Maximum likelihood vs. minimum chi-square

In minimum chi-square logit or probit analysis of quantal bioassay data, a requirement for proper asymptotic behavior of the estimates made is that test-group sizes get indefinitely large. Inconsistent estimates result if group sizes are small, however numerous the groups. Maximum likelihood estimates do not show this inconsistent behavior, even if all the many group sizes are only unity. The inconsistent behavior for minimum chi-square results from a bias toward 0.5 for response probabilities. At 0.5 the binomial variance is at a maximum of 0.25, so tending to minimize the calculated value of chi square. The principle of minimum chi-square should not be confused with the principle of least squares.     

Joint estimation of recombination fractions and interference coefficients in multilocus linkage analysis.

Estimation of recombination fractions and interference coefficients is of importance in multilocus linkage analysis. With the development of molecular genetic technologies such as RFLP, multilocus data are readily available to researchers. Several methods have been developed to analyze such data, and each performs well under restrictive conditions. The present paper proposes a method based on a multiplicative model and maximum-likelihood estimation of recombination fractions and interference coefficients. The estimators are consistent regardless of the model assumptions and are efficient if the model is a good approximation. The estimators are tractable even when there are incomplete observations. Furthermore, the interference between nonadjacent chromosomal regions or those among three chromosomal regions can be modeled and tested by a simple Z-test. The proposed method was applied to linkage analysis of four-locus data obtained from Drosophila and that of seven-locus data obtained again from Drosophila. Reanalysis of the first example revealed that there is interference between chromosomal regions 2 and 3. Analysis of the second example suggested that there is triple interference as well as pairwise interference between nonadjacent chromosomal regions; the genetic interpretation of these findings remains to be developed.     

Maximum likelihood estimation of linkage disequilibrium in half-sib families

Maximum likelihood methods for the estimation of linkage disequilibrium between biallelic DNA-markers in half-sib families (half-sib method) are developed for single and multifamily situations. Monte Carlo computer simulations were carried out for a variety of scenarios regarding sire genotypes, linkage disequilibrium, recombination fraction, family size, and number of families. A double heterozygote sire was simulated with recombination fraction of 0.00, linkage disequilibrium among dams of δ=0.10, and alleles at both markers segregating at intermediate frequencies for a family size of 500. The average estimates of δ were 0.17, 0.25, and 0.10 for Excoffier and Slatkin (1995), maternal informative haplotypes, and the half-sib method, respectively. A multifamily EM algorithm was tested at intermediate frequencies by computer simulation. The range of the absolute difference between estimated and simulated δ was between 0.000 and 0.008. A cattle half-sib family was genotyped with the Illumina 50K BeadChip. There were 314,730 SNP pairs for which the sire was a homo-heterozygote with average estimates of r2 of 0.115, 0.067, and 0.111 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. There were 208,872 SNP pairs for which the sire was double heterozygote with average estimates of r2 across the genome of 0.100, 0.267, and 0.925 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. Genome analyses for all possible sire genotypes with 829,042 tests showed that ignoring half-sib family structure leads to upward biased estimates of linkage disequilibrium. Published inferences on population structure and evolution of cattle should be revisited after accommodating existing half-sib family structure in the estimation of linkage disequilibrium.     

Maximum likelihood estimation of recombination rates from population data

We describe a method for co-estimating r = C/mu (where C is the per-site recombination rate and mu is the per-site neutral mutation rate) and Theta = 4N(e)mu (where N(e) is the effective population size) from a population sample of molecular data. The technique is Metropolis-Hastings sampling: we explore a large number of possible reconstructions of the recombinant genealogy, weighting according to their posterior probability with regard to the data and working values of the parameters. Different relative rates of recombination at different locations can be accommodated if they are known from external evidence, but the algorithm cannot itself estimate rate differences. The estimates of Theta are accurate and apparently unbiased for a wide range of parameter values. However, when both Theta and r are relatively low, very long sequences are needed to estimate r accurately, and the estimates tend to be biased upward. We apply this method to data from the human lipoprotein lipase locus.     

Maximum likelihood estimation of linkage and interference from tetrad data

Maximum likelihood equations have been derived for estimation of map distance and interference from two-point and ranked tetrad data. The estimators have been applied to data from Saccharomyces cerevisiae and Schizosaccharomyces pombe. S. cerevisiae consistently shows quite strong interference over the mapped genome. In striking contrast, S. pombe consistently shows much weaker interference and many crosses exhibit negative interference. In neither species was there a conspicuous tendency for intervals spanning a centromere to show less interference than those that did not. Since the amount of recombination per microgram of DNA in the two species is similar, the difference in interference characteristics seems to be a reflection of some fundamental difference in the recombination process of the two species.     

Age trends in human chiasma frequencies and recombination fractions. II. Method for analyzing recombination fractions and applications to the ABO:Nail-Patella linkage

A new method is presented for studying the relationship between human recombination fractions and parental age at the time of conception. Assuming the sex specific recombination fraction to be a linear function of age, a feasible computer algorithm is described whereby the likelihood of multigenerational families can be calculated. Using this method and the likelihood ratio test, it is found that for the ABO:nail-patella linkage age (P= .17)is more significant than sex (p= .23) in its effect on the recombination fraction. The age effect, if it is real, appears to be limited to males: the paternal recombination fraction decreases by .0062(+/- .0036) per year.     

Multilocus linkage analysis in humans: detection of linkage and estimation of recombination.

Multilocus linkage analysis is investigated from the viewpoint of the efficiency of recombination estimates under different strategies for detecting linkage and determining gene order within a linkage group. We consider the appropriateness of assuming no interference with data available in human genetic studies. Examples are given to show the significance of multilocus analysis in humans. A computer program package, LINKAGE, for multilocus linkage analysis is described.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

Polylink: to support two-point linkage analysis in autotetraploids

SUMMARY: Polylink runs under Microsoft Windows (95 or later). It performs various calculations that are useful for investigating two-point linkage analysis for autopolyploids, based on the random chromosome pairing model. These include calculation of offspring phenotypic probabilities as functions of the recombination fraction, calculation of theoretical standard errors for the maximum likelihood estimator of and numerical computation of maximum likelihood estimates. It also includes simulation facilities. AVAILABILITY: Polylink is free and available from Xiangming Xu via email     

Maximum likelihood analysis of a general latent variable model with hierarchically mixed data

A general two-level latent variable model is developed to provide a comprehensive framework for model comparison of various submodels. Nonlinear relationships among the latent variables in the structural equations at both levels, as well as the effects of fixed covariates in the measurement and structural equations at both levels, can be analyzed within the framework. Moreover, the methodology can be applied to hierarchically mixed continuous, dichotomous, and polytomous data. A Monte Carlo EM algorithm is implemented to produce the maximum likelihood estimate. The E-step is completed by approximating the conditional expectations through observations that are simulated by Markov chain Monte Carlo methods, while the M-step is completed by conditional maximization. A procedure is proposed for computing the complicated observed-data log likelihood and the BIC for model comparison. The methods are illustrated by using a real data set.     

Maximum likelihood estimation in dynamical models of HIV

The study of dynamical models of HIV infection, based on a system of nonlinear ordinary differential equations (ODE), has considerably improved the knowledge of its pathogenesis. While the first models used simplified ODE systems and analyzed each patient separately, recent works dealt with inference in non-simplified models borrowing strength from the whole sample. The complexity of these models leads to great difficulties for inference and only the Bayesian approach has been attempted by now. We propose a full likelihood inference, adapting a Newton-like algorithm for these particular models. We consider a relatively complex ODE model for HIV infection and a model for the observations including the issue of detection limits. We apply this approach to the analysis of a clinical trial of antiretroviral therapy (ALBI ANRS 070) and we show that the whole algorithm works well in a simulation study.     

Simultaneous maximum likelihood estimation of linkage and linkage phases in outcrossing species

With the advent of new molecular marker technologies, it is now feasible to initiate genome projects for outcrossing plant species, which have not received much attention in genetic research, despite their great agricultural and environmental value. Because outcrossing species typically have heterogeneous genomes, data structure for molecular markers representing an entire genome is complex: some markers may have more alleles than others, some markers are codominant whereas others are dominant, and some markers are heterozygous in one parent but fixed in the other parent whereas the opposite can be true for other markers. A major difficulty in analyzing these different types of marker at the same time arises from uncertainty about parental linkage phases over markers. In this paper, we present a general maximum-likelihood-based algorithm for simultaneously estimating linkage and linkage phases for a mixed set of different marker types containing fully informative markers (segregating 1:1:1:1) and partially informative markers (or missing markers, segregating 1:2:1, 3:1, and 1:1) in a full-sib family derived from two outbred parent plants. The characterization of linkage phases is based on the posterior probability distribution of the assignment of alternative alleles at given markers to two homologous chromosomes of each parent, conditional on the observed phenotypes of the markers. Two- and multi-point analyses are performed to estimate the recombination fraction and determine the most likely linkage phase between different types of markers. A numerical example is presented to demonstrate the statistical properties of the model for characterizing the linkage phase between markers.     

Maximum likelihood estimation of population growth rates based on the coalescent.

We describe a method for co-estimating 4Nemu (four times the product of effective population size and neutral mutation rate) and population growth rate from sequence samples using Metropolis-Hastings sampling. Population growth (or decline) is assumed to be exponential. The estimates of growth rate are biased upwards, especially when 4Nemu is low; there is also a slight upwards bias in the estimate of 4Nemu itself due to correlation between the parameters. This bias cannot be attributed solely to Metropolis-Hastings sampling but appears to be an inherent property of the estimator and is expected to appear in any approach which estimates growth rate from genealogy structure. Sampling additional unlinked loci is much more effective in reducing the bias than increasing the number or length of sequences from the same locus.