Interval estimation of the median lethal dose

Assuming a logistic dose-response curve, one can construct a confidence interval for the LD50 from the asymptotic likelihood ratio test. Reasons are given for preferring this likelihood ratio interval to the established interval calculated by applying Fieller's theorem to the maximum likelihood estimates.     

The Asymptotic Covariance Matrix of the Maximum Likelihood Parameter Estimator in Conditional Poisson Log-linear Models

The asymptotic covariance matrix of the maximum likelihood estimator for the log-linear model is given for a general class of conditional Poisson distributions which include the unconditional Poisson, multinomial and product-multinomial, as special cases. The general conditions are given under which the maximum likelihood covariance matrix is equal to the covariance matrix of an equivalent closed-form weighted least squares estimator.     

Distribution-free regression analysis of grouped survival data

Methods based on regression models for logarithmic hazard functions, Cox models, are given for analysis of grouped and censored survival data. By making an approximation it is possible to obtain explicitly a maximum likelihood function involving only the regression parameters. This likelihood function is a convenient analog to Cox's partial likelihood for ungrouped data. The method is applied to data from a toxicological experiment.     

Nonparametric estimation and testing in a cure model.

Nonparametric generalized maximum likelihood product limit point estimators and confidence intervals are given for a cure model with random censorship. One-, two-, and K-sample likelihood ratio tests for inference on the cure rates are developed. In the two-sample case its power is compared to the power of several alternatives, including the log-rank and Gray and Tsiatis (1989, Biometrics 45, 899-904) tests. Implications for the use of the likelihood ratio test in a clinical trial designed to compare cure rates are discussed.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

HAPLOFREQ--estimating haplotype frequencies efficiently.

A commonly used tool in disease association studies is the search for discrepancies between the haplotype distribution in the case and control populations. In order to find this discrepancy, the haplotypes frequency in each of the populations is estimated from the genotypes. We present a new method HAPLOFREQ to estimate haplotype frequencies over a short genomic region given the genotypes or haplotypes with missing data or sequencing errors. Our approach incorporates a maximum likelihood model based on a simple random generative model which assumes that the genotypes are independently sampled from the population. We first show that if the phased haplotypes are given, possibly with missing data, we can estimate the frequency of the haplotypes in the population by finding the global optimum of the likelihood function in polynomial time. If the haplotypes are not phased, finding the maximum value of the likelihood function is NP-hard. In this case, we define an alternative likelihood function which can be thought of as a relaxed likelihood function. We show that the maximum relaxed likelihood can be found in polynomial time and that the optimal solution of the relaxed likelihood approaches asymptotically to the haplotype frequencies in the population. In contrast to previous approaches, our algorithms are guaranteed to converge in polynomial time to a global maximum of the different likelihood functions. We compared the performance of our algorithm to the widely used program PHASE, and we found that our estimates are at least 10% more accurate than PHASE and about ten times faster than PHASE. Our techniques involve new algorithms in convex optimization. These algorithms may be of independent interest. Particularly, they may be helpful in other maximum likelihood problems arising from survey sampling.     

Exact computation of coalescent likelihood for panmictic and subdivided populations under the infinite sites model

Coalescent likelihood is the probability of observing the given population sequences under the coalescent model. Computation of coalescent likelihood under the infinite sites model is a classic problem in coalescent theory. Existing methods are based on either importance sampling or Markov chain Monte Carlo and are inexact. In this paper, we develop a simple method that can compute the exact coalescent likelihood for many data sets of moderate size, including real biological data whose likelihood was previously thought to be difficult to compute exactly. Our method works for both panmictic and subdivided populations. Simulations demonstrate that the practical range of exact coalescent likelihood computation for panmictic populations is significantly larger than what was previously believed. We investigate the application of our method in estimating mutation rates by maximum likelihood. A main application of the exact method is comparing the accuracy of approximate methods. To demonstrate the usefulness of the exact method, we evaluate the accuracy of program Genetree in computing the likelihood for subdivided populations.     

Use of the Maximum Likelihood Method in the Analysis of Phenotypic Stability

The stability variance is an important estimator of phenotypic stability of genotypes. It may be estimated by method of moments and by maximum likelihood. We demonstrate by Monte Carlo simulation that, given a sufficient number of environments, maximum likelihood estimates (MLE's) are slightly better if ranking of genotypes is the experimenter's major aim. A likelihood ratio test is available for different hypotheses.     

An approximation to the likelihood for a pedigree with loops

This paper presents a new approximation to the likelihood for a pedigree with loops, based on cutting all loops and extending the pedigree at the cuts. An opimum loop-cutting strategy and an iterative extension technique are presented. The likelihood for a pedigree with loops is then approximated by the conditional likelihood for the entire cut-extended pedigree given the extended part. The approximate likelihoods are compared with the exact likelihoods obtained using the program MENDEL for several small pedigrees with loops. The approximation is efficient for large pedigrees with complex loops in terms of computing speed and memory requirements.     

Proportionality of Diagonal Blocks with Blockwise Independence

In this paper, the tests of similarities among group covariance matrices and the differences among block covariance matrices within a group under repeated measurement model are studied. There are nine hierarchical nested structures of covariance matrices which have been tested. The likelihood ratio tests have been derived for these nine hierarchically structured models. An algorithm for determining the numerical solution of the corresponding maximum likelihood equations is also given.     

Computer software for performing likelihood tests of pedigree relationship using genetic markers

Molecular techniques are making ever more genetic markers available for use in parentage assignment, and measures of relatedness. We present a program, Kinship, designed to use likelihood techniques to test for any non-inbred pedigree relationship between pairs of individuals, using single-locus codominant genetic markers. Kinship calculates the likelihood that each pair of individuals in a data set are related by a given pedigree hypothesis, and likelihood ratios for any pair of hypotheses. The program also uses a simulation routine to attach statistical significance to its results.     

Tests for association of gene frequencies at several loci in random mating diploid populations.

Methods are outlined for analyzing data on genotype frequencies at several codominant loci in random mating diploid populations. Maximum likelihood (ML) methods are given for estimating chromosomal frequencies. Using these, a succession of models of assumed independence of gene frequency are fitted. These are based on those used in multi-dimensional contigency tables, and tests for association (linkage disequilibrium), made using likelihood ratios. The methods are illustrated with an example.     

Maximum likelihood for interval censored data: Consistency and computation

Standard convex optimization techniques are applied to the analysisof interval censored data. These methods provide easily verifiableconditions for the self-consistent estimator proposed by Turnbull(1976) to be a maximum likelihood estimator and for checkingwhether the maximum likelihood estimate is unique. A sufficientcondition is given for the almost sure convergence of the maximumlikelihood estimator to the true underlying distribution function.     

A Life Table Regression Analysis for Complex Survey Data

A methodology for obtaining maximum likelihood estimates of life table regression coefficients from complex survey data is presented. Certain of the issues of writing a likelihood for survey data are presented and discussed. The proposed methodology includes consideration of the sampling design in any inference by using design based variance estimates for the parameters. An example is given using data from the 1973 United States National Survey of Family Growth.     

A likelihood approach for functional response models

Functional response is an important determinant of community dynamics, and thus empirical methods for characterizing functional responses are as important in understanding ecological processes. The most commonly used method is based on the sum of squares, and the maximum likelihood method is rarely used. When the likelihood method is used, potentially inappropriate probability distributions such as binomial distributions are typically assumed for the number of prey eaten in experiments. In this study, I present a likelihood approach in which the probability distributions are generated by mechanistic understanding of predation processes using Monte Carlo simulations. An example is given on the Holling type II functional response model, but the method is flexible and allows characterization of a wide variety of functional response models. In the example, the likelihood method consistently resulted in superior estimates than the least squares method.     

Simplified point and interval estimation for removal trapping

A regression technique, based on the limiting normal distribution of the multinomial, is given for point and interval estimation of the parameters in the removal trapping method of determining animal and insect populations. Comparisons are made with maximum likelihood estimates. Two examples of estimating spider populations are given.     

A two-stage pruning algorithm for likelihood computation for a population tree

We have developed a pruning algorithm for likelihood estimation of a tree of populations. This algorithm enables us to compute the likelihood for large trees. Thus, it gives an efficient way of obtaining the maximum-likelihood estimate (MLE) for a given tree topology. Our method utilizes the differences accumulated by random genetic drift in allele count data from single-nucleotide polymorphisms (SNPs), ignoring the effect of mutation after divergence from the common ancestral population. The computation of the maximum-likelihood tree involves both maximizing likelihood over branch lengths of a given topology and comparing the maximum-likelihood across topologies. Here our focus is the maximization of likelihood over branch lengths of a given topology. The pruning algorithm computes arrays of probabilities at the root of the tree from the data at the tips of the tree; at the root, the arrays determine the likelihood. The arrays consist of probabilities related to the number of coalescences and allele counts for the partially coalesced lineages. Computing these probabilities requires an unusual two-stage algorithm. Our computation is exact and avoids time-consuming Monte Carlo methods. We can also correct for ascertainment bias.     

Restricted BLUP for Mixed Linear Models

A new estimation procedure for mixed regression models is introduced. It is a development of Henderson's best linear unbiased prediction procedure which uses the joint distribution of the observed dependent random variables and the unknown realisations of the random components of the model. It is proposed to replace the likelihood of the observations given the random components by the asymptotic likelihood of the maximum likelihood estimators and the prior distribution of the random components by a restricted prior distribution which is consistent with the usual restrictions placed on the random components when they are considered conditionally fixed.     

Maximum likelihood vs. minimum chi-square--a general comparison with applications to the estimation of recombination fractions in two-point linkage analysis.

Some relationships between the estimates of recombination fraction in two-point linkage analysis obtained by maximum likelihood, minimum chi-square, and general least squares are derived. These theoretical results are based on an approximation for the multinomial distribution. Applications (theoretical and experimental) with RFLP (restriction fragment length polymorphism) markers for a segregating F2 population are given. The minimum chi-square estimate is slightly larger than the maximum likelihood estimate. For applications, however, both estimates must be considered to be approximately equal. The least squares estimates are slightly different (larger or smaller) from these estimates.     

MLEP: an R package for exploring the maximum likelihood estimates of penetrance parameters

ABSTRACT: BACKGROUND: Linkage analysis is a useful tool for detecting genetic variants that regulate a trait of interest, especially genes associated with a given disease. Although penetrance parameters play an important role in determining gene location, they are assigned arbitrary values according to the researcher's intuition or as estimated by the maximum likelihood principle. Several methods exist by which to evaluate the maximum likelihood estimates of penetrance, although not all of these are supported by software packages and some are biased by marker genotype information, even when disease development is due solely to the genotype of a single allele. FINDINGS: Programs for exploring the maximum likelihood estimates of penetrance parameters were developed using the R statistical programming language supplemented by external C functions. The software returns a vector of polynomial coefficients of penetrance parameters, representing the likelihood of pedigree data. From the likelihood polynomial supplied by the proposed method, the likelihood value and its gradient can be precisely computed. To reduce the effect of the supplied dataset on the likelihood function, feasible parameter constraints can be introduced into maximum likelihood estimates, thus enabling flexible exploration of the penetrance estimates. An auxiliary program generates a perspective plot allowing visual validation of the model's convergence. The functions are collectively available as the MLEP R package. CONCLUSIONS: Linkage analysis using penetrance parameters estimated by the MLEP package enables feasible localization of a disease locus. This is shown through a simulation study and by demonstrating how the package is used to explore maximum likelihood estimates. Although the input dataset tends to bias the likelihood estimates, the method yields accurate results superior to the analysis using intuitive penetrance values for disease with low allele frequencies. MLEP is part of the Comprehensive R Archive Network and is freely available at http://cran.r-project.org/web/packages/MLEP/index.html.