Digging through model complexity: using hierarchical models to uncover evolutionary processes in the wild

The growing interest for studying questions in the wild requires acknowledging that eco-evolutionary processes are complex, hierarchically structured and often partially observed or with measurement error. These issues have long been ignored in evolutionary biology, which might have led to flawed inference when addressing evolutionary questions. Hierarchical modelling (HM) has been proposed as a generic statistical framework to deal with complexity in ecological data and account for uncertainty. However, to date, HM has seldom been used to investigate evolutionary mechanisms possibly underlying observed patterns. Here, we contend the HM approach offers a relevant approach for the study of eco-evolutionary processes in the wild by confronting formal theories to empirical data through proper statistical inference. Studying eco-evolutionary processes requires considering the complete and often complex life histories of organisms. We show how this can be achieved by combining sequentially all life-history components and all available sources of information through HM. We demonstrate how eco-evolutionary processes may be poorly inferred or even missed without using the full potential of HM. As a case study, we use the Atlantic salmon and data on wild marked juveniles. We assess a reaction norm for migration and two potential trade-offs for survival. Overall, HM has a great potential to address evolutionary questions and investigate important processes that could not previously be assessed in laboratory or short time-scale studies.     

Quantifying the effects of temperature and flow regime on the abundance of 0+ cyprinids in the upper River Rhone using Bayesian hierarchical modelling

1. Assuming that recruitment variation is one of the main sources of fish population and assemblage changes, it is necessary to understand how natural variations in the environment influence 0+ fish abundance. Temperature regimes play an important role in enhancing both spawning activity and survival during early larval fish development. Flow regime variation, which is a powerful source of stream disturbance, is another factor to be taken into account. 2. Responses to these variables need to be assessed using long‐term datasets, since standard statistical approaches fail to provide a causal structure or to quantify the different effects. We therefore used a 26‐year dataset to evaluate the respective effects of seven derived independent variables describing the effects of temperature and flow regimes on the 0+ juvenile abundance of eight fish species in the River Rhone. 3. A clustering procedure using the Kendall tau rank correlation coefficient was implemented and identified three groups of fish according to their synchronic variations in juvenile abundance; i.e. varying with decreasing juvenile abundance, slightly increasing juvenile abundance and increasing juvenile abundance. These clusters provided the basis for building hierarchical log‐Poisson generalized linear models. The Bayesian paradigm gives a reliable framework for model selection, and the best model was determined using the Bayes Factor. 4. The posterior distribution of the regression parameters was coherent with what was expected based on knowledge of the biology of the different species. It indicates that temperature regime drives 0+ juvenile abundance but that flow regime also plays an important regulating role. The models thus detected evidence of the consequences of specific flow events such as larval drift and an increase in available habitat during higher flow discharges. 5. Our study illustrates the advantages of using a hierarchical modelling approach to quantify ecological effects by improving discrimination between the different sources of uncertainty, leading to better precision when estimating regression parameters.     

A generalized calibrated Bayesian hierarchical modeling approach to basket trials with multiple endpoints

A basket trial simultaneously evaluates a treatment in multiple cancer subtypes, offering an effective way to accelerate drug development in multiple indications. Many basket trials are designed and monitored based on a single efficacy endpoint, primarily the tumor response. For molecular targeted or immunotherapy agents, however, a single efficacy endpoint cannot adequately characterize the treatment effect. It is increasingly important to use more complex endpoints to comprehensively assess the risk–benefit profile of such targeted therapies. We extend the calibrated Bayesian hierarchical modeling approach to monitor phase II basket trials with multiple endpoints. We propose two generalizations, one based on the latent variable approach and the other based on the multinomial–normal hierarchical model, to accommodate different types of endpoints and dependence assumptions regarding information sharing. We introduce shrinkage parameters as functions of statistics measuring homogeneity among subgroups and propose a general calibration approach to determine the functional forms. Theoretical properties of the generalized hierarchical models are investigated. Simulation studies demonstrate that the monitoring procedure based on the generalized approach yields desirable operating characteristics.     

Bayesian hierarchical models suggest oldest known plant-visiting bat was omnivorous

The earliest record of plant visiting in bats dates to the Middle Miocene of La Venta, the world''s most diverse tropical palaeocommunity. Palynephyllum antimaster is known from molars that indicate nectarivory. Skull length, an important indicator of key traits such as body size, bite force and trophic specialization, remains unknown. We developed Bayesian models to infer skull length based on dental measurements. These models account for variation within and between species, variation between clades, and phylogenetic error structure. Models relating skull length to trophic level for nectarivorous bats were then used to infer the diet of the fossil. The skull length estimate for Palynephyllum places it among the larger lonchophylline bats. The inferred diet suggests Palynephyllum fed on nectar and insects, similar to its living relatives. Omnivory has persisted since the mid-Miocene. This is the first study to corroborate with fossil data that highly specialized nectarivory in bats requires an omnivorous transition.     

Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models

The problem of evaluating the goodness of the predictive distributionsof hierarchical Bayesian and empirical Bayes models is investigated.A Bayesian predictive information criterion is proposed as anestimator of the posterior mean of the expected loglikelihoodof the predictive distribution when the specified family ofprobability distributions does not contain the true distribution.The proposed criterion is developed by correcting the asymptoticbias of the posterior mean of the loglikelihood as an estimatorof its expected loglikelihood. In the evaluation of hierarchicalBayesian models with random effects, regardless of our parametricfocus, the proposed criterion considers the bias correctionof the posterior mean of the marginal loglikelihood becauseit requires a consistent parameter estimator. The use of thebootstrap in model evaluation is also discussed.     

Test of an environmental flow release in a British Columbia river: does more water mean more fish?

1. Water managers must make difficult decisions about the allocation of streamflows between out‐of‐channel human uses and environmental flows for aquatic resources. However, the effects environmental flows on stream ecosystems are infrequently evaluated. 2. We used a 13‐year experiment in the regulated Bridge River, British Columbia, Canada, to determine whether an environmental flow release designed to increase salmonid productivity was successful. A hierarchical Bayesian model was used to compare juvenile Pacific salmon (Oncorhynchus spp.) abundance before and after the flow release. 3. We found that the total number of salmonids did increase after the release, but most of the gains could be attributed to the rewatering of a previously dry channel located immediately below the dam. In reaches that had flowing water during the baseline period, the response of individual salmon species to the increase in flow was variable, and there was little change in total abundance after the flow release. Our results were inconsistent with both habitat modelling, which predicted a decrease in habitat quality with increasing flow, and holistic instream flow approaches, which imply greater benefits with larger flows. 4. We question whether biotic responses to flow changes can be predicted reliably with currently available methods and suggest that adaptive management or the use of decision tools that account for the uncertainty in the biotic response is required for instream flow decisions when the competing demands for water are great.     

Evaluating scaling models in biology using hierarchical Bayesian approaches

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.     

Scales of association: hierarchical linear models and the measurement of ecological systems

A fundamental challenge to understanding patterns in ecological systems lies in employing methods that can analyse, test and draw inference from measured associations between variables across scales. Hierarchical linear models (HLM) use advanced estimation algorithms to measure regression relationships and variance–covariance parameters in hierarchically structured data. Although hierarchical models have occasionally been used in the analysis of ecological data, their full potential to describe scales of association, diagnose variance explained, and to partition uncertainty has not been employed. In this paper we argue that the use of the HLM framework can enable significantly improved inference about ecological processes across levels of organization. After briefly describing the principals behind HLM, we give two examples that demonstrate a protocol for building hierarchical models and answering questions about the relationships between variables at multiple scales. The first example employs maximum likelihood methods to construct a two-level linear model predicting herbivore damage to a perennial plant at the individual- and patch-scale; the second example uses Bayesian estimation techniques to develop a three-level logistic model of plant flowering probability across individual plants, microsites and populations. HLM model development and diagnostics illustrate the importance of incorporating scale when modelling associations in ecological systems and offer a sophisticated yet accessible method for studies of populations, communities and ecosystems. We suggest that a greater coupling of hierarchical study designs and hierarchical analysis will yield significant insights on how ecological processes operate across scales.     

Estimating seed and pollen movement in a monoecious plant: a hierarchical Bayesian approach integrating genetic and ecological data

The scale of seed and pollen movement in plants has a critical influence on population dynamics and interspecific interactions, as well as on their capacity to respond to environmental change through migration or local adaptation. However, dispersal can be challenging to quantify. Here, we present a Bayesian model that integrates genetic and ecological data to simultaneously estimate effective seed and pollen dispersal parameters and the parentage of sampled seedlings. This model is the first developed for monoecious plants that accounts for genotyping error and treats dispersal from within and beyond a plot in a fully consistent manner. The flexible Bayesian framework allows the incorporation of a variety of ecological variables, including individual variation in seed production, as well as multiple sources of uncertainty. We illustrate the method using data from a mixed population of red oak (Quercus rubra, Q. velutina, Q. falcata) in the NC piedmont. For simulated test data sets, the model successfully recovered the simulated dispersal parameters and pedigrees. Pollen dispersal in the example population was extensive, with an average father-mother distance of 178 m. Estimated seed dispersal distances at the piedmont site were substantially longer than previous estimates based on seed-trap data (average 128 m vs. 9.3 m), suggesting that, under some circumstances, oaks may be less dispersal-limited than is commonly thought, with a greater potential for range shifts in response to climate change.     

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.     

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.     

Modelling the impacts of flow regulation on fish distributions in naturally intermittent lowland streams: an approach for predicting restoration responses

1. Understanding the relationships between flow regime and the distribution of biota is critical for managing flows in regulated rivers. In northern Victoria, Australia, efforts are presently underway to restore a natural, intermittent flow regime to several streams which, for over 100 years, have received perennial diversions as part of a stock, irrigation and domestic water supply. 2. Bayesian, model‐averaged, binomial regression was used to predict probabilities of occurrence for 13 fish species, including five non‐native species, based on hydrologic variables characterising both the current and modelled future flow regimes at 10 sites representing a range of hydrologic regimes (categorised here as heavily regulated, moderately regulated and unregulated). 3. Regression models accurately predicted present probabilities of occurrence for most species across all sites. The models predicted a reduced likelihood of large, native, flow‐dependent species occurring at regulated sites following flow restoration. Predictions regarding the future distribution of widespread species including two small‐bodied native and four exotic species were less certain as current distributions of these widespread species were unrelated to hydrologic variables we examined and thus unlikely to be significantly affected by flow restoration. The distributions of two small native species currently restricted to unregulated sites are predicted to increase throughout the system. 4. This study illustrates the effects of artificially induced perennial flow on lowland fish distributions. Furthermore, the combination of pre‐restoration data together with predictive modelling provides valuable insights into the likely outcomes of flow regime shifts. 5. This study clearly demonstrates the value of combining empirical research and modelling in guiding environmental flow and ecosystem restoration decisions. Knowledge from the study is now helping guide management decisions and the development of mitigation strategies to protect highly valued species in the system from potential future threats.     

Ecoacoustics can detect ecosystem responses to environmental water allocations

1. Globally, water abstraction for human consumption and irrigated agriculture leads to significant changes in aquatic ecosystems. To counter these detrimental effects, water releases—often termed environmental water allocations—restore overbank flow or are delivered to artificially disconnected wetlands. While a suite of monitoring methods is available, few programmes track continuous change in biota, mainly because repeated remote site visits can be prohibitively expensive.
2. In this paper, we propose a new approach to environmental flow monitoring, using ecoacoustic methods. We test acoustic monitoring of frog and waterbird responses to environmental water deliveries in the Goulburn Broken, a valley in the southern Murray–Darling river system. Response to three major environmental water deliveries within 2 years was monitored at four sites along Reedy swamp. Every 2 weeks, 30 s were recorded every 30 min, for a total of 24 hr.
3. We used two analysis strategies—manual counts of bird calls, as well as ecoacoustic indices, which describe the sonic properties of the acoustic spectrum at a site. Manual counts demonstrated that water-dependent birds were clearly responding to environmental water deliveries, whereas non-water-dependent species did not show any increases in activity. After restricting the analysis to the dawn chorus of birds and frogs, two acoustic indices (the median amplitude and the acoustic complexity index) showed responses to watering events.
4. Ecoacoustic methods show promise for continuous response monitoring to environmental water allocations. However, the first strategy—manual annotation of calls—might be too labour intensive for standard monitoring programmes. The second strategy—index-based approaches—can also detect ecological responses, although further investigation using control sites is needed. Automated call classifiers are an alternative that is currently being developed for endangered species. We also encourage simultaneous monitoring of the soundscape above and under water.