Pearson’s chi-square statistic is well established for testing goodness-of-fit of various hypotheses about observed frequency distributions in contingency tables. A general formula for ANOVA-like decompositions of Pearson’s statistic is given under the independence assumption along with their extensions to higher-order tables. Mathematically, it makes the terms in the partitions and orthogonality among them obvious. Practically, it enables simultaneous analyses of marginal and joint probabilities in contingency tables under a variety of hypotheses about the marginal probabilities. Specifically, this framework accommodates the specification of theoretically driven probabilities as well as the well known cases in which the marginal probabilities are fixed or estimated from the data. The former allows tests of prescribed marginal probabilities, while the latter allows tests of the associations among variables after eliminating the marginal effects. Mixtures of these two cases are also permitted. Examples are given to illustrate the tests.

中文翻译：

---

列联表分析中Pearson卡方统计量的家庭分解

培生的卡方统计量已经很好地建立，可以用来测试关于列联表中观察到的频率分布的各种假设的拟合优度。在独立性假设下以及对高阶表的扩展下，给出了Pearson统计量的类ANOVA分解的一般公式。从数学上讲，它使分区中的术语和正交性变得显而易见。实际上，它可以在有关边际概率的各种假设下同时分析列联表中的边际和联合概率。具体而言，此框架适用于理论驱动的概率的规范以及边际概率是固定的或根据数据估算的众所周知的情况。前者允许测试规定的边际概率，而后者允许在消除边际效应后测试变量之间的关联。也可以将这两种情况混合使用。给出示例以说明测试。