|Genetic diversity issues in animal populations in the genomic era
Woolliams, J.A. ; Oldenbroek, J.K. - \ 2017
In: Genomic management of animal genetic diversity / Oldenbroek, Kor, Wageningen Academic Publishers - ISBN 9789086862979 - p. 13 - 47.
Genetic diversity is the set of differences between species, breeds within species, and individuals within breeds present in their DNA or observed in animals as a consequence. Nowadays, genetic diversity can be measured directly on the DNA itself and it accurately presents the genetic variations between breeds, within breeds, and within half- and full-sib groups. The existence of genetic diversity is a prerequisite for natural selection directed to adaptation and articial selection directed to improved performance. Genetic diversity measured at DNA-level can be used for a description of the genetic history of the breed, for genomic selection, for the management of genetic diversity in populations and in genebanks, for the introgression of desired traits and for the elimination of genetic defects. Genomic management becomes an important tool in breeding programmes for livestock and companion animals in large as well as in small populations.
Unravelling the contribution of host genetics to infectious disease outbreaks
Doeschl-Wilson, A. ; Anacleto, O. ; Tsairidou, S. ; Lough, G. ; Houston, R.D. ; Woolliams, J.A. ; Mulder, H.A. ; Cabaleiro, S. ; Saura, M. ; Villanueva, B. - \ 2016
In: Book of Abstracts of the 67th Annual Meeting of the European Federation of Animal Science. - Wageningen : Wageningen Academic Publishers (Book of abstracts 22) - ISBN 9789086862849 - p. 189 - 189.
Bias, accuracy, and impact of indirect genetic effects in infectious diseases
Lipschutz-Powell, D. ; Woolliams, J.A. ; Bijma, P. ; Pong-Wong, R. ; Bermingham, M.L. ; Doeschl-Wilson, A.B. - \ 2012
Frontiers in Genetics Livestock Genomics 3 (2012). - ISSN 1664-8021
Selection for improved host response to infectious disease offers a desirable alternative to chemical treatment but has proven difficult in practice, due to low heritability estimates of disease traits. Disease data from field studies is often binary, indicating whether an individual has become infected or not following exposure to an infectious disease. Numerous studies have shown that from this data one can infer genetic variation in individuals’ underlying susceptibility. In a previous study, we showed that with an indirect genetic effect (IGE) model it is possible to capture some genetic variation in infectivity, if present, as well as in susceptibility. Infectivity is the propensity of transmitting infection upon contact with a susceptible individual. It is an important factor determining the severity of an epidemic. However, there are severe shortcomings with the Standard IGE models as they do not accommodate the dynamic nature of disease data. Here we adjust the Standard IGE model to (1) make expression of infectivity dependent on the individuals’ disease status (Case Model) and (2) to include timing of infection (Case-ordered Model). The models are evaluated by comparing impact of selection, bias, and accuracy of each model using simulated binary disease data. These were generated for populations with known variation in susceptibility and infectivity thus allowing comparisons between estimated and true breeding values. Overall the Case Model provided better estimates for host genetic susceptibility and infectivity compared to the Standard Model in terms of bias, impact, and accuracy. Furthermore, these estimates were strongly influenced by epidemiological characteristics. However, surprisingly, the Case-Ordered model performed considerably worse than the Standard and the Case Models, pointing toward limitations in incorporating disease dynamics into conventional variance component estimation methodology and software used in animal breeding. - See more at: http://journal.frontiersin.org/Journal/10.3389/fgene.2012.00215/full#h1
Indirect Genetic Effects and the Spread of Infectious Disease: Are We Capturing the Full Heritable Variation Underlying Disease Prevalence?
Lipschutz-Powell, D. ; Woolliams, J.A. ; Bijma, P. ; Doeschl-Wilson, A.B. - \ 2012
PLoS ONE 7 (2012)6. - ISSN 1932-6203
biological groups - selection - resistance - emergence - parameters - evolution - dynamics - programs - humans - models
Reducing disease prevalence through selection for host resistance offers a desirable alternative to chemical treatment. Selection for host resistance has proven difficult, however, due to low heritability estimates. These low estimates may be caused by a failure to capture all the relevant genetic variance in disease resistance, as genetic analysis currently is not taylored to estimate genetic variation in infectivity. Host infectivity is the propensity of transmitting infection upon contact with a susceptible individual, and can be regarded as an indirect effect to disease status. It may be caused by a combination of physiological and behavioural traits. Though genetic variation in infectivity is difficult to measure directly, Indirect Genetic Effect (IGE) models, also referred to as associative effects or social interaction models, allow the estimation of this variance from more readily available binary disease data (infected/non-infected). We therefore generated binary disease data from simulated populations with known amounts of variation in susceptibility and infectivity to test the adequacy of traditional and IGE models. Our results show that a conventional model fails to capture the genetic variation in infectivity inherent in populations with simulated infectivity. An IGE model, on the other hand, does capture some of the variation in infectivity. Comparison with expected genetic variance suggests that there is scope for further methodological improvement, and that potential responses to selection may be greater than values presented here. Nonetheless, selection using an index of estimated direct and indirect breeding values was shown to have a greater genetic selection differential and reduced future disease risk than traditional selection for resistance only. These findings suggest that if genetic variation in infectivity substantially contributes to disease transmission, then breeding designs which explicitly incorporate IGEs might help reduce disease prevalence.
Imputation of Missing Genotypes From Sparse to High Density Using Long-Range Phasing
Daetwyler, H.D. ; Wiggans, G.R. ; Hayes, B.J. ; Woolliams, J.A. ; Goddard, M.E. - \ 2011
Genetics 189 (2011)1. - ISSN 0016-6731 - p. 317 - 327.
genetic-linkage maps - haplotype imputation - marker maps - accuracy - genome - pedigrees - inference - selection - populations - descent
Related individuals share potentially long chromosome segments that trace to a common ancestor. We describe a phasing algorithm (ChromoPhase) that utilizes this characteristic of finite populations to phase large sections of a chromosome. In addition to phasing, our method imputes missing genotypes in individuals genotyped at lower marker density when more densely genotyped relatives are available. ChromoPhase uses a pedigree to collect an individual's (the proband) surrogate parents and offspring and uses genotypic similarity to identify its genomic surrogates. The algorithm then cycles through the relatives and genomic surrogates one at a time to find shared chromosome segments. Once a segment has been identified, any missing information in the proband is filled in with information from the relative. We tested ChromoPhase in a simulated population consisting of 400 individuals at a marker density of 1500/M, which is approximately equivalent to a 50K bovine single nucleotide polymorphism chip. In simulated data, 99.9% loci were correctly phased and, when imputing from 100 to 1500 markers, more than 87% of missing genotypes were correctly imputed. Performance increased when the number of generations available in the pedigree increased, but was reduced when the sparse genotype contained fewer loci. However, in simulated data, ChromoPhase correctly imputed at least 12% more genotypes than fastPHASE, depending on sparse marker density. We also tested the algorithm in a real Holstein cattle data set to impute 50K genotypes in animals with a sparse 3K genotype. In these data 92% of genotypes were correctly imputed in animals with a genotyped sire. We evaluated the accuracy of genomic predictions with the dense, sparse, and imputed simulated data sets and show that the reduction in genomic evaluation accuracy is modest even with imperfectly imputed genotype data. Our results demonstrate that imputation of missing genotypes, and potentially full genome sequence, using long-range phasing is feasible.
The Impact of Genetic Architecture on Genome-Wide Evaluation Methods
Daetwyler, H.D. ; Pong-Wong, R. ; Villanueva, B. ; Woolliams, J.A. - \ 2010
Genetics 185 (2010)3. - ISSN 0016-6731 - p. 1021 - 1031.
breeding values - linkage disequilibrium - selection - prediction - accuracy - information - algorithm - variance - models - lasso
The rapid increase in high-throughput single-nucleotide polymorphism data has led to a great interest in applying genome-wide evaluation methods to identify an individual's genetic merit. Genome-wide evaluation combines statistical methods with genomic data to predict genetic values for complex traits. Considerable uncertainty currently exists in determining which genome-wide evaluation method is the most appropriate. We hypothesize that genome-wide methods deal differently with the genetic architecture of quantitative traits and genomes. A genomic linear method (GBLUP), and a genomic nonlinear Bayesian variable selection method (BayesB) are compared using stochastic simulation across three effective population sizes and a wide range of numbers of quantitative trait loci (N-QTL). GBLUP had a constant accuracy, for a given heritability and sample size, regardless of NQTL. BayesB had a higher accuracy than GBLUP when N-QTL was low, but this advantage diminished as N-QTL increased and when N-QTL became large, GBLUP slightly outperformed BayesB. In addition, deterministic equations are extended to predict the accuracy of both methods and to estimate the number of independent chromosome segments (Me) and N-QTL. The predictions of accuracy and estimates of Me and N-QTL were generally in good agreement with results from simulated data. We conclude that the relative accuracy of GBLUP and BayesB for a given number of records and heritability are highly dependent on Me, which is a property of the target genome, as well as the architecture of the trait (N-QTL).
The importance of associative effects in the control of infectious disease through selection
Lipschutz-Powell, D. ; Woolliams, J.A. ; Bijma, P. ; Doeschl-Wilson, A. - \ 2010
Genome-wide evaluation of populations
Daetwyler, H.D. - \ 2009
Wageningen University. Promotor(en): Johan van Arendonk; J.A. Woolliams, co-promotor(en): B. Villanueva. - [S.l. : S.n. - ISBN 9789085855286 - 187
dierveredeling - loci voor kwantitatief kenmerk - genomen - rundvee - inteelt - dierziekten - stamboom - genomica - diergenetica - genotyping - animal breeding - quantitative trait loci - genomes - cattle - inbreeding - animal diseases - pedigree - genomics - animal genetics - genotyping
Dit proefschrift onderzoekt het gebruik van moleculaire merkers voor genetische evaluatie van populaties. Moleculaire merkers kunnen worden gebruikt om de nauwkeurigheid van geschatte fokwaardes te verhogen. In het verleden was men gericht op het opsporen van een beperkt aantal zogenaamde QTL, delen van het genoom, die direct in verband staan met een kenmerk. Het doel was om deze QTL te benutten in fokprogramma’s met behulp van merker-ondersteunde selectie. Met het beschikbaar komen van grote hoeveelheden SNP-merkers kan gebruik worden gemaakt van een methode die gericht is op het gehele genoom, en bekend staat als “genome-wide evaluation” (GWE). Dit proefschrift presenteert resultaten van zowel QTL-detectie als GWE. Deterministische voorspellingen van nauwkeurigheid worden gepresenteerd en getest, en de invloed van de genetische structuur op nauwkeurigheid wordt onderzocht. Een methode wordt gepresenteerd voor het berekenen van missende genotypes, met als doel merkerdichtheid en nauwkeurigheid van GWE te verhogen. Daarnaast worden praktische toepassing van GWE en manieren om ontbrekende genetische variatie te kwantificeren bediscussieerd.
Accuracy of Predicting the Genetic Risk of Disease Using a Genome-Wide Approach.
Daetwyler, H.D. ; Villanueva, B. ; Woolliams, J.A. - \ 2008
PLoS ONE 3 (2008)10. - ISSN 1932-6203 - 8 p.
Background - The prediction of the genetic disease risk of an individual is a powerful public health tool. While predicting risk has been successful in diseases which follow simple Mendelian inheritance, it has proven challenging in complex diseases for which a large number of loci contribute to the genetic variance. The large numbers of single nucleotide polymorphisms now available provide new opportunities for predicting genetic risk of complex diseases with high accuracy. Methodology/Principal Findings - We have derived simple deterministic formulae to predict the accuracy of predicted genetic risk from population or case control studies using a genome-wide approach and assuming a dichotomous disease phenotype with an underlying continuous liability. We show that the prediction equations are special cases of the more general problem of predicting the accuracy of estimates of genetic values of a continuous phenotype. Our predictive equations are responsive to all parameters that affect accuracy and they are independent of allele frequency and effect distributions. Deterministic prediction errors when tested by simulation were generally small. The common link among the expressions for accuracy is that they are best summarized as the product of the ratio of number of phenotypic records per number of risk loci and the observed heritability. Conclusions/Significance - This study advances the understanding of the relative power of case control and population studies of disease. The predictions represent an upper bound of accuracy which may be achievable with improved effect estimation methods. The formulae derived will help researchers determine an appropriate sample size to attain a certain accuracy when predicting genetic risk
Inbreeding in genome-wide selection
Daetwyler, H.D. ; Villanueva, B. ; Bijma, P. ; Woolliams, J.A. - \ 2007
Journal of Animal Breeding and Genetics 124 (2007)6. - ISSN 0931-2668 - p. 369 - 376.
marker-assisted selection - complex vertebral malformation - mendelian sampling terms - cattle breeding schemes - dairy-cattle - dynamic selection - genetic-markers - prediction - programs - populations
Traditional selection methods, such as sib and best linear unbiased prediction (BLUP) selection, which increased genetic gain by increasing accuracy of evaluation have also led to an increased rate of inbreeding per generation (¿FG). This is not necessarily the case with genome-wide selection, which also increases genetic gain by increasing accuracy. This paper explains why genome-wide selection reduces ¿FG when compared with sib and BLUP selection. Genome-wide selection achieves high accuracies of estimated breeding values through better prediction of the Mendelian sampling term component of breeding values. This increases differentiation between sibs and reduces coselection of sibs and ¿FG. The high accuracy of genome-wide selection is expected to reduce the between family variance and reweigh the emphasis of estimated breeding values of individuals towards the Mendelian sampling term. Moreover, estimation induced intraclass correlations of sibs are expected to be lower in genome-wide selection leading to a further decrease of coselection of sibs when compared with BLUP. Genome-wide prediction of breeding values, therefore, enables increased genetic gain while at the same time reducing ¿FG when compared with sib and BLUP selection.
Marker densities and the mapping of ancestral junctions
Macleod, A.K. ; Haley, C.S. ; Woolliams, J.A. ; Stam, P. - \ 2005
Genetical Research 85 (2005)1. - ISSN 0016-6723 - p. 69 - 79.
random mating populations - linkage disequilibrium - haplotype blocks - human genome - recombination - segments - descent - model
In any partially inbred population, `junctions` are the loci that form boundaries between segments of ancestral chromosomes. Here we show that the expected number of junctions per Morgan in such a population is linearly related to the inbreeding coefficient of the population, with a maximum in a completely inbred population corresponding to the prediction given by Stam (1980). We further show that high-density marker maps (fully informative markers with average densities of up to 200 per cM) will fail to detect a significant proportion of the junctions present in highly inbred populations. The number of junctions detected is lower than that which would be expected if junctions were distributed randomly along the chromosome, and we show that junctions are not, in fact randomly spaced. This non-random spacing of junctions significantly increases the number of markers that is required to detect 90% of the junctions present on any chromosome: a marker count of at least 12 times the number of junctions present will be needed to detect this proportion
Criteria to assess the degree of endangerment of livestock breeds in Europe
Gandini, G.C. ; Ollivier, L. ; Danell, B. ; Distl, O. ; Georgoudis, A. ; Groeneveld, E. ; Martyniuk, E. ; Arendonk, J.A.M. van; Woolliams, J.A. - \ 2004
Livestock Production Science 91 (2004)1-2. - ISSN 0301-6226 - p. 173 - 182.
dierveredeling - rundvee - rundveerassen - bedreigde rassen - uitsterven - conservering - genetische bronnen van diersoorten - genetische variatie - genetische bronnen - veredelingsprogramma's - fokwaarde - demografie - populatiegroei - europa - animal breeding - cattle - cattle breeds - endangered breeds - extinction - conservation - animal genetic resources - genetic variation - genetic resources - breeding programmes - breeding value - demography - population growth - europe - overlapping generations - predicting rates - populations - selection - diversity
The degree to which a breed is exposed to becoming extinct, i.e. its degree of endangerment (DE), is an essential information to orient conservation policies. Assessing DE properly is a difficult task, as numerous factors are involved. Several methods are currently used in Europe and the paper first discusses the development of some objective criteria to promote the creation of a uniform system. Both demographic and genetic aspects of population decline are considered. It is proposed to estimate the number of years needed to reach a critical population size, which is also a measure of time available to evaluate options and undertake action before extinction. Thresholds of endangerment for both the demographic and genetic aspects are discussed. In addition, the population growth rate of 110 European cattle breeds is analysed. Growth rate is normally distributed with a mean of 1.00 (S.D. 0.09, range 0.77-1.27). Population size at the beginning of the analysed period and country of breeding affect growth rate significantly
The degree to which a breed is exposed to becoming extinct, i.e. its degree of endangerment (DE), is an essential information to orient conservation policies. Assessing DE properly is a difficult task, as numerous factors are involved. Several methods are currently used in Europe and the paper first discusses the development of some objective criteria to promote the creation of a uniform system. Both demographic and genetic aspects of population decline are considered. It is proposed to estimate the number of years needed to reach a critical population size, which is also a measure of time available to evaluate options and undertake action before extinction. Thresholds of endangerment for both the demographic and genetic aspects are discussed. In addition, the population growth rate of 110 European cattle breeds is analysed. Growth rate is normally distributed with a mean of 1.00 (S.D. 0.09, range 0.77-1.27). Population size at the beginning of the analysed period and country of breeding affect growth rate significantly. (C) 2004 Elsevier B.V. All rights reserved.
Minimizing inbreeding by managing genetic contributions across generations
Sanchez-Rodriquez, L. ; Bijma, P. ; Woolliams, J.A. - \ 2003
Genetics 164 (2003). - ISSN 0016-6731 - p. 1589 - 1595.
effective population-size - overlapping generations - selection - depression - extinction - systems - herd
Here we present the strategy that achieves the lowest possible rate of inbreeding (DeltaF) for a population with unequal members of sires and dams with random mating. This new strategy results in a DeltaF as much as 10% lower than previously achieved. A simple and efficient approach to reducing inbreeding in small populations with sexes of unequal census numbers is to impose a breeding structure where parental success is controlled in each generation. This approach led to the development of strategies for selecting replacements each generation that were based upon parentage, e.g., a son replacing its sire. This study extends these strategies to a multigeneration round robin scheme where genetic contributions of ancestors to descendants are managed to remove all uncertain tics about breeding roles over generations; i.e., male descendants are distributed as equally as possible among dams. In doing so, the sampling variance of genetic contributions within each breeding category is eliminated and consequently DeltaF is minimized. Using the concept of long-term genetic contributions, the asymptotic DeltaF of the new strategy for random mating, M sires and d dams per sire, is phi/(12M), where phi = [1 + 2((1)/(4))(d)]. Predictions were validated using Monte Carlo simulations. The scheme was shown to achieve the lowest possible DeltaF using pedigree alone and showed that further reductions in DeltaF below that obtained from mating arise from preferential mating of relatives and not front their avoidance.
Selection: Software to predict selection response and rate of inbreeding in livestock breeding programs
Rutten, M.J.M. ; Bijma, P. ; Woolliams, J.A. ; Arendonk, J.A.M. van - \ 2002
Journal of Heredity 93 (2002)6. - ISSN 0022-1503 - p. 456 - 458.
asymptotic rates - populations - multivariate
|Design of sustainable breeding programs in developed countries
Bijma, P. ; Meuwissen, T.H.E. ; Woolliams, J.A. - \ 2002
In: Proceedings 7th World Congress on Genetics Applied to Livestock Production, Montpellier, 19-23 August. Vol. 30 [S.l.] : S.n. - p. 133 - 133.
Precision of methods for calculating identity-by-descent matrices using multiple markers
Sorensen, A.C. ; Pong Wong, R. ; Windig, J.J. ; Woolliams, J.A. - \ 2002
Genetics, Selection, Evolution 34 (2002)5. - ISSN 0999-193X - p. 557 - 579.
A rapid, deterministic method (DET) based on a recursive algorithm and a stochastic method based on Markov Chain Monte Carlo (MCMC) for calculating identity-by-descent (IBD) matrices conditional on multiple markers were compared using stochastic simulation. Precision was measured by the mean squared error (MSE) of the relationship coefficients in predicting the true IBD relationships, relative to MSE obtained from using pedigree only. Comparisons were made when varying marker density, allele numbers, allele frequencies, and the size of full-sib families. The precision of DET was 75-99% relative to MCMC, but was not simply related to the informativeness of individual loci. For situations mimicking microsatellite markers or dense SNP, the precision of DET was $\\geq 95\\%$ relative to MCMC. Relative precision declined for the SNP, but not microsatellites as marker density decreased. Full-sib family size did not affect the precision. The methods were tested in interval mapping and marker assisted selection, and the performance was very largely determined by the MSE. A multi-locus information index considering the type, number, and position of markers was developed to assess precision. It showed a marked empirical relationship with the observed precision for DET and MCMC and explained the complex relationship between relative precision and the informativeness of individual loci.
Genetic gain of pure line selection and combined crossbred purebred selection with constrained inbreeding
Bijma, P. ; Woolliams, I.A. ; Arendonk, J.A.M. van - \ 2001
Animal Science 72 (2001). - ISSN 1357-7298 - p. 225 - 232.
Using deterministic methods, rates of genetic gain (ýG) and inbreeding (ýF) were compared between pure line selection (PLS) and combined crossbred purebred selection (CCPS), for the sire line of a three-way crossbreeding scheme. Purebred performance and crossbred performance were treated as genetically correlated traits assuming the infinitesimal model. Breeding schemes were compared at a fixed total number of purebred selection candidates, i.e. including crossbred information did not affect the size of the purebred nucleus. Selection was by truncation on estimated breeding values for crossbred performance. Rates of genetic gain were predicted using a pseudo-BLUP selection index. Rates of inbreeding were predicted using recently developed methods based on long-term genetic contributions. Results showed that changing from PLS to CCPS may increase ýF by a factor of 2·14. In particular with high heritabilities and low purebred-crossbred genetic correlations, CCPS requires a larger number of parents than PLS, to avoid excessive ýF. The superiority of CCPS over PLS was judged by comparing ýG from both selection strategies at the same ýF. At the same ýF, CCPS was superior to PLS and the superiority of CCPS was only moderately reduced compared with the situation without a restriction on ýF. This paper shows that the long-term genetic contribution theory can be used to balance ýF and ýG in animal breeding schemes within very limited computing time.
|The development of criteria for evaluating the degree of endangerment of livestock breeds in Europe
Banell, B. ; Distl, O. ; Georgoudis, A. ; Groeneveld, E. ; Martyniuk, E. ; Ollivier, L. ; Arendonk, J.A.M. van; Woolliams, J. - \ 2001
In: Book of abstracts no. 7 : 52nd Meeting of the European Association for Animal Production, Budapest, 2001. - Budapest : [s.n.], 2001 - p. 76 - 76.
Predicting rates of inbreeding for livestock improvement schemes
Bijma, P. ; Arendonk, J.A.M. van; Woolliams, J.A. - \ 2001
Journal of Animal Science 79 (2001). - ISSN 0021-8812 - p. 840 - 853.
This article presents a deterministic method to predict rates of inbreeding (ΔF) for typical livestock improvement schemes. The method is based on a recently developed general theory to predict rates of inbreeding, which uses the concept of long-term genetic contributions. A typical livestock breeding population was modeled, with overlapping generations, BLUP selection, and progeny testing of male selection candidates. Two types of selection were practiced: animals were either selected by truncation on estimated breeding values (EBV) across age classes, or the number of parents selected from each age class was set to a fixed value and truncation selection was practiced within age classes. Bulmer's equilibrium genetic parameters were obtained by iterating on a pseudo-BLUP selection index and ΔF was predicted for the equilibrium situation. Predictions were substantially more accurate than predictions from other available methods, which ignore the effect of selection on ΔF. Predictions were accurate for schemes with up to 20 sires. Predicted ΔF was somewhat too low for schemes with more than 20 sires, which was due to the use of simple linear models to predict genetic contributions. The present method provides a computationally feasible (i.e., deterministic) tool to consider both the rate of inbreeding and the rate of genetic gain when optimizing livestock improvement schemes.
Predicting rates of inbreeding in populations undergoing selection
Woolliams, J.A. ; Bijma, P. - \ 2000
Genetics 154 (2000). - ISSN 0016-6731 - p. 1851 - 1864.
Tractable forms of predicting rates of inbreeding (F) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. F was shown to be ~1/4(1 - ) times the expected sum of squared lifetime contributions, where is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express F in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing 1/4 (since = 0) was increased to 1/2. Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting F with sib indices in discrete generations since previously published solutions had proved complex.