Abstract The isometric logratio transformation, in the form of a so-called “balance”, has been promoted as a way to contrast two groups of parts in a compositional data set by forming ratios of their respective geometric means. This transformation has attractive theoretical properties and hence provides a useful reference, but geometric means are highly affected by parts with small relative values. When a comparison between two groups of parts is required in practical applications, such as the investigation and construction of models, while making use of substantive domain knowledge, it is demonstrated that the logratio of two amalgamations serves as an alternative, interpretable form of balance. A geochemical data set is considered, which has been analyzed previously by transforming to a set of isometric logratio balances. An alternative approach, using a reduced set of pairwise logratios of parts, optionally involving prescribed amalgamations, is very close to optimal in accounting for the variance in this compositional data set. These simpler transformations also have an exact back-transformation to the original parts. This approach highlights for this dataset which compositional parts are driving the data structure, using variables that are easy to interpret and that map well to research-driven objectives.

中文翻译：

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成分数据分析中等距和合并对数比平衡的比较

摘要 等距对数变换，以所谓的“平衡”的形式，已被推广为一种通过形成它们各自几何平均值的比率来对比组成数据集中两组零件的方法。这种变换具有吸引人的理论特性，因此提供了有用的参考，但几何平均值受相对值较小的零件的影响很大。当在实际应用中需要对两组部分进行比较时，例如模型的调查和构建，同时利用实质性领域知识，证明了两个合并的对数比作为一种替代的、可解释的平衡形式。考虑了地球化学数据集，该数据集之前已通过转换为一组等距对数比平衡进行了分析。另一种方法，使用一组减少的部分成对对数比，可选地涉及规定的合并，在解释这个组成数据集中的方差时非常接近最佳。这些更简单的转换也对原始部分进行了精确的反向转换。这种方法为该数据集突出显示了哪些组成部分正在驱动数据结构，使用易于解释并很好地映射到研究驱动目标的变量。