Use of hidden Markov capture–recapture models to estimate abundance in the presence of uncertainty: Application to the estimation of prevalence of hybrids in animal populations

Estimating the relative abundance (prevalence) of different population segments is a key step in addressing fundamental research questions in ecology, evolution, and conservation. The raw percentage of individuals in the sample (naive prevalence) is generally used for this purpose, but it is likely to be subject to two main sources of bias. First, the detectability of individuals is ignored; second, classification errors may occur due to some inherent limits of the diagnostic methods. We developed a hidden Markov (also known as multievent) capture–recapture model to estimate prevalence in free‐ranging populations accounting for imperfect detectability and uncertainty in individual's classification. We carried out a simulation study to compare naive and model‐based estimates of prevalence and assess the performance of our model under different sampling scenarios. We then illustrate our method with a real‐world case study of estimating the prevalence of wolf (Canis lupus) and dog (Canis lupus familiaris) hybrids in a wolf population in northern Italy. We showed that the prevalence of hybrids could be estimated while accounting for both detectability and classification uncertainty. Model‐based prevalence consistently had better performance than naive prevalence in the presence of differential detectability and assignment probability and was unbiased for sampling scenarios with high detectability. We also showed that ignoring detectability and uncertainty in the wolf case study would lead to underestimating the prevalence of hybrids. Our results underline the importance of a model‐based approach to obtain unbiased estimates of prevalence of different population segments. Our model can be adapted to any taxa, and it can be used to estimate absolute abundance and prevalence in a variety of cases involving imperfect detection and uncertainty in classification of individuals (e.g., sex ratio, proportion of breeders, and prevalence of infected individuals).     

Detectability in Audio-Visual Surveys of Tropical Rainforest Birds: The Influence of Species,Weather and Habitat Characteristics

Indices of relative abundance do not control for variation in detectability, which can bias density estimates such that ecological processes are difficult to infer. Distance sampling methods can be used to correct for detectability, but in rainforest, where dense vegetation and diverse assemblages complicate sampling, information is lacking about factors affecting their application. Rare species present an additional challenge, as data may be too sparse to fit detection functions. We present analyses of distance sampling data collected for a diverse tropical rainforest bird assemblage across broad elevational and latitudinal gradients in North Queensland, Australia. Using audio and visual detections, we assessed the influence of various factors on Effective Strip Width (ESW), an intuitively useful parameter, since it can be used to calculate an estimate of density from count data. Body size and species exerted the most important influence on ESW, with larger species detectable over greater distances than smaller species. Secondarily, wet weather and high shrub density decreased ESW for most species. ESW for several species also differed between summer and winter, possibly due to seasonal differences in calling behavior. Distance sampling proved logistically intensive in these environments, but large differences in ESW between species confirmed the need to correct for detection probability to obtain accurate density estimates. Our results suggest an evidence-based approach to controlling for factors influencing detectability, and avenues for further work including modeling detectability as a function of species characteristics such as body size and call characteristics. Such models may be useful in developing a calibration for non-distance sampling data and for estimating detectability of rare species.     

Modelling occurrence and abundance of species when detection is imperfect

Relationships between species abundance and occupancy are of considerable interest in metapopulation biology and in macroecology. Such relationships may be described concisely using probability models that characterize variation in abundance of a species. However, estimation of the parameters of these models in most ecological problems is impaired by imperfect detection. When organisms are detected imperfectly, observed counts are biased estimates of true abundance, and this induces bias in stated occupancy or occurrence probability. In this paper we consider a class of models that enable estimation of abundance/occupancy relationships from counts of organisms that result from surveys in which detection is imperfect. Under such models, parameter estimation and inference are based on conventional likelihood methods. We provide an application of these models to geographically extensive breeding bird survey data in which alternative models of abundance are considered that include factors that influence variation in abundance and detectability. Using these models, we produce estimates of abundance and occupancy maps that honor important sources of spatial variation in avian abundance and provide clearly interpretable characterizations of abundance and occupancy adjusted for imperfect detection.     

Estimating equations with nonignorably missing response data

Troxel, Lipsitz, and Brennan (1997, Biometrics 53, 857-869) considered parameter estimation from survey data with nonignorable nonresponse and proposed weighted estimating equations to remove the biases in the complete-case analysis that ignores missing observations. This paper suggests two alternative modifications for unbiased estimation of regression parameters when a binary outcome is potentially observed at successive time points. The weighting approach of Robins, Rotnitzky, and Zhao (1995, Journal of the American Statistical Association 90, 106-121) is also modified to obtain unbiased estimating functions. The suggested estimating functions are unbiased only when the missingness probability is correctly specified, and misspecification of the missingness model will result in biases in the estimates. Simulation studies are carried out to assess the performance of different methods when the covariate is binary or normal. For the simulation models used, the relative efficiency of the two new methods to the weighting methods is about 3.0 for the slope parameter and about 2.0 for the intercept parameter when the covariate is continuous and the missingness probability is correctly specified. All methods produce substantial biases in the estimates when the missingness model is misspecified or underspecified. Analysis of data from a medical survey illustrates the use and possible differences of these estimating functions.     

Ignoring Imperfect Detection in Biological Surveys Is Dangerous: A Response to ‘Fitting and Interpreting Occupancy Models'

In a recent paper, Welsh, Lindenmayer and Donnelly (WLD) question the usefulness of models that estimate species occupancy while accounting for detectability. WLD claim that these models are difficult to fit and argue that disregarding detectability can be better than trying to adjust for it. We think that this conclusion and subsequent recommendations are not well founded and may negatively impact the quality of statistical inference in ecology and related management decisions. Here we respond to WLD''s claims, evaluating in detail their arguments, using simulations and/or theory to support our points. In particular, WLD argue that both disregarding and accounting for imperfect detection lead to the same estimator performance regardless of sample size when detectability is a function of abundance. We show that this, the key result of their paper, only holds for cases of extreme heterogeneity like the single scenario they considered. Our results illustrate the dangers of disregarding imperfect detection. When ignored, occupancy and detection are confounded: the same naïve occupancy estimates can be obtained for very different true levels of occupancy so the size of the bias is unknowable. Hierarchical occupancy models separate occupancy and detection, and imprecise estimates simply indicate that more data are required for robust inference about the system in question. As for any statistical method, when underlying assumptions of simple hierarchical models are violated, their reliability is reduced. Resorting in those instances where hierarchical occupancy models do no perform well to the naïve occupancy estimator does not provide a satisfactory solution. The aim should instead be to achieve better estimation, by minimizing the effect of these issues during design, data collection and analysis, ensuring that the right amount of data is collected and model assumptions are met, considering model extensions where appropriate.     

Raising the bar for the next generation of biological atlases: using existing data to inform the design and implementation of atlas monitoring

Selecting a sampling design to monitor multiple species across a broad geographical region can be a daunting task and often involves tradeoffs between limited resources and the accurate estimation of population abundance and occurrence. Since the 1950s, biological atlases have been implemented in various regions to document the occurrence of plant and animal species. As next‐generation atlases repeat original surveys, investigators often seek to raise the rigour of atlases by incorporating species abundances. We present a repeatable framework that incorporates existing monitoring data, hierarchical modelling and sampling simulations to augment existing atlas occurrence and breeding status maps with a secondary sampling of species abundances. Using existing information on three bird species with varying abundance and detectability, we evaluated several sampling scenarios for the 2nd Wisconsin Breeding Bird Atlas. In general, we found that most sampling schemes produced accurate mean statewide abundance estimates for species with medium to high abundance and detection probability, but estimates varied significantly for species with low abundance and low detection probability. Our approach provided a statewide point‐count sampling design that: provided precise and unbiased abundance estimates for species of varied prevalence and detectability; ensured suitable spatial coverage across the state and its habitats; and reduced spending on total survey costs. Our framework could benefit investigators conducting atlases and other broad‐scale avian surveys that seek to add systematic, multi‐species sampling for estimating density and abundance across broad geographical regions.     

Estimation and Correction of Visibility Bias in Aerial Surveys of Wintering Ducks

Abstract: Incomplete detection of all individuals leading to negative bias in abundance estimates is a pervasive source of error in aerial surveys of wildlife, and correcting that bias is a critical step in improving surveys. We conducted experiments using duck decoys as surrogates for live ducks to estimate bias associated with surveys of wintering ducks in Mississippi, USA. We found detection of decoy groups was related to wetland cover type (open vs. forested), group size (1–100 decoys), and interaction of these variables. Observers who detected decoy groups reported counts that averaged 78% of the decoys actually present, and this counting bias was not influenced by either covariate cited above. We integrated this sightability model into estimation procedures for our sample surveys with weight adjustments derived from probabilities of group detection (estimated by logistic regression) and count bias. To estimate variances of abundance estimates, we used bootstrap resampling of transects included in aerial surveys and data from the bias-correction experiment. When we implemented bias correction procedures on data from a field survey conducted in January 2004, we found bias-corrected estimates of abundance increased 36–42%, and associated standard errors increased 38–55%, depending on species or group estimated. We deemed our method successful for integrating correction of visibility bias in an existing sample survey design for wintering ducks in Mississippi, and we believe this procedure could be implemented in a variety of sampling problems for other locations and species. (JOURNAL OF WILDLIFE MANAGEMENT 72(3):808–813; 2008)     

Bias and Sensitivity Analysis When Estimating Treatment Effects from the Cox Model with Omitted Covariates

Summary

Omission of relevant covariates can lead to bias when estimating treatment or exposure effects from survival data in both randomized controlled trials and observational studies. This paper presents a general approach to assessing bias when covariates are omitted from the Cox model. The proposed method is applicable to both randomized and non‐randomized studies. We distinguish between the effects of three possible sources of bias: omission of a balanced covariate, data censoring and unmeasured confounding. Asymptotic formulae for determining the bias are derived from the large sample properties of the maximum likelihood estimator. A simulation study is used to demonstrate the validity of the bias formulae and to characterize the influence of the different sources of bias. It is shown that the bias converges to fixed limits as the effect of the omitted covariate increases, irrespective of the degree of confounding. The bias formulae are used as the basis for developing a new method of sensitivity analysis to assess the impact of omitted covariates on estimates of treatment or exposure effects. In simulation studies, the proposed method gave unbiased treatment estimates and confidence intervals with good coverage when the true sensitivity parameters were known. We describe application of the method to a randomized controlled trial and a non‐randomized study.     

Estimating allelic richness: effects of sample size and bottlenecks

Although differences in sampling intensity can bias comparisons of allelic richness (A) among populations, investigators often fail to correct estimates of A for differences in sample size. Methods that standardize A on the basis of the size of the smallest number of samples in a comparison are preferable to other approaches. Rarefaction and repeated random subsampling provide unbiased estimates of A with the greatest precision and thus provide greatest statistical power to detect differences in variation. Less promising approaches, in terms of bias or precision, include single random subsampling, eliminating very small samples, using sample size as a covariate or extrapolating estimates obtained from small samples to a larger number of individuals.     

Improving removal-based estimates of abundance by sampling a population of spatially distinct subpopulations

A statistical modeling framework is described for estimating the abundances of spatially distinct subpopulations of animals surveyed using removal sampling. To illustrate this framework, hierarchical models are developed using the Poisson and negative-binomial distributions to model variation in abundance among subpopulations and using the beta distribution to model variation in capture probabilities. These models are fitted to the removal counts observed in a survey of a federally endangered fish species. The resulting estimates of abundance have similar or better precision than those computed using the conventional approach of analyzing the removal counts of each subpopulation separately. Extension of the hierarchical models to include spatial covariates of abundance is straightforward and may be used to identify important features of an animal's habitat or to predict the abundance of animals at unsampled locations.     

Addressing Temporal Variability in Bird Calling with Design and Estimation: A Northern Bobwhite Example

Imprecise or biased density estimates can lead to inadequate conservation action, overexploitation of game species, or lost recreational opportunities. Common approaches to estimating density of avian populations often either ignore the probability that an individual is present within the sampling area but is not available to be sampled (e.g., not vocalizing), or do not consider covariates that could influence availability. Additionally, management decisions made at the management unit scale are often informed by inadequate monitoring practices, such as limited sampling intensity. In such cases, management agencies calculate density by applying correction factors (e.g., detection probabilities estimated using empirical data from a different study system) to count data, rather than estimating a detection function directly using statistical models. We conducted a simulation study using northern bobwhite (Colinus virginianus; bobwhite) as a model species to quantify the consequences of mis-specifying avian point count models on bias and precision of density estimates. We compared bias and precision of estimates from a fully specified distance-sampling model that estimates availability and detection to 4 different mis-specified approaches, including 2 approaches to calculating density using correction factors. Using correction factors to calculate density produced estimates with low bias but relatively lower precision compared to the fully specified model (CV of density estimates at 35 sites over 5 years: fully specified = 10%, correction factors = 25% and 30%). Although the mean precision and bias of the fully specified model improved with more data (70 sites over 5 years, CV = 9%; 35 sites over 10 years, CV = 9%), precision of correction factors did not (70 sites over 5 years, CV = 22% and 27%; 35 sites over 10 years, CV = 24% and 29%). The fully specified model captured the underlying temporal variation in detection and availability. Increasing sampling duration from 5 to 10 years improved modeled estimates of growth rate, even for mis-specified models, but not derived growth rates using pre-determined detection functions. We demonstrated that conducting point counts 3 times/year at a feasible number of sites can produce relatively unbiased estimates of bobwhite density. Pre-determined detection functions can be fortuitously unbiased for certain years, but they are not a reliable method for determining density or identifying trends in density over time. © 2020 The Wildlife Society.     

Accounting for weather and time-of-day parameters when analysing count data from monitoring programs

Problems induced by heterogeneity in species and individuals detectability are now well recognized when analysing count data. Yet, most recent techniques developed to handle this problem are still hardly applicable to many monitoring schemes, and do not provide abundance estimates at the point count scale. Here, we show how using simple weather variables can be a useful surrogate to detect variability in species detectability. We further look for a potential bias or loss in statistical power based on count data while ignoring weather and time-of-day variables. We first used the French Breeding Bird Survey to test how each of the counts of the 97 most common breeding species was influenced by weather and time-of-day variables. We assessed how the estimation of each species response to fragmentation could be influenced by correcting counts with such variables. Among 97 species, 75 were affected by at least one of the five weather and time-of-day variables considered. Despite these strong influences, the relationship between species abundance and fragmentation was not biased when not controlling counts for weather and time-of-day variables and further found no improvement in statistical power when accounting for these variables. Our results show that simple variables can be very powerful to assess how species detectability is influenced by weather conditions but they are inconsistent with any specific bias due to heterogeneous detectability. We suggest that raw count data can be used without any correction in case the sources of variation in detectability could be considered independent to the factor of interest.     

Does the punishment fit the crime? Consequences and diagnosis of misspecified detection functions in Bayesian spatial capture–recapture modeling

Spatial capture–recapture (SCR) analysis is now used routinely to inform wildlife management and conservation decisions. It is therefore imperative that we understand the implications of and can diagnose common SCR model misspecifications, as flawed inferences could propagate to policy and interventions. The detection function of an SCR model describes how an individual''s detections are distributed in space. Despite the detection function''s central role in SCR, little is known about the robustness of SCR‐derived abundance estimates and home range size estimates to misspecifications. Here, we set out to (a) determine whether abundance estimates are robust to a wider range of misspecifications of the detection function than previously explored, (b) quantify the sensitivity of home range size estimates to the choice of detection function, and (c) evaluate commonly used Bayesian p‐values for detecting misspecifications thereof. We simulated SCR data using different circular detection functions to emulate a wide range of space use patterns. We then fit Bayesian SCR models with three detection functions (half‐normal, exponential, and half‐normal plateau) to each simulated data set. While abundance estimates were very robust, estimates of home range size were sensitive to misspecifications of the detection function. When misspecified, SCR models with the half‐normal plateau and exponential detection functions produced the most and least reliable home range size, respectively. Misspecifications with the strongest impact on parameter estimates were easily detected by Bayesian p‐values. Practitioners using SCR exclusively for density estimation are unlikely to be impacted by misspecifications of the detection function. However, the choice of detection function can have substantial consequences for the reliability of inferences about space use. Although Bayesian p‐values can aid the diagnosis of detection function misspecification under certain conditions, we urge the development of additional custom goodness‐of‐fit diagnostics for Bayesian SCR models to identify a wider range of model misspecifications.     

Patterns of treatment effects in subsets of patients in clinical trials

We discuss the practice of examining patterns of treatment effects across overlapping patient subpopulations. In particular, we focus on the case in which patient subgroups are defined to contain patients having increasingly larger (or smaller) values of one particular covariate of interest, with the intent of exploring the possible interaction between treatment effect and that covariate. We formalize these subgroup approaches (STEPP: subpopulation treatment effect pattern plots) and implement them when treatment effect is defined as the difference in survival at a fixed time point between two treatment arms. The joint asymptotic distribution of the treatment effect estimates is derived, and used to construct simultaneous confidence bands around the estimates and to test the null hypothesis of no interaction. These methods are illustrated using data from a clinical trial conducted by the International Breast Cancer Study Group, which demonstrates the critical role of estrogen receptor content of the primary breast cancer for selecting appropriate adjuvant therapy. The considerations are also relevant for general subset analysis, since information from the same patients is typically used in the estimation of treatment effects within two or more subgroups of patients defined with respect to different covariates.     

The importance of incorporating imperfect detection in biodiversity assessments: a case study of small mammals in an Australian region

Aim Assessments of biodiversity are an essential requirement of conservation management planning. Species distributional modelling is a popular approach to quantifying biodiversity whereby occurrence data are related to environmental covariates. An important confounding factor that is often overlooked in the development of such models is uncertainty due to imperfect detection. Here, I demonstrate how an analytical approach that accounts for the bias due to imperfect detection can be applied retrospectively to an existing biodiversity survey data set to produce more realistic estimates of species distributions and unbiased covariate relationships. Location Pilbara biogeographic region, Australia. Methods As a component of the Pilbara survey, presence/absence (i.e. undetected) data on small ground‐dwelling mammals were collected. I applied a multiseason occupancy modelling approach to six of the most common species encountered during this survey. Detection and occupancy rates, as well as extinction and colonization probabilities, were determined, and the influence of covariates on these parameters was examined using the multi‐model inference approach. Results Detection probabilities for all six species were considerably lower than 1.0 and varied across time and species. Naïve estimates of occupancy underestimated occupancy rates corrected for species detectability by up to 45%. Seasonal variation in occupancy status was attributed to changes in detection for two of the focal species, while reproductive events explained variation in occupancy in three others. Covariates describing the substrate strongly influenced site occupancy for most of the species modelled. A comparison of the occupancy model with a generalized linear model, assuming perfect detection, showed that the effects of the covariates were underestimated in the latter model. Main conclusions The application of the multiseason occupancy modelling approach to the Pilbara mammal data set demonstrated a powerful framework for examining changes in site occupancy, as well as local colonization and extinction rates of species which are not confounded by variable species detection rates.     

Detection heterogeneity in underwater visual‐census data

This study shows how capture–mark–recapture (CMR) models can provide robust estimates of detection heterogeneity (sources of bias) in underwater visual‐census data. Detection biases among observers and fish family groups were consistent between fished and unfished reef sites in Kenya, even when the overall level of detection declined between locations. Species characteristics were the greatest source of detection heterogeneity and large, highly mobile species were found to have lower probabilities of detection than smaller, site‐attached species. Fish family and functional‐group detectability were also found to be lower at fished locations, probably due to differences in local abundance. Because robust CMR models deal explicitly with sampling where not all species are detected, their use is encouraged for studies addressing reef‐fish community dynamics.     

Genetic variation,population structure and identification of yellow catfish, <Emphasis Type="Italic">Mystus nemurus</Emphasis> (C&;V) in Thailand using RAPD,ISSR and SCAR marker

Random amplified polymorphic DNA (RAPD) and inter-simple sequence repeat (ISSR) markers were used to investigate the genetic structure of four subpopulations of Mystus nemurus in Thailand. The 7 RAPD and 7 ISSR primers were selected. Of 83 total RAPD fragments, 80 (96.39%) were polymorphic loci, and of 81 total ISSR fragments, 75 (92.59%) were polymorphic loci. Genetic variation and genetic differentiation obtained from RAPD fragments or ISSR fragments showed similar results. Percentage of polymorphic loci (%P), observed number of alleles, effective number of alleles, Nei’s gene diversity (H) and Shannon’s information index revealed moderate to high level of genetic variations within each M. nemurus subpopulation and overall population. High levels of genetic differentiations were received from pairwise unbiased genetic distance (D) and coefficient of differentiation. Mantel test between D or gene flow and geographical distance showed a low to moderate correlation. Analysis of molecular variance indicated that variations among subpopulations were higher than those within subpopulations. The UPGMA dendrograms, based on RAPD and ISSR, showing the genetic relationship among subpopulations are grouped into three clusters; Songkhla (SK) subpopulation was separated from the other subpopulations. The candidate species-specific and subpopulation-specific RAPD fragments were sequenced and used to design sequence-characterized amplified region primers which distinguished M. nemurus from other species and divided SK subpopulation from the other subpopulations. The markers used in this study should be useful for breeding programs and future aquacultural development of this species in Thailand.     

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models

Line transect sampling is one of the most widely used methods for animal abundance assessment. Standard estimation methods assume certain detection on the transect, no animal movement, and no measurement errors. Failure of the assumptions can cause substantial bias. In this work, the effect of error measurement on line transect estimators is investigated. Based on considerations of the process generating the errors, a multiplicative error model is presented and a simple way of correcting estimates based on knowledge of the error distribution is proposed. Using beta models for the error distribution, the effect of errors and of the proposed correction is assessed by simulation. Adequate confidence intervals for the corrected estimates are obtained using a bootstrap variance estimate for the correction and the delta method. As noted by Chen (1998, Biometrics 54, 899-908), even unbiased estimators of the distances might lead to biased density estimators, depending on the actual error distribution. In contrast with the findings of Chen, who used an additive model, unbiased estimation of distances, given a multiplicative model, lead to overestimation of density. Some error distributions result in observed distance distributions that make efficient estimation impossible, by removing the shoulder present in the original detection function. This indicates the need to improve field methods to reduce measurement error. An application of the new methods to a real data set is presented.