Bayesian influence measures for joint models for longitudinal and survival data

Summary This article develops a variety of influence measures for carrying out perturbation (or sensitivity) analysis to joint models of longitudinal and survival data (JMLS) in Bayesian analysis. A perturbation model is introduced to characterize individual and global perturbations to the three components of a Bayesian model, including the data points, the prior distribution, and the sampling distribution. Local influence measures are proposed to quantify the degree of these perturbations to the JMLS. The proposed methods allow the detection of outliers or influential observations and the assessment of the sensitivity of inferences to various unverifiable assumptions on the Bayesian analysis of JMLS. Simulation studies and a real data set are used to highlight the broad spectrum of applications for our Bayesian influence methods.     

Local Influence for Generalized Linear Models with Missing Covariates

Summary In the analysis of missing data, sensitivity analyses are commonly used to check the sensitivity of the parameters of interest with respect to the missing data mechanism and other distributional and modeling assumptions. In this article, we formally develop a general local influence method to carry out sensitivity analyses of minor perturbations to generalized linear models in the presence of missing covariate data. We examine two types of perturbation schemes (the single‐case and global perturbation schemes) for perturbing various assumptions in this setting. We show that the metric tensor of a perturbation manifold provides useful information for selecting an appropriate perturbation. We also develop several local influence measures to identify influential points and test model misspecification. Simulation studies are conducted to evaluate our methods, and real datasets are analyzed to illustrate the use of our local influence measures.     

Heterogeneity and concordance in locus‐specific differentiation and introgression between species of towhees

The maintenance or breakdown of reproductive isolation is an observable outcome of secondary contact between species. In cases where hybrids beyond the F1 are formed, the representation of each species' ancestry can vary dramatically among genomic regions. This genomic heterogeneity in ancestry and introgression can offer insight into evolutionary processes, particularly if introgression is compared in multiple hybrid zones. Similarly, considerable heterogeneity exists across the genome in the extent to which populations and species have diverged, reflecting the combined effects of different evolutionary processes on genetic variation. We studied hybridization across two hybrid zones of two phenotypically well‐differentiated bird species in Mexico (Pipilo maculatus and P. ocai), to investigate genomic heterogeneity in differentiation and introgression. Using genotyping‐by‐sequencing (GBS) and hierarchical Bayesian models, we genotyped 460 birds at over 41 000 single nucleotide polymorphism (SNP) loci. We identified loci exhibiting extreme introgression relative to the genome‐wide expectation using a Bayesian genomic cline model. We also estimated locus‐specific F_ST and identified loci with exceptionally high genetic divergence between the parental species. We found some concordance of locus‐specific introgression in the two independent hybrid zones (6–20% of extreme loci shared across zones), reflecting areas of the genome that experience similar gene flow when the species interact. Additionally, heterogeneity in introgression and divergence across the genome revealed another subset of loci under the influence of locally specific factors. These results are consistent with a history in which reproductive isolation has been influenced by a common set of loci in both hybrid zones, but where local environmental and stochastic factors also lead to genomic differentiation.     

RECOMMENDATIONS FOR USING MSBAYES TO INCORPORATE UNCERTAINTY IN SELECTING AN ABC MODEL PRIOR: A RESPONSE TO OAKS ET AL.

Prior specification is an essential component of parameter estimation and model comparison in Approximate Bayesian computation (ABC). Oaks et al. present a simulation‐based power analysis of msBayes and conclude that msBayes has low power to detect genuinely random divergence times across taxa, and suggest the cause is Lindley's paradox. Although the predictions are similar, we show that their findings are more fundamentally explained by insufficient prior sampling that arises with poorly chosen wide priors that critically undersample nonsimultaneous divergence histories of high likelihood. In a reanalysis of their data on Philippine Island vertebrates, we show how this problem can be circumvented by expanding upon a previously developed procedure that accommodates uncertainty in prior selection using Bayesian model averaging. When these procedures are used, msBayes supports recent divergences without support for synchronous divergence in the Oaks et al. data and we further present a simulation analysis that demonstrates that msBayes can have high power to detect asynchronous divergence under narrower priors for divergence time. Our findings highlight the need for exploration of plausible parameter space and prior sampling efficiency for ABC samplers in high dimensions. We discus potential improvements to msBayes and conclude that when used appropriately with model averaging, msBayes remains an effective and powerful tool.     

Molecular dating for phylogenies containing a mix of populations and species by using Bayesian and RelTime approaches

Simultaneous molecular dating of population and species divergences is essential in many biological investigations, including phylogeography, phylodynamics and species delimitation studies. In these investigations, multiple sequence alignments consist of both intra‐ and interspecies samples (mixed samples). As a result, the phylogenetic trees contain interspecies, interpopulation and within‐population divergences. Bayesian relaxed clock methods are often employed in these analyses, but they assume the same tree prior for both inter‐ and intraspecies branching processes and require specification of a clock model for branch rates (independent vs. autocorrelated rates models). We evaluated the impact of a single tree prior on Bayesian divergence time estimates by analysing computer‐simulated data sets. We also examined the effect of the assumption of independence of evolutionary rate variation among branches when the branch rates are autocorrelated. Bayesian approach with coalescent tree priors generally produced excellent molecular dates and highest posterior densities with high coverage probabilities. We also evaluated the performance of a non‐Bayesian method, RelTime, which does not require the specification of a tree prior or a clock model. RelTime's performance was similar to that of the Bayesian approach, suggesting that it is also suitable to analyse data sets containing both populations and species variation when its computational efficiency is needed.     

Multiple approaches to detect outliers in a genome scan for selection in ocellated lizards (Lacerta lepida) along an environmental gradient

Identification of loci with adaptive importance is a key step to understand the speciation process in natural populations, because those loci are responsible for phenotypic variation that affects fitness in different environments. We conducted an AFLP genome scan in populations of ocellated lizards (Lacerta lepida) to search for candidate loci influenced by selection along an environmental gradient in the Iberian Peninsula. This gradient is strongly influenced by climatic variables, and two subspecies can be recognized at the opposite extremes: L. lepida iberica in the northwest and L. lepida nevadensis in the southeast. Both subspecies show substantial morphological differences that may be involved in their local adaptation to the climatic extremes. To investigate how the use of a particular outlier detection method can influence the results, a frequentist method, DFDIST, and a Bayesian method, BayeScan, were used to search for outliers influenced by selection. Additionally, the spatial analysis method was used to test for associations of AFLP marker band frequencies with 54 climatic variables by logistic regression. Results obtained with each method highlight differences in their sensitivity. DFDIST and BayeScan detected a similar proportion of outliers (3–4%), but only a few loci were simultaneously detected by both methods. Several loci detected as outliers were also associated with temperature, insolation or precipitation according to spatial analysis method. These results are in accordance with reported data in the literature about morphological and life‐history variation of L. lepida subspecies along the environmental gradient.     

Influence functions and robust Bayes and empirical Bayes small area estimation

We introduce new robust small area estimation procedures basedon area-level models. We first find influence functions correspondingto each individual area-level observation by measuring the divergencebetween the posterior density functions of regression coefficientswith and without that observation. Next, based on these influencefunctions, properly standardized, we propose some new robustBayes and empirical Bayes small area estimators. The mean squarederrors and estimated mean squared errors of these estimatorsare also found. A small simulation study compares the performanceof the robust and the regular empirical Bayes estimators. Whenthe model variance is larger than the sample variance, the proposedrobust empirical Bayes estimators are superior.     

Local Influence in Comparative Calibration Models

In this paper, the local influence approach for detecting the effect of small perturbations of the model or data is applied in the context of comparative calibration models. Such models are typically used for comparing several measuring instruments and can be considered in a functional version as well as in a structural version as is the case with ordinary measurement error models. Different perturbation schemes are considered and some real data applications illustrate the usefulness of the approach.     

Local influence in comparative calibration models under elliptical t-distributions

In this paper we consider applications of local influence (Cook, 1986) to evaluate small perturbations in the model or data set in the context of structural comparative calibration (Bolfarine and Galea, 1995) assuming that the measurements obtained follow a multivariate elliptical distribution. Different perturbation schemes are investigated and an application is considered to a real data set, using the elliptical t-distribution.     

An MLP training algorithm taking into account known errors on inputs and outputs

A training algorithm is introduced that takes into account a priori known errors on both inputs and outputs in an MLP network. The new cost function introduced for this case is based on a linear approximation of the network function over the input distribution for a given input pattern. Update formulas, in the form of the gradient of the new cost function, is given for a MLP network, together with expressions for the Hessian matrix. This is later used to calculate error bars in a Bayesian framework. The error bars thus derived are discussed in relation to the more commonly used width of the target posterior predictive distribution. It will also be shown that the taking into account of known input uncertainties in the way suggested in this article will have a strong regularizing effect on the solution.     

Bayesian estimation of species divergence times under a molecular clock using multiple fossil calibrations with soft bounds

We implement a Bayesian Markov chain Monte Carlo algorithm for estimating species divergence times that uses heterogeneous data from multiple gene loci and accommodates multiple fossil calibration nodes. A birth-death process with species sampling is used to specify a prior for divergence times, which allows easy assessment of the effects of that prior on posterior time estimates. We propose a new approach for specifying calibration points on the phylogeny, which allows the use of arbitrary and flexible statistical distributions to describe uncertainties in fossil dates. In particular, we use soft bounds, so that the probability that the true divergence time is outside the bounds is small but nonzero. A strict molecular clock is assumed in the current implementation, although this assumption may be relaxed. We apply our new algorithm to two data sets concerning divergences of several primate species, to examine the effects of the substitution model and of the prior for divergence times on Bayesian time estimation. We also conduct computer simulation to examine the differences between soft and hard bounds. We demonstrate that divergence time estimation is intrinsically hampered by uncertainties in fossil calibrations, and the error in Bayesian time estimates will not go to zero with increased amounts of sequence data. Our analyses of both real and simulated data demonstrate potentially large differences between divergence time estimates obtained using soft versus hard bounds and a general superiority of soft bounds. Our main findings are as follows. (1) When the fossils are consistent with each other and with the molecular data, and the posterior time estimates are well within the prior bounds, soft and hard bounds produce similar results. (2) When the fossils are in conflict with each other or with the molecules, soft and hard bounds behave very differently; soft bounds allow sequence data to correct poor calibrations, while poor hard bounds are impossible to overcome by any amount of data. (3) Soft bounds eliminate the need for "safe" but unrealistically high upper bounds, which may bias posterior time estimates. (4) Soft bounds allow more reliable assessment of estimation errors, while hard bounds generate misleadingly high precisions when fossils and molecules are in conflict.     

Phylogenomic datasets provide both precision and accuracy in estimating the timescale of placental mammal phylogeny

The fossil record suggests a rapid radiation of placental mammals following the Cretaceous-Paleogene (K-Pg) mass extinction 65 million years ago (Ma); nevertheless, molecular time estimates, while highly variable, are generally much older. Early molecular studies suffer from inadequate dating methods, reliance on the molecular clock, and simplistic and over-confident interpretations of the fossil record. More recent studies have used Bayesian dating methods that circumvent those issues, but the use of limited data has led to large estimation uncertainties, precluding a decisive conclusion on the timing of mammalian diversifications. Here we use a powerful Bayesian method to analyse 36 nuclear genomes and 274 mitochondrial genomes (20.6 million base pairs), combined with robust but flexible fossil calibrations. Our posterior time estimates suggest that marsupials diverged from eutherians 168-178 Ma, and crown Marsupialia diverged 64-84 Ma. Placentalia diverged 88-90 Ma, and present-day placental orders (except Primates and Xenarthra) originated in a ～20 Myr window (45-65 Ma) after the K-Pg extinction. Therefore we reject a pre K-Pg model of placental ordinal diversification. We suggest other infamous instances of mismatch between molecular and palaeontological divergence time estimates will be resolved with this same approach.     

A phylogeny of anisopterous dragonflies (Insecta,Odonata) using mtRNA genes and mixed nucleotide/doublet models

The application of mixed nucleotide/doublet substitution models has recently received attention in RNA‐based phylogenetics. Within a Bayesian approach, it was shown that mixed models outperformed analyses relying on simple nucleotide models. We analysed an mt RNA data set of dragonflies representing all major lineages of Anisoptera plus outgroups, using a mixed model in a Bayesian and parsimony (MP) approach. We used a published mt 16S rRNA secondary consensus structure model and inferred consensus models for the mt 12S rRNA and tRNA valine. Secondary structure information was used to set data partitions for paired and unpaired sites on which doublet or nucleotide models were applied, respectively. Several different doublet models are currently available of which we chose the most appropriate one by a Bayes factor test. The MP reconstructions relied on recoded data for paired sites in order to account for character covariance and an application of the ratchet strategy to find most parsimonious trees. Bayesian and parsimony reconstructions are partly differently resolved, indicating sensitivity of the reconstructions to model specification. Our analyses depict a tree in which the damselfly family Lestidae is sister group to a monophyletic clade Epiophlebia + Anisoptera, contradicting recent morphological and molecular work. In Bayesian analyses, we found a deep split between Libelluloidea and a clade ‘Aeshnoidea’ within Anisoptera largely congruent with Tillyard’s early ideas of anisopteran evolution, which had been based on evidently plesiomorphic character states. However, parsimony analysis did not support a clade ‘Aeshnoidea’, but instead, placed Gomphidae as sister taxon to Libelluloidea. Monophyly of Libelluloidea is only modestly supported, and many inter‐family relationships within Libelluloidea do not receive substantial support in Bayesian and parsimony analyses. We checked whether high Bayesian node support was inflated owing to either: (i) wrong secondary consensus structures; (ii) under‐sampling of the MCMC process, thereby missing other local maxima; or (iii) unrealistic prior assumptions on topologies or branch lengths. We found that different consensus structure models exert strong influence on the reconstruction, which demonstrates the importance of taxon‐specific realistic secondary structure models in RNA phylogenetics.     

主流分子钟定年方法的原理、误差来源和使用建议

近年来, 分子钟定年方法(molecular dating methods)得以广泛运用, 为宏观进化研究尤其是生物多样性及其格局形成历史的相关研究提供了不可或缺且十分详尽的进化时间框架。贝叶斯方法(Bayesian methods)和马尔可夫链蒙特卡罗方法 (Markov chain Monte Carlo)可容纳多维度、多类型的数据和参数设置, 因此以BEAST、PAML-MCMCTree等软件为代表的贝叶斯节点标记法(Bayesian node-dating methods)逐渐成为分子钟定年方法中最为广泛使用的类型。贝叶斯框架的优势之一在于其可以利用复杂模型考虑各种不确定性因素, 但是该类方法中各类模型和参数的设置都可能引入误差, 从而影响进化分化时间估算的可靠性。本文介绍了贝叶斯分子钟定年方法的原理和主要类型, 并以贝叶斯节点标记法为例, 重点讨论了分子钟模型、化石标记的选择与放置、采样频率及化石标记点年龄先验分布等因素对节点定年的影响; 提供了贝叶斯时间树构建软件的使用建议、节点年龄的讨论原则和不同模型下时间树的比较方法, 针对常见的引起节点年龄潜在高估和低估风险的情况作了分析并给出了合理化建议。我们认为, 合理整合多种贝叶斯方法和模型得出的结果并从中择优, 能够提高定年结果的可靠性; 研究人员应对时间树构建结果与其参数设置的关系开展讨论, 从而为其他学者提供参考; 化石记录的更新与分子钟定年方法的改进应同步不断跟进。     

Slower Visuomotor Corrections with Unchanged Latency are Consistent with Optimal Adaptation to Increased Endogenous Noise in the Elderly

We analyzed age-related changes in motor response in a visuomotor compensatory tracking task. Subjects used a manipulandum to attempt to keep a displayed cursor at the center of a screen despite random perturbations to its location. Cross-correlation analysis of the perturbation and the subject response showed no age-related increase in latency until the onset of response to the perturbation, but substantial slowing of the response itself. Results are consistent with age-related deterioration in the ratio of signal to noise in visuomotor response. The task is such that it is tractable to use Bayesian and quadratic optimality assumptions to construct a model for behavior. This model assumes that behavior resembles an optimal controller subject to noise, and parametrizes response in terms of latency, willingness to expend effort, noise intensity, and noise bandwidth. The model is consistent with the data for all young (n = 12, age 20–30) and most elderly (n = 12, age 65–92) subjects. The model reproduces the latency result from the cross-correlation method. When presented with increased noise, the computational model reproduces the experimentally observed age-related slowing and the observed lack of increased latency. The model provides a precise way to quantitatively formulate the long-standing hypothesis that age-related slowing is an adaptation to increased noise.     

Identifying differentially expressed genes in meta-analysis via Bayesian model-based clustering

A Bayesian model-based clustering approach is proposed for identifying differentially expressed genes in meta-analysis. A Bayesian hierarchical model is used as a scientific tool for combining information from different studies, and a mixture prior is used to separate differentially expressed genes from non-differentially expressed genes. Posterior estimation of the parameters and missing observations are done by using a simple Markov chain Monte Carlo method. From the estimated mixture model, useful measure of significance of a test such as the Bayesian false discovery rate (FDR), the local FDR (Efron et al., 2001), and the integration-driven discovery rate (IDR; Choi et al., 2003) can be easily computed. The model-based approach is also compared with commonly used permutation methods, and it is shown that the model-based approach is superior to the permutation methods when there are excessive under-expressed genes compared to over-expressed genes or vice versa. The proposed method is applied to four publicly available prostate cancer gene expression data sets and simulated data sets.     

Historical connectivity,contemporary isolation and local adaptation in a widespread but discontinuously distributed species endemic to Taiwan,Rhododendron oldhamii (Ericaceae)

Elucidation of the evolutionary processes that constrain or facilitate adaptive divergence is a central goal in evolutionary biology, especially in non-model organisms. We tested whether changes in dynamics of gene flow (historical vs contemporary) caused population isolation and examined local adaptation in response to environmental selective forces in fragmented Rhododendron oldhamii populations. Variation in 26 expressed sequence tag-simple sequence repeat loci from 18 populations in Taiwan was investigated by examining patterns of genetic diversity, inbreeding, geographic structure, recent bottlenecks, and historical and contemporary gene flow. Selection associated with environmental variables was also examined. Bayesian clustering analysis revealed four regional population groups of north, central, south and southeast with significant genetic differentiation. Historical bottlenecks beginning 9168–13,092 years ago and ending 1584–3504 years ago were revealed by estimates using approximate Bayesian computation for all four regional samples analyzed. Recent migration within and across geographic regions was limited. However, major dispersal sources were found within geographic regions. Altitudinal clines of allelic frequencies of environmentally associated positively selected outliers were found, indicating adaptive divergence. Our results point to a transition from historical population connectivity toward contemporary population isolation and divergence on a regional scale. Spatial and temporal dispersal differences may have resulted in regional population divergence and local adaptation associated with environmental variables, which may have played roles as selective forces at a regional scale.     

A Bayesian basket trial design that borrows information across strata based on the similarity between the posterior distributions of the response probability

Basket trials simultaneously evaluate the effect of one or more drugs on a defined biomarker, genetic alteration, or molecular target in a variety of disease subtypes, often called strata. A conventional approach for analyzing such trials is an independent analysis of each of the strata. This analysis is inefficient as it lacks the power to detect the effect of drugs in each stratum. To address these issues, various designs for basket trials have been proposed, centering on designs using Bayesian hierarchical models. In this article, we propose a novel Bayesian basket trial design that incorporates predictive sample size determination, early termination for inefficacy and efficacy, and the borrowing of information across strata. The borrowing of information is based on the similarity between the posterior distributions of the response probability. In general, Bayesian hierarchical models have many distributional assumptions along with multiple parameters. By contrast, our method has prior distributions for response probability and two parameters for similarity of distributions. The proposed design is easier to implement and less computationally demanding than other Bayesian basket designs. Through a simulation with various scenarios, our proposed design is compared with other designs including one that does not borrow information and one that uses a Bayesian hierarchical model.     

Bayesian case influence diagnostics for survival models

We propose Bayesian case influence diagnostics for complex survival models. We develop case deletion influence diagnostics for both the joint and marginal posterior distributions based on the Kullback-Leibler divergence (K-L divergence). We present a simplified expression for computing the K-L divergence between the posterior with the full data and the posterior based on single case deletion, as well as investigate its relationships to the conditional predictive ordinate. All the computations for the proposed diagnostic measures can be easily done using Markov chain Monte Carlo samples from the full data posterior distribution. We consider the Cox model with a gamma process prior on the cumulative baseline hazard. We also present a theoretical relationship between our case-deletion diagnostics and diagnostics based on Cox's partial likelihood. A simulated data example and two real data examples are given to demonstrate the methodology.     

Population structure and spatial influence of agricultural variables on Hessian fly populations in the southeastern United States

Population structure dictates the evolution of each population, and thus, the species as a whole. Incorporating spatial variables with population genetic statistics allows for greater discovery beyond traditional population genetics alone and can inform management decisions. The understanding of population structure in Hessian fly, Mayetiola destructor (Say), a pest of wheat, has been limited in the past. We scored 14 microsatellite loci from 12 collections of Hessian fly in the southeastern United States. Through Bayesian clustering analysis, we found two major populations of Hessian fly covering the entire southeastern United States. We evaluated correlations between agriculturally significant spatial variables and population genetic differentiation to test if genetic structure has an ecological component in a wheat agro-ecosystem. Our results suggest the total amount of alternative host plants in the county may be driving some genetic differentiation. Although planting date may also be influential, geographic distance, mean annual temperature, and harvested wheat for grain do not seem to be contributing factors. The ecological or spatial component to population structure, however, may be minimal compared to factors such as genetic drift.