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: (1) 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; and (2) 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.