Incorporating uncertainty and prior information into stable isotope mixing models

Stable isotopes are a powerful tool for ecologists, often used to assess contributions of different sources to a mixture (e.g. prey to a consumer). Mixing models use stable isotope data to estimate the contribution of sources to a mixture. Uncertainty associated with mixing models is often substantial, but has not yet been fully incorporated in models. We developed a Bayesian-mixing model that estimates probability distributions of source contributions to a mixture while explicitly accounting for uncertainty associated with multiple sources, fractionation and isotope signatures. This model also allows for optional incorporation of informative prior information in analyses. We demonstrate our model using a predator–prey case study. Accounting for uncertainty in mixing model inputs can change the variability, magnitude and rank order of estimates of prey (source) contributions to the predator (mixture). Isotope mixing models need to fully account for uncertainty in order to accurately estimate source contributions.     

Estimating dengue transmission intensity from serological data: A comparative analysis using mixture and catalytic models

BackgroundDengue virus (DENV) infection is a global health concern of increasing magnitude. To target intervention strategies, accurate estimates of the force of infection (FOI) are necessary. Catalytic models have been widely used to estimate DENV FOI and rely on a binary classification of serostatus as seropositive or seronegative, according to pre-defined antibody thresholds. Previous work has demonstrated the use of thresholds can cause serostatus misclassification and biased estimates. In contrast, mixture models do not rely on thresholds and use the full distribution of antibody titres. To date, there has been limited application of mixture models to estimate DENV FOI.MethodsWe compare the application of mixture models and time-constant and time-varying catalytic models to simulated data and to serological data collected in Vietnam from 2004 to 2009 (N ≥ 2178) and Indonesia in 2014 (N = 3194).ResultsThe simulation study showed larger mean FOI estimate bias from the time-constant and time-varying catalytic models (-0.007 (95% Confidence Interval (CI): -0.069, 0.029) and -0.006 (95% CI -0.095, 0.043)) than from the mixture model (0.001 (95% CI -0.036, 0.065)). Coverage of the true FOI was > 95% for estimates from both the time-varying catalytic and mixture model, however the latter had reduced uncertainty. When applied to real data from Vietnam, the mixture model frequently produced higher FOI and seroprevalence estimates than the catalytic models.ConclusionsOur results suggest mixture models represent valid, potentially less biased, alternatives to catalytic models, which could be particularly useful when estimating FOI from data with largely overlapping antibody titre distributions.     

Using dynamic N‐mixture models to test cavity limitation on northern flying squirrel demographic parameters using experimental nest box supplementation

Dynamic N‐mixture models have been recently developed to estimate demographic parameters of unmarked individuals while accounting for imperfect detection. We propose an application of the Dail and Madsen ( 2011 : Biometrics, 67 , 577–587) dynamic N‐mixture model in a manipulative experiment using a before‐after control‐impact design (BACI). Specifically, we tested the hypothesis of cavity limitation of a cavity specialist species, the northern flying squirrel, using nest box supplementation on half of 56 trapping sites. Our main purpose was to evaluate the impact of an increase in cavity availability on flying squirrel population dynamics in deciduous stands in northwestern Québec with the dynamic N‐mixture model. We compared abundance estimates from this recent approach with those from classic capture–mark–recapture models and generalized linear models. We compared apparent survival estimates with those from Cormack–Jolly–Seber (CJS) models. Average recruitment rate was 6 individuals per site after 4 years. Nevertheless, we found no effect of cavity supplementation on apparent survival and recruitment rates of flying squirrels. Contrary to our expectations, initial abundance was not affected by conifer basal area (food availability) and was negatively affected by snag basal area (cavity availability). Northern flying squirrel population dynamics are not influenced by cavity availability at our deciduous sites. Consequently, we suggest that this species should not be considered an indicator of old forest attributes in our study area, especially in view of apparent wide population fluctuations across years. Abundance estimates from N‐mixture models were similar to those from capture–mark–recapture models, although the latter had greater precision. Generalized linear mixed models produced lower abundance estimates, but revealed the same relationship between abundance and snag basal area. Apparent survival estimates from N‐mixture models were higher and less precise than those from CJS models. However, N‐mixture models can be particularly useful to evaluate management effects on animal populations, especially for species that are difficult to detect in situations where individuals cannot be uniquely identified. They also allow investigating the effects of covariates at the site level, when low recapture rates would require restricting classic CMR analyses to a subset of sites with the most captures.     

Assessing variation in life‐history tactics within a population using mixture regression models: a practical guide for evolutionary ecologists

Mixed models are now well‐established methods in ecology and evolution because they allow accounting for and quantifying within‐ and between‐individual variation. However, the required normal distribution of the random effects can often be violated by the presence of clusters among subjects, which leads to multi‐modal distributions. In such cases, using what is known as mixture regression models might offer a more appropriate approach. These models are widely used in psychology, sociology, and medicine to describe the diversity of trajectories occurring within a population over time (e.g. psychological development, growth). In ecology and evolution, however, these models are seldom used even though understanding changes in individual trajectories is an active area of research in life‐history studies. Our aim is to demonstrate the value of using mixture models to describe variation in individual life‐history tactics within a population, and hence to promote the use of these models by ecologists and evolutionary ecologists. We first ran a set of simulations to determine whether and when a mixture model allows teasing apart latent clustering, and to contrast the precision and accuracy of estimates obtained from mixture models versus mixed models under a wide range of ecological contexts. We then used empirical data from long‐term studies of large mammals to illustrate the potential of using mixture models for assessing within‐population variation in life‐history tactics. Mixture models performed well in most cases, except for variables following a Bernoulli distribution and when sample size was small. The four selection criteria we evaluated [Akaike information criterion (AIC), Bayesian information criterion (BIC), and two bootstrap methods] performed similarly well, selecting the right number of clusters in most ecological situations. We then showed that the normality of random effects implicitly assumed by evolutionary ecologists when using mixed models was often violated in life‐history data. Mixed models were quite robust to this violation in the sense that fixed effects were unbiased at the population level. However, fixed effects at the cluster level and random effects were better estimated using mixture models. Our empirical analyses demonstrated that using mixture models facilitates the identification of the diversity of growth and reproductive tactics occurring within a population. Therefore, using this modelling framework allows testing for the presence of clusters and, when clusters occur, provides reliable estimates of fixed and random effects for each cluster of the population. In the presence or expectation of clusters, using mixture models offers a suitable extension of mixed models, particularly when evolutionary ecologists aim at identifying how ecological and evolutionary processes change within a population. Mixture regression models therefore provide a valuable addition to the statistical toolbox of evolutionary ecologists. As these models are complex and have their own limitations, we provide recommendations to guide future users.     

Uncertainty in source partitioning using stable isotopes

Stable isotope analyses are often used to quantify the contribution of multiple sources to a mixture, such as proportions of food sources in an animal's diet, or C₃ and C₄ plant inputs to soil organic carbon. Linear mixing models can be used to partition two sources with a single isotopic signature (e.g., '¹³C) or three sources with a second isotopic signature (e.g., '¹⁵N). Although variability of source and mixture signatures is often reported, confidence interval calculations for source proportions typically use only the mixture variability. We provide examples showing that omission of source variability can lead to underestimation of the variability of source proportion estimates. For both two- and three-source mixing models, we present formulas for calculating variances, standard errors (SE), and confidence intervals for source proportion estimates that account for the observed variability in the isotopic signatures for the sources as well as the mixture. We then performed sensitivity analyses to assess the relative importance of: (1) the isotopic signature difference between the sources, (2) isotopic signature standard deviations (SD) in the source and mixture populations, (3) sample size, (4) analytical SD, and (5) the evenness of the source proportions, for determining the variability (SE) of source proportion estimates. The proportion SEs varied inversely with the signature difference between sources, so doubling the source difference from 2‰ to 4‰ reduced the SEs by half. Source and mixture signature SDs had a substantial linear effect on source proportion SEs. However, the population variability of the sources and the mixture are fixed and the sampling error component can be changed only by increasing sample size. Source proportion SEs varied inversely with the square root of sample size, so an increase from 1 to 4 samples per population cut the SE in half. Analytical SD had little effect over the range examined since it was generally substantially smaller than the population SDs. Proportion SEs were minimized when sources were evenly divided, but increased only slightly as the proportions varied. The variance formulas provided will enable quantification of the precision of source proportion estimates. Graphs are provided to allow rapid assessment of possible combinations of source differences and source and mixture population SDs that will allow source proportion estimates with desired precision. In addition, an Excel spreadsheet to perform the calculations for the source proportions and their variances, SEs, and 95% confidence intervals for the two-source and three-source mixing models can be accessed at http://www.epa.gov/wed/pages/models.htm.     

A bias in ML estimates of branch lengths in the presence of multiple signals

Sequence data often have competing signals that are detected by network programs or Lento plots. Such data can be formed by generating sequences on more than one tree, and combining the results, a mixture model. We report that with such mixture models, the estimates of edge (branch) lengths from maximum likelihood (ML) methods that assume a single tree are biased. Based on the observed number of competing signals in real data, such a bias of ML is expected to occur frequently. Because network methods can recover competing signals more accurately, there is a need for ML methods allowing a network. A fundamental problem is that mixture models can have more parameters than can be recovered from the data, so that some mixtures are not, in principle, identifiable. We recommend that network programs be incorporated into best practice analysis, along with ML and Bayesian trees.     

Distinguishing between Selective Sweeps from Standing Variation and from a De Novo Mutation

An outstanding question in human genetics has been the degree to which adaptation occurs from standing genetic variation or from de novo mutations. Here, we combine several common statistics used to detect selection in an Approximate Bayesian Computation (ABC) framework, with the goal of discriminating between models of selection and providing estimates of the age of selected alleles and the selection coefficients acting on them. We use simulations to assess the power and accuracy of our method and apply it to seven of the strongest sweeps currently known in humans. We identify two genes, ASPM and PSCA, that are most likely affected by selection on standing variation; and we find three genes, ADH1B, LCT, and EDAR, in which the adaptive alleles seem to have swept from a new mutation. We also confirm evidence of selection for one further gene, TRPV6. In one gene, G6PD, neither neutral models nor models of selective sweeps fit the data, presumably because this locus has been subject to balancing selection.     

Estimating the prevalence of infections in vector populations using pools of samples

Several statistical methods have been proposed for estimating the infection prevalence based on pooled samples, but these methods generally presume the application of perfect diagnostic tests, which in practice do not exist. To optimize prevalence estimation based on pooled samples, currently available and new statistical models were described and compared. Three groups were tested: (a) Frequentist models, (b) Monte Carlo Markov‐Chain (MCMC) Bayesian models, and (c) Exact Bayesian Computation (EBC) models. Simulated data allowed the comparison of the models, including testing the performance under complex situations such as imperfect tests with a sensitivity varying according to the pool weight. In addition, all models were applied to data derived from the literature, to demonstrate the influence of the model on real‐prevalence estimates. All models were implemented in the freely available R and OpenBUGS software and are presented in Appendix S1. Bayesian models can flexibly take into account the imperfect sensitivity and specificity of the diagnostic test (as well as the influence of pool‐related or external variables) and are therefore the method of choice for calculating population prevalence based on pooled samples. However, when using such complex models, very precise information on test characteristics is needed, which may in general not be available.     

A multispecies dependent double‐observer model: A new method for estimating multispecies abundance

Conservation of biological communities requires accurate estimates of abundance for multiple species. Recent advances in estimating abundance of multiple species, such as Bayesian multispecies N‐mixture models, account for multiple sources of variation, including detection error. However, false‐positive errors (misidentification or double counts), which are prevalent in multispecies data sets, remain largely unaddressed. The dependent‐double observer (DDO) method is an emerging method that both accounts for detection error and is suggested to reduce the occurrence of false positives because it relies on two observers working collaboratively to identify individuals. To date, the DDO method has not been combined with advantages of multispecies N‐mixture models. Here, we derive an extension of a multispecies N‐mixture model using the DDO survey method to create a multispecies dependent double‐observer abundance model (MDAM). The MDAM uses a hierarchical framework to account for biological and observational processes in a statistically consistent framework while using the accurate observation data from the DDO survey method. We demonstrate that the MDAM accurately estimates abundance of multiple species with simulated and real multispecies data sets. Simulations showed that the model provides both precise and accurate abundance estimates, with average credible interval coverage across 100 repeated simulations of 94.5% for abundance estimates and 92.5% for detection estimates. In addition, 92.2% of abundance estimates had a mean absolute percent error between 0% and 20%, with a mean of 7.7%. We present the MDAM as an important step forward in expanding the applicability of the DDO method to a multispecies setting. Previous implementation of the DDO method suggests the MDAM can be applied to a broad array of biological communities. We suggest that researchers interested in assessing biological communities consider the MDAM as a tool for deriving accurate, multispecies abundance estimates.     

Mixture models for estimating the size of a closed population when capture rates vary among individuals

We develop a parameterization of the beta-binomial mixture that provides sensible inferences about the size of a closed population when probabilities of capture or detection vary among individuals. Three classes of mixture models (beta-binomial, logistic-normal, and latent-class) are fitted to recaptures of snowshoe hares for estimating abundance and to counts of bird species for estimating species richness. In both sets of data, rates of detection appear to vary more among individuals (animals or species) than among sampling occasions or locations. The estimates of population size and species richness are sensitive to model-specific assumptions about the latent distribution of individual rates of detection. We demonstrate using simulation experiments that conventional diagnostics for assessing model adequacy, such as deviance, cannot be relied on for selecting classes of mixture models that produce valid inferences about population size. Prior knowledge about sources of individual heterogeneity in detection rates, if available, should be used to help select among classes of mixture models that are to be used for inference.     

New technologies in the mix: Assessing N‐mixture models for abundance estimation using automated detection data from drone surveys

1. Reliable estimates of abundance are critical in effectively managing threatened species, but the feasibility of integrating data from wildlife surveys completed using advanced technologies such as remotely piloted aircraft systems (RPAS) and machine learning into abundance estimation methods such as N‐mixture modeling is largely unknown due to the unique sources of detection errors associated with these technologies.
2. We evaluated two modeling approaches for estimating the abundance of koalas detected automatically in RPAS imagery: (a) a generalized N‐mixture model and (b) a modified Horvitz–Thompson (H‐T) estimator method combining generalized linear models and generalized additive models for overall probability of detection, false detection, and duplicate detection. The final estimates from each model were compared to the true number of koalas present as determined by telemetry‐assisted ground surveys.
3. The modified H‐T estimator approach performed best, with the true count of koalas captured within the 95% confidence intervals around the abundance estimates in all 4 surveys in the testing dataset (n = 138 detected objects), a particularly strong result given the difficulty in attaining accuracy found with previous methods.
4. The results suggested that N‐mixture models in their current form may not be the most appropriate approach to estimating the abundance of wildlife detected in RPAS surveys with automated detection, and accurate estimates could be made with approaches that account for spurious detections.

     

Interval mapping of quantitative trait loci for time-to-event data with the proportional hazards mixture cure model

Interval mapping using normal mixture models has been an important tool for analyzing quantitative traits in experimental organisms. When the primary phenotype is time-to-event, it is natural to use survival models such as Cox's proportional hazards model instead of normal mixtures to model the phenotype distribution. An extra challenge for modeling time-to-event data is that the underlying population may consist of susceptible and nonsusceptible subjects. In this article, we propose a semiparametric proportional hazards mixture cure model which allows missing covariates. We discuss applications to quantitative trait loci (QTL) mapping when the primary trait is time-to-event from a population of mixed susceptibility. This model can be used to characterize QTL effects on both susceptibility and time-to-event distribution, and to estimate QTL location. The model can naturally incorporate covariate effects of other risk factors. Maximum likelihood estimates for the parameters in the model as well as their corresponding variance estimates can be obtained numerically using an EM-type algorithm. The proposed methods are assessed by simulations under practical settings and illustrated using a real data set containing survival times of mice after infection with Listeria monocytogenes. An extension to multiple intervals is also discussed.     

Density influences accuracy of model‐based estimates for a forest songbird

Golden‐cheeked Warblers (Setophaga chrysoparia) are endangered songbirds that breed exclusively in the Ashe juniper (Juniperus ashei) and oak (Quercus spp.) woodlands of central Texas. Despite being the focus of numerous studies, we still know little about the size of the range‐wide breeding population and how density varies across the spectrum of juniper co‐dominated woodlands. Models that have been tested and shown to be accurate are needed to help develop management and conservation guidelines. We evaluated the accuracy and bias of density estimates from binomial mixture models, the dependent double‐observer method, and distance sampling by comparing them to actual densities determined by intensive territory monitoring on plots in the Balcones Canyonlands Preserve, Austin, Texas. We found that the binomial mixture models consistently overestimated density by 1.1–3.2 times (actual density = 0.07–0.46 males/ha), and the other two models overestimated by 1.1–29.8 times at low density and underestimated by 0.5–0.9 times at high density plots (actual density = 0.01–0.46 males/ha). The magnitude of error for all models was greatest at sites with few or no birds (<0.15 males/ha), with model performance improving as actual density increased. These non‐linear relationships indicate a lack of sensitivity with respect to true changes in density. Until systematic evaluation demonstrates that models such as those we tested provide accurate and unbiased density estimates for a given species over space and time, we recommend additional field tests to validate model‐based estimates. Continued model validation and refinement of point‐count methods are needed until accurate estimates are obtained across the density spectrum for Golden‐cheeked Warblers and other songbird species.     

Mixture modeling of progression pathways of heterogeneous breast tumors

Uncovering pathways of tumor progression is an important topic in cancer research that has led to numerous studies, including several pathway models proposed and investigated by Sontag and Axelrod [Progression of heterogeneous breast tumors. J. Theoret. Biol. 210, 107-119, 2005]. In their comparative studies, the authors focused on relative goodness of fits of the various pathways, but a simple test revealed that even the "best" model did not provide an adequate explanation for the observed breast tumor data. The heterogeneous nature of breast tumors leads to the question of whether more than one (i.e., a combination of) pathway models are needed in order to explain the observed data. In the current paper, we address this question based on the finite mixture modeling framework and utilizing the four pathways proposed in Sontag and Axelrod as our individual pathway models. The expectation-maximization algorithm was used to derive estimates for the mixing proportions of the mixture models. Indeed, a two-pathway mixture provides a dramatic improvement over any of the single pathway models for explaining the data derived under either the Van Nuys or the Holland system. In particular, for data graded under the Van Nuys system, the mixture model was shown to be consistent with the observed data at the 1% significant level.     

Generalized parametric cure models for relative survival

Cure models are used in time-to-event analysis when not all individuals are expected to experience the event of interest, or when the survival of the considered individuals reaches the same level as the general population. These scenarios correspond to a plateau in the survival and relative survival function, respectively. The main parameters of interest in cure models are the proportion of individuals who are cured, termed the cure proportion, and the survival function of the uncured individuals. Although numerous cure models have been proposed in the statistical literature, there is no consensus on how to formulate these. We introduce a general parametric formulation of mixture cure models and a new class of cure models, termed latent cure models, together with a general estimation framework and software, which enable fitting of a wide range of different models. Through simulations, we assess the statistical properties of the models with respect to the cure proportion and the survival of the uncured individuals. Finally, we illustrate the models using survival data on colon cancer, which typically display a plateau in the relative survival. As demonstrated in the simulations, mixture cure models which are not guaranteed to be constant after a finite time point, tend to produce accurate estimates of the cure proportion and the survival of the uncured. However, these models are very unstable in certain cases due to identifiability issues, whereas LC models generally provide stable results at the price of more biased estimates.     

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.     

On classifying observations when one population is a mixture of normals

Some simple models for the mixture of distribution problems are considered. Two possible alternatives to the rather complex optimal discriminant function rule are mentioned. The performance of the BC method is never as satisfactory as the QDF method and in some cases it is far worse than the QDF method. The QDF can be used whenever sufficient numbers of observations are available to provide reasonably good estimates of means and covariances.     

Peak quantification in surface-enhanced laser desorption/ionization by using mixture models

Surface-enhanced laser desorption/ionization (SELDI) time of flight (TOF) is a mass spectrometry technology for measuring the composition of a sampled protein mixture. A mass spectrum contains peaks corresponding to proteins in the sample. The peak areas are proportional to the measured concentrations of the corresponding proteins. Quantifying peak areas is difficult for existing methods because peak shapes are not constant across a spectrum and because peaks often overlap. We present a new method for quantifying peak areas. Our method decomposes a spectrum into peaks and a baseline using so-called statistical finite mixture models. We illustrate our method in detail on 8 samples from culture media of adipose tissue and globally on 64 samples from serum to compare our method to the standard Ciphergen method. Both methods give similar estimates for singleton peaks, but not for overlapping peaks. The Ciphergen method overestimates the heights of such peaks while our method still gives appropriate estimates. Peak quantification is an important step in pre-processing SELDI-TOF data and improvements therein will pay off in the later biomarker discovery phase.     

Partitioning Detectability Components in Populations Subject to Within-Season Temporary Emigration Using Binomial Mixture Models

Detectability of individual animals is highly variable and nearly always < 1; imperfect detection must be accounted for to reliably estimate population sizes and trends. Hierarchical models can simultaneously estimate abundance and effective detection probability, but there are several different mechanisms that cause variation in detectability. Neglecting temporary emigration can lead to biased population estimates because availability and conditional detection probability are confounded. In this study, we extend previous hierarchical binomial mixture models to account for multiple sources of variation in detectability. The state process of the hierarchical model describes ecological mechanisms that generate spatial and temporal patterns in abundance, while the observation model accounts for the imperfect nature of counting individuals due to temporary emigration and false absences. We illustrate our model’s potential advantages, including the allowance of temporary emigration between sampling periods, with a case study of southern red-backed salamanders Plethodon serratus. We fit our model and a standard binomial mixture model to counts of terrestrial salamanders surveyed at 40 sites during 3–5 surveys each spring and fall 2010–2012. Our models generated similar parameter estimates to standard binomial mixture models. Aspect was the best predictor of salamander abundance in our case study; abundance increased as aspect became more northeasterly. Increased time-since-rainfall strongly decreased salamander surface activity (i.e. availability for sampling), while higher amounts of woody cover objects and rocks increased conditional detection probability (i.e. probability of capture, given an animal is exposed to sampling). By explicitly accounting for both components of detectability, we increased congruence between our statistical modeling and our ecological understanding of the system. We stress the importance of choosing survey locations and protocols that maximize species availability and conditional detection probability to increase population parameter estimate reliability.