Departamento de Metodología, Facultad de Psicología, Universidad Complutense, Madrid, Spain.
Departamento de Econometría y Estadística (E.A. III), Universidad del País Vasco UPV/EHU, Bilbao, Spain.
Stat Methods Med Res. 2020 Sep;29(9):2569-2582. doi: 10.1177/0962280220902480. Epub 2020 Feb 5.
Controversy over the validity of significance tests in the analysis of contingency tables is motivated by the disagreement between asymptotic and exact values and its dependence on the magnitude of expected frequencies. Variants of Pearson's statistic and their asymptotic distributions were proposed to overcome the difficulties, but several approaches also exist to conduct exact tests. This paper shows that discrepant asymptotic and exact results may or may not occur whether expected frequencies are large or small: Eventual inaccuracy of asymptotic values is instead caused by idiosyncrasies of the discrete distribution of . More importantly, discrepancies are also artificially created by the hypergeometric sampling model used to perform exact tests. Exact computations under the alternative full-multinomial or product-multinomial models require eliminating nuisance parameters and we propose a novel method that integrates them out. The resultant exact distributions are very accurately approximated by the asymptotic distribution, which eliminates concerns about the accuracy of the latter. We also discuss that the two-stage approach that tests for significance of residuals conditional on a significant test is inadvisable and that an alternative single-stage test preserves Type-I error rates and further eliminates concerns about asymptotic accuracy.
对列联表分析中显著性检验有效性的争议源于渐近值和精确值之间的不一致，以及其对期望频率大小的依赖性。为了克服这些困难，提出了皮尔逊统计量的变体及其渐近分布，但也存在几种进行精确检验的方法。本文表明，无论期望频率大小如何，渐近值和精确值的不一致可能会出现，也可能不会出现：渐近值的最终不准确性是由离散 的分布特征引起的。更重要的是，使用超几何抽样模型进行精确检验也会人为地产生差异。在备择的完全多项或乘积多项模型下进行精确计算需要消除无关参数，我们提出了一种新的方法来消除它们。通过渐近分布可以非常准确地近似化精确分布，从而消除了对后者准确性的担忧。我们还讨论了在显著 检验的条件下对残差进行显著性检验的两阶段方法是不可取的，而替代的单阶段检验保留了第一类错误率，并进一步消除了对渐近准确性的担忧。
