Niwen Zhou and Xu Guo’s Contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes
Niwen Zhou, Xu Guo · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 2022
We thank Professors Vansteelandt and Dukes for their innovative and stimulating paper. They introduce a new estimand which reduces to the exposure effect parameter when the (semi)parametric model holds, is generic for continuous or discrete exposure, and still captures conditional association when model assumption fails. We make some additional understanding of the new estimand and introduce a related new estimand. Here we consider the expectation of the ratio. Here β· and ω· are unknown functions. Different from the model (4) in Vansteelandt and Dukes (2021), we now allow A–L interaction, and extend to the exposure effect heterogeneity setting, with respect to L. While under model (1), γ=Eβ(L), which measures the average exposure effect. When there is no exposure effect heterogeneity, i.e. β(L)≡β, γ reduces to β as the estimand in Vansteelandt and Dukes (2021). Even when the model fails, the estimand γ is still meaningful to capture conditional association. Clearly, the new estimand in Vansteelandt and Dukes (2021) can be viewed as an extension of weighted average treatment effect; while our new estimand is an extension of the classical average treatment effect. Lastly, current literature only focus on the conditional expectation of the exposures, which is sensitive to outliers. Conditional quantile can provide a more robust and complete view of the association between response and exposure. Thus it would be very interesting to extend the insightful idea in Vansteelandt and Dukes (2021) to quantile setting.