Standard errors for EM estimates in generalized linear models with random effects

A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulas are used to approximate the integrals in the EM algorithm, where two different approaches are pursued, i.e., Gauss-Hermite quadrature in the case of Gaussian random effects and nonparametric maximum likelihood estimation for an unspecified random effect distribution. An approximation of the expected Fisher information matrix is derived from an expansion of the EM estimating equations. This allows for inferential arguments based on EM estimates, as demonstrated by an example and simulations.     

On testing departure from the binomial and multinomial assumptions

This paper is concerned with testing the multinomial (binomial) assumption against the Dirichlet-multinomial (beta-binomial) alternatives. In particular, we discuss the distribution of the asymptotic likelihood ratio (LR) test and obtain the C(alpha) goodness-of-fit test statistic. The inadequacy of the regular chi-square approximation to the LR test is supported by some Monte Carlo experiments. The C(alpha) test is recommended based on empirical significance level and power and also computational simplicity. Two examples are given.     

Estimation of evolutionary parameters with phylogenetic trees

An important issue in the phylogenetic analysis of nucleotide sequence data using the maximum likelihood (ML) method is the underlying evolutionary model employed. We consider the problem of simultaneously estimating the tree topology and the parameters in the underlying substitution model and of obtaining estimates of the standard errors of these parameter estimates. Given a fixed tree topology and corresponding set of branch lengths, the ML estimates of standard evolutionary model parameters are asymptotically efficient, in the sense that their joint distribution is asymptotically normal with the variance–covariance matrix given by the inverse of the Fisher information matrix. We propose a new estimate of this conditional variance based on estimation of the expected information using a Monte Carlo sampling (MCS) method. Simulations are used to compare this conditional variance estimate to the standard technique of using the observed information under a variety of experimental conditions. In the case in which one wishes to estimate simultaneously the tree and parameters, we provide a bootstrapping approach that can be used in conjunction with the MCS method to estimate the unconditional standard error. The methods developed are applied to a real data set consisting of 30 papillomavirus sequences. This overall method is easily incorporated into standard bootstrapping procedures to allow for proper variance estimation.     

Inference on Risk Rates Based on Mortality Data Under Censoring and Competing Risks Using Parametric Models

In this paper we consider the competing risks model where the risks may not be independent. We assume both fixed and random censoring. The random censoring mechanism could have either a parametric or a non-parametric form. The life distributions and the parametric censoring distribution considered are exponential or Weibull. The expressions for the asymptotic confidence intervals for various parameters of interest under different models, using the estimated Fisher information matrix and parametric bootstrap techniques have been derived. Monte Carlo simulation studies for some of these cases have been carried out.     

New algorithms for Luria-Delbrück fluctuation analysis

Fluctuation analysis is the most widely used approach in estimating microbial mutation rates. Development of methods for point and interval estimation of mutation rates has long been hampered by lack of closed form expressions for the probability mass function of the number of mutants in a parallel culture. This paper uses sequence convolution to derive exact algorithms for computing the score function and observed Fisher information, leading to efficient computation of maximum likelihood estimates and profile likelihood based confidence intervals for the expected number of mutations occurring in a test tube. These algorithms and their implementation in SALVADOR 2.0 facilitate routine use of modern statistical techniques in fluctuation analysis by biologists engaged in mutation research.     

The impact of litter effects on dose-response modeling in teratology

The fitting of dose-response models to teratology data involving littermates in order to generate estimates of teratogenic risk is receiving increasing attention as a potential alternative to the "safety-factor" approach to risk estimation. In this paper, we utilize the beta-binomial distribution to introduce varying degrees of intralitter correlation, and, for purposes of illustration, consider a logistic dose-response model that describes the logit of risk as a straight-line function of ln(dose). The biases and (exact and asymptotic) variances of the maximum likelihood estimators of the intercept and slope are studied by simulation as a function of the intralitter correlation structure.     

On symmetric semiparametric two-sample problem

We consider a two-sample problem where data come from symmetric distributions. Usual two-sample data with only magnitudes recorded, arising from case-control studies or logistic discriminant analyses, may constitute a symmetric two-sample problem. We propose a semiparametric model such that, in addition to symmetry, the log ratio of two unknown density functions is modeled in a known parametric form. The new semiparametric model, tailor-made for symmetric two-sample data, can also be viewed as a biased sampling model subject to symmetric constraint. A maximum empirical likelihood estimation approach is adopted to estimate the unknown model parameters, and the corresponding profile empirical likelihood ratio test is utilized to perform hypothesis testing regarding the two population distributions. Symmetry, however, comes with irregularity. It is shown that, under the null hypothesis of equal symmetric distributions, the maximum empirical likelihood estimator has degenerate Fisher information, and the test statistic has a mixture of χ²-type asymptotic distribution. Extensive simulation studies have been conducted to demonstrate promising statistical powers under correct and misspecified models. We apply the proposed methods to two real examples.     

An Evaluation of Beta-Binomial and Dirichlet-Trinomial Models for the Analysis of Reproductive and Developmental Effects

The common endpoints for the evaluation of reproductive and developmental toxic effects are the number of dead/resorbed fetuses, the number of malformed fetuses, and the number of normal fetuses for each litter. The joint distribution of the three endpoints could be modelled by a Dirichlettrinomial distribution or by a product of two-beta-binomial distributions. A simulation experiment is used to investigate the biases of the maximum likelihood estimate (MLE) for the probability of adverse effects under the Dirichlet-trinomial model and the beta-binomial model. Also, the type I errors and powers of the likelihood ratio test for comparing the difference between treatment and control are evaluated for the two underlying models. In estimation, the two MLE's are comparable, the bias estimates are small. In testing, the likelihood ratio test is generally more powerful under the Dirichlet-trinomial model than the beta-binomial model. The type I error rate is greater than the nominal level using the Dirichlet-trinomial model in some cases, when the data are generated from the two-beta-binomial model, and it is less than the nominal level using the beta-binomial model in other cases, when the data are generated from the Dirichlet-trinomial model.     

A stabilized moment estimator for the beta-binomial distribution

The beta-binomial distribution has been proposed as a model for the incorporation of historical control data in the analysis of rodent carcinogenesis bioassays. Low spontaneous tumor incidences along with the small number and sizes of historical control groups combine to make the moment and maximum likelihood estimates of the beta-binomial parameters deficient. We therefore propose a stabilized moment estimator for one of the parameters. The stabilized moment estimator is similar to the ridge regression estimator and introduces a shrinkage parameter. Computer simulations were run to examine the behavior of the stabilized moment estimator. The effect of the stabilized moment estimator on the score test for dose-related trend is considered both on simulated data and on an example from the literature.     

On the Fisher Information Matrix in Type II Censored Data from the Exponentiated Exponential Family

In this note, we express, in a general setting, the Fisher information matrix under Type II censoring in terms of the hazard function and then obtain the Fisher information matrix under Type II censoring as a single integral for the exponentiated exponential family, which can be easily evaluated. The Fisher information under Type II censoring can also be used to characterize the exponential distribution among the exponentiated exponential family.     

A Mixture Model for Quantum Dot Images of Kinesin Motor Assays

Summary We introduce a nearly automatic procedure to locate and count the quantum dots in images of kinesin motor assays. Our procedure employs an approximate likelihood estimator based on a two‐component mixture model for the image data; the first component has a normal distribution, and the other component is distributed as a normal random variable plus an exponential random variable. The normal component has an unknown variance, which we model as a function of the mean. We use B‐splines to estimate the variance function during a training run on a suitable image, and the estimate is used to process subsequent images. Parameter estimates are generated for each image along with estimates of standard errors, and the number of dots in the image is determined using an information criterion and likelihood ratio tests. Realistic simulations show that our procedure is robust and that it leads to accurate estimates, both of parameters and of standard errors.     

Estimating sampling error of evolutionary statistics based on genetic covariance matrices using maximum likelihood

We explore the estimation of uncertainty in evolutionary parameters using a recently devised approach for resampling entire additive genetic variance–covariance matrices ( G ). Large‐sample theory shows that maximum‐likelihood estimates (including restricted maximum likelihood, REML) asymptotically have a multivariate normal distribution, with covariance matrix derived from the inverse of the information matrix, and mean equal to the estimated G . This suggests that sampling estimates of G from this distribution can be used to assess the variability of estimates of G , and of functions of G . We refer to this as the REML‐MVN method. This has been implemented in the mixed‐model program WOMBAT. Estimates of sampling variances from REML‐MVN were compared to those from the parametric bootstrap and from a Bayesian Markov chain Monte Carlo (MCMC) approach (implemented in the R package MCMCglmm). We apply each approach to evolvability statistics previously estimated for a large, 20‐dimensional data set for Drosophila wings. REML‐MVN and MCMC sampling variances are close to those estimated with the parametric bootstrap. Both slightly underestimate the error in the best‐estimated aspects of the G matrix. REML analysis supports the previous conclusion that the G matrix for this population is full rank. REML‐MVN is computationally very efficient, making it an attractive alternative to both data resampling and MCMC approaches to assessing confidence in parameters of evolutionary interest.     

Fast inference in generalized linear models via expected log-likelihoods

Generalized linear models play an essential role in a wide variety of statistical applications. This paper discusses an approximation of the likelihood in these models that can greatly facilitate computation. The basic idea is to replace a sum that appears in the exact log-likelihood by an expectation over the model covariates; the resulting “expected log-likelihood” can in many cases be computed significantly faster than the exact log-likelihood. In many neuroscience experiments the distribution over model covariates is controlled by the experimenter and the expected log-likelihood approximation becomes particularly useful; for example, estimators based on maximizing this expected log-likelihood (or a penalized version thereof) can often be obtained with orders of magnitude computational savings compared to the exact maximum likelihood estimators. A risk analysis establishes that these maximum EL estimators often come with little cost in accuracy (and in some cases even improved accuracy) compared to standard maximum likelihood estimates. Finally, we find that these methods can significantly decrease the computation time of marginal likelihood calculations for model selection and of Markov chain Monte Carlo methods for sampling from the posterior parameter distribution. We illustrate our results by applying these methods to a computationally-challenging dataset of neural spike trains obtained via large-scale multi-electrode recordings in the primate retina.     

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     

Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

In this article we study some properties of a new family of distributions, namely Exponentiated Exponential distribution, discussed in Gupta , Gupta , and Gupta (1998). The Exponentiated Exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed that many properties of this new family are quite similar to those of a Weibull or a gamma family, therefore this distribution can be used as a possible alternative to a Weibull or a gamma distribution. We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.     

A multiple imputation approach to Cox regression with interval-censored data

We propose a general semiparametric method based on multiple imputation for Cox regression with interval-censored data. The method consists of iterating the following two steps. First, from finite-interval-censored (but not right-censored) data, exact failure times are imputed using Tanner and Wei's poor man's or asymptotic normal data augmentation scheme based on the current estimates of the regression coefficient and the baseline survival curve. Second, a standard statistical procedure for right-censored data, such as the Cox partial likelihood method, is applied to imputed data to update the estimates. Through simulation, we demonstrate that the resulting estimate of the regression coefficient and its associated standard error provide a promising alternative to the nonparametric maximum likelihood estimate. Our proposal is easily implemented by taking advantage of existing computer programs for right-censored data.     

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.     

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.     

FGX: a frequentist gene expression index for Affymetrix arrays

We consider a new frequentist gene expression index for Affymetrix oligonucleotide DNA arrays, using a similar probe intensity model as suggested by Hein and others (2005), called the Bayesian gene expression index (BGX). According to this model, the perfect match and mismatch values are assumed to be correlated as a result of sharing a common gene expression signal. Rather than a Bayesian approach, we develop a maximum likelihood algorithm for estimating the underlying common signal. In this way, estimation is explicit and much faster than the BGX implementation. The observed Fisher information matrix, rather than a posterior credibility interval, gives an idea of the accuracy of the estimators. We evaluate our method using benchmark spike-in data sets from Affymetrix and GeneLogic by analyzing the relationship between estimated signal and concentration, i.e. true signal, and compare our results with other commonly used methods.     

An asymptotic theory for model selection inference in general semiparametric problems

Hjort & Claeskens (2003) developed an asymptotic theoryfor model selection, model averaging and subsequent inferenceusing likelihood methods in parametric models, along with associatedconfidence statements. In this article, we consider a semiparametricversion of this problem, wherein the likelihood depends on parametersand an unknown function, and model selection/averaging is tobe applied to the parametric parts of the model. We show thatall the results of Hjort & Claeskens hold in the semiparametriccontext, if the Fisher information matrix for parametric modelsis replaced by the semiparametric information bound for semiparametricmodels, and if maximum likelihood estimators for parametricmodels are replaced by semiparametric efficient profile estimators.Our methods of proof employ Le Cam's contiguity lemmas, leadingto transparent results. The results also describe the behaviourof semiparametric model estimators when the parametric componentis misspecified, and also have implications for pointwise-consistentmodel selectors.