Rank-based tests are alternatives to likelihood-based tests popularised by their relative robustness and underlying elegant mathematical theory. There has been a surge in research activities in this area in recent years since a number of researchers are working to develop and extend rank-based procedures to clustered-dependent data which include situations with known correlation structures (e.g. as in mixed effects models) as well as more general form of dependence. The purpose of this paper is to test the symmetry of a marginal distribution under clustered data. However, unlike most other papers in the area, we consider the possibility that the cluster size is a random variable whose distribution is dependent on the distribution of the variable of interest within a cluster. This situation typically arises when the clusters are defined in a natural way (e.g. not controlled by the experimenter or statistician) and in which the size of the cluster may carry information about the distribution of data values within a cluster. Under the scenario of an informative cluster size, attempts to use some form of variance-adjusted sign or signed-rank tests would fail since they would not maintain the correct size under the distribution of marginal symmetry. To overcome this difficulty, Datta and Satten [2008, ‘A Signed-Rank Test for Clustered Data’, Biometrics, 64, 501–507] proposed a Wilcoxon-type signed-rank test based on the principle of within-cluster resampling. In this paper, we study this problem in more generality by introducing a class of valid tests employing a general score function. Asymptotic null distribution of these tests is obtained. A simulation study shows that a more general choice of the score function can sometimes result in greater power than the Datta and Satten test; furthermore, this development offers the user a wider choice. We illustrate our tests using a real data example on spinal cord injury (SCI) patients.

中文翻译：

---

当簇大小可能提供信息时，对簇数据进行一类通用的符号秩测试


基于排名的测试是基于可能性的测试的替代方案，因其相对稳健性和基础优雅的数学理论而广受欢迎。近年来，该领域的研究活动激增，因为许多研究人员正在致力于开发基于排名的程序并将其扩展到依赖于聚类的数据，其中包括具有已知相关结构的情况（例如，在混合效应模型中）：以及更一般形式的依赖。本文的目的是测试聚类数据下边缘分布的对称性。然而，与该领域的大多数其他论文不同，我们考虑簇大小是随机变量的可能性，其分布取决于簇内感兴趣变量的分布。当以自然方式定义聚类（例如，不受实验者或统计学家控制）并且其中聚类的大小可以携带关于聚类内数据值的分布的信息时，通常会出现这种情况。在信息丰富的簇大小的情况下，尝试使用某种形式的方差调整符号或符号秩检验将会失败，因为它们在边缘对称分布下无法保持正确的大小。为了克服这个困难，Datta 和 Satten [2008, 'A Signed-Rank Test for Clustered Data', Biometrics, 64, 501–507] 提出了基于簇内重采样原理的 Wilcoxon 型符号秩检验。在本文中，我们通过引入一类采用通用评分函数的有效测试来更普遍地研究这个问题。获得这些检验的渐近零分布。 模拟研究表明，更通用的评分函数选择有时可以产生比 Datta 和 Satten 检验更大的功效；此外，这一发展为用户提供了更广泛的选择。我们使用脊髓损伤 (SCI) 患者的真实数据示例来说明我们的测试。