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Complementary hypotheses in safety surveillance 安全监测中的互补假设
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-07-02 DOI: 10.1080/07474946.2020.1823195
A. Walker
Abstract Postmarketing safety surveillance studies address two actionable questions: (1) Is the test product riskier than a standard? (2) Is the risk associated with the test product within some tolerable margin by comparison to the standard? Established techniques, not commonly applied to the setting of such complementary one-sided hypotheses, lead to useful conclusions in practice. For two-group studies, a search over possible one-sided binomial test results yields sample sizes that guarantee that the confidence bounds exclude one or the other of the hypotheses. With continuous monitoring, simple curtailment reduces the sample size. Point and interval estimates follow from the binomial distribution of events at the end of the study or from component negative binomials for crossing a bound of simple curtailment with continuous monitoring and earlier stopping. An asymptotic derivation corresponds to the problem of constructing a confidence interval that is smaller than the distance between the parameter values for tolerable excess and the absence of excess risk. Studies with guaranteed rejection of one of the pair of complementary hypotheses are somewhat larger than corresponding studies of a single hypothesis under usual power requirements, but the increase may be tolerable in return for certainty that there will be an actionable conclusion.
摘要上市后安全监督研究解决了两个可操作的问题:(1)测试产品的风险是否高于标准?(2) 与标准相比,与测试产品相关的风险是否在一定的可容忍范围内?既定的技术通常不适用于这种互补的片面假设,但在实践中会得出有用的结论。对于两组研究,对可能的单侧二项式检验结果的搜索产生了样本量,保证置信区间排除了一个或另一个假设。通过连续监测,简单的缩减可以减少样本量。点和区间估计值来自研究结束时事件的二项式分布,或来自通过连续监测和提前停止的简单缩减界限的成分负二元数。渐近推导对应于构造置信区间的问题,该置信区间小于可容忍超额和不存在超额风险的参数值之间的距离。在通常的功率要求下,保证拒绝一对互补假设中的一个的研究比对单个假设的相应研究要大一些,但这种增加可能是可以容忍的,以换取肯定会有一个可行的结论。
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引用次数: 0
On Wald’s sequential test for vector mean under multivariate normal distribution and correlated data 多元正态分布和相关数据下向量均值的Wald序列检验
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-07-02 DOI: 10.1080/07474946.2020.1823196
S. Mingoti, G. A. Diniz
Abstract In this article Wald’s sequential probability ratio test (SPRT) is implemented for multivariate normal distribution, for independent and autocorrelated data and known covariance matrix. The methodology based on residuals from the vector autoregressive moving average (VARMA) class models is presented for autocorrelated data. In this approach the average sample size required to take a decision in the sequential test is based on the Mahalanobis distance between the vectors of the intercept constants with respect to the error covariance matrix. Monte Carlo simulations were performed considering different scenarios for bivariate normal distribution. For fixed probabilities of type I and II errors, the results showed that the estimated average sample sizes to stop the sequential test were a little larger than those expected by Wald’s theory for autocorrelated and independent data. Under independence assumption the SPRT estimated sample sizes were also smaller than the sample sizes required by Hotelling’s test. It was shown that the omission of the correlation structure of the data strongly affects the type I and II errors of the sequential test. An example in the quality control field is presented using real data from a pig iron production process and the multivariate VAR(1) model.
摘要本文对多元正态分布、独立自相关数据和已知协方差矩阵进行了Wald序列概率比检验。针对自相关数据,提出了一种基于向量自回归移动平均(VARMA)类模型残差的方法。在这种方法中,在顺序测试中做出决定所需的平均样本量是基于截距常数的向量之间相对于误差协方差矩阵的Mahalanobis距离。考虑到二元正态分布的不同情况,进行了蒙特卡罗模拟。对于I型和II型错误的固定概率,结果表明,停止序列检验的估计平均样本量略大于Wald理论对自相关和独立数据的预期。在独立性假设下,SPRT估计的样本量也小于霍特林检验所需的样本量。结果表明,数据相关性结构的遗漏强烈影响序列检验的I型和II型误差。利用生铁生产过程的实际数据和多元VAR(1)模型,给出了质量控制领域的一个例子。
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引用次数: 0
Sequential design and analysis in the randomized clinical trial: A historical perspective 随机临床试验的顺序设计和分析:历史视角
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-07-02 DOI: 10.1080/07474946.2020.1823190
W. Rosenberger
Abstract Sequential analysis, as invented by Wald, was not targeted to clinical trials; in fact, the randomized clinical trial was being invented nearly simultaneously by Hill on the other side of the Atlantic. The connection to clinical trials was established by Bross and Armitage in the early 1950s and applied in a number of trials during that decade. Restrictions in its applicability in practice led to a dry spell until its resurgence with group sequential methods. This article reviews the historical context of the use of sequential analysis in actual randomized clinical trials. It does not review methodological developments, except as they relate to the historical and philosophical setting.
摘要瓦尔德发明的序列分析并没有针对临床试验;事实上,这项随机临床试验几乎是由大西洋彼岸的希尔同时发明的。Bross和Armitage在20世纪50年代初建立了与临床试验的联系,并在这十年中应用于多项试验。它在实践中的适用性受到限制,导致了一段干旱期,直到它用分组顺序方法死灰复燃。本文回顾了序列分析在实际随机临床试验中使用的历史背景。它不回顾方法论的发展,除非它们与历史和哲学背景有关。
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引用次数: 0
EDA on the asymptotic normality of the standardized sequential stopping times, Part-II: Distribution-free models 标准化顺序停车时间渐近正态性的EDA,第二部分:无分布模型
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-07-02 DOI: 10.1080/07474946.2020.1823193
N. Mukhopadhyay, Chen Zhang
Abstract In sequential analysis, an experimenter gathers information regarding an unknown functional (parameter) by observing random samples in successive steps. We discuss a number of distribution-free scenarios under a variety of loss functions. The number of observations gathered upon termination is a positive integer-valued random variable, customarily denoted by N. Often, a standardized version of N would follow an approximate normal distribution in the asymptotic sense. We provide exploratory data analysis (EDA) with the help of a number of interesting illustrations. We do so via large-scale simulation studies to demonstrate broad applicability of the purely sequential methodologies along with the appropriateness of asymptotic normality of the standardized stopping variables as a practical and useful guideline.
在序列分析中,实验者通过连续观察随机样本来收集关于未知功能(参数)的信息。我们讨论了在各种损失函数下的一些无分布情形。在终止时收集到的观测值的数量是一个正整数随机变量,通常用N表示。通常,N的标准化版本将遵循渐进意义上的近似正态分布。我们在一些有趣的插图的帮助下提供探索性数据分析(EDA)。我们通过大规模的模拟研究来证明纯序列方法的广泛适用性,以及标准化停止变量的渐近正态性作为实用和有用的指导方针的适当性。
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引用次数: 3
Power of the MOSUM test for online detection of a transient change in mean 在线检测瞬态平均值变化的MOSUM测试的功率
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1767406
J. Noonan, A. Zhigljavsky
Abstract In this article we discuss an online moving sum (MOSUM) test for detection of a transient change in the mean of a sequence of independent and identically distributed (i.i.d.) normal random variables. By using a well-developed theory for continuous time Gaussian processes and subsequently correcting the results for discrete time, we provide accurate approximations for the average run length (ARL) and power of the test. We check theoretical results against simulations, compare the power of the MOSUM test with that of the cumulative sum (CUSUM), and briefly consider the cases of nonnormal random variables and weighted sums.
摘要本文讨论了一种用于检测独立同分布(i.i.d)正态随机变量序列均值暂态变化的在线移动和检验方法。通过使用成熟的连续时间高斯过程理论,并随后对离散时间的结果进行校正,我们提供了测试的平均运行长度(ARL)和功率的精确近似值。我们将理论结果与模拟结果进行了对比,比较了MOSUM检验与累积和检验(CUSUM)的有效性,并简要考虑了非正态随机变量和加权和的情况。
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引用次数: 11
Sequential hypothesis tests under random horizon 随机视野下的序列假设检验
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1766875
A. Novikov, Juan Luis Palacios-Soto
Abstract We consider a problem of sequential testing a simple hypothesis against a simple alternative, based on observations of a discrete-time stochastic process in the presence of a random horizon H. At any time n of the experiment, the statistician is only informed whether H > n or not. In this latter case, the experiment should be terminated and the final decision on the acceptance or rejection of the hypothesis should be taken on the basis of the available observations ( ). H is assumed to be independent of the observations, and its distribution is known to the statistician. Under the random horizon, we consider a variant of the modified Kiefer-Weiss problem: given restrictions on the probabilities of errors, minimize the average sample size calculated under the assumption that the observations follow a fixed distribution, not necessarily one of those hypothesized. Under suitable conditions on the process and/or the horizon, we characterize the structure of all optimal sequential tests in this problem. Then, we apply these results to characterize optimal tests in the case of independent observations. On the basis of the general theory, more specific results are obtained for independent and identically distributed (i.i.d.) observations with a geometrically distributed horizon. In a simple sampling model, we solve the Kiefer-Weiss problem under the random horizon model. We also discuss the questions of Wald-Wolfowitz optimality in the presence of the random horizon. In particular, we show that the stopping rules of the optimal tests, minimizing the average sample size under one of the hypotheses, are randomized versions of those of Wald’s sequential probability ratio tests.
摘要我们考虑了一个问题,基于在随机视界H存在的情况下对离散时间随机过程的观察,将一个简单假设与一个简单的替代方案进行顺序检验。在实验的任何时间n,统计学家只被告知是否H > n。在后一种情况下,应终止实验,并根据现有观察结果做出接受或拒绝假设的最终决定()。假设H与观测值无关,统计学家已知其分布。在随机水平下,我们考虑修正的Kiefer-Weiss问题的一个变体:在对误差概率的限制下,最小化在假设观测遵循固定分布的情况下计算的平均样本量,而不一定是假设的分布之一。在过程和/或范围上的适当条件下,我们刻画了该问题中所有最优序列测试的结构。然后,我们将这些结果应用于独立观测情况下的最优检验。在一般理论的基础上,对具有几何分布视界的独立同分布(i.i.d.)观测得到了更具体的结果。在一个简单的采样模型中,我们解决了随机水平模型下的Kiefer-Weiss问题。我们还讨论了随机视界存在下的Wald-Wolfowitz最优性问题。特别地,我们证明了最优检验的停止规则,在其中一个假设下最小化平均样本量,是Wald序列概率比检验的随机版本。
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引用次数: 2
Optimizing sample size in relative bioavailability trials using a Bayesian decision-theoretic framework 利用贝叶斯决策理论框架优化相对生物利用度试验的样本量
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1766887
P. Meyvisch
Abstract Bioequivalence (BE) trials are sometimes preceded by a pilot relative bioavailability (BA) trial to investigate whether the test formulation is sufficiently similar to the reference. The geometric mean ratio and its confidence bounds provide guidance as to how the BE trial can be appropriately sized to attain sufficient power. The aim of this work is to optimize the sample size of a pilot BA trial in order to minimize the overall sample size for the combination of pilot and pivotal trials. This is done through specification of a gain function associated with any of two possible outcomes of the trial; that is, abandon further development of the test formulation or proceed to a pivotal BE trial. The gain functions will be constructed on the basis of sample size considerations only, because subject numbers are indicative of both the cost and the feasibility of a clinical trial. Using simulations, it is demonstrated that for drugs with high intrasubject variability, the BA trial should be sufficiently sized to avoid erroneous decision making and to control the overall cost. In contrast, when the intrasubject variability of the pharmacokinetic (PK) parameters is low, not conducting the BA trial should be considered. It is concluded that the rather typical practice of conducting small pilot trials is unlikely to be a cost-effective approach.
摘要生物等效性(BE)试验之前有时会进行中试相对生物利用度(BA)试验,以调查试验制剂是否与参考制剂足够相似。几何平均比及其置信区间为如何适当确定BE试验的大小以获得足够的功率提供了指导。这项工作的目的是优化试点BA试验的样本量,以最大限度地减少试点和关键试验组合的总体样本量。这是通过指定与试验的两种可能结果中的任何一种相关的增益函数来实现的;即放弃测试配方的进一步开发或进行关键的BE试验。增益函数将仅基于样本量考虑进行构建,因为受试者数量指示了临床试验的成本和可行性。通过模拟,证明了对于受试者内部变异性高的药物,BA试验的规模应该足够大,以避免错误的决策并控制总体成本。相反,当受试者体内药代动力学(PK)参数的变异性较低时,应考虑不进行BA试验。得出的结论是,进行小型试点试验的相当典型的做法不太可能是一种具有成本效益的方法。
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引用次数: 0
Quickest detection of an accumulated state-dependent change point 对累积状态相关变化点的最快检测
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1766928
Liang Cai
Abstract Motivated by the practical investigation of a state-dependent quickest detection problem in continuous time, especially for Brownian observations, we propose an asymptotic scheme in discrete time called a quickest detection scheme of an accumulated state-dependent change point. Here the state-dependent means that the priori probability of the change point depends on the current state. We reduce the problem to finding an optimal stopping time of a vector-valued Markov process. We illustrate the scheme via a numerical example.
摘要基于对连续时间中状态相关最快检测问题的实际研究,特别是对于布朗观测,我们提出了一种离散时间中的渐近方案,称为累积状态相关变化点的最快检测方案。这里,状态相关意味着改变点的先验概率取决于当前状态。我们将问题简化为寻找向量值马尔可夫过程的最优停止时间。我们通过一个数值例子来说明这个方案。
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引用次数: 0
Purely sequential FWCI and MRPE problems for the mean of a normal population by sampling in groups with illustrations using breast cancer data 纯顺序的FWCI和MRPE问题的平均正常人群的抽样与插图组使用乳腺癌数据
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1766893
N. Mukhopadhyay, Zhe Wang
Abstract Two fundamental problems on purely sequential estimation are revisited—(i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem—in the context of estimating an unknown mean (μ) in a normal population having an unknown variance ( ). We begin by laying down general frameworks for the second-order asymptotic analyses, in both problems, under sequential sampling of one observation at a time. Then, instead of gathering one observation at a time, we consider sequentially sampling k observations at a time in defining our proposed estimation strategies. We replace the customary sample standard deviation as an estimator for σ with a number of other pertinent estimators to come up with new and more appropriate stopping rules to suit the occasion. We do so because in real life we know that packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This article builds the whole array of estimation methodologies in order to address both FWCI and MRPE problems with appropriate first-order and second-order asymptotic analyses. These are followed by extensive sets of carefully laid out data analyses assisted via large-scale computer simulations. These are wrapped up with illustrations using breast cancer data.
摘要在估计具有未知方差的正态总体的未知均值(μ)的情况下,重新研究了纯序列估计的两个基本问题——(i)固定宽度置信区间(FWCI)估计问题和(ii)最小风险点估计问题。我们首先为这两个问题的二阶渐近分析,在一次一个观测值的顺序抽样下,建立一般框架。然后,在定义我们提出的估计策略时,我们考虑一次顺序采样k个观测值,而不是一次收集一个观测值。我们用一些其他相关的估计量来代替习惯的样本标准差作为σ的估计量,以提出新的和更合适的停止规则来适应这种情况。我们这样做是因为在现实生活中,我们知道批量购买的包装商品的单位样品成本通常低于单个商品的成本。本文构建了整个估计方法阵列,以便通过适当的一阶和二阶渐近分析来解决FWCI和MRPE问题。随后是通过大规模计算机模拟辅助的大量精心布置的数据分析。这些都附有使用乳腺癌数据的插图。
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引用次数: 5
Minimum risk point estimation (MRPE) of the mean in an exponential distribution under powered absolute error loss (PAEL) due to estimation plus cost of sampling 在幂绝对误差损失(PAEL)下的指数分布中均值的最小风险点估计(MRPE),这是由于估计加上采样成本造成的
IF 0.8 4区 数学 Q3 Mathematics Pub Date : 2020-04-02 DOI: 10.1080/07474946.2020.1766930
N. Mukhopadhyay, Ya. G. Khariton
Abstract We begin with a review of asymptotic properties of a purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean in a one-parameter exponential distribution under a class of generalized loss functions. This class of powered absolute error loss (PAEL) includes both squared error loss (SEL) and absolute error loss (AEL) plus cost of sampling. We prove the asymptotic second-order efficiency property and asymptotic first-order risk efficiency property associated with the purely sequential MRPE problem. For operational convenience, we then move to implement an accelerated sequential MRPE methodology and prove the analogous asymptotic second-order efficiency property and asymptotic first-order risk efficiency property. We follow up with extensive data analysis from simulations and provide illustrations using cancer data.
摘要我们首先回顾了一类广义损失函数下单参数指数分布中未知均值的纯序列最小风险点估计(MRPE)方法的渐近性质。这类功率绝对误差损失(PAEL)包括平方误差损失(SEL)和绝对误差损耗(AEL)加上采样成本。我们证明了与纯序列MRPE问题相关的渐近二阶效率性质和渐近一阶风险效率性质。为了操作方便,我们随后实现了一种加速序列MRPE方法,并证明了类似的渐近二阶效率性质和渐近一阶风险效率性质。我们通过模拟进行了广泛的数据分析,并使用癌症数据进行了说明。
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引用次数: 8
期刊
Sequential Analysis-Design Methods and Applications
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