Estimating the Frequency Distribution of Crossovers during Meiosis from Recombination Data

Estimation of tetrad crossover frequency distributions from genetic recombination data is a classic problem dating back to Weinstein (1936, Genetics 21, 155-199). But a number of important issues, such as how to specify the maximum number of crossovers, how to construct confidence intervals for crossover probabilities, and how to obtain correct p-values for hypothesis tests, have never been adequately addressed. In this article, we obtain some properties of the maximum likelihood estimate (MLE) for crossover probabilities that imply guidelines for choosing the maximum number of crossovers. We give these results for both normal meiosis and meiosis with nondisjunction. We also develop an accelerated EM algorithm to find the MLE more efficiently. We propose bootstrap-based methods to find confidence intervals and p-values and conduct simulation studies to check the validity of the bootstrap approach.     

Estimation of unbiased recombination values in the presence of misclassification of a trait using RFLP maps

The effect of misclassification of phenotypes of a trait on the estimation of recombination value was investigated. The effect was larger for closer linkage. If a locus is dominant and linked with the misclassfied trait locus in the repulsion phase, then the effect on the recombination value between the two loci is largest. A method for estimating the unbiased recombination value and the misclassification rate using maximum likelihood associated with an EM algorithm is also presented. This method was applied to a numerical example from rice genome data. It was concluded that the present method combined with the metric multi-dimensional scaling method is useful for the detection of misclassified markers and for the estimation of unbiased recombination values.     

Effects of constitutive heterochromatin and genotype on frequency and distribution of chiasmata in the seven individual rye bivalents

Summary Number and distribution of chiasmata were studied in the single pair of homologous rye chromosomes in 29 chromosomal F₁ hybrids between the seven disomic wheat rye addition lines of Chinese Spring/ Imperial and five selected inbred genotypes of cultivated rye by using the differential Giemsa staining technique. The results indicate that the number and position of chiasmata is independent from the amount and position of C-heterochromatin. Genotype had an effect on chiasma number, whereas chiasma distribution within bivalents appeared to be determined by morphological features of chromosomes. Late replicating DNA in constitutive heterochromatin may delay the separation of half bivalents if chiasmata are formed between them and the centromere.Supported by Deutsche Forschungsgemeinschaft, Bonn     

Maximum Likelihood Estimation of Simultaneous Pairwise Linear Structural Relationships

The problem of assessing the relative calibrations and relative accuracies of a set of p instruments, each designed to measure the same characteristic on a common group of individuals is considered by using the EM algorithm. As shown, the EM algorithm provides a general solution for this problem. Its implementation is simple and in its most general form requires no extra iterative procedures within the M step. One important feature of the algorithm in this set up is that the error variance estimates are always positive. Thus, it can be seen as a kind of restricted maximization procedure. The expected information matrix for the maximum likelihood estimators is derived, upon which the large sample estimated covariance matrix for the maximum likelihood estimators can be computed. The problem of testing hypothesis about the calibration lines can be approached by using the Wald statistics. The approach is illustrated by re-analysing two data sets in the literature.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

A Mixture‐of‐Genotypes Model for the Distribution of Thermostable Phenol Sulfotransferase Activity

A statistical method for parametric density estimation based upon a mixture‐of‐genotypes model is developed for the thermostable phenol sulfotransferase (SULT1A1) activity which has a putative role in modifying risk for colon and prostate cancer/polyps. The EM algorithm for the general mixture model is modified to accommodate the genetic constraints and is used to estimate genotype frequencies from the distribution of the SULT1A1 phenotype. A parametric bootstrap likelihood ratio test is considered as a testing method for the number of mixing components. The size and power of the test is then investigated and compared with the conventional chi‐squared test. The relative risk associated with genotypes defined by this model is also investigated through the generalized linear model. This analysis revealed that a genotype with the highest mean value of SULT1A1 activity has greater impact on cancer risk than others. This result suggests that the phenotype with a higher SULT1A1 activity might be important in studying the association between the cancer risk and SULT1A1 activity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plant cover estimation based on the beta distribution in grassland vegetation

Cover is the most frequently used measure of abundance in vegetation surveys of grasslands, and various qualitative and semi-quantitative methods have been developed for visual estimation of this metric. Field survey is usually made with a point-grid plate. The frequency distributions of cover derived from point-grid counts follow a beta distribution. Combining point-grid counts from a field survey and the beta distribution for a statistical analysis, we developed an effort-saving cover-measurement method. Cover is measured with a transparent plastic plate on which, for example, 10 × 10 = 100 points are arranged in a lattice with 1-cm grid spacing (thus, one point count represents 1 cm² of cover). N quadrats are set out at randomly dispersed sites in a grassland, and, in each, the plastic plate is used for making counts. The number of grid points located above a given species is counted in every quadrat until the number of counted points reaches a given value c, which is determined in advance. If the number of counted points reaches c in a quadrat, the count is stopped and the quadrat is classified in the category “>c”. In quadrats where c is not attained, full point counts above the species bodies are made. Let g be the number of observed quadrats whose cover is ≤c. Using these g cover measurements and the number of quadrats (N − g) with cover >c, we can quantitatively estimate cover for each species and the spatial pattern index value based on the maximum likelihood method. In trial counts using this method, the time savings varied between 5% and 41%, depending on the shape of the cover frequency distribution. The mean cover value estimates agreed well with conventional measures without a stopping point (i.e., based on full counts of all points in each quadrat).     

Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Bias Reduction and a Solution for Separation of Logistic Regression with Missing Covariates

Summary Logistic regression is an important statistical procedure used in many disciplines. The standard software packages for data analysis are generally equipped with this procedure where the maximum likelihood estimates of the regression coefficients are obtained iteratively. It is well known that the estimates from the analyses of small‐ or medium‐sized samples are biased. Also, in finding such estimates, often a separation is encountered in which the likelihood converges but at least one of the parameter estimates diverges to infinity. Standard approaches of finding such estimates do not take care of these problems. Moreover, the missingness in the covariates adds an extra layer of complexity to the whole process. In this article, we address these three practical issues—bias, separation, and missing covariates by means of simple adjustments. We have applied the proposed technique using real and simulated data. The proposed method always finds a solution and the estimates are less biased. A SAS macro that implements the proposed method can be obtained from the authors.     

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.     

Statistical inference for serial dilution assay data

Serial dilution assays are widely employed for estimating substance concentrations and minimum inhibitory concentrations. The Poisson-Bernoulli model for such assays is appropriate for count data but not for continuous measurements that are encountered in applications involving substance concentrations. This paper presents practical inference methods based on a log-normal model and illustrates these methods using a case application involving bacterial toxins.