Comparison of methods in analysis of the tree spatial distribution pattern

The L-function based on K-function, the pair-correlation function and the uniform angle index used in analyzing the tree spatial distribution pattern were compared on the basis of the data from 5 actual plots and 30 simulated plots. It was concluded that the pair-correlation function and the uniform angle index were more accurate than the L-function; the uniform angle index was more effective and feasible than the L-function and the pair-correlation function; the uniform angle index has an additional advantage that it could combine qualitative and quantitative analyses by using the distribution diagram of the uniform angle index and the average valueW .     

Comparison of methods in analysis of the tree spatial distribution pattern

The L-function based on K-function, the pair-correlation function and the uniform angle index used in analyzing the tree spatial distribution pattern were compared on the basis of the data from 5 actual plots and 30 simulated plots. It was concluded that the pair-correlation function and the uniform angle index were more accurate than the L-function; the uniform angle index was more effective and feasible than the L-function and the pair-correlation function; the uniform angle index has an additional advantage that it could combine qualitative and quantitative analyses by using the distribution diagram of the uniform angle index and the average valueW .     

The supramolecular organization of oxyhemoglobin in erythrocytes. A small-angle x-ray scattering study]

From X-ray scattering diagrams of concentrated solutions of hemoglobin the pair-correlation function of the molecules is calculated. At a concentration of 324 g/l the distance between neighbouring molecules amounts to 65 A. The number of direct neighbours of one molecule is 9. The pair-correlation function cannot be described by the assumption of a lattice-cell-model; therefore, a lattice-vacancy-model with fluid order is proposed.     

Bias correction in the hierarchical likelihood approach to the analysis of multivariate survival data

Frailty models are useful for measuring unobserved heterogeneity in risk of failures across clusters, providing cluster-specific risk prediction. In a frailty model, the latent frailties shared by members within a cluster are assumed to act multiplicatively on the hazard function. In order to obtain parameter and frailty variate estimates, we consider the hierarchical likelihood (H-likelihood) approach (Ha, Lee and Song, 2001. Hierarchical-likelihood approach for frailty models. Biometrika 88, 233-243) in which the latent frailties are treated as "parameters" and estimated jointly with other parameters of interest. We find that the H-likelihood estimators perform well when the censoring rate is low, however, they are substantially biased when the censoring rate is moderate to high. In this paper, we propose a simple and easy-to-implement bias correction method for the H-likelihood estimators under a shared frailty model. We also extend the method to a multivariate frailty model, which incorporates complex dependence structure within clusters. We conduct an extensive simulation study and show that the proposed approach performs very well for censoring rates as high as 80%. We also illustrate the method with a breast cancer data set. Since the H-likelihood is the same as the penalized likelihood function, the proposed bias correction method is also applicable to the penalized likelihood estimators.     

林木空间分布格局分析方法

采用5块实地调查样地和30块模拟样地,通过基于Ripley K-函数的L-函数、双相关函数和角尺度方法在林木空间分布格局研究中的比较分析,得出如下结论:双相关函数和角尺度在判断的准确性方面优于L-函数;角尺度在有效性和可行性方面比L-函数和双相关函数更强,并且它能利用角尺度分布图和均值同时作定性和定量分析,从而使其优势更加明显。     

Generalization of the Mantel-Haenszel estimating function for sparse clustered binary data

We extend the Mantel-Haenszel estimating function to estimate both the intra-cluster pairwise correlation and the main effects for sparse clustered binary data. We propose both a composite likelihood approach and an estimating function approach for the analysis of such data. The proposed estimators are consistent and asymptotically normally distributed. Simulation results demonstrate that the two approaches are comparable in terms of bias and efficiency; however, the estimating equation approach is computationally simpler. Analysis of the Georgia High Blood Pressure survey is used for illustration.     

A comparison of several point estimators of the odds ratio in a single 2 x 2 contingency table.

The relative performance of the unconditioned maximum likelihood estimators (UMLEs), conditional MLEs (CMLEs), and Jewell-type estimators of the odds ratio (OR) and its logarithm were investigated in sets of single 2 x 2 contingency tables. The tables were generated by complete enumeration of all possible cell frequencies consistent with a single fixed margin. The bias, mean squared error (MSE), and average absolute error (AAE) were computed for all estimators using the individual table probabilities as weights. The results showed that, for the OR, Jewell's estimator usually had smaller bias, MSE, and AAE than either of the MLEs. While the differences were often slight for MSE and AAE, for bias it was sometimes substantial. For the log(OR), the UMLE usually had the lowest bias, and its MSE and AAE were only slightly greater than those for the other estimators. Overall, we recommend estimation on the log scale using the UMLE. If OR is to be estimated, Jewell's method had strong merit, although it is nonsymmetric with respect to the table orientation. In view of this, the UMLE may again be favoured in some situations.     

Properties of bias and variance of two multiallelic estimators of F(ST)

This study compares theoretical and simulated properties of two estimators of fixation indices F(ST) for multiallelic data from a single locus. The estimators are due to Weir and Cockerham (straight theta(WC)) and to Robertson and Hill (;straight theta(RH)), respectively. Both estimators are linear combinations of biallelic estimators which differ in the way frequent and rare alleles are weighted. Consequently, their sampling properties differ as far as bias and variance are concerned. In the infinite island model at migration-drift equilibrium, in the case of one multiallelic locus, we analytically approximate the bias of the two estimators and show that;straight theta(WC) is nearly unbiased, whereas;straight theta(RH) is negatively biased. Hence, we propose a correction of bias of the latter. Moreover, we reexamine the properties of variance of the initial estimators: due to their construction, their variances are minimal, each over different parameter ranges,;straight theta(RH) performing better for low differentiation and;straight theta(WC) for high differentiation. These theoretical properties are confirmed by simulations, which show that our correction of bias of;straight theta(RH) performs well and does not affect its property of minimal variance for low differentiation. Hence, we advocate the use of;straight theta(WC) for high values of differentiation, and the use of;straight theta(RH), with our correction of bias, for low or moderate differentiation.     

Ratio estimation with measurement error in the auxiliary variate

Summary .  With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate.     

Statistical analysis of binary data generated from multilocus dominant DNA markers

The use of methodologies such as RAPD and AFLP for studying genetic variation in natural populations is widespread in the ecology community. Because data generated using these methods exhibit dominance, their statistical treatment is less straightforward. Several estimators have been proposed for estimating population genetic parameters, assuming simple random sampling and the Hardy-Weinberg (HW) law. The merits of these estimators remain unclear because no comparative studies of their theoretical properties have been carried out. Furthermore, ascertainment bias has not been explicitly modelled. Here, we present a comparison of a set of candidate estimators of null allele frequency (q), locus-specific heterozygosity (h) and average heterozygosity () in terms of their bias, standard error, and root mean square error (RMSE). For estimating q and h, we show that none of the estimators considered has the least RMSE over the parameter space. Our proposed zero-correction procedure, however, generally leads to estimators with improved RMSE. Assuming a beta model for the distribution of null homozygote proportions, we show how correction for ascertainment bias can be carried out using a linear transform of the sample average of h and the truncated beta-binomial likelihood. Simulation results indicate that the maximum likelihood and empirical Bayes estimator of have negligible bias and similar RMSE. Ascertainment bias in estimators of is most pronounced when the beta distribution is J-shaped and negligible when the latter is inverse J-shaped. The validity of the current findings depends importantly on the HW assumption-a point that we illustrate using data from two published studies.     

Model misspecification and bias for inverse probability weighting estimators of average causal effects

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood lead levels.     

Reduction of bias in estimating the frequency of recessive genes.

The standard approach to estimating the frequency of a completely recessive autosomal gene is to use the maximum-likelihood estimator (MLE), q = square root q2. Since the expectation oof Q using MLE is systematically less than the true value, this estimator always gives a negatively biased estimate of q. Here we describe the bias associated the MLE over a range of q and N values, explore some of the properties of this estimator, and propose new estimators which reduce the bias. We also describe some of the new estimators' properties, as well as the remaining bias associated with them for varying q and N values. We further propose one of these estimators as the one which most effectively reduces bias over a specific q value range of approximately .005 to .05, and which is less biased than JLE over essentially all q and N values. The proposed estimator also is directly compared with MLE in calculating various available estimates of q, demonstrating the percentage of reduction in bias achieved. This reduction varies from negligible for estimates of q above .3 and N greater than 100, to a 23% reduction in bias for a q value of .09 and an N value of 215.     

Resampling methods to reduce the selection bias in genetic effect estimation in genome-wide scans

Using the simulated data of Problem 2 for Genetic Analysis Workshop 14 (GAW14), we investigated the ability of three bootstrap-based resampling estimators (a shrinkage, an out-of-sample, and a weighted estimator) to reduce the selection bias for genetic effect estimation in genome-wide linkage scans. For the given marker density in the preliminary genome scans (7 cM for microsatellite and 3 cM for SNP), we found that the two sets of markers produce comparable results in terms of power to detect linkage, localization accuracy, and magnitude of test statistic at the peak location. At the locations detected in the scan, application of the three bootstrap-based estimators substantially reduced the upward selection bias in genetic effect estimation for both true and false positives. The relative effectiveness of the estimators depended on the true genetic effect size and the inherent power to detect it. The shrinkage estimator is recommended when the power to detect the disease locus is low. Otherwise, the weighted estimator is recommended.     

Estimating the encounter rate variance in distance sampling

Summary .  The dominant source of variance in line transect sampling is usually the encounter rate variance. Systematic survey designs are often used to reduce the true variability among different realizations of the design, but estimating the variance is difficult and estimators typically approximate the variance by treating the design as a simple random sample of lines. We explore the properties of different encounter rate variance estimators under random and systematic designs. We show that a design-based variance estimator improves upon the model-based estimator of Buckland et al. (2001, Introduction to Distance Sampling. Oxford: Oxford University Press, p. 79) when transects are positioned at random. However, if populations exhibit strong spatial trends, both estimators can have substantial positive bias under systematic designs. We show that poststratification is effective in reducing this bias.     

Maximum-likelihood estimation of relatedness

Relatedness between individuals is central to many studies in genetics and population biology. A variety of estimators have been developed to enable molecular marker data to quantify relatedness. Despite this, no effort has been given to characterize the traditional maximum-likelihood estimator in relation to the remainder. This article quantifies its statistical performance under a range of biologically relevant sampling conditions. Under the same range of conditions, the statistical performance of five other commonly used estimators of relatedness is quantified. Comparison among these estimators indicates that the traditional maximum-likelihood estimator exhibits a lower standard error under essentially all conditions. Only for very large amounts of genetic information do most of the other estimators approach the likelihood estimator. However, the likelihood estimator is more biased than any of the others, especially when the amount of genetic information is low or the actual relationship being estimated is near the boundary of the parameter space. Even under these conditions, the amount of bias can be greatly reduced, potentially to biologically irrelevant levels, with suitable genetic sampling. Additionally, the likelihood estimator generally exhibits the lowest root mean-square error, an indication that the bias in fact is quite small. Alternative estimators restricted to yield only biologically interpretable estimates exhibit lower standard errors and greater bias than do unrestricted ones, but generally do not improve over the maximum-likelihood estimator and in some cases exhibit even greater bias. Although some nonlikelihood estimators exhibit better performance with respect to specific metrics under some conditions, none approach the high level of performance exhibited by the likelihood estimator across all conditions and all metrics of performance.     

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.     

Small-sample bias of point estimators of the odds ratio from matched sets

The bias of several point estimators of the odds ratio arising from matched-pair data is investigated for small samples. Simple alternatives to the traditional maximum likelihood estimator are suggested, on both the original scale and the logarithm scale. In each case the suggested estimators possess a superior performance in terms of mean square error. Generalizations are given for 1:R matched data sets.     

Bias-reduced estimators and confidence intervals for odds ratios in genome-wide association studies

Genome-wide association studies (GWAS) provide an important approach to identifying common genetic variants that predispose to human disease. A typical GWAS may genotype hundreds of thousands of single nucleotide polymorphisms (SNPs) located throughout the human genome in a set of cases and controls. Logistic regression is often used to test for association between a SNP genotype and case versus control status, with corresponding odds ratios (ORs) typically reported only for those SNPs meeting selection criteria. However, when these estimates are based on the original data used to detect the variant, the results are affected by a selection bias sometimes referred to the "winner's curse" (Capen and others, 1971). The actual genetic association is typically overestimated. We show that such selection bias may be severe in the sense that the conditional expectation of the standard OR estimator may be quite far away from the underlying parameter. Also standard confidence intervals (CIs) may have far from the desired coverage rate for the selected ORs. We propose and evaluate 3 bias-reduced estimators, and also corresponding weighted estimators that combine corrected and uncorrected estimators, to reduce selection bias. Their corresponding CIs are also proposed. We study the performance of these estimators using simulated data sets and show that they reduce the bias and give CI coverage close to the desired level under various scenarios, even for associations having only small statistical power.     

Empirical Likelihood Semiparametric Regression Analysis for Longitudinal Data

A semiparametric regression model for longitudinal data is considered.The empirical likelihood method is used to estimate the regressioncoefficients and the baseline function, and to construct confidenceregions and intervals. It is proved that the maximum empiricallikelihood estimator of the regression coefficients achievesasymptotic efficiency and the estimator of the baseline functionattains asymptotic normality when a bias correction is made.Two calibrated empirical likelihood approaches to inferencefor the baseline function are developed. We propose a groupwiseempirical likelihood procedure to handle the inter-series dependencefor the longitudinal semiparametric regression model, and employbias correction to construct the empirical likelihood ratiofunctions for the parameters of interest. This leads us to provea nonparametric version of Wilks' theorem. Compared with methodsbased on normal approximations, the empirical likelihood doesnot require consistent estimators for the asymptotic varianceand bias. A simulation compares the empirical likelihood andnormal-based methods in terms of coverage accuracies and averageareas/lengths of confidence regions/intervals.