between these methods. We also estimate effects for three QTL previously identified in those populations, obtaining posterior intervals that describe how the phenotype might be affected by diplotype substitutions at the modeled locus.     

Bayesian outbreak detection algorithm for monitoring reported cases of campylobacteriosis in Germany

In infectious disease epidemiology, statistical methods are an indispensable component for the automated detection of outbreaks in routinely collected surveillance data. So far, methodology in this area has been largely of frequentist nature and has increasingly been taking inspiration from statistical process control. The present work is concerned with strengthening Bayesian thinking in this field. We extend the widely used approach of Farrington et al. and Heisterkamp et al. to a modern Bayesian framework within a time series decomposition context. This approach facilitates a direct calculation of the decision‐making threshold while taking all sources of uncertainty in both prediction and estimation into account. More importantly, with the methodology it is now also possible to integrate covariate processes, e.g. weather influence, into the outbreak detection. Model inference is performed using fast and efficient integrated nested Laplace approximations, enabling the use of this method in routine surveillance at public health institutions. Performance of the algorithm was investigated by comparing simulations with existing methods as well as by analysing the time series of notified campylobacteriosis cases in Germany for the years 2002–2011, which include absolute humidity as a covariate process. Altogether, a flexible and modern surveillance algorithm is presented with an implementation available through the R package ‘surveillance’.     

Estimating phylogenetic trees from pairwise likelihoods and posterior probabilities of substitution counts

The field of phylogenetic tree estimation has been dominated by three broad classes of methods: distance-based approaches, parsimony and likelihood-based methods (including maximum likelihood (ML) and Bayesian approaches). Here we introduce two new approaches to tree inference: pairwise likelihood estimation and a distance-based method that estimates the number of substitutions along the paths through the tree. Our results include the derivation of the formulae for the probability that two leaves will be identical at a site given a number of substitutions along the path connecting them. We also derive the posterior probability of the number of substitutions along a path between two sequences. The calculations for the posterior probabilities are exact for group-based, symmetric models of character evolution, but are only approximate for more general models.     

Bayesian variable selection for latent class models

In this article, we develop a latent class model with class probabilities that depend on subject-specific covariates. One of our major goals is to identify important predictors of latent classes. We consider methodology that allows estimation of latent classes while allowing for variable selection uncertainty. We propose a Bayesian variable selection approach and implement a stochastic search Gibbs sampler for posterior computation to obtain model-averaged estimates of quantities of interest such as marginal inclusion probabilities of predictors. Our methods are illustrated through simulation studies and application to data on weight gain during pregnancy, where it is of interest to identify important predictors of latent weight gain classes.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence

Summary A Bayesian method was developed for identifying genetic markers linked to quantitative trait loci (QTL) by analyzing data from daughter or granddaughter designs and single markers or marker pairs. Traditional methods may yield unrealistic results because linkage tests depend on number of markers and QTL gene effects associated with selected markers are overestimated. The Bayesian or posterior probability of linkage combines information from a daughter or granddaughter design with the prior probability of linkage between a marker locus and a QTL. If the posterior probability exceeds a certain quantity, linkage is declared. Upon linkage acceptance, Bayesian estimates of marker-QTL recombination rate and QTL gene effects and frequencies are obtained. The Bayesian estimates of QTL gene effects account for different amounts of information by shrinking information from data toward the mean or mode of a prior exponential distribution of gene effects. Computation of the Bayesian analysis is feasible. Exact results are given for biallelic QTL, and extensions to multiallelic QTL are suggested.     

Nonparametric Bayesian density estimation on manifolds with applications to planar shapes

Statistical analysis on landmark-based shape spaces has diverse applications in morphometrics, medical diagnostics, machine vision and other areas. These shape spaces are non-Euclidean quotient manifolds. To conduct nonparametric inferences, one may define notions of centre and spread on this manifold and work with their estimates. However, it is useful to consider full likelihood-based methods, which allow nonparametric estimation of the probability density. This article proposes a broad class of mixture models constructed using suitable kernels on a general compact metric space and then on the planar shape space in particular. Following a Bayesian approach with a nonparametric prior on the mixing distribution, conditions are obtained under which the Kullback-Leibler property holds, implying large support and weak posterior consistency. Gibbs sampling methods are developed for posterior computation, and the methods are applied to problems in density estimation and classification with shape-based predictors. Simulation studies show improved estimation performance relative to existing approaches.     

A closer look at the distribution of number needed to treat (NNT): a Bayesian approach

In this paper, I present a Bayesian approach to estimation of the number needed to treat (NNT). The use of NNT as a measure of clinical benefit is now becoming commonplace. Various methods of estimation have been proposed, but none of them seem to provide entirely good estimates. Very little has been done to understand the statistical properties of NNT. Here, I derive the posterior distribution of NNT and use simulations to investigate the general behaviour of the distribution. The posterior mode of the distribution is proposed as a point estimate and results are compared with the conventional method of estimation of NNT done by inversion.     

Bayesian inference for small-sample capture-recapture data

We consider data on the survival of a population of Cephalorhynchus hectori, Hector's dolphins, in a marine area of New Zealand. To estimate survival probabilities of animal populations, a multiple capture-recapture sampling scheme can be used. In this paper, we propose a practical methodology to derive approximations to posterior distributions based on Laplace methods. We show how to calculate Bayes estimates and credible intervals in this setting.     

Inferring speciation times under an episodic molecular clock

We extend our recently developed Markov chain Monte Carlo algorithm for Bayesian estimation of species divergence times to allow variable evolutionary rates among lineages. The method can use heterogeneous data from multiple gene loci and accommodate multiple fossil calibrations. Uncertainties in fossil calibrations are described using flexible statistical distributions. The prior for divergence times for nodes lacking fossil calibrations is specified by use of a birth-death process with species sampling. The prior for lineage-specific substitution rates is specified using either a model with autocorrelated rates among adjacent lineages (based on a geometric Brownian motion model of rate drift) or a model with independent rates among lineages specified by a log-normal probability distribution. We develop an infinite-sites theory, which predicts that when the amount of sequence data approaches infinity, the width of the posterior credibility interval and the posterior mean of divergence times form a perfect linear relationship, with the slope indicating uncertainties in time estimates that cannot be reduced by sequence data alone. Simulations are used to study the influence of among-lineage rate variation and the number of loci sampled on the uncertainty of divergence time estimates. The analysis suggests that posterior time estimates typically involve considerable uncertainties even with an infinite amount of sequence data, and that the reliability and precision of fossil calibrations are critically important to divergence time estimation. We apply our new algorithms to two empirical data sets and compare the results with those obtained in previous Bayesian and likelihood analyses. The results demonstrate the utility of our new algorithms.     

Early estimation of the reproduction number in the presence of imported cases: pandemic influenza H1N1-2009 in New Zealand

We analyse data from the early epidemic of H1N1-2009 in New Zealand, and estimate the reproduction number R. We employ a renewal process which accounts for imported cases, illustrate some technical pitfalls, and propose a novel estimation method to address these pitfalls. Explicitly accounting for the infection-age distribution of imported cases and for the delay in transmission dynamics due to international travel, R was estimated to be (95% confidence interval: 107,1.47). Hence we show that a previous study, which did not account for these factors, overestimated R. Our approach also permitted us to examine the infection-age at which secondary transmission occurs as a function of calendar time, demonstrating the downward bias during the beginning of the epidemic. These technical issues may compromise the usefulness of a well-known estimator of R--the inverse of the moment-generating function of the generation time given the intrinsic growth rate. Explicit modelling of the infection-age distribution among imported cases and the examination of the time dependency of the generation time play key roles in avoiding a biased estimate of R, especially when one only has data covering a short time interval during the early growth phase of the epidemic.     

Gender heterogeneity in self-reported hypertension

We investigate the gender gap in hypertension misreporting using the French Constances cohort. We show that false negative reporting of hypertension is more frequent among men than among women, even after conditioning on a series of individual characteristics. As a second step, we investigate the causes of the gender gap in hypertension misreporting. We show that women go to the doctor more often than men do and that they have better knowledge of their family medical history. Once these differences are taken into account, the gender gap in false negative reporting of hypertension is reversed. This suggests that information acquisition and healthcare utilisation are crucial ingredients in fighting undiagnosed male hypertension.