Employing nested sequences of models is a common practice when exploring the extent to which one set of variables mediates the impact of another set. Such an analysis in the context of logistic regression models confronts two challenges: (i) direct comparisons of coefficients across models are generally biased due to the changes in scale that accompany the changes in the set of explanatory variables, (ii) conducting a large number of tests induces a problem of multiplicity that can lead to spurious findings of significance if not heeded. This article aims to illustrate a practical strategy for conducting analyses in the face of these challenges. The challenges—and how to address them—are illustrated using a subset of the findings reported by Braun (Large-scale Assess Educ 6(4):1–52, 2018. 10.1186/s40536-018-0058-x), drawn from the Programme for the International Assessment of Adult Competencies (PIAAC), an international, large-scale assessment of adults. For each country in the dataset, a nested pair of logistic regression models was fit in order to investigate the role of Educational Attainment and Cognitive Skills in mediating the impact of family background and demographic characteristics on the location of an individual’s annual income in the national income distribution. A modified version of the Karlson–Holm–Breen (KHB) method was employed to obtain an unbiased estimate of the true differences in the coefficients between nested logistic models. In order to address the issue of multiplicity, a recent generalization of the Benjamini–Hochberg (BH) False Discovery Rate (FDR)-controlling procedure to hierarchically structured hypotheses was employed and compared to two conventional methods. The differences between the changes in coefficients calculated conventionally and with the KHB adjustment varied from negligible to very substantial. When combined with the actual magnitudes of the coefficients, we concluded that the more proximal factors indeed act as strong mediators for the background factors, but less so for Age, and hardly at all for Gender. With respect to multiplicity, applying the FDR-controlling procedure yielded results very similar to those obtained by applying a standard per-comparison procedure, but quite a few more discoveries in comparison to the Bonferroni procedure. The KHB methodology illustrated here can be applied wherever there is interest in comparing nested logistic regressions. Modifications to account for probability sampling are practicable. The categorization of variables and the order of entry should be determined by substantive considerations. On the other hand, the BH procedure is perfectly general and can be implemented to address multiplicity issues in a broad range of settings.

在探索一组变量在多大程度上调节另一组变量的影响时，使用嵌套的模型序列是一种常见的做法。这种在逻辑回归模型背景下的分析面临两个挑战：(i) 模型间系数的直接比较通常是有偏差的，因为伴随解释变量集变化的规模变化，(ii) 进行大量测试会引发多重性问题，如果不加以注意，可能会导致虚假的重要发现。本文旨在说明在面对这些挑战时进行分析的实用策略。使用 Braun 报告的调查结果的子集说明了这些挑战以及如何解决这些挑战（Large-scale Assess Educ 6(4):1–52, 2018. 10.1186/s40536-018-0058-x），来自成人能力国际评估计划 (PIAAC)，这是一项针对成人的国际大规模评估。对于数据集中的每个国家，拟合一对嵌套的逻辑回归模型，以研究教育程度和认知技能在调节家庭背景和人口特征对个人年收入在国民收入中的位置的影响中的作用分配。使用 Karlson-Holm-Breen (KHB) 方法的修改版本来获得嵌套逻辑模型之间系数的真实差异的无偏估计。为了解决多重性问题，最近将 Benjamini-Hochberg (BH) 错误发现率 (FDR) 控制程序推广到分层结构的假设，并与两种传统方法进行了比较。传统计算的系数变化和 KHB 调整之间的差异从可以忽略不计到非常显着。当与系数的实际大小相结合时，我们得出结论，更接近的因素确实作为背景因素的强中介，但对年龄的作用较小，对性别几乎没有。关于多重性，应用 FDR 控制程序产生的结果与应用标准的每个比较程序获得的结果非常相似，但与 Bonferroni 程序相比，发现更多。此处说明的 KHB 方法可以应用于对比较嵌套逻辑回归感兴趣的任何地方。考虑概率抽样的修改是可行的。变量的分类和条目的顺序应根据实质性考虑来确定。另一方面，BH 程序是完全通用的，可以实施以解决广泛设置中的多重性问题。