Hidden population size estimation and diagnostics using two respondent-driven samples with applications in Armenia

Estimating the size of hidden populations is essential to understand the magnitude of social and healthcare needs, risk behaviors, and disease burden. However, due to the hidden nature of these populations, they are difficult to survey, and there are no gold standard size estimation methods. Many different methods and variations exist, and diagnostic tools are needed to help researchers assess method-specific assumptions as well as compare between methods. Further, because many necessary mathematical assumptions are unrealistic for real survey implementation, assessment of how robust methods are to deviations from the stated assumptions is essential. We describe diagnostics and assess the performance of a new population size estimation method, capture–recapture with successive sampling population size estimation (CR-SS-PSE), which we apply to data from 3 years of studies from three cities and three hidden populations in Armenia. CR-SS-PSE relies on data from two sequential respondent-driven sampling surveys and extends the successive sampling population size estimation (SS-PSE) framework by using the number of individuals in the overlap between the two surveys and a model for the successive sampling process to estimate population size. We demonstrate that CR-SS-PSE is more robust to violations of successive sampling assumptions than SS-PSE. Further, we compare the CR-SS-PSE estimates to population size estimations using other common methods, including unique object and service multipliers, wisdom of the crowd, and two-source capture–recapture to illustrate volatility across estimation methods.     

How emergence and death assumptions affect count-based estimates of butterfly abundance and lifespan

Transect count data form the basis of many butterfly and other insect monitoring programs worldwide. A clear understanding of the limitations of such datasets, including the potential for biases in the statistical methods used to analyze them, is therefore crucial. The classical Zonneveld model (CZ) can extract estimates of a suite of demographic parameters from transect count datasets, and has also been used in theoretical analyses of protandry and reproductive asynchrony. The CZ relies on strong assumptions about the emergence and death processes underlying observed transect count datasets. Though reasonable as a starting place, a growing body of empirical evidence suggests these assumptions will, in many cases, not hold. Here, I explore how violations of these assumptions bias CZ-based estimates of two key population parameters: total population size and mean individual lifespan. To do this, I generalize the Zonneveld model by relaxing the symmetrical emergence distribution and constant death rate assumptions such that the generalized models contain the CZ as a special case. Using the generalized models as data generating processes, I then show that the CZ is able to closely mimic the shape of the abundance time course produced by either variant of the generalized model under a wide range of conditions, but produces highly biased estimates of population size and mean lifespan in doing so. My analysis therefore demonstrates both that the CZ is not robust to violations of its emergence and death assumptions, and that a good observed fit to transect count data does not mean these assumptions are satisfied.     

Understanding implications of detection heterogeneity in wildlife abundance estimation

Negative bias in mark-recapture abundance estimators due to heterogeneity in detection (capture) probability is a well-known problem, but we believe most biologists do not understand why heterogeneity causes bias and how bias can be reduced. We demonstrate how heterogeneity creates dependence and bias in mark-recapture approaches to abundance estimation. In comparison, heterogeneity, and hence estimator bias, is not as problematic for distance sampling and mark-resight methods because both techniques estimate detection probabilities based on a known quantity. We show how the introduction of a known number of individuals planted into a study population prior to a mark-recapture survey can reduce bias from heterogeneity in detection probability. We provide examples with simulation and an analysis of motion-sensitive camera data from a study population of introduced eastern wild turkeys (Meleagris gallopavo silvestris) of known size with a subset of telemetered birds. In choosing a method for abundance estimation, careful consideration should be given to assumptions and how heterogeneity in detection probability can be accommodated for each application.     

Noninvasive Hair Sampling and Genetic Tagging of Co-Distributed Fishers and American Martens

ABSTRACT Estimation of abundance is important for assessing population responses to management actions. Accurate abundance estimates are particularly critical for monitoring temporal variation following reintroductions when the management goal is to attain population sizes capable of sustaining harvest. Numerous reintroductions have taken place in the Great Lakes region of North America, including efforts to restore extirpated fishers (Martes pennanti) and American martens (M. americana). We used a DNA-based noninvasive hair-snaring method based on one trap design and trapping -grid configuration, and evaluated capture—mark—recapture (CMR) analytical approaches to simultaneously estimate population size for co-distributed fishers and American martens in a 671-km² area of the Ottawa National Forest in the western Upper Peninsula of Michigan, USA. We included harvest as a final recapture period to increase probability of recapture and to evaluate potential violations of geographic closure assumptions. We used microsatellite markers to identify target species, eliminate congener species, and provide individual identity for estimation of abundance. Population estimates for fishers and martens on the study area ranged from 35 to 60 and 8 to 28, respectively. Estimators incorporating harvest data resulted in up to a 40% increase in abundance estimates relative to estimators without harvest. We considered population estimates not including harvest data the most appropriate for the study due to timing of sampling and environmental factors, but inclusion of harvested individuals was shown to be useful as a means to detect violations of the assumption of geographic closure. We suggest improvements on future CMR sampling designs for larger landscape scales of relevance to management through incorporation of habitat or historical harvest data. Noninvasive genetic methods that simultaneously estimate the numerical abundance of co-distributed species can greatly decrease assessment costs relative to traditional methods, and increase resulting demographic and ecological information.     

Inferring species richness using multispecies occupancy modeling: Estimation performance and interpretation

Multispecies occupancy models can estimate species richness from spatially replicated multispecies detection/non‐detection survey data, while accounting for imperfect detection. A model extension using data augmentation allows inferring the total number of species in the community, including those completely missed by sampling (i.e., not detected in any survey, at any site). Here we investigate the robustness of these estimates. We review key model assumptions and test performance via simulations, under a range of scenarios of species characteristics and sampling regimes, exploring sensitivity to the Bayesian priors used for model fitting. We run tests when assumptions are perfectly met and when violated. We apply the model to a real dataset and contrast estimates obtained with and without predictors, and for different subsets of data. We find that, even with model assumptions perfectly met, estimation of the total number of species can be poor in scenarios where many species are missed (>15%–20%) and that commonly used priors can accentuate overestimation. Our tests show that estimation can often be robust to violations of assumptions about the statistical distributions describing variation of occupancy and detectability among species, but lower‐tail deviations can result in large biases. We obtain substantially different estimates from alternative analyses of our real dataset, with results suggesting that missing relevant predictors in the model can result in richness underestimation. In summary, estimates of total richness are sensitive to model structure and often uncertain. Appropriate selection of priors, testing of assumptions, and model refinement are all important to enhance estimator performance. Yet, these do not guarantee accurate estimation, particularly when many species remain undetected. While statistical models can provide useful insights, expectations about accuracy in this challenging prediction task should be realistic. Where knowledge about species numbers is considered truly critical for management or policy, survey effort should ideally be such that the chances of missing species altogether are low.     

Comparing mark-recapture and constant removal protocols for estimating forager population size of the subterranean termite Coptotermes lacteus (Isoptera: Rhinotermitidae)

Forager population sizes of colonies of Coptotermes lacteus(Froggatt) in New South Wales were estimated using two methods: mark-recapture and constant removal, in two disturbed habitats: a pine plantation and cleared farmland. Mark-recapture population estimates were unrealistic and unreliable: they could be improbably large, over 200 million foragers, and they varied enormously between samples for each colony without any pattern. The constant removal population estimates could also be unrealistic: they could be negative or quite different when calculated using regression and maximum likelihood methods. However, the unrealistic results could be predicted reliably, and explained by the lack of re-contact with the sampling devices (bait stations) - a violation of an assumption of the method. This happened more frequently in the plantation than in the farmland, probably because of the greater abundance of alternative food sources in the plantation. Of the two methods, constant removal provided reasonable forager population estimates, relative to direct counts, at least some of the time, plus a mechanism by which reliability could be tested, whereas mark-recapture provided neither. Further refinement and testing of constant removal methods are urged to provide a more reliable population estimation technique for termites.     

Parentage versus two-generation analyses for estimating pollen-mediated gene flow in plant populations

Assessment of contemporary pollen-mediated gene flow in plants is important for various aspects of plant population biology, genetic conservation and breeding. Here, through simulations we compare the two alternative approaches for measuring pollen-mediated gene flow: (i) the NEIGHBORHOOD model--a representative of parentage analyses, and (ii) the recently developed TWOGENER analysis of pollen pool structure. We investigate their properties in estimating the effective number of pollen parents (N(ep)) and the mean pollen dispersal distance (delta). We demonstrate that both methods provide very congruent estimates of N(ep) and delta, when the methods' assumptions considering the shape of pollen dispersal curve and the mating system follow those used in data simulations, although the NEIGHBORHOOD model exhibits generally lower variances of the estimates. The violations of the assumptions, especially increased selfing or long-distance pollen dispersal, affect the two methods to a different degree; however, they are still capable to provide comparable estimates of N(ep). The NEIGHBORHOOD model inherently allows to estimate both self-fertilization and outcrossing due to the long-distance pollen dispersal; however, the TWOGENER method is particularly sensitive to inflated selfing levels, which in turn may confound and suppress the effects of distant pollen movement. As a solution we demonstrate that in case of TWOGENER it is possible to extract the fraction of intraclass correlation that results from outcrossing only, which seems to be very relevant for measuring pollen-mediated gene flow. The two approaches differ in estimation precision and experimental efforts but they seem to be complementary depending on the main research focus and type of a population studied.     

Joint estimation of migration rate and effective population size using the island model

Using the island model of population demography, I report that the demographic parameters migration rate and effective population size can be jointly estimated with equilibrium probabilities of identity in state calculated using a sample of genotypes collected at a single point in time from a single generation. The method, which uses moment-type estimators, applies to dioecious populations in which females and males have identical demography and monoecious populations with no selfing and requires that offspring genotypes are sampled following reproduction and prior to migration. I illustrate the estimation procedure using the infinite-island model with no mutation and the finite-island model with three kinds of mutation models. In the infinite-island model with no mutation, the estimators can be expressed as simple functions of estimates of the F-statistic parameters F(IT) and F(ST). In the finite-island model with mutation among k alleles, mutation rate, migration rate, and effective population size can be simultaneously estimated. The estimates of migration rate and effective population size are somewhat robust to violations in assumptions that may arise in empirical applications such as different kinds of mutation models and deviations from temporal equilibrium.     

The effect of proband designation on segregation analysis

In many family studies, it is often difficult to know exactly how the families were ascertained. Even if known, the circumstances under which the families came to the attention of the study may violate the assumptions of classical ascertainment bias correction. The purpose of this work was to investigate the effect on segregation analysis of violations of the assumptions of the classical ascertainment model. We simulated family data generated under a simple recessive model of inheritance. We then ascertained families under different "scenarios." These scenarios were designed to simulate actual conditions under which families come to the attention of-and then interact with-a clinic or genetic study. We show that how one designates probands, which one must do under the classical ascertainment model, can influence parameter estimation and hypothesis testing. We demonstrate that, in some cases, there may be no "correct" way to designate probands. Further, we show that interactions within the family, the conditions under which the genetic study must function, and even social influences can have a profound effect on segregation analysis. We also propose a method for dealing with the ascertainment problem that is applicable to almost any study situation.     

Phylogenetic estimation under codon models can be biased by codon usage heterogeneity

In theory, codon models that account for the dependence of nucleotide substitutions between codon positions as well as differences between synonymous and non-synonymous changes best describe the sequence evolution in protein coding genes. However, in practice we know little about the degree to which violations of the assumptions of codon model-based estimates occur, and how significant these artifacts may be. In nucleotide-based phylogenies from first and second codon positions in a concatenated plastid gene data set, two distantly related taxa--dinoflagellate and haptophyte plastids--were robustly grouped together. This artifactual grouping is attributed to the parallel heterogeneity in leucine (Leu) and serine (Ser) codon usages in the data set. Here, by using this data set, we demonstrated that codon-based phylogenetic estimations are seriously biased, robustly uniting the dinoflagellate and haptophyte plastids into a monophyletic clade, when the model assumption of homogeneity of codon composition was violated. Our results suggest that similar phylogenetic artifacts may occur via codon usage heterogeneity in any amino acids in codon model-based estimations. We advise that homogeneity in codon usage across taxa in a data set be confirmed before codon model-based phylogenetic estimation is attempted.     

Monte carlo studies of plant mating system estimation models: the one-pollen parent and mixed mating models

Estimation of mating system parameters in plant populations typically employs family-structured samples of progeny genotypes. These estimation models postulate a mixture of self-fertilization and random outcrossing. One assumption of such models concerns the distribution of pollen genotypes among eggs within single maternal families. Previous applications of the mixed mating model to mating system estimation have assumed that pollen genotypes are sampled randomly from the total population in forming outcrossed progeny within families. In contrast, the one-pollen parent model assumes that outcrossed progeny within a family share a single-pollen parent genotype. Monte Carlo simulations of family-structured sampling were carried out to examine the consequences of violations of the different assumptions of the two models regarding the distribution of pollen genotypes among eggs. When these assumptions are violated, estimates of mating system parameters may be significantly different from their true values and may exhibit distributions which depart from normality. Monte Carlo methods were also used to examine the utility of the bootstrap resampling algorithm for estimating the variances of mating system parameters. The bootstrap method gives variance estimates that approximate empirically determined values. When applied to data from two plant populations which differ in pollen genotype distributions within families, the two estimation procedures exhibit the same behavior as that seen with the simulated data.     

Capture–recapture analysis with a latent class model allowing for local dependence and observed heterogeneity

In epidemiology, capture–recapture models are commonly used to estimate the size of an unknown population based on several incomplete lists of individuals. The method operates under two main assumptions: independence between the lists (local independence) and homogeneity of capture probabilities of individuals. In practice, these assumptions are rarely satisfied. We introduce a multinomial latent class model that can account for both list dependence and heterogeneity. Parameter estimation is performed by maximizing the conditional likelihood function with the use of the EM algorithm. In addition, a new approach for evaluating the standard errors of the parameter estimates is discussed, which considerably reduces the computational burden associated with the evaluation of the variance of the population size estimate.     

METHODS FOR JOINT INFERENCE FROM MULTIPLE DATA SOURCES FOR IMPROVED ESTIMATES OF POPULATION SIZE AND SURVIVAL RATES

Critical conservation decisions often hinge on estimates of population size, population growth rate, and survival rates, but as a practical matter it is difficult to obtain enough data to provide precise estimates. Here we discuss Bayesian methods for simultaneously drawing on the information content from multiple sorts of data to get as much precision as possible for the estimates. The basic idea is that an underlying population model can connect the various sorts of observations, so this can be elaborated into a joint likelihood function for joint estimation of the respective parameters. The potential for improved estimates derives from the potentially greater effective sample size of the aggregate of data, even though some of the data types may only bear directly on a subset of the parameters. The achieved improvement depends on specifics of the interactions among parameters in the underlying model, and on the actual content of the data. Assuming the respective data sets are unbiased, notwithstanding the fact that they may be noisy, we may gauge the average improvement in the estimates of the parameters of interest from the reduction, if any, in the standard deviations of their posterior marginal distributions. Prospective designs may be evaluated from analysis of simulated data. Here this approach is illustrated with an assessment of the potential value in various ways of merging mark-resight and carcass-survey data for the Florida manatee, as could be made possible by various modifications in the data collection protocols in both programs.     

Effects of sampling regime on the mean and variance of home range size estimates

1. Although the home range is a fundamental ecological concept, there is considerable debate over how it is best measured. There is a substantial literature concerning the precision and accuracy of all commonly used home range estimation methods; however, there has been considerably less work concerning how estimates vary with sampling regime, and how this affects statistical inferences. 2. We propose a new procedure, based on a variance components analysis using generalized mixed effects models to examine how estimates vary with sampling regime. 3. To demonstrate the method we analyse data from one study of 32 individually marked roe deer and another study of 21 individually marked kestrels. We subsampled these data to simulate increasingly less intense sampling regimes, and compared the performance of two kernel density estimation (KDE) methods, of the minimum convex polygon (MCP) and of the bivariate ellipse methods. 4. Variation between individuals and study areas contributed most to the total variance in home range size. Contrary to recent concerns over reliability, both KDE methods were remarkably efficient, robust and unbiased: 10 fixes per month, if collected over a standardized number of days, were sufficient for accurate estimates of home range size. However, the commonly used 95% isopleth should be avoided; we recommend using isopleths between 90 and 50%. 5. Using the same number of fixes does not guarantee unbiased home range estimates: statistical inferences differ with the number of days sampled, even if using KDE methods. 6. The MCP method was highly inefficient and results were subject to considerable and unpredictable biases. The bivariate ellipse was not the most reliable method at low sample sizes. 7. We conclude that effort should be directed at marking more individuals monitored over long periods at the expense of the sampling rate per individual. Statistical results are reliable only if the whole sampling regime is standardized. We derive practical guidelines for field studies and data analysis.     

An Evaluation of Sex-Age-Kill (SAK) Model Performance

ABSTRACT The sex-age-kill (SAK) model is widely used to estimate abundance of harvested large mammals, including white-tailed deer (Odocoileus virginianus). Despite a long history of use, few formal evaluations of SAK performance exist. We investigated how violations of the stable age distribution and stationary population assumption, changes to male or female harvest, stochastic effects (i.e., random fluctuations in recruitment and survival), and sampling efforts influenced SAK estimation. When the simulated population had a stable age distribution and λ > 1, the SAK model underestimated abundance. Conversely, when λ < 1, the SAK overestimated abundance. When changes to male harvest were introduced, SAK estimates were opposite the true population trend. In contrast, SAK estimates were robust to changes in female harvest rates. Stochastic effects caused SAK estimates to fluctuate about their equilibrium abundance, but the effect dampened as the size of the surveyed population increased. When we considered both stochastic effects and sampling error at a deer management unit scale the resultant abundance estimates were within ±121.9% of the true population level 95% of the time. These combined results demonstrate extreme sensitivity to model violations and scale of analysis. Without changes to model formulation, the SAK model will be biased when λ ≠ 1. Furthermore, any factor that alters the male harvest rate, such as changes to regulations or changes in hunter attitudes, will bias population estimates. Sex-age-kill estimates may be precise at large spatial scales, such as the state level, but less so at the individual management unit level. Alternative models, such as statistical age-at-harvest models, which require similar data types, might allow for more robust, broad-scale demographic assessments.     

Evaluating the performance of a multilocus Bayesian method for the estimation of migration rates

Bayesian methods have become extremely popular in molecular ecology studies because they allow us to estimate demographic parameters of complex demographic scenarios using genetic data. Articles presenting new methods generally include sensitivity studies that evaluate their performance, but they tend to be limited and need to be followed by a more thorough evaluation. Here we evaluate the performance of a recent method, bayesass , which allows the estimation of recent migration rates among populations, as well as the inbreeding coefficient of each local population. We expand the simulation study of the original publication by considering multi-allelic markers and scenarios with varying number of populations. We also investigate the effect of varying migration rates and F _ST more thoroughly in order to identify the region of parameter space where the method is and is not able to provide accurate estimates of migration rate. Results indicate that if the demographic history of the species being studied fits the assumptions of the inference model, and if genetic differentiation is not too low ( F _ST ≥ 0.05), then the method can give fairly accurate estimates of migration rates even when they are fairly high (about 0.1). However, when the assumptions of the inference model are violated, accurate estimates are obtained only if migration rates are very low ( m  = 0.01) and genetic differentiation is high ( F _ST ≥ 0.10). Our results also show that using posterior assignment probabilities as an indication of how much confidence we can place on the assignments is problematical since the posterior probability of assignment can be very high even when the individual assignments are very inaccurate.     

Surveying an endangered saproxylic beetle, Osmoderma eremita, in Mediterranean woodlands: a comparison between different capture methods

Measuring population size is riddled with difficulties for wildlife biologists and managers, and in the case of rare species, it is sometimes practically impossible to estimate abundance, whereas estimation of occupancy is possible. Furthermore, obtaining reliable population size estimates is not straightforward, as different sampling techniques can give misleading results. A mark-recapture study of the endangered saproxylic beetle Osmoderma eremita was performed in central Italy by applying four independent capture methods within a study area where 116 hollow trees were randomly selected to set traps. Detection probability and population size estimates were drawn from each of these four capture methods. There were strong differences in detection probability among methods. Despite using pheromone and beetle manipulation, capture histories were not affected by trap-happiness or trap-shyness. Population size estimates varied considerably in both abundance and precision by capture method. A number of 0.5 and 0.2 adult beetles per tree was estimated using the whole data set by closed and open population models, respectively. Pitfall trap appeared the optimal method to detect the occurrence of this species. Since in the southern part of its distribution range, a single population of O. eremita is widespread in the landscape, and includes beetles from more than one hollow tree, conservation efforts should focus not only on preserving few and isolated monumental hollow trees, but should be extended to large stands.     

大中型兽类种群数量估算的研究进展

大中型兽类种群数量的估算是动物生态学中重要的基本问题, 受到研究者、管理者和公众的共同关注。国际上从20世纪中期开始研究该问题, 已出现了多种研究方法和相应案例, 且还在快速发展, 但世界各地仍有很多物种的种群数量尚未知晓。在我国, 从20世纪80年代开始调查大中型兽类种群数量, 取得了重要进展, 也还有很多物种的种群数量尚不清楚。因此, 有必要归纳国际上种群数量估算的研究进展, 同时, 总结国内研究的现状、优势和趋势, 供研究者参考。本文首先选择估算大中型兽类种群数量的原理、数据来源和模型这3个要素归纳出简明的研究框架, 将现有的多种方法置于其中予以阐述。在该框架下, 根据估算原理分为4大类方法, 为距离取样法、标志重捕法、基于遇见率法和遥感影像直接计数法。针对每一大类方法, 论述其基本原理模型和模型假设, 说明能实现该原理的相应数据来源(视觉观测、红外相机拍摄、DNA微卫星识别、卫星定位跟踪、声音监测或遥感影像)的特点及如何实现该原理, 评价其适用性及优缺点, 并选择其中具有可比性的方法予以比较评价。其次, 参照该研究框架, 总结我国的研究现状, 分析未来发展的优势和趋势: 我国的红外相机数据积累充分, 可以发展以此为数据源的距离取样法、标志重捕法和基于遇见率法; 发展以粪便样品为数据来源的距离取样法和粪便DNA标志重捕法; 相比地面调查数据, 获取高分辨率遥感影像数据更容易, 尽量以此估算符合适用条件的大中型兽类的种群数量。最后, 本文提出了适用于我国大中型兽类种群数量的估算方法的选择流程, 供研究者参考。     

Short-term genetic changes: evaluating effective population size estimates in a comprehensively described brown trout (Salmo trutta) population

The effective population size (N(e)) is notoriously difficult to accurately estimate in wild populations as it is influenced by a number of parameters that are difficult to delineate in natural systems. The different methods that are used to estimate N(e) are affected variously by different processes at the population level, such as the life-history characteristics of the organism, gene flow, and population substructure, as well as by the frequency patterns of genetic markers used and the sampling design. Here, we compare N(e) estimates obtained by different genetic methods and from demographic data and elucidate how the estimates are affected by various factors in an exhaustively sampled and comprehensively described natural brown trout (Salmo trutta) system. In general, the methods yielded rather congruent estimates, and we ascribe that to the adequate genotyping and exhaustive sampling. Effects of violating the assumptions of the different methods were nevertheless apparent. In accordance with theoretical studies, skewed allele frequencies would underestimate temporal allele frequency changes and thereby upwardly bias N(e) if not accounted for. Overlapping generations and iteroparity would also upwardly bias N(e) when applied to temporal samples taken over short time spans. Gene flow from a genetically not very dissimilar source population decreases temporal allele frequency changes and thereby acts to increase estimates of N(e). Our study reiterates the importance of adequate sampling, quantification of life-history parameters and gene flow, and incorporating these data into the N(e) estimation.     

Problems in the definition, interpretation, and evaluation of genetic heterogeneity

Suppose that we wish to classify families with multiple cases of disease into one of three categories: those that segregate mutations of a gene of interest, those which segregate mutations of other genes, and those whose disease is due to nonhereditary factors or chance. Among families in the first two categories (the hereditary families), we wish to estimate the proportion, p, of families that segregate mutations of the gene of interest. Although this proportion is a commonly accepted concept, it is well defined only with an unambiguous definition of "family." Even then, extraneous factors such as family sizes and structures can cause p to vary across different populations and, within a population, to be estimated differently by different studies. Restrictive assumptions about the disease are needed, in order to avoid this undesirable variation. The assumptions require that mutations of all disease-causing genes (i) have no effect on family size, (ii) have very low frequencies, and (iii) have penetrances that satisfy certain constraints. Despite the unverifiability of these assumptions, linkage studies often invoke them to estimate p, using the admixture likelihood introduced by Smith and discussed by Ott. We argue against this common practice, because (1) it also requires the stronger assumption of equal penetrances for all etiologically relevant genes; (2) even if all assumptions are met, estimates of p are sensitive to misspecification of the unknown phenocopy rate; (3) even if all the necessary assumptions are met and the phenocopy rate is correctly specified, estimates of p that are obtained by linkage programs such as HOMOG and GENEHUNTER are based on the wrong likelihood and therefore are biased in the presence of phenocopies. We show how to correct these estimates; but, nevertheless, we do not recommend the use of parametric heterogeneity models in linkage analysis, even merely as a tool for increasing the statistical power to detect linkage. This is because the assumptions required by these models cannot be verified, and their violation could actually decrease power. Instead, we suggest that estimation of p be postponed until the relevant genes have been identified. Then their frequencies and penetrances can be estimated on the basis of population-based samples and can be used to obtain more-robust estimates of p for specific populations.