Bootstrap方法在Cox模型参数估计中的应用

目的：探讨自变量分布极偏时Cox模型的参数以及区间估计是否准确，引入Bootstrap法对其加以验证。方法：拟合Cox模型研究痰检结果与肺癌发病风险的关系，使用非参数Bootstrap法验证Cox模型的参数估计值、可信区间是否有效。结果：两的参数估计值和大部分可信区间基本一致，个别变量的可信区间不一致，Bootstrap可信区间的下界更低，甚至变为负值。结论：当自变量分布极偏时，Cox模型所估计的参数可信区间可能有偏，Bootstrap可信区间更准确。     

层次结构定量数据组内相关系数的非参数Bootstrap估计

探讨Bootstrap自抽样如何在层次结构数据中实现，为组内相关系数(ICC)可信区间的计算方法提供选择。文中利用混合效应模型估计重复测量数据和两阶段抽样数据的ICC及利用Bootstrap法估计ICC的可信区间，比较不同的自抽样模式下ICC可信区间结果。重复测量实例 结果显示Bootstrap整群抽样估计的可信区间包含ICC真值，如忽视数据的层次结构特征， Bootstrap方法得到无效的可信区间估计；两阶段抽样实例结果显示整群Bootstrap自抽样方法估计的ICC均数与原样本ICC偏差最小，可信区间宽泛。表明对层次结构数据进行Bootstrap自抽样，需考虑数据的产生机制，即高水平Bootstrap自抽样的统计量估计更接近原样本统计量。     

一种估计小样本总体率可信区间的校正方法

[目的]对基于二项分布的小样本总体率可信区间估计方法-Clopper-Pearson提出的精确法进行校正,以减少精确法的保守性,提高可信区间的精密度.[方法]运用SAS软件,编制二项分布的Monte Carlo模拟抽样程序,通过计算95%可信区间的实际可信度寻找合适的校正系数k.[结果]校正系数k=0.5时校正法所估计的95%可信区间的实际可信度比精确法更接近期望可信度（95%）,其可信区间宽度比精确法更窄;而当样本含量小于15时,k=0.6的校正结果最理想.[结论]校正法可以减少精确法的保守性,提高可信区间的精密度.     

正态近似法计算二项分布总体率95%可信区间的应用条件研究

目的对目前惯用的正态近似法计算总体率可信区间的应用条件进行评价,为正确应用该法提供理论基础和应用指导.方法应用二项分布原理计算总体率精确可信区间并与正态近似法计算结果相比较;采用蒙特卡洛模拟抽样评价可信度;应用SAS和Excel软件绘制二项分类数据概率分布图.结果以n×p=5作为近似条件,用正态近似法计算总体率可信区间可造成显著的相对误差.当n×p为常数时,随着p减小,相对误差在一定范围内呈线性增加;随着n增加,相对误差呈非线性增加.结论目前惯用的估计总体率可信区间的正态近似法应用条件并不能保证总体率估计的可信度和准确度.根据实验结果,提出了使用正态近似法估计总体率95%可信区间一套新的应用条件.     

配对病例对照研究比数比的确切可信区间

配对病例对照研究中需估计比数比及其可信区间，本文提供了比数比确切可信区间的计算方法。     

线性模型中自变量相对重要性优势分析法估计及其应用

目的在多元线性回归模型中,估计各自变量的相对重要性,并探索区间估计方法。方法在自变量间存在相关时,运用Budescu(1993),Azen(2003)提出的优势分析法估计肝手术病例预计存活时间的影响因素重要性,并运用Bootstrap法探索区间估计方法以此来评价估计结果的变异性。结果血凝素、预后指数、酶功能对预计存活时间的相对贡献分别为0.1415、0.3408和0.490,其Bootstrap法95%可信区间分别为(0.0573,0.2744)、(0.2359,0.4545)和(0.3411,0.6090)。结论酶功能对肝手术病例预计存活时间的影响最大,预后指数次之,血凝素最小。当自变量间存在相关时,优势分析法估计的自变量相对重要性结果更精确稳定,值得推广应用。     

核平滑半参数回归模型在重复测量资料中的应用

目的探讨应用核平滑的半参数模型处理重复测量资料。方法利用核平滑和带约束的最小二乘法估计参数,并采用Bootstrap建立可信区间。结果用Matlab程序应用模拟数据得到相应结果。结论核平滑半参数模型可作为处理重复测量资料的一种方法。     

隐函数Delta法调整人群归因危险度可信区间的估计及其应用

目的介绍隐函数Delta法计算调整人群归因危险度(PAR)可信区间的基本原理和SAS编程方法.方法实例介绍用SAS PROC IML过程,实现病例-对照研究中调整PAR及其可信区间的估计.结果利用此程序计算了上海市男性食管癌病例-对照研究资料吸烟、饮酒和水果摄入过少因素的调整PAR及其95%可信区间分别46.59%(34.92X.26%),20.87%(10.281.47%)34.54%(19.81%～49.29%).结论对符合条件的病例-对照资料,隐函数Delta法可以对暴露因素调整PAR及其可信区间进行较为准确的估计.     

成组病例对照研究优势比的几种可信区间估计方法比较

在成组病例对照研究中,可通过计算优势比(OR)对相对危险度(RR)作出估计[1],常涉及到优势比(OR)的可信区间估计,其中常用的方法有直接计算概率法、Woolf法、Cornfield法和Test-based法(Miettinen 法)[2].本文利用Stata7.0软件[3-5]进行模拟实验,比较这四种方法的适用范围及检验效能.     

一组构成比资料的多元统计分析

以医学实例介绍了一组构成比资料的多元统计分析。文中对联合可信区间法如某类总体构成比可信区间估计、总体两类构成比差的可信区间估计，与χ２检验作了必要的讨论。     

Uncertainty of incremental cost-effectiveness ratios. A comparison of Fieller and bootstrap confidence intervals

OBJECTIVE: To compare different methods to estimate the confidence interval of the incremental cost-effectiveness ratio (ICER). METHODS: The adequacy of Fieller intervals and three methods for calculating bootstrap intervals are compared based on a simulation of 10,000 trials, using data from one trial. RESULTS: Both Fieller and bootstrap methods lead to unsatisfactory results when the difference in effectiveness is approximately zero. Where this difference is significant, the four methods for calculating confidence intervals for ICER do not give very different results, but Fieller's interval performs best. CONCLUSIONS: Since Fieller's confidence limits are relatively easy to compute compared with bootstrap simulations, we recommend using this method.     

The effect of outliers on confidence interval procedures for cost-effectiveness ratios

Cost-effectiveness ratio (CER) is defined as the ratio of the difference in cost between a test and standard health care programme to the difference in benefit, respectively. Methods to obtain confidence intervals for CERs are either variants of Fieller's method (1954), or bootstrap methods. We study the effect of outliers in cost measures on the precision of confidence interval procedures for CERs. In particular the performance of the procedures under single and multiple case influential deletion diagnostics, respectively, are evaluated. Simulation studies suggest that the bias-corrected percentile bootstrap procedure gives better precision and coverage under either diagnostic.     

A note on the estimation of confidence intervals for cost-effectiveness when costs and effects are censored.

BACKGROUND: The relation between methodological advances in estimation of confidence intervals (CIs) for incremental cost-effectiveness ratios (ICER) and estimation of cost effectiveness in the presence of censoring has not been explored. The authors address the joint problem of estimating ICER precision in the presence of censoring. METHODS: Using patient-level data (n = 168) on cost and survival from a published placebo-controlled trial, the authors compared 2 methods of measuring uncertainty with censored data: 1) Bootstrap with censor adjustment (BCA); 2) Fieller's method with censor adjustment (FCA). The authors estimate the FCA over all possible values for the correlation (rho) between costs and effects (range = -1 to + 1) and also examine the use of the correlation between cases without censoring adjustment (i.e., simple time-on-study) for costs and effects as an approximation for rho. RESULTS: Using time-on-study, which considers all censored observations as responders (deaths), yields 0.64 life-years gained at an additional cost of 87.9 for a cost per life-year of 137 (95% CI by bootstrap -5.9 to 392). Censoring adjustment corrects for the bias in the time-on-study approach and reduces the cost per life-year estimate to 132 (=72/0.54). Confidence intervals with censor adjustment were approximately 40% wider than the base-case without adjustment. Using the Fieller method with an approximation of rho based on the uncensored cost and effect correlation provides a 95% CI of (-48 to 529), which is very close to the BCA interval of (-52 to 504). CONCLUSIONS: Adjustment for censoring is necessary in cost-effectiveness studies to obtain unbiased estimates of ICER with appropriate uncertainty limits. In this study, BCA and FCA methods, the latter with approximated covariance, are simple to compute and give similar confidence intervals.     

Confidence interval estimation for the changepoint of treatment stratification in the presence of a qualitative covariate-treatment interaction

The goal in stratified medicine is to administer the “best” treatment to a patient. Not all patients might benefit from the same treatment; the choice of best treatment can depend on certain patient characteristics. In this article, it is assumed that a time-to-event outcome is considered as a patient-relevant outcome and a qualitative interaction between a continuous covariate and treatment exists, ie, that patients with different values of one specific covariate should be treated differently. We suggest and investigate different methods for confidence interval estimation for the covariate value, where the treatment recommendation should be changed based on data collected in a randomized clinical trial. An adaptation of Fieller's theorem, the delta method, and different bootstrap approaches (normal, percentile-based, wild bootstrap) are investigated and compared in a simulation study. Extensions to multivariable problems are presented and evaluated. We observed appropriate confidence interval coverage following Fieller's theorem irrespective of sample size but at the cost of very wide or even infinite confidence intervals. The delta method and the wild bootstrap approach provided the smallest intervals but inadequate coverage for small to moderate event numbers, also depending on the location of the true changepoint. For the percentile-based bootstrap, wide intervals were observed, and it was slightly conservative regarding coverage, whereas the normal bootstrap did not provide acceptable results for many scenarios. The described methods were also applied to data from a randomized clinical trial comparing two treatments for patients with symptomatic, severe carotid artery stenosis, considering patient's age as predictive marker.     

Fieller's method and net health benefits

Statistical and conceptual difficulties complicate the estimation of the incremental cost-effectiveness ratio (ICER). An alternative approach is to measure cost-effectiveness by the incremental net health benefit (INHB), defined as the difference in mean effectiveness of a new treatment compared with a standard, adjusted for cost difference by subtracting the health foregone if purchasing care at the rate of a marginally cost-effective therapy. Because net health benefit (NHB) is dependent on this threshold rate, one can construct confidence intervals for the INHB at various values of the rate. It turns out that the set of rates where new and standard are not significantly different is equal to the Fieller's method confidence set for the ICER. We review the derivation of the Fieller's method confidence set, present numerical examples, and discuss the implications of our result for the calculation and interpretation of NHB analyses.     

Confidence intervals for the risk ratio under inverse sampling

In this paper, we investigate various confidence intervals for the risk ratio under inverse sampling (also known as negative binomial sampling). Three existing confidence intervals (namely, the confidence intervals that are based on Fieller's theorem, the delta method and the F-statistic) are reviewed and three new confidence intervals (namely, the score, likelihood ratio and saddlepoint approximation (SA)-based confidence intervals) are developed. Comparative studies among these confidence intervals through Monte Carlo simulations are evaluated in terms of their coverage probabilities and expected interval widths under different settings. Our simulation results suggest that the SA-based confidence interval is generally more appealing. We illustrate these confidence interval construction methods with real data sets from a drug comparison study and a congenital heart disease study.     

Statistical determination of cost-effectiveness frontier based on net health benefits

Statistical methods are given for producing a cost-effectiveness frontier for an arbitrary number of programs. In the deterministic case, the net health benefit (NHB) decision rule is optimal; the rule funds the program with the largest positive NHB at each lambda, the amount a decision-maker is willing to pay for an additional unit of effectiveness. For bivariate normally distributed cost and effectiveness variables and a specified lambda, a statistical procedure is presented, based on the method of constrained multiple comparisons with the best (CMCB), for determining the program with the largest NHB. A one-tailed t test is used to determine if the NHB is positive. To obtain a statistical frontier in the lambda-NHB plane, we develop a method to produce the region in which each program has the largest NHB, by pivoting a CMCB confidence interval. A one-sided version of Fieller's theorem is used to determine the region where the NHB of each program is positive. At each lambda, the pointwise error rate is bounded by a prespecified alpha. Upper bounds on the familywise error rate, the probability of an error at any value of lambda, are given. The methods are applied to a hypothetical clinical trial of antipsychotic agents.     

Interval estimation of generalized odds ratio in data with repeated measurements

When the underlying responses are on an ordinal scale, the generalized odds ratio (GOR), defined as the ratio of the proportions of concordant and discordant pairs, is a useful index to summarize the difference between two stochastically ordered distributions of an ordinal categorical variable. We discuss interval estimation of the GOR for ordinal data with repeated measurements. On the basis of the Dirichlet-multinomial model, we develop three asymptotic interval estimators of the GOR using Wald's test statistic, a logarithmic transformation, and a method analogous to Fieller's theorem, respectively. To evaluate and compare the finite-sample performance of these estimators, we apply Monte Carlo simulation. We find that when the number of subjects per group is not large, the coverage probability of interval estimator using Wald's test statistic is likely to be less than the desired confidence level. By contrast, the coverage probability of the other two estimators are approximately equal to or larger than the desired confidence level. When the number of subjects per group is small and the intraclass correlation between repeated measurements within subjects is large, we note that applying the interval estimator derived from a method analogous to Fieller's theorem can lose efficiency. We also note that the interval estimator using the logarithmic transformation is generally preferable to the other two estimators with respect to both the coverage probability and the average length. Finally, on the basis of a few preliminary simulations, we do find some robustness for all the estimators developed here. We include an example comparing the inflammation grade after lung transplant between surgeries to illustrate the use of the proposed interval estimators.     

Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo

In non-inferiority trials, where non-inferiority of a new experimental drug compared to an active control has to be shown, it may be advisable to use an additional placebo group for internal validation if ethically justifiable. The focus of this paper is on such designs. Assuming normality and homogeneity of variances we will derive a statistical test procedure which turns out to be equivalent to the assessment based on Fieller's confidence interval. Based on the power function of this test, sample size calculations are carried out to achieve a given power. Additionally, the optimal allocation of the total sample size is derived. As an alternative to this parametric procedure, the bootstrap percentile interval is discussed and finally compared with Fieller's confidence interval in a study on mildly asthmatic patients.     

The origin of DNA replication and Fieller's problem

The paper considers the statistical problem of estimating the origin of DNA replication within the human ribosomal DNA (rDNA) locus and the issue of assessing the standard error of the estimate. Based on mapping the cumulative replication index (CRI), two different modelling schemes are suggested and investigated. The statistical problem of constructing a confidence interval for the origin of DNA replication is related to Fieller's problem of obtaining a confidence interval for the ratio of two normal means. Standard normal theory, the delta and bootstrap methods are used to estimate the standard error of the estimate of the origin of DNA replication, as well as the variation of the replication rate.