Bayesian inference in ecology

Bayesian inference is an important statistical tool that is increasingly being used by ecologists. In a Bayesian analysis, information available before a study is conducted is summarized in a quantitative model or hypothesis: the prior probability distribution. Bayes’ Theorem uses the prior probability distribution and the likelihood of the data to generate a posterior probability distribution. Posterior probability distributions are an epistemological alternative to P‐values and provide a direct measure of the degree of belief that can be placed on models, hypotheses, or parameter estimates. Moreover, Bayesian information‐theoretic methods provide robust measures of the probability of alternative models, and multiple models can be averaged into a single model that reflects uncertainty in model construction and selection. These methods are demonstrated through a simple worked example. Ecologists are using Bayesian inference in studies that range from predicting single‐species population dynamics to understanding ecosystem processes. Not all ecologists, however, appreciate the philosophical underpinnings of Bayesian inference. In particular, Bayesians and frequentists differ in their definition of probability and in their treatment of model parameters as random variables or estimates of true values. These assumptions must be addressed explicitly before deciding whether or not to use Bayesian methods to analyse ecological data.     

Frequentist properties of Bayesian posterior probabilities of phylogenetic trees under simple and complex substitution models

What does the posterior probability of a phylogenetic tree mean?This simulation study shows that Bayesian posterior probabilities have the meaning that is typically ascribed to them; the posterior probability of a tree is the probability that the tree is correct, assuming that the model is correct. At the same time, the Bayesian method can be sensitive to model misspecification, and the sensitivity of the Bayesian method appears to be greater than the sensitivity of the nonparametric bootstrap method (using maximum likelihood to estimate trees). Although the estimates of phylogeny obtained by use of the method of maximum likelihood or the Bayesian method are likely to be similar, the assessment of the uncertainty of inferred trees via either bootstrapping (for maximum likelihood estimates) or posterior probabilities (for Bayesian estimates) is not likely to be the same. We suggest that the Bayesian method be implemented with the most complex models of those currently available, as this should reduce the chance that the method will concentrate too much probability on too few trees.     

An empirical examination of the standard errors of maximum likelihood phylogenetic parameters under the molecular clock via bootstrapping

The molecular clock theory has greatly enlightened our understanding of macroevolutionary events. Maximum likelihood (ML) estimation of divergence times involves the adoption of fixed calibration points, and the confidence intervals associated with the estimates are generally very narrow. The credibility intervals are inferred assuming that the estimates are normally distributed, which may not be the case. Moreover, calculation of standard errors is usually carried out by the curvature method and is complicated by the difficulty in approximating second derivatives of the likelihood function. In this study, a standard primate phylogeny was used to examine the standard errors of ML estimates via the bootstrap method. Confidence intervals were also assessed from the posterior distribution of divergence times inferred via Bayesian Markov Chain Monte Carlo. For the primate topology under evaluation, no significant differences were found between the bootstrap and the curvature methods. Also, Bayesian confidence intervals were always wider than those obtained by ML.     

Choice of number of doses for maximum likelihood estimation of the ED50 for quantal dose-response data

We address the question of how to choose the number of doses when estimating the median effective dose (ED50) of a symmetric dose-response curve by the maximum likelihood method. One criterion for this choice here is the asymptotic mean squared error (determined by the asymptotic variance) of the estimated ED50 of a dose-response relationship with qualitative responses. The choice is based on an analysis of the inverse of the information matrix. We find that in many cases, assuming various symmetric dose-response curves and various design densities, choice of as many doses as possible, i.e., the allocation of one subject per dose, is optimal. The theoretical and numerical results are supported by simulations and by an example concerning choice of design in an adolescence study.     

Effect of GABA agonists on the neurotoxicity and anticonvulsant activity of benzodiazepines

Progabide (50 mg/kg, i.p.), a GABA receptor agonist, significantly decreases the median minimal neurotoxic dose (TD50) of clobazam, chlordiazepoxide, and diazepam; the receptor binding of these substances is highly enhanced by muscimol. Progabide has no significant effect on the TD50 of clonazepam and triazolam; the receptor bindings of these substances is either only slightly enhanced or not altered by muscimol. Progabide also significantly decreases the median antimaximal electroshock dose (MES ED50) of all the benzodiazepines tested. However, progabide has no effect on the median antipentylenetetrazol dose (PTZ ED50) of the benzodiazepines. Likewise, THIP (2.5 mg/kg, i.p.) significantly decreases the TD50 of chlordiazepoxide but not that of triazolam. THIP significantly decreases the MES ED50 of chlordiazepoxide and triazolam but has no effect on the PTZ ED50 of these two substances. The above data suggest that benzodiazepine receptors linked to GABA receptors contribute to the minimal neurotoxicity and anti-MES activity but not to the anti-PTZ activity of benzodiazepines.     

Critical Assessment of the C-Optimality Design Criteria for Estimating the Median Effective Dose in Quantal Dose-Response Curves

In this paper the properties of C-optimal designs constructed for estimating the median effective dose within the framework of two-parametric linear logistic models are critically assessed. It is well known that this design criterion which is based on the first-order variance approximation of the exact variance of the maximum likelihood estimate of the ED50 leads to a one-point design where the maximum likelihood theory breaks down. The single dose used in this design is identical with the true but unknown value of the ED50. It will be shown, that at this one-point design the asymptotic variance does not exist. A two-point design in the neighbourhood of the one-point design which is symmetrical about the ED50 and associated with a small dose-distance would be nearly optimal, but extremely nonrobust if the best guess of the ED50 differs from the true value. In this situation the asymptotic variance of the two-point design converging towards the one-point design tends to infinity. Moreover, taking in consideration, that for searching an optimal design the exact variance is of primary interest and the asymptotic variance serves only as an approximation of the exact variance, we calculate the exact variance of the estimator from balanced, symmetric 2-point designs in the neighbourhood of the limiting 1-point design for various dose distances and initial best guesses of the ED50. We compare the true variance of the estimate of the ED50 with the asymptotic variance and show that the approximations generally do not represent suitable substitutes for the exact variance even in case of unrealistically large sample sizes. Kalish (1990) proposed a criterion based on the second-order asymptotic variance of the maximum likelihood estimate of the ED50 to overcome the degenerated 1-point design as the solution of the optimization procedure. In fact, we are able to show that this variance approximation does not perform substantially better than the first–order variance. From these considerations it follows, that the C-optimality criterion is not useful in this estimation problem. Other criteria like the F-optimality should be used.     

On the Bayesian analysis of ring-recovery data

Vounatsou and Smith (1995, Biometrics 51, 687-708) describe the modern Bayesian analysis of ring-recovery data. Here we discuss and extend their work. We draw different conclusions from two major data analyses. We emphasize the extreme sensitivity of certain parameter estimates to the choice of prior distribution and conclude that naive use of Bayesian methods in this area can be misleading. Additionally, we explain the discrepancy between the Bayesian and classical analyses when the likelihood surface has a flat ridge. In this case, when there is no unique maximum likelihood estimate, the Bayesian estimators are remarkably precise.     

Tail paradox, partial identifiability, and influential priors in Bayesian branch length inference

Recent studies have observed that Bayesian analyses of sequence data sets using the program MrBayes sometimes generate extremely large branch lengths, with posterior credibility intervals for the tree length (sum of branch lengths) excluding the maximum likelihood estimates. Suggested explanations for this phenomenon include the existence of multiple local peaks in the posterior, lack of convergence of the chain in the tail of the posterior, mixing problems, and misspecified priors on branch lengths. Here, we analyze the behavior of Bayesian Markov chain Monte Carlo algorithms when the chain is in the tail of the posterior distribution and note that all these phenomena can occur. In Bayesian phylogenetics, the likelihood function approaches a constant instead of zero when the branch lengths increase to infinity. The flat tail of the likelihood can cause poor mixing and undue influence of the prior. We suggest that the main cause of the extreme branch length estimates produced in many Bayesian analyses is the poor choice of a default prior on branch lengths in current Bayesian phylogenetic programs. The default prior in MrBayes assigns independent and identical distributions to branch lengths, imposing strong (and unreasonable) assumptions about the tree length. The problem is exacerbated by the strong correlation between the branch lengths and parameters in models of variable rates among sites or among site partitions. To resolve the problem, we suggest two multivariate priors for the branch lengths (called compound Dirichlet priors) that are fairly diffuse and demonstrate their utility in the special case of branch length estimation on a star phylogeny. Our analysis highlights the need for careful thought in the specification of high-dimensional priors in Bayesian analyses.     

An examination of the monophyly of morning glory taxa using Bayesian phylogenetic inference

The objective of this study was to obtain a quantitative assessment of the monophyly of morning glory taxa, specifically the genus Ipomoea and the tribe Argyreieae. Previous systematic studies of morning glories intimated the paraphyly of Ipomoea by suggesting that the genera within the tribe Argyreieae are derived from within Ipomoea; however, no quantitative estimates of statistical support were developed to address these questions. We applied a Bayesian analysis to provide quantitative estimates of monophyly in an investigation of morning glory relationships using DNA sequence data. We also explored various approaches for examining convergence of the Markov chain Monte Carlo (MCMC) simulation of the Bayesian analysis by running 18 separate analyses varying in length. We found convergence of the important components of the phylogenetic model (the tree with the maximum posterior probability, branch lengths, the parameter values from the DNA substitution model, and the posterior probabilities for clade support) for these data after one million generations of the MCMC simulations. In the process, we identified a run where the parameter values obtained were often outside the range of values obtained from the other runs, suggesting an aberrant result. In addition, we compared the Bayesian method of phylogenetic analysis to maximum likelihood and maximum parsimony. The results from the Bayesian analysis and the maximum likelihood analysis were similar for topology, branch lengths, and parameters of the DNA substitution model. Topologies also were similar in the comparison between the Bayesian analysis and maximum parsimony, although the posterior probabilities and the bootstrap proportions exhibited some striking differences. In a Bayesian analysis of three data sets (ITS sequences, waxy sequences, and ITS + waxy sequences) no supoort for the monophyly of the genus Ipomoea, or for the tribe Argyreieae, was observed, with the estimate of the probability of the monophyly of these taxa being less than 3.4 x 10(-7).     

Bayesian Estimation of Small Effects in Exercise and Sports Science

The aim of this paper is to provide a Bayesian formulation of the so-called magnitude-based inference approach to quantifying and interpreting effects, and in a case study example provide accurate probabilistic statements that correspond to the intended magnitude-based inferences. The model is described in the context of a published small-scale athlete study which employed a magnitude-based inference approach to compare the effect of two altitude training regimens (live high-train low (LHTL), and intermittent hypoxic exposure (IHE)) on running performance and blood measurements of elite triathletes. The posterior distributions, and corresponding point and interval estimates, for the parameters and associated effects and comparisons of interest, were estimated using Markov chain Monte Carlo simulations. The Bayesian analysis was shown to provide more direct probabilistic comparisons of treatments and able to identify small effects of interest. The approach avoided asymptotic assumptions and overcame issues such as multiple testing. Bayesian analysis of unscaled effects showed a probability of 0.96 that LHTL yields a substantially greater increase in hemoglobin mass than IHE, a 0.93 probability of a substantially greater improvement in running economy and a greater than 0.96 probability that both IHE and LHTL yield a substantially greater improvement in maximum blood lactate concentration compared to a Placebo. The conclusions are consistent with those obtained using a ‘magnitude-based inference’ approach that has been promoted in the field. The paper demonstrates that a fully Bayesian analysis is a simple and effective way of analysing small effects, providing a rich set of results that are straightforward to interpret in terms of probabilistic statements.     

Comparison of Bayesian and maximum-likelihood inference of population genetic parameters

Comparison of the performance and accuracy of different inference methods, such as maximum likelihood (ML) and Bayesian inference, is difficult because the inference methods are implemented in different programs, often written by different authors. Both methods were implemented in the program MIGRATE, that estimates population genetic parameters, such as population sizes and migration rates, using coalescence theory. Both inference methods use the same Markov chain Monte Carlo algorithm and differ from each other in only two aspects: parameter proposal distribution and maximization of the likelihood function. Using simulated datasets, the Bayesian method generally fares better than the ML approach in accuracy and coverage, although for some values the two approaches are equal in performance. MOTIVATION: The Markov chain Monte Carlo-based ML framework can fail on sparse data and can deliver non-conservative support intervals. A Bayesian framework with appropriate prior distribution is able to remedy some of these problems. RESULTS: The program MIGRATE was extended to allow not only for ML(-) maximum likelihood estimation of population genetics parameters but also for using a Bayesian framework. Comparisons between the Bayesian approach and the ML approach are facilitated because both modes estimate the same parameters under the same population model and assumptions.     

Direct analysis of quantal radiation response data

A direct analysis is proposed for quantal (all-or-nothing) responses to fractionated radiation and endpoint-dilution assays of cell survival. As opposed to two-step methods such as the reciprocal-dose technique, in which ED50 values are first estimated for different fractionation schemes and then fit (as reciprocals) against dose per fraction, all raw data are included in a single maximum-likelihood treatment. The method accommodates variations such as short-interval fractionation regimens designed to determine tissue repair kinetics, tissue response to continuous exposures, and data obtained using endpoint-dilution assays of cell survival after fractionated doses. Monte-Carlo techniques were used to compare the direct and reciprocal-dose methods for analysis of small-scale and large-scale studies of response to fractionated doses. Both methods tended toward biased estimates in the analysis of the small-scale (3 fraction numbers) studies. The alpha/beta ratios showed less scatter when estimated by the direct method. Most important, the 95 per cent confidence intervals determined by the direct method were more appropriate than those determined by reciprocal-dose analysis, for which 18 per cent (small-scale study) or 8 per cent (large-scale study) of the confidence intervals did not include the 'true' value of alpha/beta.     

Symmetric Intervals and Confidence Intervals

We show that the symmetric “confidence intervals” of WESTLAKE (1972, 1976), widely used and referred to in bioequivalence studies, are not confidence intervals in any accepted sense. Nevertheless, meaningful symmetric intervals can be constructed in the context of Bayesian or fiducial inference.     

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.     

Empirical Bayes interval estimates that are conditionally equal to unadjusted confidence intervals or to default prior credibility intervals

Problems involving thousands of null hypotheses have been addressed by estimating the local false discovery rate (LFDR). A previous LFDR approach to reporting point and interval estimates of an effect-size parameter uses an estimate of the prior distribution of the parameter conditional on the alternative hypothesis. That estimated prior is often unreliable, and yet strongly influences the posterior intervals and point estimates, causing the posterior intervals to differ from fixed-parameter confidence intervals, even for arbitrarily small estimates of the LFDR. That influence of the estimated prior manifests the failure of the conditional posterior intervals, given the truth of the alternative hypothesis, to match the confidence intervals. Those problems are overcome by changing the posterior distribution conditional on the alternative hypothesis from a Bayesian posterior to a confidence posterior. Unlike the Bayesian posterior, the confidence posterior equates the posterior probability that the parameter lies in a fixed interval with the coverage rate of the coinciding confidence interval. The resulting confidence-Bayes hybrid posterior supplies interval and point estimates that shrink toward the null hypothesis value. The confidence intervals tend to be much shorter than their fixed-parameter counterparts, as illustrated with gene expression data. Simulations nonetheless confirm that the shrunken confidence intervals cover the parameter more frequently than stated. Generally applicable sufficient conditions for correct coverage are given. In addition to having those frequentist properties, the hybrid posterior can also be motivated from an objective Bayesian perspective by requiring coherence with some default prior conditional on the alternative hypothesis. That requirement generates a new class of approximate posteriors that supplement Bayes factors modified for improper priors and that dampen the influence of proper priors on the credibility intervals. While that class of posteriors intersects the class of confidence-Bayes posteriors, neither class is a subset of the other. In short, two first principles generate both classes of posteriors: a coherence principle and a relevance principle. The coherence principle requires that all effect size estimates comply with the same probability distribution. The relevance principle means effect size estimates given the truth of an alternative hypothesis cannot depend on whether that truth was known prior to observing the data or whether it was learned from the data.     

Model parameterization, prior distributions, and the general time-reversible model in Bayesian phylogenetics

Bayesian phylogenetic methods require the selection of prior probability distributions for all parameters of the model of evolution. These distributions allow one to incorporate prior information into a Bayesian analysis, but even in the absence of meaningful prior information, a prior distribution must be chosen. In such situations, researchers typically seek to choose a prior that will have little effect on the posterior estimates produced by an analysis, allowing the data to dominate. Sometimes a prior that is uniform (assigning equal prior probability density to all points within some range) is chosen for this purpose. In reality, the appropriate prior depends on the parameterization chosen for the model of evolution, a choice that is largely arbitrary. There is an extensive Bayesian literature on appropriate prior choice, and it has long been appreciated that there are parameterizations for which uniform priors can have a strong influence on posterior estimates. We here discuss the relationship between model parameterization and prior specification, using the general time-reversible model of nucleotide evolution as an example. We present Bayesian analyses of 10 simulated data sets obtained using a variety of prior distributions and parameterizations of the general time-reversible model. Uniform priors can produce biased parameter estimates under realistic conditions, and a variety of alternative priors avoid this bias.     

Statistics of the log-det estimator

The log-det estimator is a measure of divergence (evolutionary distance) between sequences of biological characters, DNA or amino acids, for example, and has been shown to be robust to biases in composition that can cause problems for other estimators. We provide a statistical framework to construct high-accuracy confidence intervals for log-det estimates and compare the efficiency of the estimator to that of maximum likelihood using time-reversible Markov models. The log-det estimator is found to have good statistical properties under such general models.     

Analysis of multiple tumor data from a rodent carcinogenicity experiment

Rodent tumorigenicity experiments are conducted to determine the safety of substances for human exposure. The carcinogenicity of a substance is generally determined by statistical tests that compare the effects of treatment on the rate of tumor development at several body sites. The statistical analysis of such studies often includes hypothesis testing of the dose effect at each of the sites. However, the multiplicity of the significance tests may cause an excess overall false positive rate. In consideration of this problem, recent interest has focused on developing methods to test simultaneously for the treatment effect at multiple sites. In this paper, we propose a test that is based on the count of tumor-bearing sites. The test is appropriate regardless of tumor lethality or of treatment-related differences in the underlying mortality. Simulations are given which compare the performance of the proposed test to several other tests including a Bonferroni adjustment of site-specific tests, and the test is illustrated using the data from the large ED01 experiment.     

Bayesian and influence function-based empirical likelihoods for inference of sensitivity to the early diseased stage in diagnostic tests

In practice, a disease process might involve three ordinal diagnostic stages: the normal healthy stage, the early stage of the disease, and the stage of full development of the disease. Early detection is critical for some diseases since it often means an optimal time window for therapeutic treatments of the diseases. In this study, we propose a new influence function-based empirical likelihood method and Bayesian empirical likelihood methods to construct confidence/credible intervals for the sensitivity of a test to patients in the early diseased stage given a specificity and a sensitivity of the test to patients in the fully diseased stage. Numerical studies are performed to compare the finite sample performances of the proposed approaches with existing methods. The proposed methods are shown to outperform existing methods in terms of coverage probability. A real dataset from the Alzheimer's Disease Neuroimaging Initiative (ANDI) is used to illustrate the proposed methods.     

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.