On the sampling variance of intraclass correlations and genetic correlations

Review of sample size determination methods for the intraclass correlation coefficient in the one-way analysis of variance model

Reliability of measurement instruments providing quantitative outcomes is usually assessed by an intraclass correlation coefficient. When participants are repeatedly measured by a single rater or device, or, are each rated by a different group of raters, the intraclass correlation coefficient is based on a one-way analysis of variance model. When planning a reliability study, it is essential to determine the number of participants and measurements per participant (i.e. number of raters or number of repeated measurements). Three different sample size determination approaches under the one-way analysis of variance model were identified in the literature, all based on a confidence interval for the intraclass correlation coefficient. Although eight different confidence interval methods can be identified, Wald confidence interval with Fisher's large sample variance approximation remains most commonly used despite its well-known poor statistical properties. Therefore, a first objective of this work is comparing the statistical properties of all identified confidence interval methods-including those overlooked in previous studies. A second objective is developing a general procedure to determine the sample size using all approaches since a closed-form formula is not always available. This procedure is implemented in an R Shiny app. Finally, we provide advice for choosing an appropriate sample size determination method when planning a reliability study.

Estimating the heritability of nitrogen and carbon isotopes in the tail hair of beef cattle

BACKGROUND

The natural abundance of nitrogen (δ15N) and carbon (δ13C) isotopes in animal tissues are used to estimate an animal's efficiency in nitrogen utilization, and their feed conversion efficiency, especially in tropical grazing systems with prolonged protein restriction. It is postulated that selection for improving these two characteristics (δ15N and δ13C) would assist the optimisation of the adaptation in ever-changing environments, particularly in response to climate change. The aim of this study was to determine the heritability of δ15N and δ13C in the tail hair of tropically adapted beef cattle to validate their inclusion in genetic breeding programs.

METHODS

In total, 492 steers from two breeds, Brahman (n = 268) and Droughtmaster (n = 224) were used in this study. These steers were managed in two mixed breed contemporary groups across two years (year of weaning): steers weaned in 2019 (n = 250) and 2020 (n = 242). Samples of tail switch hair representing hair segments grown during the dry season were collected and analysed for δ15N and δ13C with isotope-ratio mass spectrometry. Heritability and variance components were estimated in a univariate multibreed (and single breed) animal model in WOMBAT and ASReml using three generations of full pedigree.

RESULTS

The estimated heritability of both traits was significantly different from 0, i.e. 0.43 ± 0.14 and 0.41 ± 0.15 for δ15N and δ13C, respectively. These traits had favourable moderate to high genetic and phenotypic correlations (- 0.78 ± 0.16 and - 0.40 ± 0.04, respectively). The study also provides informative single-breed results in spite of the limited sample size, with estimated heritability values of 0.37 ± 0.19 and 0.19 ± 0.17 for δ15N and δ13C in Brahman, and 0.36 ± 0.21 and 0.46 ± 0.22 for δ15N and δ13C in Droughtmaster, respectively.

CONCLUSIONS

The findings of this study show, for the first time, that the natural abundances of both nitrogen and carbon isotopes in the tail hair in cattle may be moderately heritable. With further research and validation, tail hair isotopes can become a practical tool for the large-scale selection of more efficient cattle.

Relationship between milk urea content and important milk traits in Holstein cattle

Breeding cattle with low nitrogen emissions has been proposed as a countermeasure against eutrophication due to dairy production. Milk urea content (MU) could potentially serve as a new readily measured indicator trait for nitrogen emissions by cows. Therefore, we estimated genetic parameters related to MU and its relationship with other milk traits. We analysed 4 178 735 milk samples collected between January 2008 and June 2019 from 261 866 German Holstein dairy cows during their first, second, and third lactations. Restricted maximum likelihood estimation was conducted using univariate and bivariate random regression sire models in WOMBAT. We obtained moderate average daily heritability estimates for the daily MU of 0.24 in first lactation cows, 0.23 in second lactation cows, and 0.21 in third lactation cows with average daily genetic SDs of 25.16 mg/kg, 24.93 mg/kg, and 23.75 mg/kg, respectively. Averaged over days in milk, the repeatability estimates were low at 0.41 in first, second, and third lactation cows. A strong positive genetic correlation was found between MU and milk urea yield (MUY; 0.72 on average). In addition, 305-day heritabilities were estimated as 0.50, 0.52, and 0.50 in first, second, and third lactation cows, respectively, with genetic correlations of 0.94 or higher for MU in different lactations. By contrast, the averaged estimates of the genetic correlations between MU and other milk traits were low (-0.07 to 0.15). Moderate heritability estimates clearly allow the possible selection for MU, and the near-zero estimates of genetic correlations indicate no risk of undesired correlated selection responses in other milk traits. However, a relationship still needs to be established between MU as an indicator trait and the target trait, defined as total individual nitrogen emissions.

Variability in Enteric Methane Emissions among Dairy Cows during Lactation

The aim of this study was to investigate variability in enteric CH₄ emission rate and emissions per unit of milk across lactations among dairy cows on commercial farms in the UK. A total of 105,701 CH₄ spot measurements were obtained from 2206 mostly Holstein-Friesian cows on 18 dairy farms using robotic milking stations. Eleven farms fed a partial mixed ration (PMR) and 7 farms fed a PMR with grazing. Methane concentrations (ppm) were measured using an infrared CH₄ analyser at 1s intervals in breath samples taken during milking. Signal processing was used to detect CH₄ eructation peaks, with maximum peak amplitude being used to derive CH₄ emission rate (g/min) during each milking. A multiple-experiment meta-analysis model was used to assess effects of farm, week of lactation, parity, diet, and dry matter intake (DMI) on average CH₄ emissions (expressed in g/min and g/kg milk) per individual cow. Estimated mean enteric CH₄ emissions across the 18 farms was 0.38 (s.e. 0.01) g/min, ranging from 0.2 to 0.6 g/min, and 25.6 (s.e. 0.5) g/kg milk, ranging from 15 to 42 g/kg milk. Estimated dry matter intake was positively correlated with emission rate, which was higher in grazing cows, and negatively correlated with emissions per kg milk and was most significant in PMR-fed cows. Mean CH₄ emission rate increased over the first 9 weeks of lactation and then was steady until week 70. Older cows were associated with lower emissions per minute and per kg milk. Rank correlation for CH₄ emissions among weeks of lactation was generally high. We conclude that CH₄ emissions appear to change across and within lactations, but ranking of a herd remains consistent, which is useful for obtaining CH₄ spot measurements.

Utilizing a Population-Genetic Framework to Test for Gene-Environment Interactions between Zebrafish Behavior and Chemical Exposure

Individuals within genetically diverse populations display broad susceptibility differences upon chemical exposures. Understanding the role of gene-environment interactions (GxE) in differential susceptibility to an expanding exposome is key to protecting public health. However, a chemical's potential to elicit GxE is often not considered during risk assessment. Previously, we've leveraged high-throughput zebrafish (Danio rerio) morphology screening data to reveal patterns of potential GxE effects. Here, using a population genetics framework, we apportioned variation in larval behavior and gene expression in three different PFHxA environments via mixed-effect modeling to assess significance of GxE term. We estimated the intraclass correlation (ICC) between full siblings from different families using one-way random-effects model. We found a significant GxE effect upon PFHxA exposure in larval behavior, and the ICC of behavioral responses in the PFHxA exposed population at the lower concentration was 43.7%, while that of the control population was 14.6%. Considering global gene expression data, a total of 3746 genes showed statistically significant GxE. By showing evidence that heritable genetics are directly affecting gene expression and behavioral susceptibility of individuals to PFHxA exposure, we demonstrate how standing genetic variation in a heterogeneous population such as ours can be leveraged to test for potential GxE.

High-density genetic linkage mapping reveals low stability of QTLs across environments for economic traits in Eucalyptus

Introduction

Eucalyptus urophylla, E. tereticornis and their hybrids are the most important commercial forest tree species in South China where they are grown for pulpwood and solid wood production. Construction of a fine-scale genetic linkage map and detecting quantitative trait loci (QTL) for economically important traits linked to these end-uses will facilitate identification of the main candidate genes and elucidate the regulatory mechanisms.

Method

A high-density consensus map (a total of 2754 SNPs with 1359.18 cM) was constructed using genotyping by sequencing (GBS) on clonal progenies of E. urophylla × tereticornis hybrids. QTL mapping of growth and wood property traits were conducted in three common garden experiments, resulting in a total of 108 QTLs. A total of 1052 candidate genes were screened by the efficient combination of QTL mapping and transcriptome analysis.

Results

Only ten QTLs were found to be stable across two environments, and only one (qSG10Stable mapped on chromosome 10, and associated with lignin syringyl-to-guaiacyl ratio) was stable across all three environments. Compared to other QTLs, qSG10Stable explained a very high level of phenotypic variation (18.4-23.6%), perhaps suggesting that QTLs with strong effects may be more stably inherited across multiple environments. Screened candidate genes were associated with some transcription factor families, such as TALE, which play an important role in the secondary growth of plant cell walls and the regulation of wood formation.

Discussion

While QTLs such as qSG10Stable, found to be stable across three sites, appear to be comparatively uncommon, their identification is likely to be a key to practical QTL-based breeding. Further research involving clonally-replicated populations, deployed across multiple target planting sites, will be required to further elucidate QTL-by-environment interactions.

A unified framework for association and prediction from vertex-wise grey-matter structure

The recent availability of large‐scale neuroimaging cohorts facilitates deeper characterisation of the relationship between phenotypic and brain architecture variation in humans. Here, we investigate the association (previously coined morphometricity) of a phenotype with all 652,283 vertex‐wise measures of cortical and subcortical morphology in a large data set from the UK Biobank (UKB; N = 9,497 for discovery, N = 4,323 for replication) and the Human Connectome Project (N = 1,110). We used a linear mixed model with the brain measures of individuals fitted as random effects with covariance relationships estimated from the imaging data. We tested 167 behavioural, cognitive, psychiatric or lifestyle phenotypes and found significant morphometricity for 58 phenotypes (spanning substance use, blood assay results, education or income level, diet, depression, and cognition domains), 23 of which replicated in the UKB replication set or the HCP. We then extended the model for a bivariate analysis to estimate grey‐matter correlation between phenotypes, which revealed that body size (i.e., height, weight, BMI, waist and hip circumference, body fat percentage) could account for a substantial proportion of the morphometricity (confirmed using a conditional analysis), providing possible insight into previous MRI case–control results for psychiatric disorders where case status is associated with body mass index. Our LMM framework also allowed to predict some of the associated phenotypes from the vertex‐wise measures, in two independent samples. Finally, we demonstrated additional new applications of our approach (a) region of interest (ROI) analysis that retain the vertex‐wise complexity; (b) comparison of the information retained by different MRI processings.

Heritable genetic variation but no local adaptation in a pine-ectomycorrhizal interaction

Local adaptation of plants to mycorrhizal fungi helps determine the outcome of mycorrhizal interactions. However, there is comparatively little work exploring the potential for evolution in interactions with ectomycorrhizal fungi, and fewer studies have explored the heritability of mycorrhizal responsiveness, which is required for local adaptation to occur. We set up a reciprocal inoculation experiment using seedlings and soil from four populations of Scots pine (Pinus sylvestris) from Scotland, measuring seedling response to mycorrhizal inoculation after 4 months. We estimated heritability for the response traits and tested for genotype × environment interactions. While we found that ectomycorrhizal responsiveness was highly heritable, we found no evidence that pine populations were locally adapted to fungal communities. Instead, we found a complex suite of interactions between pine population and soil inoculum. Our results suggest that, while Scots pine has the potential to evolve in response to mycorrhizal fungi, evolution in Scotland has not resulted in local adaptation. Long generation times and potential for rapid shifts in fungal communities in response to environmental change may preclude the opportunity for such adaptation in this species, and selection for other factors such as resistance to fungal pathogens may explain the pattern of interactions found.

Assessment of sire contribution and breed-of-origin of alleles in a three-way crossbred broiler dataset

Broiler breeding programs rely on crossbreeding. With genomic selection, widespread use of crossbred performance in breeding programs comes within reach. Commercial crossbreds, however, may have unknown pedigrees and their genomes may include DNA from 2 to 4 different breeds. Our aim was, for a broiler dataset with a limited number of sires having both purebred and crossbred offspring generated using natural mating, to rapidly derive parentage, assess the distribution of the sire contribution to the offspring generation, and to assess breed-of-origin of alleles in crossbreds. The dataset contained genotypes for 56,075 SNPs for 5,882 purebred and 10,943 3-way crossbred offspring generated by natural mating of 164 purebred sires to 1,016 purebred and 1,386 F1 crossbred hens. Using our algorithm FindParents, joint parentage derivation for the offspring and parent generations required only 1 m 29 s to retrieve parentage for 20,253 animals considering 4,504 possible parents. FindParents was similarly accurate as a maximum likelihood based method, apart from situations where settings of FindParents did not match the genotyping error rate in the data. Numbers of offspring per sire had a very skewed distribution, ranging from 1 to 270 crossbreds and 1 to 154 purebreds. Derivation of breed-of-origin of alleles relied on phasing all genotypes, including 8,205, 372, and 720 animals from the 3 pure lines involved, and allocating haplotypes in the crossbreds to purebred lines based on observed frequencies in the purebred lines. Breed-of-origin could be derived for 96.94% of the alleles of the 1,386 F1 crossbred hens and for 91.88% of the alleles of the 10,943 3-way crossbred offspring, of which 49.49% to the sire line. The achieved percentage of assignment to the sire line was sufficient to proceed with subsequent analyses requiring only the breed-of-origin of the paternal alleles to be known. Although required number of animals may be population dependent, to increase the total percentage of assigned alleles, it seems advisable to use at least approx. 1,000 genotyped purebred animals for each of the lines involved.

Fisher's sons' effect in sexual selection: absent, intermittent or just low experimental power?

The Fisherian sexual selection paradigm has been called the null model of sexual selection. At its heart is the expectation of a genetic correlation (r_G ) between female preference and male trait. However, recent meta-analysis has shown estimated correlations are often extremely weak and not statistically significant. We show here that systematic failure of studies to reject the null hypothesis that r_G  = 0 is almost certainly due to the low power of most experimental designs used. We provide an easy way to assess experimental power a priori and suggest that current data make it difficult to definitively test a key component of the Fisher effect.

Has Scots pine (Pinus sylvestris) co-evolved with Dothistroma septosporum in Scotland? Evidence for spatial heterogeneity in the susceptibility of native provenances

Spatial heterogeneity in pathogen pressure leads to genetic variation in, and evolution of, disease-related traits among host populations. In contrast, hosts are expected to be highly susceptible to exotic pathogens as there has been no evolution of defence responses. Host response to pathogens can therefore be an indicator of a novel or endemic pathosystem. Currently, the most significant threat to native British Scots pine (Pinus sylvestris) forests is Dothistroma needle blight (DNB) caused by the foliar pathogen Dothistroma septosporum which is presumed to be exotic. A progeny-provenance trial of 6-year-old Scots pine, comprising eight native provenances each with four families in six blocks, was translocated in April 2013 to a clear-fell site in Galloway adjacent to a DNB-infected forest. Susceptibility to D. septosporum, measured as DNB severity (estimated percentage nongreen current-year needles), was assessed visually over 2 years (2013-2014 and 2014-2015; two assessments per year). There were highly significant differences in susceptibility among provenances but not among families for each annual assessment. Provenance mean susceptibility to D. septosporum was negatively and significantly associated with water-related variables at site of origin, potentially due to the evolution of low susceptibility in the host in response to high historical pathogen pressure.

Substantial heritable variation for susceptibility to Dothistroma septosporum within populations of native British Scots pine (Pinus sylvestris)

The threat from pests and pathogens to native and commercially planted forest trees is unprecedented and expected to increase under climate change. The degree to which forests respond to threats from pathogens depends on their adaptive capacity, which is determined largely by genetically controlled variation in susceptibility of the individual trees within them and the heritability and evolvability of this trait. The most significant current threat to the economically and ecologically important species Scots pine (Pinus sylvestris) is dothistroma needle blight (DNB), caused by the foliar pathogen Dothistroma septosporum. A progeny-population trial of 4-year-old Scots pine trees, comprising six populations from native Caledonian pinewoods each with three to five families in seven blocks, was artificially inoculated using a single isolate of D. septosporum. Susceptibility to D. septosporum, assessed as the percentage of non-green needles, was measured regularly over a period of 61 days following inoculation, during which plants were maintained in conditions ideal for DNB development (warm; high humidity; high leaf wetness). There were significant differences in susceptibility to D. septosporum among families indicating that variation in this trait is heritable, with high estimates of narrow-sense heritability (0.38-0.75) and evolvability (genetic coefficient of variation, 23.47). It is concluded that native Scots pine populations contain sufficient genetic diversity to evolve lower susceptibility to D. septosporum through natural selection in response to increased prevalence of this pathogen.

Standard error of the genetic correlation: how much data do we need to estimate a purebred-crossbred genetic correlation?

BACKGROUND

The additive genetic correlation (rg) is a key parameter in livestock genetic improvement. The standard error (SE) of an estimate of rg, r^g, depends on whether both traits are recorded on the same individual or on distinct individuals. The genetic correlation between traits recorded on distinct individuals is relevant as a measure of, e.g., genotype-by-environment interaction and for traits expressed in purebreds vs. crossbreds. In crossbreeding schemes, rg between the purebred and crossbred trait is the key parameter that determines the need for crossbred information. This work presents a simple equation to predict the SE of r^g between traits recorded on distinct individuals for nested full-half sib schemes with common-litter effects, using the purebred-crossbred genetic correlation as an example. The resulting expression allows a priori optimization of designs that aim at estimating rg. An R-script that implements the expression is included.

RESULTS

The SE of r^g is determined by the true value of rg, the number of sire families (N), and the reliabilities of sire estimated breeding values (EBV):SEr^g≈1ρx2ρy2+1+0.5ρx4+0.5ρy4-2ρx2-2ρy2rg2+rg4N-1,where ρx2 and ρy2 are the reliabilities of the sire EBV for both traits. Results from stochastic simulation show that this equation is accurate since the average absolute error of the prediction across 320 alternative breeding schemes was 3.2%. Application to typical crossbreeding schemes shows that a large number of sire families is required, usually more than 100. Since SEr^g is a function of reliabilities of EBV, the result probably extends to other cases such as repeated records, but this was not validated by simulation.

CONCLUSIONS

This work provides an accurate tool to determine a priori the amount of data required to estimate a genetic correlation between traits measured on distinct individuals, such as the purebred-crossbred genetic correlation.

Statistical power to detect genetic (co)variance of complex traits using SNP data in unrelated samples

We have recently developed analysis methods (GREML) to estimate the genetic variance of a complex trait/disease and the genetic correlation between two complex traits/diseases using genome-wide single nucleotide polymorphism (SNP) data in unrelated individuals. Here we use analytical derivations and simulations to quantify the sampling variance of the estimate of the proportion of phenotypic variance captured by all SNPs for quantitative traits and case-control studies. We also derive the approximate sampling variance of the estimate of a genetic correlation in a bivariate analysis, when two complex traits are either measured on the same or different individuals. We show that the sampling variance is inversely proportional to the number of pairwise contrasts in the analysis and to the variance in SNP-derived genetic relationships. For bivariate analysis, the sampling variance of the genetic correlation additionally depends on the harmonic mean of the proportion of variance explained by the SNPs for the two traits and the genetic correlation between the traits, and depends on the phenotypic correlation when the traits are measured on the same individuals. We provide an online tool for calculating the power of detecting genetic (co)variation using genome-wide SNP data. The new theory and online tool will be helpful to plan experimental designs to estimate the missing heritability that has not yet been fully revealed through genome-wide association studies, and to estimate the genetic overlap between complex traits (diseases) in particular when the traits (diseases) are not measured on the same samples.

Bias-corrected estimator for intraclass correlation coefficient in the balanced one-way random effects model

Background

Intraclass correlation coefficients (ICCs) are used in a wide range of applications. However, most commonly used estimators for the ICC are known to be subject to bias.

Methods

Using second order Taylor series expansion, we propose a new bias-corrected estimator for one type of intraclass correlation coefficient, for the ICC that arises in the context of the balanced one-way random effects model. A simulation study is performed to assess the performance of the proposed estimator. Data have been generated under normal as well as non-normal scenarios.

Results

Our simulation results show that the new estimator has reduced bias compared to the least square estimator which is often referred to as the conventional or analytical estimator. The results also show marked bias reduction both in normal and non-normal data scenarios. In particular, our estimator outperforms the analytical estimator in a non-normal setting producing estimates that are very close to the true ICC values.

Conclusions

The proposed bias-corrected estimator for the ICC from a one-way random effects analysis of variance model appears to perform well in the scenarios we considered in this paper and can be used as a motivation to construct bias-corrected estimators for other types of ICCs that arise in more complex scenarios. It would also be interesting to investigate the bias-variance trade-off.

Phenotypic plasticity, QTL mapping and genomic characterization of bud set in black poplar

BACKGROUND

The genetic control of important adaptive traits, such as bud set, is still poorly understood in most forest trees species. Poplar is an ideal model tree to study bud set because of its indeterminate shoot growth. Thus, a full-sib family derived from an intraspecific cross of P. nigra with 162 clonally replicated progeny was used to assess the phenotypic plasticity and genetic variation of bud set in two sites of contrasting environmental conditions.

RESULTS

Six crucial phenological stages of bud set were scored. Night length appeared to be the most important signal triggering the onset of growth cessation. Nevertheless, the effect of other environmental factors, such as temperature, increased during the process. Moreover, a considerable role of genotype × environment (G × E) interaction was found in all phenological stages with the lowest temperature appearing to influence the sensitivity of the most plastic genotypes.Descriptors of growth cessation and bud onset explained the largest part of phenotypic variation of the entire process. Quantitative trait loci (QTL) for these traits were detected. For the four selected traits (the onset of growth cessation (date2.5), the transition from shoot to bud (date1.5), the duration of bud formation (subproc1) and bud maturation (subproc2)) eight and sixteen QTL were mapped on the maternal and paternal map, respectively. The identified QTL, each one characterized by small or modest effect, highlighted the complex nature of traits involved in bud set process. Comparison between map location of QTL and P. trichocarpa genome sequence allowed the identification of 13 gene models, 67 bud set-related expressional and six functional candidate genes (CGs). These CGs are functionally related to relevant biological processes, environmental sensing, signaling, and cell growth and development. Some strong QTL had no obvious CGs, and hold great promise to identify unknown genes that affect bud set.

CONCLUSIONS

This study provides a better understanding of the physiological and genetic dissection of bud set in poplar. The putative QTL identified will be tested for associations in P. nigra natural populations. The identified QTL and CGs will also serve as useful targets for poplar breeding.

Functional and genetic characterization of gas exchange and intrinsic water use efficiency in a full-sib family of Pinus pinaster Ait. in response to drought

Drought is an important environmental factor in Mediterranean ecosystems affecting seedling recruitment, productivity or susceptibility to fires and pathogens. Studying water use efficiency in these environments is crucial due to its adaptive value allowing trees to cope with low water availability. We studied the phenotypic variability and genetic control of intrinsic water use efficiency (WUE(i)) and related traits in a full-sib family of Pinus pinaster under drought imposition. We detected significant differences in WUE(i) between clones of the same family and moderate heritability estimates that indicate some degree of genetic control over this trait. Stomatal conductance to water vapor was the trait most affected by drought imposition and it showed the strongest influence in WUE(i). Stomatal conductance to water vapor and specific leaf area (SLA) were the traits with highest heritabilities and they showed a significant genetic correlation with WUE(i), suggesting that selection of needles with low SLA values will improve WUE(i) in this species by reducing water losses through stomatal control.

Reliable confidence intervals in quantitative genetics: narrow-sense heritability

Many quantitative genetic statistics are functions of variance components, for which a large number of replicates is needed for precise estimates and reliable measures of uncertainty, on which sound interpretation depends. Moreover, in large experiments the deaths of some individuals can occur, so methods for analysing such data need to be robust to missing values. We show how confidence intervals for narrow-sense heritability can be calculated in a nested full-sib/half-sib breeding design (males crossed with several females) in the presence of missing values. Simulations indicate that the method provides accurate results, and that estimator uncertainty is lowest for sampling designs with many males relative to the number of females per male, and with more females per male than progenies per female. Missing data generally had little influence on estimator accuracy, thus suggesting that the overall number of observations should be increased even if this results in unbalanced data. We also suggest the use of parametrically simulated data for prior investigation of the accuracy of planned experiments. Together with the proposed confidence intervals an informed decision on the optimal sampling design is possible, which allows efficient allocation of resources.

Full-genome scan for linkage in 50 families segregating the bipolar affective disease phenotype

A genome scan of approximately 12-cM initial resolution was done on 50 of a set of 51 carefully ascertained unilineal multiplex families segregating the bipolar affective disorder phenotype. In addition to standard multipoint linkage analysis methods, a simultaneous-search algorithm was applied in an attempt to surmount the problem of genetic heterogeneity. The results revealed no linkage across the genome. The results exclude monogenic models and make it unlikely that two genes account for the disease in this sample. These results support the conclusion that at least several hundred kindreds will be required in order to establish linkage of susceptibility loci to bipolar disorder in heterogeneous populations.