Unmodeled Heterogeneity Research Articles

Accounting for misidentification and heterogeneity in occupancy studies using hidden Markov models

Occupancy models allow assessing species occurrence while accounting for imperfect detection. As with any statistical models, occupancy models rely on several assumptions amongst which (i) there should be no unmodelled heterogeneity in the detection probability and (ii) the species should not be detected when absent from a site, in other words there should be no false positives (e.g., due to misidentification). In the real world, these two assumptions are often violated. To date, models accounting simultaneously for both detection heterogeneity and false positives are yet to be developed. Here, we first show how occupancy models with false positives can be formulated as hidden Markov models (HMM). Second, benefiting from the HMM framework flexibility, we extend models with false positives to account for heterogeneity with finite mixtures. First, using simulations, we demonstrate that, as the level of heterogeneity increases, occupancy models accounting for both heterogeneity and misidentification perform better in terms of bias and precision than models accounting for misidentification only. Next, we illustrate the implementation of our new model to a real case study with grey wolves (Canis lupus) in France. We demonstrate that heterogeneity in wolf detection (false negatives) is mainly due to a heterogeneous sampling effort across space. In addition to providing a novel modeling formulation, this work illustrates the flexibility of HMM framework to formulate complex ecological models and relax important assumptions that are not always likely to hold. In particular, we show how to decompose the model structure in several simple components, in a way that provides much clearer ecological interpretation.

Beyond path diagrams: Enhancing applied structural equation modeling research through data visualization

IntroductionStructural equation modeling (SEM) is a multivariate data analytic technique used in many domains of addictive behaviors research. SEM results are usually summarized and communicated through statistical tables and path diagrams, which emphasize path coefficients and global fit without showing specific quantitative values of data points that underlie the model results. Data visualization methods are often absent in SEM research, which may limit the quality and impact of SEM research by reducing data transparency, obscuring unexpected data anomalies and unmodeled heterogeneity, and inhibiting the communication of SEM research findings to research stakeholders who do not have advanced statistical training in SEM. Methods and resultsIn this report, we show how data visualization methods can address these limitations and improve the quality of SEM-based addictive behaviors research. We first introduce SEM and data visualization methodologies and differentiate data visualizations from model visualizations that are commonly used in SEM, such as path diagrams. We then discuss ways researchers may utilize data visualization in SEM research, including by obtaining estimates of latent variables and by visualizing multivariate relations in two-dimensional figures. R syntax is provided to help others generate data visualizations for several types of effects commonly modeled in SEM, including correlation, regression, moderation, and simple mediation. DiscussionThe techniques outlined here may help spur the use of data visualization in SEM-based addictive behaviors research. Using data visualization in SEM may enhance methodological transparency and improve communication of research findings.

The importance of core habitat for a threatened species in changing landscapes

Abstract Habitat loss, fragmentation, and alteration of the landscape matrix are interdependent processes, collectively responsible for most recent species extinctions. Thus, determining the extent to which these landscape processes affect animals is critical for conservation. However, researchers have often assumed that interdependent effects are independently related to animals’ responses, underestimating the importance of one or several landscape processes in driving species declines. We demonstrate how to disentangle the interdependent effects of habitat amount, fragmentation, and edge context on population size by assessing abundance of a rapidly declining grassland songbird species (grasshopper sparrow Ammodramus savannarum) in eastern Kansas (USA). We conducted >7,000 point count surveys at >2,000 sites over two breeding seasons, then modelled the direct, interactive, and indirect effects of landscape factors on abundance within spatial scales (200‐, 400‐, 800‐, and 1,600‐m radii) relevant to our focal species’ dispersal behaviour. Sparrow abundance correlated most strongly with landscape structure within 400‐m radii, increasing nonlinearly with grassland area and decreasing with the proportion of grassland near cropland or woody edges. Sparrows’ negative response to cropland edges was mostly an added, indirect consequence of reduced grassland area, whereas sparrows’ stronger negative response to woody edges was not attributable to variation in grassland area. Fragmentation and edge context mattered most in landscapes comprising c. 50%–80% grassland. Synthesis and applications. In our research, abundance of a threatened grassland songbird was influenced more by core grassland area (a function of total grassland area, fragmentation, and edge context) than total grassland area per se. Moreover, a local extinction threshold of c. 50% grassland indicated that small amounts of habitat were unsuitable for our focal species regardless of habitat configuration or matrix type. Local extinction thresholds in response to habitat area provide clear baseline targets for land managers; above those thresholds, configuration and the matrix can be modified to increase abundance of edge‐sensitive animals. Conflicting evidence in the literature regarding the importance of fragmentation and matrix features could be partially explained by species‐level traits, or methodological issues such as defining landscapes at ecologically arbitrary spatial scales, assessing landscape quality using species richness, and ignoring interactive and indirect effects.

Simultaneous modelling of movement, measurement error, and observer dependence in mark-recapture distance sampling: An application to Arctic bird surveys

Mark-recapture distance sampling is a promising method for surveying bird populations from aircraft in open landscapes. However, commonly available distance sampling estimators require that distances to target animals are made without error and that animals are stationary while sampling is being conducted. Motivated by a recent bird survey where these requirements were routinely violated, we describe a marginal likelihood framework for estimating abundance from double-observer data that can accommodate movement and measurement error when observations are made consecutively (as with front and rear observers), when animals are uniformly distributed during detection by the first observer, and when detections consist of both moving and stationary animals. Assuming that all animals are subject to measurement error and that some animals can move between detections, we integrate over unknown animal locations to construct a marginal likelihood for detection, movement, and measurement error parameters. Estimates of animal abundance are then obtained using a modified Horvitz–Thompson-like estimator. In addition, unmodelled heterogeneity in detection probability can be accommodated through observer dependence parameters. Using simulation, we show that our approach yields low bias compared to approaches that ignore movement and/or measurement error, including in cases where there is considerable detection heterogeneity. Applying our approach to data from a double-observer waterfowl helicopter survey in northern Canada, we are able to estimate bird density accounting for movement and measurement error and corrected for observer heterogeneity. Our approach appears promising for generating unbiased estimates of bird abundance necessary for reliable conservation and management.

Fitting N-mixture models to count data with unmodeled heterogeneity: Bias, diagnostics, and alternative approaches

Monitoring animal populations is central to wildlife and fisheries management, and the use of N-mixture models toward these efforts has markedly increased in recent years. Nevertheless, relatively little work has evaluated estimator performance when basic assumptions are violated. Moreover, diagnostics to identify when bias in parameter estimates from N-mixture models is likely is largely unexplored. We simulated count data sets using 837 combinations of detection probability, number of sample units, number of survey occasions, and type and extent of heterogeneity in abundance or detectability. We fit Poisson N-mixture models to these data, quantified the bias associated with each combination, and evaluated if the parametric bootstrap goodness-of-fit (GOF) test can be used to indicate bias in parameter estimates. We also explored if assumption violations can be diagnosed prior to fitting N-mixture models. In doing so, we propose a new model diagnostic, which we term the quasi-coefficient of variation (QCV). N-mixture models performed well when assumptions were met and detection probabilities were moderate (i.e., ≥0.3), and the performance of the estimator improved with increasing survey occasions and sample units. However, the magnitude of bias in estimated mean abundance with even slight amounts of unmodeled heterogeneity was substantial. The parametric bootstrap GOF test did not perform well as a diagnostic for bias in parameter estimates when detectability and sample sizes were low. The results indicate the QCV is useful to diagnose potential bias and that potential bias associated with unidirectional trends in abundance or detectability can be diagnosed using Poisson regression. This study represents the most thorough assessment to date of assumption violations and diagnostics when fitting N-mixture models using the most commonly implemented error distribution. Unbiased estimates of population state variables are needed to properly inform management decision making. Therefore, we also discuss alternative approaches to yield unbiased estimates of population state variables using similar data types, and we stress that there is no substitute for an effective sample design that is grounded upon well-defined management objectives.

A Bayesian multinomial logistic exposure model for estimating probabilities of competing sources of nest failure

Understanding causes of nest loss is critical for the management of endangered bird populations. Available methods for estimating nest loss probabilities to competing sources do not allow for random effects and covariation among sources, and there are few data simulation methods or goodness‐of‐fit (GOF) tests for such models. We developed a Bayesian multinomial extension of the widely used logistic exposure (LE) nest survival model which can incorporate multiple random effects and fixed‐effect covariates for each nest loss category. We investigated the performance of this model and the accompanyingGOFtest by analysing simulated nest fate datasets with and without age‐biased discovery probability, and by comparing the estimates with those of traditional fixed‐effects estimators. We then exemplify the use of the multinomialLEmodel andGOFtest by analysing Piping PloverCharadrius melodusnest fate data (n = 443) to explore the effects of wire cages (exclosures) constructed around nests, which are used to protect nests from predation but can lead to increased nest abandonment rates. Mean parameter estimates of the random‐effects multinomialLEmodel were all within 1 sd of the true values used to simulate the datasets. Age‐biased discovery probability did not result in biased parameter estimates. Traditional fixed‐effects models provided estimates with a high bias of up to 43% with a mean of 71% smaller standard deviations. TheGOFtest identified models that were a poor fit to the simulated data. For the Piping Plover dataset, the fixed‐effects model was less well‐supported than the random‐effects model and underestimated the risk of exclosure use by 16%. The random‐effects model estimated a range of 1–6% probability of abandonment for nests not protected by exclosures across sites and 5–41% probability of abandonment for nests with exclosures, suggesting that the magnitude of exclosure‐related abandonment is site‐specific. Our results demonstrate that unmodelled heterogeneity can result in biased estimates potentially leading to incorrect management recommendations. The Bayesian multinomialLEmodel offers a flexible method of incorporating random effects into an analysis of nest failure and is robust to age‐biased nest discovery probability. This model can be generalized to other staggered‐entry, time‐to‐hazard situations.

Environmental Kuznets curve among BRICS countries: Spot lightening finance, transport, energy and growth factors

The environmental sustainability reforms considered the highly successful factor for the countries that devoted for green investment in transportation, financial activities, energy demand, and growth factors. This challenging task is monetized by fair environmental policies that integrated with the countries action plans. BRICS countries is no exception, that wider accepted this environmental task and provoked the need of healthy investment for sustainable growth. This study reviewed finance, transport, energy, and growth factors under Environmental Kuznets Curve (EKC) framework in a panel of BRICS countries by utilizing a consistent time series data from 1990 to 2015. The study employed panel fixed effect regression to observe unmodeled heterogeneity among the cross-section identifiers that vary over a period of time. The results show that transport services and industry value added (INDVAD) both escalate fossil fuel (FFUEL) energy consumption while financial factors, environmental prices, and renewable energy consumption (REC) decreases fossil energy in a region. Bank capital, broad money supply, domestic credit, railways goods transported, travel services, and INDVAD considerably increases nitrous oxide emissions (N2O) whereas; REC significantly decreases N2O emissions in a panel of countries. Greenhouse gas (GHG) emissions affected by domestic credit, railways goods transported, travel services, and INDVAD. There is a monotonic decreasing relationships between i) per capita income and FFUEL energy, ii) railways passenger carried and N2O emissions, iii) REC and GHG emissions, and iv) income and N2O emissions, whereas; there is an inverted U-shaped relationship between broad money supply and N2O emissions and U-shaped relationships between per capita income and GHG emissions. The panel causality tests confirmed the bidirectional relationship between financial indicators and railways transportation, while bank capital Granger cause FFUEL energy, per capita income, INDVAD, and REC (but not other way around) in a panel of BRICS countries. The inter-temporal relationships show that bank capital, INDVAD, and price level will largely affect FFUEL energy, N2O emissions, and GHG emissions respectively, for the next 10 years period. The results of the study emphasized the need of ‘green policy instruments’ that should be connected with the country’s policies and program for sustained growth.

Designing occupancy studies when false‐positive detections occur

Summary Recently, estimators have been developed to estimate occupancy probabilities when false‐positive detections occur during presence–absence surveys. Some of these estimators combine different types of survey data to improve estimates of occupancy. With these estimators, there is a trade‐off between the number of sample units surveyed, and the number and type of surveys at each sample unit. Guidance on efficient design of studies when false positives occur is unavailable. For a range of scenarios, I identified survey designs that minimized the mean square error of the estimate of occupancy. I considered an approach that uses one survey method and two observation states and an approach that uses two survey methods. For each approach, I used numerical methods to identify optimal survey designs when model assumptions were met and parameter values were correctly anticipated, when parameter values were not correctly anticipated and when the assumption of no unmodelled detection heterogeneity was violated. Under the approach with two observation states, false‐positive detections increased the number of recommended surveys, relative to standard occupancy models. If parameter values could not be anticipated, pessimism about detection probabilities avoided poor designs. Detection heterogeneity could require more or fewer repeat surveys, depending on parameter values. If model assumptions were met, the approach with two survey methods was inefficient. However, with poor anticipation of parameter values, with detection heterogeneity or with removal sampling schemes, combining two survey methods could improve estimates of occupancy. Ignoring false positives can yield biased parameter estimates, yet false positives greatly complicate the design of occupancy studies. Specific guidance for major types of false‐positive occupancy models, and for two assumption violations common in field data, can conserve survey resources. This guidance can be used to design efficient monitoring programmes and studies of species occurrence, species distribution or habitat selection, when false positives occur during surveys.

Capture-Recapture Estimators in Epidemiology with Applications to Pertussis and Pneumococcal Invasive Disease Surveillance.

IntroductionSurveillance networks are often not exhaustive nor completely complementary. In such situations, capture-recapture methods can be used for incidence estimation. The choice of estimator and their robustness with respect to the homogeneity and independence assumptions are however not well documented.MethodsWe investigated the performance of five different capture-recapture estimators in a simulation study. Eight different scenarios were used to detect and combine case-information. The scenarios increasingly violated assumptions of independence of samples and homogeneity of detection probabilities. Belgian datasets on invasive pneumococcal disease (IPD) and pertussis provided motivating examples.ResultsNo estimator was unbiased in all scenarios. Performance of the parametric estimators depended on how much of the dependency and heterogeneity were correctly modelled. Model building was limited by parameter estimability, availability of additional information (e.g. covariates) and the possibilities inherent to the method. In the most complex scenario, methods that allowed for detection probabilities conditional on previous detections estimated the total population size within a 20–30% error-range. Parametric estimators remained stable if individual data sources lost up to 50% of their data. The investigated non-parametric methods were more susceptible to data loss and their performance was linked to the dependence between samples; overestimating in scenarios with little dependence, underestimating in others. Issues with parameter estimability made it impossible to model all suggested relations between samples for the IPD and pertussis datasets. For IPD, the estimates for the Belgian incidence for cases aged 50 years and older ranged from 44 to58/100,000 in 2010. The estimates for pertussis (all ages, Belgium, 2014) ranged from 24.2 to30.8/100,000.ConclusionWe encourage the use of capture-recapture methods, but epidemiologists should preferably include datasets for which the underlying dependency structure is not too complex, a priori investigate this structure, compensate for it within the model and interpret the results with the remaining unmodelled heterogeneity in mind.

Accounting for Lack of Independence and Partial Overlap of Observation Zones in Line-Transect Mark-Recapture Distance Sampling

Line-transect mark-recapture distance sampling methods can be used to estimate abundance when at least two observers sight and record distances to detected groups of individuals within the survey area. However, a lack of independence between the observer’s detections will cause biased abundance estimates. Studies are also typically designed such that there is complete overlap of the regions searched by the two observers, but that may not always be possible. Here we detail an intuitive approach for line-transect distance sampling applications based upon logistic regression to account for a potential lack of independence by using the detections of one observer as a covariate in the detection function of the second observer. Partial overlap of the observer survey regions can be addressed by constraining detection probability to equal 0 for the respective observer outside of the overlap zone. We show via simulation that the method provides reliable estimates of abundance and is not affected by random unmodeled heterogeneity in detection probability. The method is illustrated by estimating abundance within the covered region of an aerial line-transect survey for New Zealand’s endemic Hector’s dolphin (Cephalorhynchus hectori hectori) conducted in the austral summer of 2013, the motivating application for this work. Supplementary materials accompanying this paper appear on-line.

Comparing spatial capture–recapture modeling and nest count methods to estimate orangutan densities in the Wehea Forest, East Kalimantan, Indonesia

Accurate information on the density and abundance of animal populations is essential for understanding species' ecology and for conservation planning, but is difficult to obtain. The endangered orangutan (Pongo spp.) is an example; due to its elusive behavior and low densities, researchers have relied on methods that convert nest counts to orangutan densities and require substantial effort for reliable results. Camera trapping and spatial capture–recapture (SCR) models could provide an alternative but have not been used for primates. We compared density estimates calculated using the two methods for orangutans in the Wehea Forest, East Kalimantan, Indonesia. Camera trapping/SCR modeling produced a density estimate of 0.16±0.09–0.29 indiv/km2, and nest counts produced a density estimate of 1.05±0.18–6.01 indiv/km2. The large confidence interval of the nest count estimate is probably due to high variance in nest encounter rates, indicating the need for larger sample size and the substantial effort required to produce reliable results using this method. The SCR estimate produced a very low density estimate and had a narrower but still fairly wide confidence interval. This was likely due to unmodeled heterogeneity and small sample size, specifically a low number of individual captures and recaptures. We propose methodological fixes that could address these issues and improve precision. A comparison of the overall costs and benefits of the two methods suggests that camera trapping/SCR modeling can potentially be a useful tool for assessing the densities of orangutans and other elusive primates, and warrant further investigation to determine broad applicability and methodological adjustments needed.

Does Cox analysis of a randomized survival study yield a causal treatment effect?

Statistical methods for survival analysis play a central role in the assessment of treatment effects in randomized clinical trials in cardiovascular disease, cancer, and many other fields. The most common approach to analysis involves fitting a Cox regression model including a treatment indicator, and basing inference on the large sample properties of the regression coefficient estimator. Despite the fact that treatment assignment is randomized, the hazard ratio is not a quantity which admits a causal interpretation in the case of unmodelled heterogeneity. This problem arises because the risk sets beyond the first event time are comprised of the subset of individuals who have not previously failed. The balance in the distribution of potential confounders between treatment arms is lost by this implicit conditioning, whether or not censoring is present. Thus while the Cox model may be used as a basis for valid tests of the null hypotheses of no treatment effect if robust variance estimates are used, modeling frameworks more compatible with causal reasoning may be preferrable in general for estimation.

Distance sampling with a random scale detection function

Distance sampling was developed to estimate wildlife abundance from observational surveys with uncertain detection in the search area. We present novel analysis methods for estimating detection probabilities that make use of random effects models to allow for unmodeled heterogeneity in detection. The scale parameter of the half-normal detection function is modeled by means of an intercept plus an error term varying with detections, normally distributed with zero mean and unknown variance. In contrast to conventional distance sampling methods, our approach can deal with long-tailed detection functions without truncation. Compared to a fixed effect covariate approach, we think of the random effect as a covariate with unknown values and integrate over the random effect. We expand the random scale to a mixed scale model by adding fixed effect covariates. We analyzed simulated data with large sample sizes to demonstrate that the code performs correctly for random and mixed effect models. We also generated replicate simulations with more practical sample sizes (\(^{\sim }100\)) and compared the random scale half-normal with the hazard rate detection function. As expected each estimation model was best for different simulation models. We illustrate the mixed effect modeling approach using harbor porpoise vessel survey data where the mixed effect model provided an improved model fit in comparison to a fixed effect model with the same covariates. We propose that a random or mixed effect model of the detection function scale be adopted as one of the standard approaches for fitting detection functions in distance sampling.

Reassessment of HIV-1 Acute Phase Infectivity: Accounting for Heterogeneity and Study Design with Simulated Cohorts

BackgroundThe infectivity of the HIV-1 acute phase has been directly measured only once, from a retrospectively identified cohort of serodiscordant heterosexual couples in Rakai, Uganda. Analyses of this cohort underlie the widespread view that the acute phase is highly infectious, even more so than would be predicted from its elevated viral load, and that transmission occurring shortly after infection may therefore compromise interventions that rely on diagnosis and treatment, such as antiretroviral treatment as prevention (TasP). Here, we re-estimate the duration and relative infectivity of the acute phase, while accounting for several possible sources of bias in published estimates, including the retrospective cohort exclusion criteria and unmeasured heterogeneity in risk.Methods and FindingsWe estimated acute phase infectivity using two approaches. First, we combined viral load trajectories and viral load-infectivity relationships to estimate infectivity trajectories over the course of infection, under the assumption that elevated acute phase infectivity is caused by elevated viral load alone. Second, we estimated the relative hazard of transmission during the acute phase versus the chronic phase (RHacute) and the acute phase duration (d acute) by fitting a couples transmission model to the Rakai retrospective cohort using approximate Bayesian computation. Our model fit the data well and accounted for characteristics overlooked by previous analyses, including individual heterogeneity in infectiousness and susceptibility and the retrospective cohort's exclusion of couples that were recorded as serodiscordant only once before being censored by loss to follow-up, couple dissolution, or study termination. Finally, we replicated two highly cited analyses of the Rakai data on simulated data to identify biases underlying the discrepancies between previous estimates and our own.From the Rakai data, we estimated RHacute = 5.3 (95% credibility interval [95% CrI]: 0.79–57) and d acute = 1.7 mo (95% CrI: 0.55–6.8). The wide credibility intervals reflect an inability to distinguish a long, mildly infectious acute phase from a short, highly infectious acute phase, given the 10-mo Rakai observation intervals. The total additional risk, measured as excess hazard-months attributable to the acute phase (EHMacute) can be estimated more precisely: EHMacute = (RHacute - 1) × d acute, and should be interpreted with respect to the 120 hazard-months generated by a constant untreated chronic phase infectivity over 10 y of infection. From the Rakai data, we estimated that EHMacute = 8.4 (95% CrI: -0.27 to 64). This estimate is considerably lower than previously published estimates, and consistent with our independent estimate from viral load trajectories, 5.6 (95% confidence interval: 3.3–9.1). We found that previous overestimates likely stemmed from failure to account for risk heterogeneity and bias resulting from the retrospective cohort study design.Our results reflect the interaction between the retrospective cohort exclusion criteria and high (47%) rates of censorship amongst incident serodiscordant couples in the Rakai study due to loss to follow-up, couple dissolution, or study termination. We estimated excess physiological infectivity during the acute phase from couples data, but not the proportion of transmission attributable to the acute phase, which would require data on the broader population's sexual network structure.ConclusionsPrevious EHMacute estimates relying on the Rakai retrospective cohort data range from 31 to 141. Our results indicate that these are substantial overestimates of HIV-1 acute phase infectivity, biased by unmodeled heterogeneity in transmission rates between couples and by inconsistent censoring. Elevated acute phase infectivity is therefore less likely to undermine TasP interventions than previously thought. Heterogeneity in infectiousness and susceptibility may still play an important role in intervention success and deserves attention in future analyses

What Are We Weighting For?

abstract:When estimating population descriptive statistics, weighting is called for if needed to make the analysis sample representative of the target population. With regard to research directed instead at estimating causal effects, we discuss three distinct weighting motives: (1) to achieve precise estimates by correcting for heteroskedasticity; (2) to achieve consistent estimates by correcting for endogenous sampling; and (3) to identify average partial effects in the presence of unmodeled heterogeneity of effects. In each case, we find that the motive sometimes does not apply in situations where practitioners often assume it does.

A hybrid double-observer sightability model for aerial surveys

Raw counts from aerial surveys make no correction for undetected animals and provide no estimate of precision with which to judge the utility of the counts. Sightability modeling and double-observer (DO) modeling are 2 commonly used approaches to account for detection bias and to estimate precision in aerial surveys. We developed a hybrid DO sightability model (model MH) that uses the strength of each approach to overcome the weakness in the other, for aerial surveys of elk (Cervus elaphus). The hybrid approach uses detection patterns of 2 independent observer pairs in a helicopter and telemetry-based detections of collared elk groups. Candidate MH models reflected hypotheses about effects of recorded covariates and unmodeled heterogeneity on the separate front-seat observer pair and back-seat observer pair detection probabilities. Group size and concealing vegetation cover strongly influenced detection probabilities. The pilot's previous experience participating in aerial surveys influenced detection by the front pair of observers if the elk group was on the pilot's side of the helicopter flight path. In 9 surveys in Mount Rainier National Park, the raw number of elk counted was approximately 80–93% of the abundance estimated by model MH. Uncorrected ratios of bulls per 100 cows generally were low compared to estimates adjusted for detection bias, but ratios of calves per 100 cows were comparable whether based on raw survey counts or adjusted estimates. The hybrid method was an improvement over commonly used alternatives, with improved precision compared to sightability modeling and reduced bias compared to DO modeling. © 2013 The Wildlife Society.

The effect of estimation method and sample size in multilevel structural equation modeling

Multilevel structural equation modeling (multilevel SEM) has become an established method to analyze multilevel multivariate data. The first useful estimation method was the pseudobalanced method. This method is approximate because it assumes that all groups have the same size, and ignores unbalance when it exists. In addition, full information maximum likelihood (ML) estimation is now available, which is often combined with robust chi‐squares and standard errors to accommodate unmodeled heterogeneity (MLR). In addition, diagonally weighted least squares (DWLS) methods have become available as estimation methods. This article compares the pseudobalanced estimation method, ML(R), and two DWLS methods by simulating a multilevel factor model with unbalanced data. The simulations included different sample sizes at the individual and group levels and different intraclass correlation (ICC). The within‐group part of the model posed no problems. In the between part of the model, the different ICC sizes had no effect. There is a clear interaction effect between number of groups and estimation method. ML reaches unbiasedness fastest, then the two DWLS methods, then MLR, and then the pseudobalanced method (which needs more than 200 groups). We conclude that both ML(R) and DWLS are genuine improvements on the pseudobalanced approximation. With small sample sizes, the robust methods are not recommended.

Double‐Observer Line Transect Methods: Levels of Independence

Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The resulting data supplement conventional distance sampling data with two-sample mark-recapture data. Like conventional mark-recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conventional mark-recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled heterogeneity in the mark-recapture component of the data. By modeling the covariance in detection probabilities with distance, we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including those currently in common use, which have intermediate levels of independence. We show how this framework can be used to provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from the North Sea and adjacent waters.

Effects of three-dimensional Earth structure on CMT earthquake parameters

We investigate errors in centroid earthquake parameters due to unmodeled structural heterogeneity. We generate a simulated dataset consisting of synthetic seismograms for 50 earthquakes and 150 stations distributed globally. To generate the synthetic seismograms we use a spectral-element wave-propagation package (SPECFEM3D_GLOBE) that accounts for the Earth’s three-dimensional structure. An established centroid–moment-tensor inversion algorithm from the Global CMT project is used to invert the synthetic dataset, with and without added noise, for earthquake source parameters. This algorithm uses a one-dimensional earth structure, together with approximate corrections for three-dimensional structure, to model the seismograms. We interpret the differences between the estimated source parameters and the parameters used to compute the synthetic dataset as errors due to unmodeled structural heterogeneity and the presence of noise. We expect that the errors obtained in this study are representative of the errors in the Global CMT catalogue. We find that the errors in scalar moment, moment-tensor elements and location are small on average. The depth and centroid time are, however, biased by a small amount. We find that the error in depth can be reduced significantly by applying corrections for the difference in the velocity structure at the source and receiver locations from the Earth’s average structure in the CMT inversions. This modification has a minimal effect on the errors in centroid time. We do not find large errors in scalar moments, even where the crustal thickness at the source is very different from the Earth’s average crustal thickness.

Open Capture–Recapture Models with Heterogeneity: II. Jolly–Seber Model

Estimation of abundance is important in both open and closed population capture-recapture analysis, but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates. This article defines and develops a suite of open population capture-recapture models using finite mixtures to model heterogeneity of capture and survival probabilities. Model comparisons and parameter estimation use likelihood-based methods. A real example is analyzed, and simulations are used to check the main features of the heterogeneous models, especially the quality of estimation of abundance, survival, recruitment, and turnover. The two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture, and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present.