Interaction Tests for 2 × s × t Contingency Tables
2 × s × t 列联表的交互作用检验AI 解读 ↓
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2 × s × t 列联表的交互作用检验
- 研究问题
- 如何将三维列联表中已有的条件交互作用检验推广到更一般的表格维度，并说明它与部分关联检验之间的联系？
- 方法与思路
- 作者推广固定边际总数参照集下的条件检验，在频数尺度构造类似方差分析平方和的统计量，并用数值例子比较分析。
- 材料陈述的贡献
- 摘要称将 Zelen 的检验推广至题名所示列联表，并揭示统计量与齐性卡方、部分关联及对数频数分析的关系。
- 阅读关注点
- 应核对固定哪些边际总数、交互作用的模型定义、统计量分解与渐近校准，同时确认摘要首句表格维度的异常字符。
- 阅读边界
- OpenAlex 摘要版本未注明，首句“2 × 8 × t”与题名及后文不一致，需查原 PDF，不能据此重构公式；数值比较细节未提供。
原文摘要
SUMMARY The interaction test for 2 x 2 x t contingency tables of Zelen (1971) is extended to 2 x 8 x t tables. This test is related to the tests of partial association, or 'main effect', of Birch (1965) and Armitage (1966). These tests are frequency analogues of the log-frequency tests con- sidered by Plackett (1962) and Goodman (1964). The various analyses are illustrated and compared in several numerical examples. Bartlett (1935) considered a model and statistical analysis for interaction in 2 x 2 x 2 tables. Norton (1945) generalized these results to 2 x 2 x t tables and Roy & Kastenbaum (1956) gave the general r x s x t results. These authors used tests based on Pearson's chi- squared statistic with parameters fitted by iterative maximum likelihood methods. Woolf (1955), using the same concept of interaction, gave a much simpler, asymptotically equiva- lent test for 2 x 2 x t tables based on the logit transformation of the data. Plackett (1962) extended Woolf's result to the general three-dimensional table and Goodman (1964) showed how Plackett's test statistics might be more easily computed, particularly for 2 x s x t tables. For further discussion and references on the various models and analyses of three-dimensional contingency tables, see Lancaster (1969, Chapter XII) and Plackett (1969). More recently, various authors have discussed simplified iterative maximum likeli- hood estimation techniques in three-dimensional tables and Goodman (1970) has extended these results to m-dimensional tables. Zelen (1971) derived a conditional test for interaction in 2 x 2 x t tables based on the reference set of all fixed marginal totals. The asymptotic version of this test may be con- sidered the analogue, in the untransformed space, of Woolf's logit test. This paper extends Zelen's test to 2 x s x t tables; the resulting test is the close analogue, in the untransformed space, of Goodman's logit analyses for such tables. The structure of the test statistic is analogous to a sum of squares in an analysis of variance. The summed terms are related to the usual homogeneity chi-squares for the s 2 x t tables and the 'correction term' is Birch's (1965) test statistic for partial association, or 'main effect'.