The Estimation of the Segregation Parameter when Ascertainment is not Complete

Procedures to estimate the genetic segregation parameter when ascertainment of families is incomplete, have previously relied on iterative computer algorithms since estimators with closed form are lacking. We now present the Minimum Variance Unbiased Estimator for the segregation parameter under any ascertainment probability. This estimator assumes a simple form when ascertainment is complete. We also present a simple estimator, akin to Li and Mantel's (1968) estimator, but without the restriction that ascertainment be complete. The performance of these estimators is compared with respect to asymptotic efficiency. We also provide tables that define the required number of families of a given size that need to be sampled to achieve a specific power for testing simple hypothesis on the segregation parameter.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

An estimator for pairwise relatedness using molecular markers

I propose a new estimator for jointly estimating two-gene and four-gene coefficients of relatedness between individuals from an outbreeding population with data on codominant genetic markers and compare it, by Monte Carlo simulations, to previous ones in precision and accuracy for different distributions of population allele frequencies, numbers of alleles per locus, actual relationships, sample sizes, and proportions of relatives included in samples. In contrast to several previous estimators, the new estimator is well behaved and applies to any number of alleles per locus and any allele frequency distribution. The estimates for two- and four-gene coefficients of relatedness from the new estimator are unbiased irrespective of the sample size and have sampling variances decreasing consistently with an increasing number of alleles per locus to the minimum asymptotic values determined by the variation in identity-by-descent among loci per se, regardless of the actual relationship. The new estimator is also robust for small sample sizes and for unknown relatives being included in samples for estimating allele frequencies. Compared to previous estimators, the new one is generally advantageous, especially for highly polymorphic loci and/or small sample sizes.     

Uniformly minimum variance unbiased estimation of gene diversity

Gene diversity is an important measure of genetic variability in inbred populations. The survival of species in changing environments depends on, among other factors, the genetic variability of the population. In this communication, I have derived the uniformly minimum variance unbiased estimator of gene diversity. The proposed estimator of gene diversity does not assume that the inbreeding coefficient is known. I have also provided the approximate variance of this estimator according to Fisher's method. In addition, I have developed a numerical resampling-based method for obtaining variances and confidence intervals based on the maximum likelihood estimator and the uniformly minimum variance unbiased estimator. Efficiency in estimation of the gene diversity based on these two estimators is discussed. In accordance with the simulation results, I found that the uniformly minimum variance estimator developed in this report is more accurate for estimation of gene diversity than the maximum likelihood estimator.     

Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Estimation of integrated transition hazards and stage occupation probabilities for non-Markov systems under dependent censoring

We propose nonparametric estimators of the stage occupation probabilities and transition hazards for a multistage system that is not necessarily Markovian, using data that are subject to dependent right censoring. We assume that the hazard of being censored at a given instant depends on a possibly time-dependent covariate process as opposed to assuming a fixed censoring hazard (independent censoring). The estimator of the integrated transition hazard matrix has a Nelson-Aalen form where each of the counting processes counting the number of transitions between states and the risk sets for leaving each stage have an IPCW (inverse probability of censoring weighted) form. We estimate these weights using Aalen's linear hazard model. Finally, the stage occupation probabilities are obtained from the estimated integrated transition hazard matrix via product integration. Consistency of these estimators under the general paradigm of non-Markov models is established and asymptotic variance formulas are provided. Simulation results show satisfactory performance of these estimators. An analysis of data on graft-versus-host disease for bone marrow transplant patients is used as an illustration.     

Two Estimators of Mean Frequency in Products of Multinormal Probability Models

Two estimators, one additive the other multiplicative, are considered for mean frequencies in a complete three-way table. Using the mean square error criterion it is shown that preference for the additive estimator can be as high 7/8 in tables with row-column independence and in homogeneous tables. Extension to other estimators are discussed.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

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.     

Generalized Mantel-Haenszel estimators for K 2 x J tables

Mickey and Elashoff (1985, Biometrics 41, 623-635) gave an extension of Mantel-Haenszel estimation to log-linear models for 2 x J x K tables. Their extension yields two generalizations of the Mantel-Haenszel odds ratio estimator to K 2 x J tables. This paper provides variance and covariance estimators for these generalized Mantel-Haenszel estimators that are dually consistent (i.e., consistent in both large strata and sparse data), and presents comparisons of the efficiency of the generalized Mantel-Haenszel estimators.     

Inferring admixture proportions from molecular data

We derive here two new estimators of admixture proportions based on a coalescent approach that explicitly takes into account molecular information as well as gene frequencies. These estimators can be applied to any type of molecular data (such as DNA sequences, restriction fragment length polymorphisms [RFLPs], or microsatellite data) for which the extent of molecular diversity is related to coalescent times. Monte Carlo simulation studies are used to analyze the behavior of our estimators. We show that one of them (mY) appears suitable for estimating admixture from molecular data because of its absence of bias and relatively low variance. We then compare it to two conventional estimators that are based on gene frequencies. mY proves to be less biased than conventional estimators over a wide range of situations and especially for microsatellite data. However, its variance is larger than that of conventional estimators when parental populations are not very differentiated. The variance of mY becomes smaller than that of conventional estimators only if parental populations have been kept separated for about N generations and if the mutation rate is high. Simulations also show that several loci should always be studied to achieve a drastic reduction of variance and that, for microsatellite data, the mean square error of mY rapidly becomes smaller than that of conventional estimators if enough loci are surveyed. We apply our new estimator to the case of admixed wolflike Canid populations tested for microsatellite data.      

An 'unconditional-like' structure for the conditional estimator of odds ratio from 2 x 2 tables

In the estimation of the odds ratio (OR), the conditional maximum-likelihood estimate (cMLE) is preferred to the more readily computed unconditional one (uMLE). However, the exact cMLE does not have a closed form to help divine it from the uMLE or to understand in what circumstances the difference between the two is appreciable. Here, the cMLE is shown to have the same 'ratio of cross-products' structure as its unconditional counterpart, but with two of the cell frequencies augmented, so as to shrink the unconditional estimator towards unity. The augmentation involves a factor, similar to the finite population correction, derived from the minimum of the marginal totals.     

Robust estimation of population size in closed animal populations from capture-recapture experiments

In this paper the problem of finding robust estimators of population size in closed K-sample capture-recapture experiments is considered. Particular attention is paid to models where heterogeneity of capture probabilities is allowed. First, a general estimation procedure is given which does not depend on any assumptions about the form of the distribution of capture probabilities. This is followed by a detailed discussion of the usefulness of the generalized jackknife technique to reduce bias. Numerical comparisons of the bias and variance of various estimators are given. Finally, a general discussion is given with several recommendations on estimators to be used in practice.     

Asymptotic Relative Efficiency of Some Jackknife Estimators of a Common Odds Ratio

In the situation of several 2 × 2 tables the asymptotic relative efficiencies of certain jackknife estimators of a common odds ratio are investigated in the case that the number of tables is fixed while the sample sizes within each table tend to infinity. The estimators show very good results over a wide range of parameters. Some situations in which the estimators have low asymptotic relative efficiency are pointed out:.     

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.     

A comparison of non-iterative and iterative estimators of heterogeneity variance for the standardized mortality ratio

This paper continues work presented in B?hning et al. (2002b, Annals of the Institute of Statistical Mathematics 54, 827-839, henceforth BMSRB) where a class of non-iterative estimators of the variance of the heterogeneity distribution for the standardized mortality ratio was discussed. Here, these estimators are further investigated by means of a simulation study. In addition, iterative estimators including the Clayton-Kaldor procedure as well as the pseudo-maximum-likelihood (PML) approach are added in the comparison. Among all candidates, the PML estimator often has the smallest mean square error, followed by the non-iterative estimator where the weights are proportional to the external expected counts. This confirms the theoretical result in BMSRB in which an asymptotic efficiency could be proved for this estimator (in the class of non-iterative estimators considered). Surprisingly, the Clayton-Kaldor iterative estimator (often recommended and used by practitioners) performed poorly with respect to the MSE. Given the widespread use of these estimators in disease mapping, medical surveillance, meta-analysis and other areas of public health, the results of this study might be of considerable interest.     

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.     

Incomplete generalized U-statistics for food risk assessment

This article proposes statistical tools for quantitative evaluation of the risk due to the presence of some particular contaminants in food. We focus on the estimation of the probability of the exposure to exceed the so-called provisional tolerable weekly intake (PTWI), when both consumption data and contamination data are independently available. A Monte Carlo approximation of the plug-in estimator, which may be seen as an incomplete generalized U-statistic, is investigated. We obtain the asymptotic properties of this estimator and propose several confidence intervals, based on two estimators of the asymptotic variance: (i) a bootstrap type estimator and (ii) an approximate jackknife estimator relying on the Hoeffding decomposition of the original U-statistics. As an illustration, we present an evaluation of the exposure to Ochratoxin A in France.     

A Phylogenetic Estimator of Effective Population Size or Mutation Rate

A new estimator of the essential parameter θ = 4N(e)μ from DNA polymorphism data is developed under the neutral Wright-Fisher model without recombination and population subdivision, where N(e) is the effective population size and μ is the mutation rate per locus per generation. The new estimator has a variance only slightly larger than the minimum variance of all possible unbiased estimators of the parameter and is substantially smaller than that of any existing estimator. The high efficiency of the new estimator is achieved by making full use of phylogenetic information in a sample of DNA sequences from a population. An example of estimating θ by the new method is presented using the mitochondrial sequences from an American Indian population.     

Nonparametric analysis of recurrent events and death

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.