Comments on “Statistical inference with non-probability survey samples”
Section 3. Weighting vs. modeling for the general user

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Wu’s paper and the above addendums tend to follow the long-trodden path regarding weighting versus modeling in the finite population inference setting, dating back at least to Hansen, Madow and Tepping (1983). In thinking about this choice I believe it is important to distinguish between models used to derive so-called descriptive parameters $–$ in the sense of Kalton (1983) $–$ and models that are of interest in and of themselves, so-called analytic parameters in regression models, latent classes analysis, etc. For the former distinguishing a descriptive target of interest $Y$ from potential modeling covariates $X$ has the advantage of creating doubly-robust estimators that are targeted to a single descriptive parameter. This also requires assumptions such as A1 in Section 2.1 (propensity score does not depend on $Y$ conditional on $X).$ When models themselves are the targets of interest, it may be that developing weights via propensity scores to account for selection bias and, as Wu notes, employing standard weighted estimating equations may be the most sensible choice, since typically a wide number of models may be considered. This comes at the cost of double robustness, since there is usually no attempt to model the analytic parameter directly. Developing ways to extend double-robustness into a broader class of model parameter estimates may be a fruitful exercise.

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