Semiparametric maximum likelihood for measurement error model regression

This paper presents an EM algorithm for semiparametric likelihood analysis of linear, generalized linear, and nonlinear regression models with measurement errors in explanatory variables. A structural model is used in which probability distributions are specified for (a) the response and (b) the measurement error. A distribution is also assumed for the true explanatory variable but is left unspecified and is estimated by nonparametric maximum likelihood. For various types of extra information about the measurement error distribution, the proposed algorithm makes use of available routines that would be appropriate for likelihood analysis of (a) and (b) if the true x were available. Simulations suggest that the semiparametric maximum likelihood estimator retains a high degree of efficiency relative to the structural maximum likelihood estimator based on correct distributional assumptions and can outperform maximum likelihood based on an incorrect distributional assumption. The approach is illustrated on three examples with a variety of structures and types of extra information about the measurement error distribution.     

Point Estimation on Relative Risk under Inverse Sampling

When the underlying disease is rare, to control the coefficient of variation for the sample proportion of cases, we may wish to apply inverse sampling. In this paper, we derive the uniformly minimum variance unbiased estimator (UMVUE) of relative risk and its variance in closed form under inverse sampling. On the basis of a Monte Carlo simulation, we demonstrate that using the UMVUE of relative risk can substantially reduce the mean-squared-error of using the maximum likelihood estimator, especially when the number of index cases in both comparison samples is small. For a given fixed total cost, we include a program that can be used to find the optimal allocation for the number of index cases to minimize the variance of the UMVUE as well.     

Comparison of recent estimators of interclass correlation from familial data

The asymptotic and finite-sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family. It is shown that Karlin's family-weighted pairwise estimator (Karlin, Cameron, and Williams, 1981, Proceedings of the National Academy of Science 78, 2664-2668) is virtually equivalent to the ensemble estimator (Rosner, Donner, and Hennekens, 1977, Applied Statistics 26, 179-187), thus suggesting an estimator of the former's asymptotic variance. Further, an estimator proposed by Srivastava (1984, Biometrika 71, 177-185) is shown to be identical to the modified sib-mean estimator (Konishi, 1982, Annals of the Institute of Statistical Mathematics 34, 505-515) when the sib-sib correlation is estimated by the method of unweighted group means. Although the estimator due to Srivastava has smaller asymptotic variance than the other two, the gain in efficiency is slight, for familial data, both asymptotically and in finite samples.     

Ranked Set Sampling: Ranking Error Models and Estimation of Visual Judgment Error Variance

Ranked set sampling where sampling is based on visual judgment of the differences between the sizes of pairs of units or on a concomitant variable is reviewed. An alternative model for judgment ranking based on ratios of sizes of pairs of units is presented. Computation of the variance of a visual ranked set sampling estimator of the mean of a distribution is enabled via maximum likelihood estimation of the visual judgment error variance. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A semiparametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome

Outcome-dependent sampling (ODS) schemes can be a cost effective way to enhance study efficiency. The case-control design has been widely used in epidemiologic studies. However, when the outcome is measured on a continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and is left unspecified. The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks-type properties in that, under the null, it follows a chi-square distribution asymptotically and is independent of the nuisance parameters. Our simulation results indicate that, for data obtained using an ODS design, the semiparametric empirical likelihood estimator is more efficient than conditional likelihood and probability weighted pseudolikelihood estimators and that ODS designs (along with the proposed estimator) can produce more efficient estimates than simple random sample designs of the same size. We apply the proposed method to analyze a data set from the Collaborative Perinatal Project (CPP), an ongoing environmental epidemiologic study, to assess the relationship between maternal polychlorinated biphenyl (PCB) level and children's IQ test performance.     

Estimation of Finite and Closed Population Size: An Inverse Sampling Approach with Unequal Recapture Probability

Based on capture-mark-recapture sampling methods the problem of estimating unknown population size was considered. The sampling started with the assumption that at the beginning of the experiment all the individuals were unmarked, and the unmarked individuals caught in each sample will be marked and returned to the original population before the next sample is drawn. It is also assumed that the population is closed by birth, death, emigration and immigration. Using a general inverse sampling approach, the unknown population size N is estimated by a maximum likelihood estimator (MLE), and a simple form for approximate MLE is obtained. The probability function for S (the minimum number of samples required to be drawn to have L (L ≥ 1) samples, each of which contains at least one marked individual) and the form for E[S] are also obtained. In addition, corrections and improvements of some previous works in this field are given.     

Semiparametric estimation exploiting covariate independence in two-phase randomized trials

Summary .  Recent results for case–control sampling suggest when the covariate distribution is constrained by gene-environment independence, semiparametric estimation exploiting such independence yields a great deal of efficiency gain. We consider the efficient estimation of the treatment–biomarker interaction in two-phase sampling nested within randomized clinical trials, incorporating the independence between a randomized treatment and the baseline markers. We develop a Newton–Raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate (SPMLE). Our algorithm accommodates both continuous phase-one outcomes and continuous phase-two biomarkers. The profile information matrix is computed explicitly via numerical differentiation. In certain situations where computing the SPMLE is slow, we propose a maximum estimated likelihood estimator (MELE), which is also capable of incorporating the covariate independence. This estimated likelihood approach uses a one-step empirical covariate distribution, thus is straightforward to maximize. It offers a closed-form variance estimate with limited increase in variance relative to the fully efficient SPMLE. Our results suggest exploiting the covariate independence in two-phase sampling increases the efficiency substantially, particularly for estimating treatment–biomarker interactions.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Unbiased and efficient log-likelihood estimation with inverse binomial sampling

The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter estimation and model evaluation. However, the log-likelihood of complex models in fields such as computational biology and neuroscience is often intractable to compute analytically or numerically. In those cases, researchers can often only estimate the log-likelihood by comparing observed data with synthetic observations generated by model simulations. Standard techniques to approximate the likelihood via simulation either use summary statistics of the data or are at risk of producing substantial biases in the estimate. Here, we explore another method, inverse binomial sampling (IBS), which can estimate the log-likelihood of an entire data set efficiently and without bias. For each observation, IBS draws samples from the simulator model until one matches the observation. The log-likelihood estimate is then a function of the number of samples drawn. The variance of this estimator is uniformly bounded, achieves the minimum variance for an unbiased estimator, and we can compute calibrated estimates of the variance. We provide theoretical arguments in favor of IBS and an empirical assessment of the method for maximum-likelihood estimation with simulation-based models. As case studies, we take three model-fitting problems of increasing complexity from computational and cognitive neuroscience. In all problems, IBS generally produces lower error in the estimated parameters and maximum log-likelihood values than alternative sampling methods with the same average number of samples. Our results demonstrate the potential of IBS as a practical, robust, and easy to implement method for log-likelihood evaluation when exact techniques are not available.     

A General Unified Framework to Assess the Sampling Variance of Heritability Estimates Using Pedigree or Marker-Based Relationships

Heritability is a population parameter of importance in evolution, plant and animal breeding, and human medical genetics. It can be estimated using pedigree designs and, more recently, using relationships estimated from markers. We derive the sampling variance of the estimate of heritability for a wide range of experimental designs, assuming that estimation is by maximum likelihood and that the resemblance between relatives is solely due to additive genetic variation. We show that well-known results for balanced designs are special cases of a more general unified framework. For pedigree designs, the sampling variance is inversely proportional to the variance of relationship in the pedigree and it is proportional to 1/N, whereas for population samples it is approximately proportional to 1/N², where N is the sample size. Variation in relatedness is a key parameter in the quantification of the sampling variance of heritability. Consequently, the sampling variance is high for populations with large recent effective population size (e.g., humans) because this causes low variation in relationship. However, even using human population samples, low sampling variance is possible with high N.     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Logistic regression in capture-recapture models

The effect of population heterogeneity in capture-recapture, or dual registration, models is discussed. An estimator of the unknown population size based on a logistic regression model is introduced. The model allows different capture probabilities across individuals and across capture times. The probabilities are estimated from the observed data using conditional maximum likelihood. The resulting population estimator is shown to be consistent and asymptotically normal. A variance estimator under population heterogeneity is derived. The finite-sample properties of the estimators are studied via simulation. An application to Finnish occupational disease registration data is presented.     

The effects of a known family-size distribution on the estimation of genetic parameters.

We consider the question: In a segregation analysis, can knowledge of the family-size distribution (FSD) in the population from which a sample is drawn improve the estimators of genetic parameters? In other words, should one incorporate the population FSD into a segregation analysis if one knows it? If so, then under what circumstances? And how much improvement may result? We examine the variance and bias of the maximum likelihood estimators both asymptotically and in finite samples. We consider Poisson and geometric FSDs, as well as a simple two-valued FSD in which all families in the population have either one or two children. We limit our study to a simple genetic model with truncate selection. We find that if the FSD is completely specified, then the asymptotic variance of the estimator may be reduced by as much as 5%-10%, especially when the FSD is heavily skewed toward small families. Results in small samples are less clear-cut. For some of the simple two-valued FSDs, the variance of the estimator in small samples of one- and two-child families may actually be increased slightly when the FSD is included in the analysis. If one knows only the statistical form of the FSD, but not its parameter, then the estimator is improved only minutely. Our study also underlines the fact that results derived from asymptotic maximum likelihood theory do not necessarily hold in small samples. We conclude that in most practical applications it is not worth incorporating the FSD into a segregation analysis. However, this practice may be justified under special circumstances where the FSD is completely specified, without error, and the population consists overwhelmingly of small families.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

On the sampling variance of intraclass correlations and genetic correlations.

Widely used standard expressions for the sampling variance of intraclass correlations and genetic correlation coefficients were reviewed for small and large sample sizes. For the sampling variance of the intraclass correlation, it was shown by simulation that the commonly used expression, derived using a first-order Taylor series performs better than alternative expressions found in the literature, when the between-sire degrees of freedom were small. The expressions for the sampling variance of the genetic correlation are significantly biased for small sample sizes, in particular when the population values, or their estimates, are close to zero. It was shown, both analytically and by simulation, that this is because the estimate of the sampling variance becomes very large in these cases due to very small values of the denominator of the expressions. It was concluded, therefore, that for small samples, estimates of the heritabilities and genetic correlations should not be used in the expressions for the sampling variance of the genetic correlation. It was shown analytically that in cases where the population values of the heritabilities are known, using the estimated heritabilities rather than their true values to estimate the genetic correlation results in a lower sampling variance for the genetic correlation. Therefore, for large samples, estimates of heritabilities, and not their true values, should be used.     

Evaluating the mean frequency of responding cells in limiting dilution assays

Summary The validity of limiting dilution assays can be compromised or negated by the use of statistical methodology which does not consider all issues surrounding the biological process. This study critically evaluates statistical methods for estimating the mean frequency of responding cells in multiple sample limiting dilution assays. We show that methods that pool limiting dilution assay data, or samples, are unable to estimate the variance appropriately. In addition, we use Monte Carlo simulations to evaluate an unweighted mean of the maximum likelihood estimator, an unweighted mean based on the jackknife estimator, and a log transform of the maximum likelihood estimator. For small culture replicate size, the log transform outperforms both unweighted mean procedures. For moderate culture replicate size, the unweighted mean based on the jackknife produces the most acceptable results. This study also addresses the important issue of experimental design in multiple sample limiting dilution assays. In particular, we demonstrate that optimization of multiple sample limiting dilution assays is achieved by increasing the number of biological samples at the expense of repeat cultures.     

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 improvement on the maximum likelihood reconstruction of pedigrees from marker data

Many methods have been proposed to reconstruct the pedigree of a sample of individuals from their multilocus marker genotypes. These methods, like those in other fields of statistical inferences, may suffer from both type I (falsely related) and type II (falsely unrelated) errors. In sibship reconstruction, type I errors come from the spurious fusion of two or more small sibships into a single sibship, and type II errors originate from the spurious splitting of a large sibship into two or more small sibships. In this study I investigate the tendencies of both types of errors made by the likelihood methods in sibship reconstruction, using both analytical and simulation approaches. I propose an improvement on the likelihood methods to reduce sibship splitting, and thus type II errors by downscaling the number of inferred siblings sharing the same genotype at a locus. Simulations are then conducted to compare the accuracy of the original and improved likelihood methods in sibship reconstruction of a large sample of individuals in full-sib families of the same small size, the same large size and highly variable sizes, using a variable number of loci with a variable number of alleles per locus. The methods were also applied to the analysis of a salmon data set. I show that my scaling scheme prevents effectively the splitting of large sibships, and reduces type II errors greatly with little increase in type I errors. As a result, it improves the overall accuracy of sibship assignments, except when sibships are expected to be uniformly small or marker information is unrealistically scarce.     

On a Simple Procedure of Inclusion Probability Proportional to Sizes (IPPS) Sampling

A simple method of inclusion probability proportional to sizes is proposed for samples of size three units. It is shown that the variance of the HORVITZ -THOMPSON estimator based on the proposed sampling scheme is uniformly smaller than that of the customary estimator used in the probability proportional to sizes with replacement sampling. Further, its performance over RAO -HARTLEY -COCHRAN and SAMPFORD sampling schemes has been studied empirically for some of the natural populations.     

Closed Form Estimators of Latent Cell Frequencies

One-stage and two-stage closed form estimators of latent cell frequencies in multidimensional contingency tables are derived from the weighted least squares criterion. The first stage estimator is asymptotically equivalent to the conditional maximum likelihood estimator and does not necessarily have minimum asymptotic variance. The second stage estimator does have minimum asymptotic variance relative to any other existing estimator. The closed form estimators are defined for any number of latent cells in contingency tables of any order under exact general linear constraints on the logarithms of the nonlatent and latent cell frequencies.