Thomas S. Richardson’s Contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes

原文摘要

I congratulate the authors on an interesting and thought provoking contribution to the literature on modelling associations. A particular strength of the approach advocated by the authors is that it may facilitate the estimation of main effects on the relative risk scale, by taking g = log. In contrast, a GLM specification such as (4) with log link presents difficulties owing to the variation dependence between β and ω(·). This dependence, also called non-congeniality, often leads to computational difficulties in practice (Lumley et al., 2006). While a partially linear GLM using the logit link function does not suffer from the problem of variation dependence it still faces the issue that, owing to the lack of collapsibility of the odds ratio, interpretation of a logistic ‘main effect’ can be challenging. In particular, even when a specification such as (4) with g = logit is correct, the main effect β will never be closer to zero than the log odds ratio for the marginal association. More generally, when (4) does not hold, the marginal (log) odds ratio may not even lie within the convex hull of the conditional (log) odds ratios as these vary over strata defined by L. In such a setting, any weighted average of the conditional associations will not reflect the marginal association. Perhaps for this reason, as noted by Lumley et al. (2006), applied researchers often express a preference for relative risks over (log) odds ratios.

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