Fully specified bootstrap confidence intervals for the difference of two independent binomial proportions based on the median unbiased estimator

In studies in which a binary response for each subject is observed, the success probability and functions of this quantity are of interest. The use of confidence intervals has been increasingly encouraged as complementary to, and indeed preferable to, p‐values as the primary expression of the impact of sampling uncertainty on the findings. The asymptotic confidence interval, based on a normal approximation, is often considered, but this interval can have poor statistical properties when the sample size is small and/or when the success probability is near 0 or 1. In this paper, an estimate of the risk difference based on median unbiased estimates (MUEs) of the two group probabilities is proposed. A corresponding confidence interval is derived using a fully specified bootstrap sample space. The proposed method is compared with Chen's quasi‐exact method, Wald intervals and Agresti and Caffo's method with regard to mean square error and coverage probability. For a variety of settings, the MUE‐based estimate of risk difference has mean square error uniformly smaller than maximum likelihood estimate within a certain range of risk difference. The fully specified bootstrap had better coverage probability in the tail area than Chen's quasi‐exact method, Wald intervals and Agresti and Caffo's intervals. Copyright © 2009 John Wiley & Sons, Ltd.     

Interval estimates for correlation coefficients corrected for within-person variation: implications for study design and hypothesis testing

It is well known that random measurement error can attenuate the correlation coefficient between two variables. One possible solution to this problem is to estimate the correlation coefficient based on an average of a large number of replicates for each individual. As an alternative, several authors have proposed an unattenuated (or corrected) correlation coefficient which is an estimate of the true correlation between two variables after removing the effect of random measurement error. In this paper, the authors obtain an estimate of the standard error for the corrected correlation coefficient and an associated 100% x (1-alpha) confidence interval. The standard error takes into account the variability of the observed correlation coefficient as well as the estimated intraclass correlation coefficient between replicates for one or both variables. The standard error is useful in hypothesis testing for comparisons of correlation coefficients based on data with different degrees of random error. In addition, the standard error can be used to evaluate the relative efficiency of different study designs. Specifically, an investigator often has the option of obtaining either a few replicates on a large number of individuals, or many replicates on a small number of individuals. If one establishes the criterion of minimizing the standard error of the corrected coefficient while fixing the total number of measurements obtained, in almost all instances it is optimal to obtain no more than five replicates per individual. If the intraclass correlation is greater than or equal to 0.5, it is usually optimal to obtain no more than two replicates per individual.     

Non-normal path analysis in the presence of measurement error and missing data: a Bayesian analysis of nursing homes' structure and outcomes

Path analytic models are useful tools in quantitative nursing research. They allow researchers to hypothesize causal inferential paths and test the significance of these paths both directly and indirectly through a mediating variable. A standard statistical method in the path analysis literature is to treat the variables as having a normal distribution and to estimate paths using several least squares regression equations. The parameters corresponding to the direct paths have point and interval estimates based on normal distribution theory. Indirect paths are a product of the direct path from the independent variable to the mediating variable and the direct path of the mediating variable to the dependent variable. However, in the case of non-normal distributions, the point and interval estimates of the indirect path become much more difficult to estimate. We address the issue of calculating indirect point and interval estimates in the case of non-normally distributed data. Our substantive application is a nursing home research problem in which the variables in the path analysis of interest involve variables with normal, Bernoulli, or Poisson distributions. Additionally, one of the Poisson variables is observed with error. This paper addresses estimating point and interval estimation of indirect paths for variables with non-normal distributions in the presence of missing data and measurement error. We handle these difficulties from a fully Bayesian point of view. We present our substantive path analysis motivated from a nursing home structure, process, and outcomes model. Our results focus on the impact job turnover in the nursing homes has on nursing home outcomes.     

Design and analysis of non-inferiority mortality trials in oncology

The recent revision of the Declaration of Helsinki and the existence of many new therapies that affect survival or serious morbidity, and that therefore cannot be denied patients, have generated increased interest in active-control trials, particularly those intended to show equivalence or non-inferiority to the active-control. A non-inferiority hypothesis has historically been formulated in terms of a fixed margin. This margin was historically designed to exclude a 'clinically meaningful difference', but has become recognized that the margin must also be no larger than the assured effect of the control in the new study. Depending on how this 'assured effect' is determined or estimated, the selected margin may be very small, leading to very large sample sizes, especially when there is an added requirement that a loss of some specified fraction of the assured effect must be ruled out. In cases where it is appropriate, this paper proposes non-inferiority analyses that do not involve a fixed margin, but can be described as a two confidence interval procedure that compares the 95 per cent two-sided CI for the difference between the treatment and the control to a confidence interval for the control effect (based on a meta-analysis of historical data comparing the control to placebo) that is chosen to preserve a study-wide type I error rate of about 0.025 (similar to the usual standard for a superiority trial) for testing for retention of a prespecified fraction of the control effect. The approach assumes that the estimate of the historical active-control effect size is applicable in the current study. If there is reason to believe that this effect size is diminished (for example, improved concomitant therapies) the estimate of this historical effect could be reduced appropriately. The statistical methodology for testing this non-inferiority hypothesis is developed for a hazard ratio (rather than an absolute difference between treatments, because a hazard ratio seems likely to be less population dependent than the absolute difference). In the case of oncology, the hazard ratio is the usual way of comparing treatments with respect to time to event (time to progression or survival) endpoints. The proportional hazards assumption is regarded as reasonable (approximately holding). The testing procedures proposed are conditionally equivalent to two confidence interval procedures that relax the conservatism of two 95 per cent confidence interval testing procedures and preserve the type I error rate at a one-sided 0.025 level. An application of this methodology to Xeloda, a recently approved drug for the treatment of metastatic colorectal cancers, is illustrated. Other methodologies are also described and assessed - including a point estimate procedure, a Bayesian procedure and two delta-method confidence interval procedures. Published in 2003 by John Wiley & Sons, Ltd.     

Intraclass correlation coefficients and bootstrap methods of hierarchical binary outcomes

Intraclass correlation coefficients are designed to assess consistency or conformity between two or more quantitative measurements. When multistage cluster sampling is implemented, no methods are readily available to estimate intraclass correlations of binomial-distributed outcomes within a cluster. Because statistical distribution of the intraclass correlation coefficients could be complicated or unspecified, we propose using a bootstrap method to estimate the standard error and confidence interval within the framework of a multilevel generalized linear model. We compared the results derived from a parametric bootstrap method with those from a non-parametric bootstrap method and found that the non-parametric method is more robust. For non-parametric bootstrap sampling, we showed that the effectiveness of sampling on the highest level is greater than that on lower levels; to illustrate the effectiveness, we analyse survey data in China and do simulation studies.     

Application of the bootstrap procedure provides an alternative to standard statistical procedures in the estimation of the vitamin B-6 requirement.

The bootstrap procedure is a versatile statistical tool for the estimation of standard errors and confidence intervals. It is useful when standard statistical methods are not available or are poorly behaved, e.g., for nonlinear functions or when assumptions of a statistical model have been violated. Inverse regression estimation is an example of a statistical tool with a wide application in human nutrition. In a recent study, inverse regression was used to estimate the vitamin B-6 requirement of young women. In the present statistical application, both standard statistical methods and the bootstrap technique were used to estimate the mean vitamin B-6 requirement, standard errors and 95% confidence intervals for the mean. The bootstrap procedure produced standard error estimates and confidence intervals that were similar to those calculated by using standard statistical estimators. In a Monte Carlo simulation exploring the behavior of the inverse regression estimators, bootstrap standard errors were found to be nearly unbiased, even when the basic assumptions of the regression model were violated. On the other hand, the standard asymptotic estimator was found to behave well when the assumptions of the regression model were met, but behaved poorly when the assumptions were violated. In human metabolic studies, which are often restricted to small sample sizes, or when statistical methods are not available or are poorly behaved, bootstrap estimates for calculating standard errors and confidence intervals may be preferred. Investigators in human nutrition may find that the bootstrap procedure is superior to standard statistical procedures in cases similar to the examples presented in this paper.     

含区间数据Gamma分布的参数估计

目的 建立含区间数据Gamma分布的参数估计方法，并用于SARS潜伏期的推算。方法 采用EM算法构造出求解含区间数据Gamma分布参数极大似然估计的迭代公式，并应用于SARS潜伏期分布的拟合。结果 基于EM算法的极大似然估计方法可以计算出含区间数据Gamma分布的两个参数，从而得到均值估计。同时，还可以根据极大似然估计的渐近性质，计算出估计量的标准误及各参数的置信区间。用于中国内地SARS爆发资料分析，发现SARS潜伏期服从Gamma(2．1，2．33)分布；潜伏期均值和方差的极大似然估计值分别为4．89天(95％CI4．43～5．35)和11．40天^2；95％的病人感染SARS-CoV后将在11．42天内发病。结论 基于EM算法的极大似然估计方法对于含区间数据Gamma分布参数的估计是强健的。可以用于含区间数据SARS潜伏期的精确估计。     

Interval estimation by simulation as an alternative to and extension of confidence intervals

There are numerous techniques for constructing confidence intervals, most of which are unavailable in standard software. Modern computing power allows one to replace these techniques with relatively simple, general simulation methods. These methods extend easily to incorporate sources of uncertainty beyond random error. The simulation concepts are explained in an example of estimating a population attributable fraction, a problem for which analytical formulas can be quite unwieldy. First, simulation of conventional intervals is illustrated and compared to bootstrapping. The simulation is then extended to include sampling of bias parameters from prior distributions. It is argued that the use of almost any neutral or survey-based prior that allows non-zero values for bias parameters will produce an interval estimate less misleading than a conventional confidence interval. Along with simplicity and generality, the ease with which simulation can incorporate these priors is a key advantage over conventional methods.     

An investigation of error sources and their impact in estimating the time to the most recent ancestor of spatially and temporally distributed HIV sequences

This is an investigation of significant error sources and their impact in estimating the time to the most recent common ancestor (MRCA) of spatially and temporally distributed human immunodeficiency virus (HIV) sequences. We simulate an HIV epidemic under a range of assumptions with known time to the MRCA (tMRCA). We then apply a range of baseline (known) evolutionary models to generate sequence data. We next estimate or assume one of several misspecified models and use the chosen model to estimate the time to the MRCA. Random effects and the extent of model misspecification determine the magnitude of error sources that could include: neglected heterogeneity in substitution rates across lineages and DNA sites; uncertainty in HIV isolation times; uncertain magnitude and type of population subdivision; uncertain impacts of host/viral transmission dynamics, and unavoidable model estimation errors. Our results suggest that confidence intervals will rarely have the nominal coverage probability for tMRCA. Neglected effects lead to errors that are unaccounted for in most analyses, resulting in optimistically narrow confidence intervals (CI). Using real HIV sequences having approximately known isolation times and locations, we present possible confidence intervals for several sets of assumptions. In general, we cannot be certain how much to broaden a stated confidence interval for tMRCA. However, we describe the impact of candidate error sources on CI width. We also determine which error sources have the most impact on CI width and demonstrate that the standard bootstrap method will underestimate the CI width.     

Multiple imputation for correcting verification bias

In the case in which all subjects are screened using a common test and only a subset of these subjects are tested using a golden standard test, it is well documented that there is a risk for bias, called verification bias. When the test has only two levels (e.g. positive and negative) and we are trying to estimate the sensitivity and specificity of the test, we are actually constructing a confidence interval for a binomial proportion. Since it is well documented that this estimation is not trivial even with complete data, we adopt multiple imputation framework for verification bias problem. We propose several imputation procedures for this problem and compare different methods of estimation. We show that our imputation methods are better than the existing methods with regard to nominal coverage and confidence interval length.     

The generalization of the odds ratio, risk ratio and risk difference to r x k tables

Familiar measures of association for 2 x 2 tables are the odds ratio, the risk ratio and the risk difference. Analagous measures of outcome-exposure association are desirable when there are several degrees of severity of both exposure and disease outcome. One such measure (alpha), which we label the general odds ratio (OR(G)), was proposed by Agresti. Convenient methods are given for calculation of both standard error and 95 per cent confidence intervals for OR(G). Other approaches to generalizing the odds ratio entail fitting statistical models which might not fit the data, and cannot handle some zero frequencies. We propose a generalization of the risk ratio (RR(G)) following the statistical approaches of Agresti, Goodman and Kruskal. A method of calculating the standard error and 95 per cent confidence interval for RR(G) is provided. A known statistic, Somers' d, fulfils the characteristics necessary for a generalized risk difference (RD(G)). These measures have straightforward interpretations, are easily computed, are at least as precise as other methods and do not require fitting statistical models to the data. We also examine the pooling of such measures as in, for example, meta-analysis.     

Evaluation of methodologies for small area life expectancy estimation

STUDY OBJECTIVE: To evaluate methods for calculating life expectancy in small areas, for example, English electoral wards. DESIGN: The Monte Carlo method was used to simulate the distribution of life expectancy (and its standard error) estimates for 10 alternative life table models. The models were combinations of Chiang or Silcocks methodology, 5 or 10 year age intervals, and a final age interval of 85+, 90+, or 95+. SETTING: A hypothetical small area experiencing the population age structure and age specific mortality rates of English men 1998-2000. PARTICIPANTS: Routine mortality and population statistics for England. MAIN RESULTS: Silcocks and Chiang based models gave similar estimates of life expectancy and its standard error. For all models, life expectancy was increasingly overestimated as the simulated population size decreased. The degree of overestimation depended largely on the final age interval chosen. Life expectancy estimates of small populations are normally distributed. The standard error estimates are normally distributed for large populations but become increasingly skewed as the population size decreases. Substitution methods to compensate for the effect of zero death counts on the standard error estimate did not improve the estimate. CONCLUSIONS: It is recommended that a population years at risk of 5000 is a reasonable point above which life expectancy calculations can be performed with reasonable confidence. Implications are discussed. Within the UK, the Chiang methodology and a five year life table to 85+ is recommended, with no adjustments to age specific death counts of zero.     

Bootstrapped confidence intervals for the Cox model using a linear relative risk form

A linear relative risk form for the Cox model is sometimes more appropriate than the usual exponential form. The usual asymptotic confidence interval may not have the appropriate coverage, however, due to flatness of the likelihood in the neighbourhood of beta. For a single continuous covariate, we derive bootstrapped confidence intervals with use of two resampling methods. The first resamples the original data and yields both one-step and fully iterated estimates of beta. The second resamples the score and information quantities at each failure time to yield a one-step estimate. We computed the bootstrapped confidence intervals by three different methods and compared these intervals to one based on the asymptotic standard error and to a likelihood-based interval. The bootstrapped intervals did not perform well and underestimated the true coverage in most cases.     

Statistical methods for estimating attributable risk from retrospective data

This paper extends Levin's measure of attributable risk¹ to adjust for confounding by aetiologic factors other than the exposure of interest. One can estimate this extended measure from case-control data provided either (i) from the control data one can estimate exposure prevalence within each stratum of the confounding factor; or (ii) one has additional information available concerning the confounder distribution and the stratum-specific disease rates. In both cases we give maximum likelihood estimates and their estimated asymptotic variances, and show them to be independent of the sampling design (matched vs. random). Computer simulations investigate the behaviour of these estimates and of three types of confidence intervals when sample size is small relative to the number of confounder strata. The simulations indicate that attributable risk estimates tend to be too low. The bias is not serious except when exposure prevalence is high among controls. In this case the estimates and their standard error estimates are also highly unstable. In general, the asymptotic standard error estimates performed quite well, even in small samples, and even when the true asymptotic standard error was too small. By contrast, the bootstrap estimate² tended to be too large. None of the three confidence intervals proved superior in accuracy to the other two. Thus there appears no advantage in using the log-based interval suggested by Walter^3′4 which is always longer than the simpler symmetric interval.     

常见结局事件的前瞻性研究中修正Poisson回归模型的应用

目的 介绍应用修正poisson回归模型计算常见结局事件的前瞻性研究中暴露因素的调整相对危险度的精确区间估计值.方法 应用稳健误差方差估计法（sandwich variance estimator）来校正相对危险度（RR）的估计方差,并通过SAS程序中GENMOD过程的REPEATED语句实现修正poisson回归.此外,采用不同的统计方法对5个虚拟的研究数据进行了分析比较.结果 以分层的Mantel-Haenszel法为标准参照,修正poisson回归对aRR点和区间估计均较为理想,普通poisson回归的aRR区间估计偏于保守.而logistic回归得到的aOR值明显偏离真实的RR值.结论 修正poisson回归模型适合于处理常见结局事件的前瞻性研究资料.     

Prevalence in two-phase surveys: accuracy of screening procedure and corrected estimates

BACKGROUND: Two-phase surveys often are used to estimate prevalence, in particular when the disease is rare or the case ascertainment procedure difficult and/or costly. However, few authors of such surveys take into account the sensitivity error associated with the use of a screening procedure in the first phase and its imprecision in correcting the prevalence estimate and confidence interval. METHODS: Two examples of two-phase surveys of rheumatic diseases (hip and knee osteoarthritis, rheumatoid arthritis and spondyloarthropathies) are used to present methodological approaches to obtain corrected prevalence estimates. Two methods for assessing the accuracy of the screening procedure are described--two-phase pilot and case-control designs--that are best suited for frequent and rare diseases, respectively, and naive and corrected estimates of prevalence compared. RESULTS: When the sensitivity error is not taken into account, prevalence is underestimated, as is, especially, the width of its confidence interval. In our examples, the corrected confidence interval width increased up to 50% as compared with na?ve one. CONCLUSIONS: The screening procedure accuracy should be thoroughly assessed in two-phase prevalence surveys and prevalence estimates and their confidence intervals corrected accordingly.     

The standard error of Cohen's Kappa

This paper gives a standard error for Cohen's Kappa, conditional on the margins of the observed r x r table. An explicit formula is given for the 2 x 2 table, and a procedure for the more general situation. A parsimonious log-linear model is suggested for the general case and an approximate confidence interval for kappa is based on that model. Numerical illustrations are given comparing the standard results with exact conditional results and those of the present paper. The conclusions are, first, that the standard error of kappa under the null hypothesis (that kappa is zero) should not be used except when the null is plausible; secondly, that the usual formula for the standard error appears appropriate except in the unusual circumstance of kappa taking large negative values; and, thirdly, in small samples the distribution of the estimate of kappa appears very non-symmetric, and it is preferable to base confidence intervals on transformations of kappa. A PC program with source code implementing the suggested procedures is available from the author.     

Correcting for bias in relative risk estimates due to exposure measurement error: a case study of occupational exposure to antineoplastics in pharmacists.

OBJECTIVES: This paper describes 2 statistical methods designed to correct for bias from exposure measurement error in point and interval estimates of relative risk. METHODS: The first method takes the usual point and interval estimates of the log relative risk obtained from logistic regression and corrects them for nondifferential measurement error using an exposure measurement error model estimated from validation data. The second, likelihood-based method fits an arbitrary measurement error model suitable for the data at hand and then derives the model for the outcome of interest. RESULTS: Data from Valanis and colleagues' study of the health effects of antineoplastics exposure among hospital pharmacists were used to estimate the prevalence ratio of fever in the previous 3 months from this exposure. For an interdecile increase in weekly number of drugs mixed, the prevalence ratio, adjusted for confounding, changed from 1.06 to 1.17 (95% confidence interval [CI] = 1.04, 1.26) after correction for exposure measurement error. CONCLUSIONS: Exposure measurement error is often an important source of bias in public health research. Methods are available to correct such biases.     

Four-fold table cell frequencies imputation in meta analysis

Meta analysis is a collection of quantitative methods devoted to combine summary information from related but independent studies. Because research reports usually present only data reductions and summary statistics rather than detailed data, the reviewer must often resort to rather crude methods for constructing summary effect estimate suitable for meta analysis pooling methods. When the studies involve a binary variable, both number of events and sample sizes are required to compute pooled estimate and its confidence interval. Sometimes, only summary statistics and related confidence intervals are provided in the publication. Although it is possible to estimate the standard error of each study's effect measure using the confidence interval from each study, this lack of detailed data compels the reviewers to use the inverse variance method to perform meta analysis, or to exclude the works with incomplete data. This paper shows three methods to reconstruct four-fold tables when summary measures for binary data and related confidence intervals and sample sizes are provided. The methods are discussed through a wider application example to assess the reconstruction precision, and the impact of using reconstructed data on meta analysis results. These methods seem to yield a correct reconstruction if original measures are reported at least with two decimal places. Meta analysis results do not seem seriously affected by the use of reconstructed data. These methods allow the reviewer to use full meta analysis statistical tools, instead of the simple inverse variance method, and can greatly contribute to the completeness of systematic reviews.     

Ensemble confidence intervals for binomial proportions

We propose two measures of performance for a confidence interval for a binomial proportion p: the root mean squared error and the mean absolute deviation. We also devise a confidence interval for p based on the actual coverage function that combines several existing approximate confidence intervals. This “Ensemble” confidence interval has improved statistical properties over the constituent confidence intervals. Software in an R package, which can be used in devising and assessing these confidence intervals, is available on CRAN.