Comparison of confidence intervals for adjusted attributable risk estimates under multinomial sampling

The epidemiologic concept of the adjusted attributable risk is a useful approach to quantitatively describe the importance of risk factors on the population level. It measures the proportional reduction in disease probability when a risk factor is eliminated from the population, accounting for effects of confounding and effect-modification by nuisance variables. The computation of asymptotic variance estimates for estimates of the adjusted attributable risk is often done by applying the delta method. Investigations on the delta method have shown, however, that the delta method generally tends to underestimate the standard error, leading to biased confidence intervals. We compare confidence intervals for the adjusted attributable risk derived by applying computer intensive methods like the bootstrap or jackknife to confidence intervals based on asymptotic variance estimates using an extensive Monte Carlo simulation and within a real data example from a cohort study in cardiovascular disease epidemiology. Our results show that confidence intervals based on bootstrap and jackknife methods outperform intervals based on asymptotic theory. Best variants of computer intensive confidence intervals are indicated for different situations.     

Estimating index of population trend by re-sampling techniques (jackknife and bootstrap) and its application to the life table study of rice leaf roller, Cnaphalocrocis medinalis (Lepidoptera: Pyralidae)

Taking a published natural population life table of rice leaf roller, Cnaphalocrocis medinalis (Lepidoptera: Pyralidae), as an example, we estimated the population trend index, I, via re‐sampling methods (jackknife and bootstrap), determined its statistical properties and illustrated the application of these methods in determining the control effectiveness of bio‐agents and chemical insecticides. Depending on the simulation outputs, the smoothed distribution pattern of the estimates of I by delete‐1 jackknife is visually distinguishable from the normal density, but the smoothed pattern produced by delete‐d jackknife, and logarithm‐transformed smoothed patterns produced by both empirical and parametric bootstraps, matched well the corresponding normal density. Thus, the estimates of I produced by delete‐1 jackknife were not used to determine the suppressive effect of wasps and insecticides. The 95% percent confidence intervals or the narrowest 95 percentiles and Z‐test criterion were employed to compare the effectiveness of Trichogramma japonicum Ashmead and insecticides (powder, 1.5% mevinphos + 3% alpha‐hexachloro cyclohexane) against the rice leaf roller based on the estimates of I produced by delete‐d jackknife and bootstrap techniques. At α= 0.05 level, there were statistical differences between wasp treatment and control, and between wasp and insecticide treatments, if the normality is ensured, or by the narrowest 95 percentiles. However, there is still no difference between insecticide treatment and control. By Z‐test criterion, wasp treatment is better than control and insecticide treatment with P‐value < 0.01. Insecticide treatment is similar to control with P‐value > 0.2 indicating that 95% confidence intervals procedure is more conservative. Although similar conclusions may be drawn by re‐sampling techniques, such as the delta method, about the suppressive effect of trichogramma and insecticides, the normality of the estimates can be checked and guaranteed, and the correlation among sequential life stages of rice leaf roller is also considered in the estimation. Judged by the P‐values from Z‐test, the delta method is more conservative.     

delrious: a computer program designed to analyse molecular marker data and calculate delta and relatedness estimates with confidence

The computer program delrious analyses molecular marker data and calculates delta and relatedness estimates. A computer simulation is presented in which delrious is used to determine relations between relatedness estimate confidence and locus number. The results obtained suggest that many kinship studies probably have been conducted at significance levels less than 95%. Confidence measures provide a means of assessing reliability of calculated parameters and, therefore, would be beneficial to kinship hypothesis testing. Consequently, resampling procedures should be conducted routinely to determine delta and relatedness estimate confidence. delrious can implement bootstrap and jackknife resampling procedures for this purpose.     

Estimating species richness using the jackknife procedure

An exact expression is given for the jackknife estimate of the number of species in a community and for the variance of this number when quadrat sampling procedures are used. The jackknife estimate is a function of the number of species that occur in one and only one quadrat. The variance of the number of species can be constructed, as can approximate two-sided confidence intervals. The behavior of the jackknife estimate, as affected by quadrat size, sample size and sampling area, is investigated by simulation.     

Nonparametric confidence interval estimators for heritability and expected selection response

Statistical methods have not been described for comparing estimates of family-mean heritability (H) or expected selection response (R), nor have consistently valid methods been described for estimating R intervals. Nonparametric methods, e.g., delete-one jackknifing, may be used to estimate variances, intervals, and hypothesis test statistics in estimation problems where parametric methods are unsuitable, nonrobust, or undefinable. Our objective was to evaluate normal-approximation jackknife interval estimators for H and R using Monte Carlo simulation. Simulations were done using normally distributed within-family effects and normally, uniformly, and exponentially distributed between-family effects. Realized coverage probabilities for jackknife interval (2) and parametric interval (5) for H were not significantly different from stated probabilities when between-family effects were normally distributed. Coverages for jackknife intervals (3) and (4) for R were not significantly different from stated coverages when between-family effects were normally distributed. Coverages for interval (3) for R were occasionally significantly less than stated when between-family effects were uniformly or exponentially distributed. Coverages for interval (2) for H were occasionally significantly less than stated when between-family effects were exponentially distributed. Thus, intervals (3) and (4) for R and (2) for H were robust. Means of analysis of variance estimates of R were often significantly less than parametric values when the number of families evaluated was 60 or less. Means of analysis of variance estimates of H were consistently significantly less than parametric values. Means of jackknife estimates of H calculated from log transformed point estimates and R calculated from untransformed or log transformed point estimates were not significantly different from parametric values. Thus, jackknife estimators of H and R were unbiased. Delete-one jackknifing is a robust, versatile, and effective statistical method when applied to estimation problems involving variance functions. Jackknifing is especially valuable in hypothesis test estimation problems where the objective is comparing estimates from different populations.     

Bias and variance in model results associated with spatial scaling of measurements for parameterization in regional assessments

Models are central to global change analyses, but they are often parameterized using data that represent only a portion of heterogeneity in a region. This creates uncertainty in the results and constrains the reliability of model inferences. Our objective was to evaluate the uncertainty associated with differential scaling of parameterization data to model soil organic carbon stock changes as a function of US agricultural land use and management. Specifically, we compared analyses in which model parameters were derived from field experimental data that were scaled to the entire US vs. the same data scaled to climate regions within the country. We evaluated the effect of differential scaling on both bias and variance in model results. Model results had less variance by scaling data to the entire country because of a larger sample size for deriving individual parameter values, although there was a relatively large bias associated with this parameterization, estimated at 2.7 Tg C yr^?1. Even with the large bias, resulting confidence intervals from the two parameterizations had considerable overlap for the estimated national rate of SOC change (i.e. 77% overlap in those intervals). Consequently, the results were relatively similar when focusing on the uncertainty rather than solely on the mean estimate. In contrast, large biases created less overlap in confidence intervals for the change rates within individual climate regions, compared with the national estimates. For example, the overlap in resulting intervals from the two parameterizations was only 32% for the warm temperate moist region, with a corresponding bias of 3.1 Tg C yr^?1. These findings demonstrate that there is a greater risk of making erroneous inferences because of large biases if models are parameterized with broader scale information, such as an entire country, and then used to address impacts at a finer spatial scale, such as sub‐regions within a country. In addition, the study demonstrates a trade‐off between variance and bias in model results that depends on the scaling of data for model parameterization.     

Evaluation of jackknife and bootstrap for defining confidence intervals for pairwise agreement measures

Several research fields frequently deal with the analysis of diverse classification results of the same entities. This should imply an objective detection of overlaps and divergences between the formed clusters. The congruence between classifications can be quantified by clustering agreement measures, including pairwise agreement measures. Several measures have been proposed and the importance of obtaining confidence intervals for the point estimate in the comparison of these measures has been highlighted. A broad range of methods can be used for the estimation of confidence intervals. However, evidence is lacking about what are the appropriate methods for the calculation of confidence intervals for most clustering agreement measures. Here we evaluate the resampling techniques of bootstrap and jackknife for the calculation of the confidence intervals for clustering agreement measures. Contrary to what has been shown for some statistics, simulations showed that the jackknife performs better than the bootstrap at accurately estimating confidence intervals for pairwise agreement measures, especially when the agreement between partitions is low. The coverage of the jackknife confidence interval is robust to changes in cluster number and cluster size distribution.     

On estimating the proportion of variance in a phenotypic trait attributable to a measured locus

The measured genotype approach can be used to estimate the variance contributions of specific candidate loci to quantitative traits of interest. We show here that both the naive estimate of measured-locus heritability, obtained by invoking infinite-sample theory, and an estimate obtained from a bias-corrected variance estimate based on finite-sample theory, produce biased estimates of heritability. We identify the sources of bias, and quantify their effects. The two sources of bias are: (1) the estimation of heritability from population samples as the ratio of two variances, and (2) the existence of sampling error. We show that neither heritability estimator is less biased (in absolute value) than the other in all situations, and the choice of an ideal estimator is therefore a function of the sample size and magnitude of the locus-specific contribution to the overall phenotypic variance. In most cases the bias is small, so that the practical implications of using either estimator are expected to be minimal.     

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.     

Adjusting for publication bias in meta-analysis via inverse probability weighting using clinical trial registries

Publication bias is a major concern in conducting systematic reviews and meta-analyses. Various sensitivity analysis or bias-correction methods have been developed based on selection models, and they have some advantages over the widely used trim-and-fill bias-correction method. However, likelihood methods based on selection models may have difficulty in obtaining precise estimates and reasonable confidence intervals, or require a rather complicated sensitivity analysis process. Herein, we develop a simple publication bias adjustment method by utilizing the information on conducted but still unpublished trials from clinical trial registries. We introduce an estimating equation for parameter estimation in the selection function by regarding the publication bias issue as a missing data problem under the missing not at random assumption. With the estimated selection function, we introduce the inverse probability weighting (IPW) method to estimate the overall mean across studies. Furthermore, the IPW versions of heterogeneity measures such as the between-study variance and the I² measure are proposed. We propose methods to construct confidence intervals based on asymptotic normal approximation as well as on parametric bootstrap. Through numerical experiments, we observed that the estimators successfully eliminated bias, and the confidence intervals had empirical coverage probabilities close to the nominal level. On the other hand, the confidence interval based on asymptotic normal approximation is much wider in some scenarios than the bootstrap confidence interval. Therefore, the latter is recommended for practical use.     

A semiparametric estimate of treatment effects with censored data

A semiparametric estimate of an average regression effect with right-censored failure time data has recently been proposed under the Cox-type model where the regression effect beta(t) is allowed to vary with time. In this article, we derive a simple algebraic relationship between this average regression effect and a measurement of group differences in k-sample transformation models when the random error belongs to the G(rho) family of Harrington and Fleming (1982, Biometrika 69, 553-566), the latter being equivalent to the conditional regression effect in a gamma frailty model. The models considered here are suitable for the attenuating hazard ratios that often arise in practice. The results reveal an interesting connection among the above three classes of models as alternatives to the proportional hazards assumption and add to our understanding of the behavior of the partial likelihood estimate under nonproportional hazards. The algebraic relationship provides a simple estimator under the transformation model. We develop a variance estimator based on the empirical influence function that is much easier to compute than the previously suggested resampling methods. When there is truncation in the right tail of the failure times, we propose a method of bias correction to improve the coverage properties of the confidence intervals. The estimate, its estimated variance, and the bias correction term can all be calculated with minor modifications to standard software for proportional hazards regression.     

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     

An estimator of number of species from quadrat sampling

We consider the problem of estimating the number of distinct species S in a study area from the recorded presence or absence of species in each of a sample of quadrats. A generalized jackknife estimator of S is derived, along with an estimate of its variance. It is compared with the jackknife estimator for S proposed by Heltshe and Forrester and the empirical Bayes estimator of Mingoti and Meeden. We show that the empirical Bayes estimator has the form of a generalized jackknife estimator under a specific model for species distribution. We compare the new estimators of S to the empirical Bayes estimator via simulation. We characterize circumstances under which each is superior.     

Small‐Sample Estimation of Species Richness Applied to Forest Communities

Summary Many well‐known methods are available for estimating the number of species in a forest community. However, most existing methods result in considerable negative bias in applications, where field surveys typically represent only a small fraction of sampled communities. This article develops a new method based on sampling with replacement to estimate species richness via the generalized jackknife procedure. The proposed estimator yields small bias and reasonably accurate interval estimation even with small samples. The performance of the proposed estimator is compared with several typical estimators via simulation study using two complete census datasets from Panama and Malaysia.     

Hierarchical Modeling for Estimating Relative Risks of Rare Genetic Variants: Properties of the Pseudo‐Likelihood Method

Summary Many major genes have been identified that strongly influence the risk of cancer. However, there are typically many different mutations that can occur in the gene, each of which may or may not confer increased risk. It is critical to identify which specific mutations are harmful, and which ones are harmless, so that individuals who learn from genetic testing that they have a mutation can be appropriately counseled. This is a challenging task, since new mutations are continually being identified, and there is typically relatively little evidence available about each individual mutation. In an earlier article, we employed hierarchical modeling ( Capanu et al., 2008 , Statistics in Medicine 27 , 1973–1992) using the pseudo‐likelihood and Gibbs sampling methods to estimate the relative risks of individual rare variants using data from a case–control study and showed that one can draw strength from the aggregating power of hierarchical models to distinguish the variants that contribute to cancer risk. However, further research is needed to validate the application of asymptotic methods to such sparse data. In this article, we use simulations to study in detail the properties of the pseudo‐likelihood method for this purpose. We also explore two alternative approaches: pseudo‐likelihood with correction for the variance component estimate as proposed by Lin and Breslow (1996, Journal of the American Statistical Association 91 , 1007–1016) and a hybrid pseudo‐likelihood approach with Bayesian estimation of the variance component. We investigate the validity of these hierarchical modeling techniques by looking at the bias and coverage properties of the estimators as well as at the efficiency of the hierarchical modeling estimates relative to that of the maximum likelihood estimates. The results indicate that the estimates of the relative risks of very sparse variants have small bias, and that the estimated 95% confidence intervals are typically anti‐conservative, though the actual coverage rates are generally above 90%. The widths of the confidence intervals narrow as the residual variance in the second‐stage model is reduced. The results also show that the hierarchical modeling estimates have shorter confidence intervals relative to estimates obtained from conventional logistic regression, and that these relative improvements increase as the variants become more rare.     

Use of the weighted jackknife method to calculate the variance in cellular-specific protein secretion rate: application to monoclonal antibody secretion rate kinetics in response to osmotic stress

The specific secretion rate (q, mug protein secreted/viable cell-h) and its variance are very useful to compare the capability of cell lines for protein secretion. An assessment of specific secretion rate variability is also beneficial and important when the specific secretion rate is to be used as an on-line process parameter to monitor culture production behavior or for in-process decisionmaking. Experimental errors in mammalian cell culture (e.g., protein concentration measurement and cell counting) and estimation error in the method of calculating q contribute to the total variance of the specific secretion rate. Although the variance of q is essential for comparing the differences between cell lines and the response of the same cell line to different nutrient or environmental conditions, few methods for calculating the variance of the specific secretion rate have been reported. As a model system, we have used the weighted jackknife method and the delta method to calculate the variance in the specific secretion rate of a murine monoclonal antibody (q(mAb)) determined by a differential method. These methods were applied to calculate q(mAb) and its standard deviation to determine the change in q(mAb) kinetics during batch culture of the 9.2.27 hybridoma in response to growth in hyperosmotic media or osmotic stress. Without osmotic stress, during exponential growth in DMEM + 5% FBS spinner culture, the estimate of q(mAb) decreases at least threefold. Results indicate that the 9.2.27 hybridoma responds to hyperosmotic media (400 mOsm, 470 mOsm) by significantly reducing the degree of q(mAb) decrease in the exponential phase, thus maintaining a higher q(mAb) through the stationary phase. The trend of q(mAb) during the batch cultures studied is further confirmed by t-test. Osmotic stress is statistically shown to be able to alter significantly the hybridoma-specific mAb secretion kinetics during batch culture. Determination of the variance of specific secretion rate using the weighted jackknife method offers a powerful approach for establishing the confidence limits of specific protein secretion rate between cell cultures in different nutritional or osmotic environments. (c) 1996 John Wiley & Sons, Inc.     

Nonparametric confidence intervals for the ratio of marginal hazard rates of paired survival times

Paired survival times with potential censoring are often observed from two treatment groups in clinical trials and other types of clinical studies. The ratio of marginal hazard rates may be used to quantify the treatment effect in these studies. In this paper, a recently proposed nonparametric kernel method is used to estimate the marginal hazard rate, and the method of variance estimates recovery (MOVER) is used for the construction of the confidence intervals of a time‐dependent hazard ratio based on the confidence limits of a single marginal hazard rate. Two methods are proposed: one uses the delta method and another adopts the transformation method to construct confidence limits for the marginal hazard rate. Simulations are performed to evaluate the performance of the proposed methods. Real data from two clinical trials are analyzed using the proposed methods.     

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.     

Incorporating Estimates of Group Size in Sightability Models for Wildlife

ABSTRACT Sightability models have been used to estimate population size of many wildlife species; however, a limitation of these models is an assumption that groups of animals observed and counted during aerial surveys are enumerated completely. Replacing these unknown counts with maximum observed counts, as is typically done, produces population size estimates that are negatively biased. This bias can be substantial depending on the degree of undercounting occurring. We first investigated a method-of-moments estimator of group sizes. We then defined a population size estimator using the method-of-moments estimator of group sizes in place of maximum counts in the traditional sightability models, thereby correcting for bias associated with undercounting group size. We also provide associated equations for calculating the variance of our estimator. This estimator is an improvement over existing sightability model techniques because it significantly reduces bias, and variance estimates provide near nominal confidence interval coverage. The data needed for this estimator can be easily collected and implemented by wildlife managers with a field crew of only 3 individuals and little additional flight or personnel time beyond the normal requirements for developing sightability models.     

Youden Index and optimal cut-point estimated from observations affected by a lower limit of detection

The receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c *) for a biomarker affected by a LOD. We develop unbiased estimators of J and c * via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities.