A Regression-Based Method for Estimating Risks and Relative Risks in Case-Base Studies

Both the absolute risk and the relative risk (RR) have a crucial role to play in epidemiology. RR is often approximated by odds ratio (OR) under the rare-disease assumption in conventional case-control study; however, such a study design does not provide an estimate for absolute risk. The case-base study is an alternative approach which readily produces RR estimation without resorting to the rare-disease assumption. However, previous researchers only considered one single dichotomous exposure and did not elaborate how absolute risks can be estimated in a case-base study. In this paper, the authors propose a logistic model for the case-base study. The model is flexible enough to admit multiple exposures in any measurement scale—binary, categorical or continuous. It can be easily fitted using common statistical packages. With one additional step of simple calculations of the model parameters, one readily obtains relative and absolute risk estimates as well as their confidence intervals. Monte-Carlo simulations show that the proposed method can produce unbiased estimates and adequate-coverage confidence intervals, for ORs, RRs and absolute risks. The case-base study with all its desirable properties and its methods of analysis fully developed in this paper may become a mainstay in epidemiology.     

Estimating divergence times in the presence of an overdispersed molecular clock

Molecular loci that fail relative-rate tests are said to be "overdispersed." Traditional molecular-clock approaches to estimating divergence times do not take this into account. In this study, a method was developed to estimate divergence times using loci that may be overdispersed. The approach was to replace the traditional Poisson process assumption with a more general stationary process assumption. A probability model was developed, and an accompanying computer program was written to find maximum-likelihood estimates of divergence times under both the Poisson process and the stationary process assumptions. In simulation, it was shown that confidence intervals under the traditional Poisson assumptions often vastly underestimate the true confidence limits for overdispersed loci. Both models were applied to two data sets: one from land plants, the other from the higher metazoans. In both cases, the traditional Poisson process model could be rejected with high confidence. Maximum-likelihood analysis of the metazoan data set under the more general stationary process suggested that their radiation occurred well over a billion years ago, but confidence intervals were extremely wide. It was also shown that a model consistent with a Cambrian (or nearly Cambrian) origination of the animal phyla, although significantly less likely than a much older divergence, fitted the data well. It is argued that without an a priori understanding of the variance in the time between substitutions, molecular data sets may be incapable of ever establishing the age of the metazoan radiation.     

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.     

Should the statistical analyses of twinning-rate data be improved?

Every statistical model is based on explicitly or implicitly formulated assumptions. In this study we address new techniques of calculation of variances and confidence intervals, analyse some statistical methods applied to modelling twinning rates, and investigate whether the improvements give more reliable results. For an observed relative frequency, the commonly used variance formula holds exactly with the assumptions that the repetitions are independent and that the probability of success is constant. The probability of a twin maternity depends not only on genetic predisposition, but also on several demographic factors, particularly ethnicity, maternal age and parity. Therefore, the assumption of constancy is questionable. The effect of grouping on the analysis of regression models for twinning rates is also considered. Our results indicate that grouping influences the efficiency of the estimates but not the estimates themselves. Recently, confidence intervals for proportions of low-incidence events have been a target for revived interest and we present the new alternatives. These confidence intervals are slightly wider and their midpoints do not coincide with the maximum-likelihood estimate of the twinning rate, but their actual coverage is closer to the nominal one than the coverage of the traditional confidence interval. In general, our findings indicate that the traditional methods are mainly satisfactorily robust and give reliable results. However, we propose that new formulae for the confidence intervals should be used. Our results are applied to twin-maternity data from Finland and Denmark.     

Using Re-Sampling Methods in Mortality Studies

Traditional methods of computing standardized mortality ratios (SMR) in mortality studies rely upon a number of conventional statistical propositions to estimate confidence intervals for obtained values. Those propositions include a common but arbitrary choice of the confidence level and the assumption that observed number of deaths in the test sample is a purely random quantity. The latter assumption may not be fully justified for a series of periodic “overlapping” studies. We propose a new approach to evaluating the SMR, along with its confidence interval, based on a simple re-sampling technique. The proposed method is most straightforward and requires neither the use of above assumptions nor any rigorous technique, employed by modern re-sampling theory, for selection of a sample set. Instead, we include all possible samples that correspond to the specified time window of the study in the re-sampling analysis. As a result, directly obtained confidence intervals for repeated overlapping studies may be tighter than those yielded by conventional methods. The proposed method is illustrated by evaluating mortality due to a hypothetical risk factor in a life insurance cohort. With this method used, the SMR values can be forecast more precisely than when using the traditional approach. As a result, the appropriate risk assessment would have smaller uncertainties.     

Confidence Intervals for Relative Risks in Disease Mapping

Several analysis of the geographic variation of mortality rates in space have been proposed in the literature. Poisson models allowing the incorporation of random effects to model extra‐variability are widely used. The typical modelling approach uses normal random effects to accommodate local spatial autocorrelation. When spatial autocorrelation is absent but overdispersion persists, a discrete mixture model is an alternative approach. However, a technique for identifying regions which have significant high or low risk in any given area has not been developed yet when using the discrete mixture model. Taking into account the importance that this information provides to the epidemiologists to formulate hypothesis related to the potential risk factors affecting the population, different procedures for obtaining confidence intervals for relative risks are derived in this paper. These methods are the standard information‐based method and other four, all based on bootstrap techniques, namely the asymptotic‐bootstrap, the percentile‐bootstrap, the BC‐bootstrap and the modified information‐based method. All of them are compared empirically by their application to mortality data due to cardiovascular diseases in women from Navarra, Spain, during the period 1988–1994. In the small area example considered here, we find that the information‐based method is sensible at estimating standard errors of the component means in the discrete mixture model but it is not appropriate for providing standard errors of the estimated relative risks and hence, for constructing confidence intervals for the relative risk associated to each region. Therefore, the bootstrap‐based methods are recommended for this matter. More specifically, the BC method seems to provide better coverage probabilities in the case studied, according to a small scale simulation study that has been carried out using a scenario as encountered in the analysis of the real data.     

Calculation of narrower confidence intervals for tree mortality rates when we know nothing but the location of the death/survival events

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A statistical method for the estimation of window-period risk of transfusion-transmitted HIV in donor screening under non-steady state

Human immunodeficiency virus (HIV) can be transmitted by transfusion of blood even if the blood unit is test-negative for HIV. This is largely due to a time period following an infection, called the window period, during which antibodies against HIV are not detectable. Window-period risk refers to the probability for a test-negative blood unit to be infectious because of its donation during the window period. Estimation of window-period risk is important in public health for evaluating the safety of donated blood. The standard method for this estimation problem has been based on so-called incidence/window-period (IWP) models in which blood-donation and HIV-infection processes are assumed to be stochastically stationary and independent. Here we propose a new approach in which we relax this key assumption of the IWP models. We estimate window-period risk for each unit of donated blood using a given distribution of window-period risk. The proposed method utilizes the actual observed donation intervals including those of seroconversions, thereby relaxing the assumption that may not be met in practice. Bootstrap is used to compute confidence intervals without specifying the complex dynamics of the donation and infection processes. A simulation study illustrates the usefulness of the proposed method over the IWP method in scenarios where the IWP assumptions do not hold. A real application of the proposed method is presented using blood bank data from a province of northern Thailand. Advantages and limitations of the proposed method are discussed and compared with the IWP models.     

Estimating population density from indirect sign: track counts and the Formozov–Malyshev–Pereleshin formula

For many purposes it is often desirable to estimate animal population densities over large areas. Where total counts are not possible and sightings are relatively rare, a range of methods exists to estimate densities from indirect sign. Such methods are frequently unreliable and usually require independent calibration or confirmation. We present an analytical method for estimating population density from track counts. The method, widely known in the Russian Federation but not in the English language scientific literature, requires counts of tracks of known age, together with estimates of animal daily travel distances. We use simulations to verify the theoretical basis of the approach and to indicate potential precision that may be achieved. We illustrate application of the approach using a large data set on ungulate track counts in the Russian Far East. We suggest that under most circumstances, nonparametric bootstrapping will be the most appropriate method for deriving estimates of confidence intervals about density estimates. As with other approaches to estimating density from indirect sign, the method that we discuss is vulnerable to violations of an array of underlying assumptions. However, it is easily applied and could represent an important method by which the relationship between indices of abundance and absolute density can be understood.     

GAMETOPHYTIC SELF-FERTILIZATION IN HOMOSPOROUS PLANTS: DEVELOPMENT,EVALUATION, AND APPLICATION OF A STATISTICAL METHOD FOR EVALUATING ITS IMPORTANCE

The methods described here make it possible to use data on sporophytic genotype frequencies to estimate the frequency of gametophytic self-fertilization in populations of homosporous plants. Bootstrap bias reduction is effective in reducing or eliminating the bias of the maximum likelihood estimate of the gametophytic selfing rate. The bias-corrected percentile method provides the most reliable confidence intervals for allele frequencies. The percentile method gives the most reliable confidence intervals for the gametophytic selfing rate when selfing is common. The maximum likelihood intervals, the percentile intervals, the bias-corrected percentile intervals, and the bootstrap t intervals are all overly conservative in their construction of confidence intervals for the gametophytic selfing rate when self-fertilization is rare. Application of the recommended methods indicates that gametophytic self-fertilization is quite rare in two sexually reproducing populations of Pellaea andromedifolia studied by Gastony and Gottlieb (1985).     

An Exact Confidence Interval for the Ratio of Two Related Proportions

I propose an exact confidence interval for the ratio of two proportions when the proportions are not independent. One application is to estimate the population prevalence using a screening test with perfect specificity but imperfect sensitivity. The population prevalence is the ratio of the observed prevalence divided by the test's sensitivity. I describe a method to calculate exact confidence intervals for this problem and compare these results with approximate confidence intervals given previously.     

Estimating lead time and overdiagnosis associated with PSA screening from prostate cancer incidence trends

Summary .   The introduction of the prostate-specific antigen (PSA) test has led to dramatic changes in the incidence of prostate cancer in the United States. In this article, we use information on the increase and subsequent decline in prostate cancer incidence following the adoption of PSA to estimate the lead time associated with PSA screening. The lead time is a key determinant of the likelihood of overdiagnosis, one of the main costs associated with the PSA test. Our approach conceptualizes observed incidence as the sum of the secular trend in incidence, which reflects incidence in the absence of PSA, and the excess incidence over and above the secular trend, which is a function of population screening patterns and the unknown lead time. We develop a likelihood model for the excess incidence given the secular trend and use it to estimate the mean lead time under specified distributional assumptions. We also develop a likelihood model for observed incidence and use it to simultaneously estimate the mean lead time together with a smooth secular trend. Variances and confidence intervals are estimated via a parametric bootstrap. Our results indicate an average lead time of approximately 4.59 years (95% confidence interval [3.24, 5.93]) for whites and 6.78 years [5.42, 8.20] for blacks with a corresponding secular trend estimate that is fairly flat after the introduction of PSA screening. These estimates correspond to overdiagnosis frequencies of approximately 22.7% and 34.4% for screen-detected whites and blacks, respectively. Our results provide the first glimpse of a plausible secular trend in prostate cancer incidence and suggest that, in the absence of PSA screening, disease incidence would not have continued its historic increase, rather it would have leveled off in accordance with changes in prostate patterns of care unrelated to PSA.     

Fish Consumption,Mercury Intake,and the Associated Risks to the Kuwaiti Population

This analysis uses recent data on fish consumption among Kuwaitis and on levels of Hg in fish collected from fish markets in Kuwait to estimate the human health risks to the Kuwaiti population due to consumption of fish containing Hg. Mercury is a known human neurotoxicant. Recent, somewhat controversial, evidence suggests that it also may play a role in cardiovascular disease. Our analysis indicates that roughly 5,000 IQ points are lost among the 30,000 Kuwaiti infants born each year and that approximately 15 fatal heart attacks each year among middle-aged and elderly Kuwaitis may be attributed to consumption of methyl Hg in fish. Approximate 90% confidence intervals are from 1,800 to 14,000 IQ points lost annually and from 0 to 72 fatal heart attacks each year. The confidence intervals for neurotoxicity are broad because of uncertainty about the slope of the dose–response and the existence of a threshold. The range of estimates for heart attacks includes zero because of residual uncertainty about the causality of observed associations between methyl Hg exposure and cardiovascular disease. These results do not imply that Kuwaitis should immediately reduce their consumption of fish, but do suggest that a careful risk–risk tradeoff analysis may be warranted.     

Phylogenetic dating with confidence intervals using mean path lengths

The mean path length (MPL) method, a simple method for dating nodes in a phylogenetic tree, is presented. For small trees the age estimates and corresponding confidence intervals, calibrated with fossil data, can be calculated by hand, and for larger trees a computer program gives the results instantaneously (a Pascal program is available upon request). Necessary input data are a rooted phylogenetic tree with edge lengths (internode lengths) approximately corresponding to the number of substitutions between the nodes. Given this, the MPL method produces relative age estimates with confidence intervals for all nodes of the tree. With the age of one or several nodes of the tree being known from reference fossils, the relative age estimates induce absolute age estimates and confidence intervals of the nodes of the tree. The MPL method relies on the assumptions that substitutions occur randomly and independently in different sites in the DNA sequence and that the substitution rates are approximately constant in time, i.e., assuming a molecular clock. A method is presented for identification of the nodes in the tree at which significant deviations from the clock assumption occur, such that dating may be done using different rates in different parts of the tree. The MPL method is illustrated with the Liliales, a group of monocot flowering plants.     

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.     

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.     

A Fast Method that Uses Polygenic Scores to Estimate the Variance Explained by Genome-wide Marker Panels and the Proportion of Variants Affecting a Trait

Several methods have been proposed to estimate the variance in disease liability explained by large sets of genetic markers. However, current methods do not scale up well to large sample sizes. Linear mixed models require solving high-dimensional matrix equations, and methods that use polygenic scores are very computationally intensive. Here we propose a fast analytic method that uses polygenic scores, based on the formula for the non-centrality parameter of the association test of the score. We estimate model parameters from the results of multiple polygenic score tests based on markers with p values in different intervals. We estimate parameters by maximum likelihood and use profile likelihood to compute confidence intervals. We compare various options for constructing polygenic scores, based on nested or disjoint intervals of p values, weighted or unweighted effect sizes, and different numbers of intervals, in estimating the variance explained by a set of markers, the proportion of markers with effects, and the genetic covariance between a pair of traits. Our method provides nearly unbiased estimates and confidence intervals with good coverage, although estimation of the variance is less reliable when jointly estimated with the covariance. We find that disjoint p value intervals perform better than nested intervals, but the weighting did not affect our results. A particular advantage of our method is that it can be applied to summary statistics from single markers, and so can be quickly applied to large consortium datasets. Our method, named AVENGEME (Additive Variance Explained and Number of Genetic Effects Method of Estimation), is implemented in R software.     

Frequentist estimation of coalescence times from nucleotide sequence data using a tree-based partition

This article proposes a method of estimating the time to the most recent common ancestor (TMRCA) of a sample of DNA sequences. The method is based on the molecular clock hypothesis, but avoids assumptions about population structure. Simulations show that in a wide range of situations, the point estimate has small bias and the confidence interval has at least the nominal coverage probability. We discuss conditions that can lead to biased estimates. Performance of this estimator is compared with existing methods based on the coalescence theory. The method is applied to sequences of Y chromosomes and mtDNAs to estimate the coalescent times of human male and female populations.     

Test-based exact confidence intervals for the difference of two binomial proportions

Confidence intervals are often provided to estimate a treatment difference. When the sample size is small, as is typical in early phases of clinical trials, confidence intervals based on large sample approximations may not be reliable. In this report, we propose test-based methods of constructing exact confidence intervals for the difference in two binomial proportions. These exact confidence intervals are obtained from the unconditional distribution of two binomial responses, and they guarantee the level of coverage. We compare the performance of these confidence intervals to ones based on the observed difference alone. We show that a large improvement can be achieved by using the standardized Z test with a constrained maximum likelihood estimate of the variance.     

Trim and fill: A simple funnel-plot-based method of testing and adjusting for publication bias in meta-analysis

We study recently developed nonparametric methods for estimating the number of missing studies that might exist in a meta-analysis and the effect that these studies might have had on its outcome. These are simple rank-based data augmentation techniques, which formalize the use of funnel plots. We show that they provide effective and relatively powerful tests for evaluating the existence of such publication bias. After adjusting for missing studies, we find that the point estimate of the overall effect size is approximately correct and coverage of the effect size confidence intervals is substantially improved, in many cases recovering the nominal confidence levels entirely. We illustrate the trim and fill method on existing meta-analyses of studies in clinical trials and psychometrics.