Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data
Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data
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离散数据的双边精确检验和匹配置信区间
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发表时间：
2010
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通讯作者：
M. Fay
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作者：
M. Fay
双边假设检验和可信区间之间存在内在联系。可以反转一系列双边假设检验以获得匹配的100%(1-α)%置信度区间，该可信区间被定义为包含在α级别不会被拒绝的所有点零参数值的最小区间。不幸的是，对于离散数据，有几种不同的方法来定义双边精确检验，而最常用的双边精确检验是以一种方式定义的，而最常用的精确可信区间是以另一种方式定义的检验的倒数。这可能会导致不一致，其中确切的测试拒绝，但确切的置信度区间包含空参数值。包精确度和精确度2x2用匹配的置信度区间提供了几个精确的测试，尽可能地避免了这些不一致。给出了二项参数和泊松参数以及成对和不成对的2×2表的例子。越来越多的应用统计学家被鼓励报告可信区间(CI)和参数估计，以及假设检验的p值。统计信息包的htest类非常适合于这些类型的分析，因为所有相关的统计信息都可以在打印结果时显示出来。对于应用于离散数据的精确的双边测试，可能会出现测试-CI不一致：p值可能指示水平α的显著结果，而相关联的100(1-α)%可信区间可能覆盖参数的空值。理想情况下，我们希望提交一份统一的报告(Hirji，2006)，从而使p值和可信区间尽可能匹配。一个鼓舞人心的例子是，我被要求帮助设计一项研究，以确定在一种寄生虫病(淋巴丝虫病)的现有治疗方案(伊维菌素)中添加一种新药(阿苯达唑)是否会增加在另一种寄生虫病(Loa Loa)地方性疾病(Loa Loa)流行的地区服用时罕见的严重不良事件的发生率。有许多与该设计相关的统计问题(Fay等人，2007年)，但这里考虑一个简单的场景来突出本文的观点。以前使用现有治疗方法的大规模治疗中，17877人中有2人经历了严重不良事件，观察到的发生率为11.2‰。假设对20,000名新受试者给予新的治疗，并假设10名受试者经历了SAE，其观察率为每100,000人中有50人。假设泊松比率，使用来自STATS包的poisson.test(c(2，10)，c(17877,20000))的精确检验(在整个过程中，我们假设STATS包的版本为2.11.0)得到的p值为p=0.0421，这意味着两个比率在0.05%水平上有显著差异，但是poisson.test还给出了95%的可信区间(0.024，1.050)，其中包含比率1，这意味着没有显著的差别。稍后，我们将回到“泊松两个样本”一节中的激励例子。双侧精确检验概述我们简要回顾了使用离散数据的p值函数的推论。详情见Hirji(2006)或Blaker(2000)。假设您有一个带有随机变量T的离散统计t，使得T的值越大，就意味着感兴趣的参数θ的值越大。设Fθ(T)=Pr[T≤t；θ]且Fθ(T)=Pr[T≥t；θ]。假设我们正在测试H0：θ≥θ0 H1：θPM，则PCα是可能的。为了计算匹配的置信度区间，我们只考虑Fθ(T)和Fθ(T)是θ的单调函数的常规情况(可能除了退化的情况，当t是最大或最小时，对于所有的θ，Fθ(T)=1或Fθ(T)=0)。在这种情况下，中心测试的匹配置信限是(θL，θU)，它是以下解的解：
There is an inherent relationship between two-sided hypothesis tests and confidence intervals. A series of two-sided hypothesis tests may be inverted to obtain the matching 100(1-α)% confidence interval defined as the smallest interval that contains all point null parameter values that would not be rejected at the α level. Unfortunately, for discrete data there are several different ways of defining two-sided exact tests and the most commonly used twosided exact tests are defined one way, while the most commonly used exact confidence intervals are inversions of tests defined another way. This can lead to inconsistencies where the exact test rejects but the exact confidence interval contains the null parameter value. The packages exactci and exact2x2 provide several exact tests with the matching confidence intervals avoiding these inconsistencies as much as possible. Examples are given for binomial and Poisson parameters and both paired and unpaired 2× 2 tables. Applied statisticians are increasingly being encouraged to report confidence intervals (CI) and parameter estimates along with p-values from hypothesis tests. The htest class of the stats package is ideally suited to these kinds of analyses, because all the related statistics may be presented when the results are printed. For exact two-sided tests applied to discrete data, a test-CI inconsistency may occur: the p-value may indicate a significant result at level α while the associated 100(1-α)% confidence interval may cover the null value of the parameter. Ideally, we would like to present a unified report (Hirji, 2006), whereby the p-value and the confidence interval match as much as possible. A motivating example I was asked to help design a study to determine if adding a new drug (albendazole) to an existing treatment regimen (ivermectin) for the treatment of a parasitic disease (lymphatic filariasis) would increase the incidence of a rare serious adverse event when given in an area endemic for another parasitic disease (loa loa). There are many statistical issues related to that design (Fay et al., 2007), but here consider a simple scenario to highlight the point of this paper. A previous mass treatment using the existing treatment had 2 out of 17877 experiencing the serious adverse event (SAE) giving an observed rate of 11.2 per 100,000. Suppose the new treatment was given to 20,000 new subjects and suppose that 10 subjects experienced the SAE giving an observed rate of 50 per 100,000. Assuming Poisson rates, an exact test using poisson.test(c(2,10),c(17877,20000)) from the stats package (throughout we assume version 2.11.0 for the stats package) gives a p-value of p = 0.0421 implying significant differences between the rates at the 0.05 level, but poisson.test also gives a 95% confidence interval of (0.024,1.050) which contains a rate ratio of 1, implying no significant difference. We return to the motivating example in the ‘Poisson two-sample’ section later. Overview of two-sided exact tests We briefly review inferences using the p-value function for discrete data. For details see Hirji (2006) or Blaker (2000). Suppose you have a discrete statistic t with random variable T such that larger values of T imply larger values of a parameter of interest, θ. Let Fθ(t) = Pr[T ≤ t;θ] and Fθ(t) = Pr[T ≥ t;θ]. Suppose we are testing H0 : θ ≥ θ0 H1 : θ pm it is possible for pc α. To calculate the matching confidence intervals, we consider only regular cases where Fθ(t) and Fθ(t) are monotonic functions of θ (except perhaps the degenerate cases where Fθ(t) = 1 or Fθ(t) = 0 for all θ when t is the maximum or minimum). In this case the matching confidence limits to the central test are (θL,θU) which are solutions to: