In the study of multivariate data, for example of change in ecological communities, ANOSIM is a robust non-parametric hypothesis-testing framework for differences in resemblances among groups of samples. RELATE is a non-parametric Mantel test of the hypothesis of no relationship between two resemblance matrices. Details are given of the explicit link between the RELATE statistic, a Spearman rank correlation (ρ) between corresponding elements in the two resemblance matrices, and the ANOSIM statistic R, a scaled contrast between the among- and within-group ranks. It is seen that R can equivalently be defined as the slope of the linear regression of ranked resemblances from observations against ranked distances among samples, the latter from a simple model matrix assigning the values 1 and 0 to between- and within-group distances, respectively. Re-defining this model matrix to represent ordered distances among groups leads naturally to a generalised ANOSIM statistic, R^O, suitable for testing, for example, ordered factor levels in space or time, or an environmental or pollution gradient. Two variants of the generalised ANOSIM statistic are described, namely R^Oc where there are replicates within groups, and R^Os where there are only single samples (no replicates) within groups, for which an ANOSIM test was not previously available. Three marine ecological examples using ANOSIM to analyse an ordered factor in one-way designs are provided. These are: (1) changes in macrofaunal composition with increasing distance from an oil rig; (2) differences in phytal meiofaunal community composition with increasing macroalgal complexity; and (3) changes in average community composition of free-living nematodes along a long-term heavy metal gradient. Incorporating knowledge of an ordering structure is seen to provide more focussed, and thus stronger, ANOSIM tests, but inevitably risks losing power if that prior knowledge is incorrect or inappropriate.

中文翻译：

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具有有序因子的设计的相似性的广义分析 (ANOSIM) 统计量

在多变量数据的研究中，例如生态群落的变化，ANOSIM 是一个强大的非参数假设检验框架，用于分析样本组之间相似性的差异。RELATE 是对两个相似矩阵之间没有关系的假设的非参数 Mantel 检验。详细介绍了 RELATE 统计量、两个相似矩阵中相应元素之间的 Spearman 等级相关 ( ρ ) 和 ANOSIM 统计量R（组间和组内等级之间的比例对比）之间的显式联系。可见，R可以等效地定义为来自观察的排序相似性的线性回归的斜率与样本之间的排序距离，后者来自简单的模型矩阵，将值 1 和 0 分别分配给组间和组内距离。重新定义该模型矩阵以表示组之间的有序距离自然会导致广义 ANOSIM 统计量R ^O，适用于测试，例如，空间或时间中的有序因子水平，或环境或污染梯度。描述了广义 ANOSIM 统计量的两种变体，即组内有重复的R ^Oc和R ^Os组内只有单个样本（无重复），之前无法进行 ANOSIM 测试。提供了三个使用 ANOSIM 分析单向设计中有序因子的海洋生态示例。它们是： (1) 随着距石油钻井平台距离的增加，大型动物组成的变化；(2) 随着大型藻类复杂性的增加，植物小型动物群落组成的差异；(3) 自由生活线虫平均群落组成沿长期重金属梯度的变化。合并排序结构的知识被视为提供更集中、因此更强大的 ANOSIM 测试，但如果先验知识不正确或不合适，则不可避免地有失去权力的风险。