The Robustness of Recombination Frequency Estimates in Intercrosses with Dominant Markers

The robustness of the maximum likelihood estimates of recombination frequencies has been investigated in double intercrosses with complete dominance at both loci. The robustness was investigated with respect to bias in the recombination frequency estimates due to: (1) limited sample sizes, (2) heterogeneity in recombination frequencies between sexes or among meioses and (3) factors that distort the segregation-misclassification or differential viability. In the coupling phase, the recombination frequency estimates are quite robust with respect to most of the investigated factors. Potentially, the most serious cause of a bias is misclassifications, which tend to increase the recombination frequency estimates. In the repulsion phase, misclassifications are particularly serious, leading to extreme discrepancies between true and observed values. In addition, limited sample size and sex differences in recombination can also bias recombination frequency estimates in repulsion. These effects may pose serious problem in genetic mapping with random amplified polymorphic DNA (RAPD) markers.     

Conclusion of LOD-score analysis for family data generated under two-locus models.

The power to detect linkage by the LOD-score method is investigated here for diseases that depend on the effects of two genes. The classical strategy is, first, to detect a major-gene (MG) effect by segregation analysis and, second, to seek for linkage with genetic markers by the LOD-score method using the MG parameters. We already showed that segregation analysis can lead to evidence for a MG effect for many two-locus models, with the estimates of the MG parameters being very different from those of the two genes involved in the disease. We show here that use of these MG parameter estimates in the LOD-score analysis may lead to a failure to detect linkage for some two-locus models. For these models, use of the sib-pair method gives a non-negligible increase of power to detect linkage. The linkage-homogeneity test among subsamples differing for the familial disease distribution provides evidence of parameter misspecification, when the MG parameters are used. Moreover, for most of the models, use of the MG parameters in LOD-score analysis leads to a large bias in estimation of the recombination fraction and sometimes also to a rejection of linkage for the true recombination fraction. A final important point is that a strong evidence of an MG effect, obtained by segregation analysis, does not necessarily imply that linkage will be detected for at least one of the two genes, even with the true parameters and with a close informative marker.     

与偏分离位点连锁的QTL作图的统计方法

提出了一种统计方法,可以估计与偏分离位点连锁的QTL的位置和效应。该方法利用回交群体中呈现偏分离的分子标记,首先用最大似然法对偏分离位点与标记位点之间的重组率和配子存活率进行估计,然后用区间作图法估计加性-显性模型下QTL的位置和效应参数。该方法可用于对常规作图研究中表现偏分离的标记进行分析,以帮助我们发现新的偏分离基因（或不育基因）和数量性状位点。     

THE RATE AND EFFECTS DISTRIBUTION OF VIABILITY MUTATION IN DROSOPHILA: MINIMUM DISTANCE ESTIMATION

The empirical distribution of the mean viability of mutation accumulation lines, obtained from three published experiments, was analyzed using minimum-distance estimation. In two cases (Mukai et al. 1972; Ohnishi 1977), mutations were allowed to accumulate in copies of chromosome II protected from natural selection and recombination. In the other one (Fernández and López-Fanjul 1996), they accumulated in inbred lines derived from an isogenic stock. In contrast with currently accepted hypotheses, we consistently estimated low (about 0.01) genomic viability mutation rates, λ, and a small kurtosis of the distribution of mutational effects on viability (a) in the three datasets. Minimum-distance estimates of the per-generation mean viability change due to mutation (λE[a]) were also obtained. These were very similar for both chromosomal datasets, their absolute values being about five times smaller than estimates obtained from the observed change in mean viability during the mutation process. It must be noted that, in both experiments, viability was measured relative to the Cy chromosome of a Cy/Pm stock. Thus, an unnoticed viability increase in this Cy chromosome may have resulted in overestimation of the mean viability reduction in the lines. In parallel, minimum-distance estimation of λE(a) from inbred lines data (where the selective pressure during the accumulation process was larger) was even somewhat smaller, in absolute value, and very close to the estimate obtained by comparing the mean viability of the lines with that of the control isogenic line. The evolutionary importance of these results, as well as their relevance to the solution of the mutational load paradox, is discussed.     

On estimating the linkage of marker genes to viability genes controlling inbreeding depression

Statistical properties and extensions of Hedrick and Muona's method for mapping viability alleles causing inbreeding depression are discussed in this paper. Their method uses the segregation ratios among selfed progeny of a marker-locus heterozygote to estimate the viability reduction, s, of an allele and its recombination fraction, c, with the marker. Explicit estimators are derived for c and s, including expressions for their variances. The degree of estimation bias is examined for cases when (1) the viability allele is partially recessive and (2) the marker locus is linked to two viability loci. If linkage or viability reduction is moderate, very large sample sizes are required to obtain reliable estimates of c and s, in part because these estimates show a statistical correlation close to unity. Power is further reduced because alleles causing viability reduction often occur at low frequency at specific loci in a population. To increase power, we present a statistical model for the joint analysis of several selfed progeny arrays selected at random from a population. Assuming a fixed total number of progeny, we determine the optimal number of progeny arrays versus number of progeny per array under this model. We also examine the increase of information provided by a second, flanking marker. Two flanking markers provide vastly superior estimation properties, reducing sample sizes by approximately 95% from those required by a single marker.     

Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes

The generalized nonepistatic selection regime encompasses combinations of multiplicative and neutral viability effects distributed across a set of loci. These subsume, in particular, mixtures of the classical modes of multiplicative and additive fitness evaluations for multilocus traits. Exact analytic conditions for existence and stability of a multilocus Hardy-Weinberg (H-W) polymorphic equilibrium configuration are ascertained. It is established that the central H-W polymorphism is stable only if the component loci are "over-dominant" and sufficient recombination is in force. The H-W central equilibrium is never stable for tight linkage whenever some multiplicative selection effects are contributed by at least two of the loci involved. In the case of additive selection expression and individual overdominant loci, the H-W polymorphism is stable independently of the level of recombination. In the context of "natural" recombination schemes, "more recombination" enhances the stability of the H-W polymorphic equilibrium.     

Interval estimation of genetic susceptibility for retrospective case-control studies

Background

This article describes classical and Bayesian interval estimation of genetic susceptibility based on random samples with pre-specified numbers of unrelated cases and controls.

Results

Frequencies of genotypes in cases and controls can be estimated directly from retrospective case-control data. On the other hand, genetic susceptibility defined as the expected proportion of cases among individuals with a particular genotype depends on the population proportion of cases (prevalence). Given this design, prevalence is an external parameter and hence the susceptibility cannot be estimated based on only the observed data. Interval estimation of susceptibility that can incorporate uncertainty in prevalence values is explored from both classical and Bayesian perspective. Similarity between classical and Bayesian interval estimates in terms of frequentist coverage probabilities for this problem allows an appealing interpretation of classical intervals as bounds for genetic susceptibility. In addition, it is observed that both the asymptotic classical and Bayesian interval estimates have comparable average length. These interval estimates serve as a very good approximation to the "exact" (finite sample) Bayesian interval estimates. Extension from genotypic to allelic susceptibility intervals shows dependency on phenotype-induced deviations from Hardy-Weinberg equilibrium.

Conclusions

The suggested classical and Bayesian interval estimates appear to perform reasonably well. Generally, the use of exact Bayesian interval estimation method is recommended for genetic susceptibility, however the asymptotic classical and approximate Bayesian methods are adequate for sample sizes of at least 50 cases and controls.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

Usefulness of single nucleotide polymorphism data for estimating population parameters

Single nucleotide polymorphism (SNP) data can be used for parameter estimation via maximum likelihood methods as long as the way in which the SNPs were determined is known, so that an appropriate likelihood formula can be constructed. We present such likelihoods for several sampling methods. As a test of these approaches, we consider use of SNPs to estimate the parameter Theta = 4N(e)micro (the scaled product of effective population size and per-site mutation rate), which is related to the branch lengths of the reconstructed genealogy. With infinite amounts of data, ML models using SNP data are expected to produce consistent estimates of Theta. With finite amounts of data the estimates are accurate when Theta is high, but tend to be biased upward when Theta is low. If recombination is present and not allowed for in the analysis, the results are additionally biased upward, but this effect can be removed by incorporating recombination into the analysis. SNPs defined as sites that are polymorphic in the actual sample under consideration (sample SNPs) are somewhat more accurate for estimation of Theta than SNPs defined by their polymorphism in a panel chosen from the same population (panel SNPs). Misrepresenting panel SNPs as sample SNPs leads to large errors in the maximum likelihood estimate of Theta. Researchers collecting SNPs should collect and preserve information about the method of ascertainment so that the data can be accurately analyzed.     

Incorporating uncertainty in spatial structure for viability predictions: a case study of California sea lions (Zalophus californianus californianus)

In recent years, population viability analysis has become a popular tool to assess the relative risk of extinction among populations. Viability estimates for spatially structured populations require movement data that are often unavailable. In this paper, a diffusion approximation model was used to explore the effects of different spatial scenarios resulting from assumptions about movement rates. Census data for 13 breeding islands occupied by California sea lions Zalophus californianus californianus in the Gulf of California were used to explore three potential scenarios: unlimited movement between sites (panmictic population), limited movement (several clusters of populations) and no movement between islands (isolated islands). Predicted viability estimates were different for each scenario, but contrary to expectations, the mean extinction risk estimates were generally lowest when movement was unlimited (panmictic scenario). However, despite an extensive dataset, the confidence of the viability predictions for each scenario was low. In some cases, uncertainty in predictions within a scenario was greater than differences between scenarios. Therefore, it is recommended that in situations where movement rates and spatial structure are unknown, extinction risk estimates should reflect both the confidence intervals for each risk estimate and the uncertainty resulting from different spatial scenarios. This study also provides the first estimate of population viability (considering spatial structure) for California sea lions in the Gulf of California and an evaluation of population status based on the IUCN criteria for species listing.     

Validation of Bayesian analysis of compartmental kinetic models in medical imaging

IntroductionKinetic compartmental analysis is frequently used to compute physiologically relevant quantitative values from time series of images. In this paper, a new approach based on Bayesian analysis to obtain information about these parameters is presented and validated.Materials and methodsThe closed-form of the posterior distribution of kinetic parameters is derived with a hierarchical prior to model the standard deviation of normally distributed noise. Markov chain Monte Carlo methods are used for numerical estimation of the posterior distribution. Computer simulations of the kinetics of F18-fluorodeoxyglucose (FDG) are used to demonstrate drawing statistical inferences about kinetic parameters and to validate the theory and implementation. Additionally, point estimates of kinetic parameters and covariance of those estimates are determined using the classical non-linear least squares approach.Results and discussionPosteriors obtained using methods proposed in this work are accurate as no significant deviation from the expected shape of the posterior was found (one-sided P > 0.08). It is demonstrated that the results obtained by the standard non-linear least-square methods fail to provide accurate estimation of uncertainty for the same data set (P < 0.0001).ConclusionsThe results of this work validate new methods for a computer simulations of FDG kinetics. Results show that in situations where the classical approach fails in accurate estimation of uncertainty, Bayesian estimation provides an accurate information about the uncertainties in the parameters. Although a particular example of FDG kinetics was used in the paper, the methods can be extended for different pharmaceuticals and imaging modalities.     

Estimation of segregation and linkage parameters in simulated data. II. Simultaneous estimation with one linked marker.

This study examined the method of simultaneous estimation of recombination frequency and parameters for a qualitative trait locus and compared the results with those of standard methods of linkage analysis. With both approaches we were able to detect linkage of an incompletely penetrant qualitative trait to highly polymorphic markers with recombination frequencies in the range of .00-.05. Our results suggest that detecting linkage at larger recombination frequencies may require larger data sets or large high-density families. When applied to all families without regard to informativeness of the family structure for linkage, analyses of simulated data could detect no advantage of simultaneous estimation over more traditional and much less time-consuming methods, either in detecting linkage, estimating frequency, refining estimates of parameters for the qualitative trait locus, or avoiding false evidence for linkage. However, the method of sampling affected results.     

Disentangling linkage disequilibrium and linkage from dense single-nucleotide polymorphism trio data

Parent-offspring trios are widely collected for disease gene-mapping studies and are being extensively genotyped as part of the International HapMap Project. With dense maps of markers on trios, the effects of LD and linkage can be separated, allowing estimation of recombination rates in a model-free setting. Here we define a model-free multipoint method on the basis of dense sequence polymorphism data from parent-offspring trios to estimate intermarker recombination rates. We use simulations to show that this method has up to 92% power to detect recombination hotspots of intensity 25 times background over a region of size 10 kb typed at density 1 marker per 2.5 kb and almost 100% power to detect large hotspots of intensity >125 times background over regions of size 10 kb typed with just 1 marker per 5 kb (alpha = 0.05). We found strong agreement at megabase scales between estimates from our method applied to HapMap trio data and estimates from the genetic map. At finer scales, using Centre d'Etude du Polymorphisme Humain (CEPH) pedigree data across a 10-Mb region of chromosome 20, a comparison of population recombination rate estimates obtained from our method with estimates obtained using a coalescent-based approximate-likelihood method implemented in PHASE 2.0 shows detection of the same coldspots and most hotspots: The Spearman rank correlation between the estimates from our method and those from PHASE is 0.58 (p < 2.2(-16)).     

Detection and localization of a single binary trait locus in experimental populations

The advancements made in molecular technology coupled with statistical methodology have led to the successful detection and location of genomic regions (quantitative trait loci; QTL) associated with quantitative traits. Binary traits (e.g. susceptibility/resistance), while not quantitative in nature, are equally important for the purpose of detecting and locating significant associations with genomic regions. Existing interval regression methods used in binary trait analysis are adapted from quantitative trait analysis and the tests for regression coefficients are tests of effect, not detection. Additionally, estimates of recombination that fail to take into account varying penetrance perform poorly when penetrance is incomplete. In this work a complete probability model for binary trait data is developed allowing for unbiased estimation of both penetrance and recombination between a genetic marker locus and a binary trait locus for backcross and F2 experimental designs. The regression model is reparameterized allowing for tests of detection. Extensive simulations were conducted to assess the performance of estimation and testing in the proposed parameterization. The proposed parameterization was compared with interval regression via simulation. The results indicate that our parameterization shows equivalent estimation capabilities, requires less computational effort and works well with only a single marker.     

On the Recombination Rate Estimation in the Presence of Population Substructure

As recombination events are not uniformly distributed along the human genome, the estimation of fine-scale recombination maps, e.g. HapMap Project, has been one of the major research endeavors over the last couple of years. For simulation studies, these estimates provide realistic reference scenarios to design future study and to develop novel methodology. To achieve a feasible framework for the estimation of such recombination maps, existing methodology uses sample probabilities for a two-locus model with recombination, with recent advances allowing for computationally fast implementations. In this work, we extend the existing theoretical framework for the recombination rate estimation to the presence of population substructure. We show under which assumptions the existing methodology can still be applied. We illustrate our extension of the methodology by an extensive simulation study.     

Methodology and Accuracy of Estimation of Quantitative Trait Loci Parameters in a Half-Sib Design Using Maximum Likelihood

Maximum likelihood methods were developed for estimation of the six parameters relating to a marker-linked quantitative trait locus (QTL) segregating in a half-sib design, namely the QTL additive effect, the QTL dominance effect, the population mean, recombination between the marker and the QTL, the population frequency of the QTL alleles, and the within-family residual variance. The method was tested on simulated stochastic data with various family structures under two genetic models. A method for predicting the expected value of the likelihood was also derived and used to predict the lower bound sampling errors of the parameter estimates and the correlations between them. It was found that standard errors and confidence intervals were smallest for the population mean and variance, intermediate for QTL effects and allele frequency, and highest for recombination rate. Correlations among standard errors of the parameter estimates were generally low except for a strong negative correlation (r = -0.9) between the QTL's dominance effect and the population mean, and medium positive and negative correlations between the QTL's additive effect and, respectively, recombination rate (r = 0.5) and residual variance (r = -0.6). The implications for experimental design and method of analysis on power and accuracy of marker-QTL linkage experiments were discussed.     

Estimation of the recombination fraction by the maximum likelihood method in mapping interacting genes relative to marker loci

For mapping nonlinked interacting genes relative to marker loci, the recombination fractions can be calculated by using the log-likelihood functions were derived that permit estimation of recombinant fractions by solving the ML equations on the basis of F2 data at various types of interaction. In some cases, the recombinant fraction estimates are obtained in the analytical form while in others they are numerically calculated from concrete experimental data. With the same type of epistasis the log-functions were shown to differ depending on the functional role (suppression or epistasis) of the mapped gene. Methods for testing the correspondence of the model and the recombination fraction estimates to the experimental data are discussed. In ambiguous cases, analysis of the linked marker behavior makes it possible to differentiate gene interaction from distorted single-locus segregation, which at some forms of interaction imitate phenotypic ratios.     

Likelihood analysis of disequilibrium mapping, and related problems.

In this paper a theory is developed that provides the sampling distribution of alleles at a diallelic marker locus closely linked to a low-frequency allele that arose as a single mutant. The sampling distribution provides a basis for maximum-likelihood estimation of either the recombination rate, the mutation rate, or the age of the allele, provided that the two other parameters are known. This theory is applied to (1) the data of Hästbacka et al., to estimate the recombination rate between a locus associated with diastrophic dysplasia and a linked RFLP marker; (2) the data of Risch et al., to estimate the age of a presumptive allele causing idiopathic distortion dystonia in Ashkenazi jews; and (3) the data of Tishkoff et al., to estimate the date at which, at the CD4 locus, non-African lineages diverged from African lineages. We conclude that the extent of linkage disequilibrium can lead to relatively accurate estimates of recombination and mutation rates and that those estimates are not very sensitive to parameters, such as the population age, whose values are not known with certainty. In contrast, we also conclude that, in many cases, linkage disequilibrium may not lead to useful estimates of allele age, because of the relatively large degree of uncertainly in those estimates.     

Allowing for random errors in radiation dose estimates for the atomic bomb survivor data

The presence of random errors in the individual radiation dose estimates for the A-bomb survivors causes underestimation of radiation effects in dose-response analyses, and also distorts the shape of dose-response curves. Statistical methods are presented which will adjust for these biases, provided that a valid statistical model for the dose estimation errors is used. Emphasis is on clarifying some rather subtle statistical issues. For most of this development the distinction between radiation dose and exposure is not critical. The proposed methods involve downward adjustment of dose estimates, but this does not imply that the dosimetry system is faulty. Rather, this is a part of the dose-response analysis required to remove biases in the risk estimates. The primary focus of this report is on linear dose-response models, but methods for linear-quadratic models are also considered briefly. Some plausible models for the dose estimation errors are considered, which have typical errors in a range of 30-40% of the true values, and sensitivity analysis of the resulting bias corrections is provided. It is found that for these error models the resulting estimates of excess cancer risk based on linear models are about 6-17% greater than estimates that make no allowance for dose estimation errors. This increase in risk estimates is reduced to about 4-11% if, as has often been done recently, survivors with dose estimates above 4 Gy are eliminated from the analysis.     

Environment dependence of mutational parameters for viability in Drosophila melanogaster

The genomic rate of mildly deleterious mutations (U) figures prominently in much evolutionary and ecological theory. In Drosophila melanogaster, estimates of U have varied widely, from <0.1 to nearly 1 per zygote. The source of this variation is unknown, but could include differences in the conditions used for assaying fitness traits. We examined how assay conditions affect estimates of the rates and effects of viability-depressing mutations in two sets of lines with accumulated spontaneous mutations on the second chromosome. In each set, the among-line variance in egg-to-adult viability was significantly greater when viability was assayed using a high parental density than when it was assayed using a low density. In contrast, the proportional decline in viability due to new mutations did not differ between densities. Two other manipulations, lowering the temperature and adding ethanol to the medium, had no significant effects on either the mean decline or among-line variance. Cross-environment genetic correlations in viability were generally close to one, implying that most mutations reduced viability in all environments. Using data from the low-density, lower-bound estimates of U approached the classic, high values of Mukai and Ohnishi; at the high density, U estimates were similar to recently reported low values. The difference in estimated mutation rates, taken at face value, would imply that many mutations affected fitness at low density but not at high density, but this is shown to be incompatible with the observed high cross-environment correlations. Possible reasons for this discrepancy are discussed. Regardless of the interpretation, the results show that assay conditions can have a large effect on estimates of mutational parameters for fitness traits.