Thanks Frank,
For illustration you could take any paper that assumes f(y_x;z,u) = E(y;x,z,u) or equivalent (e.g., that z is sufficient to “control bias” in the sense of forcing this equality) and then illustrates fitting of E(y;x,z,u) with an example. A classic used when I was a student was Cornfield’s fitting a logistic model and then using that model with x fixed at a reference level across patients to compute risk scores of patients. See for example p. 1681 of Cornfield, ‘The University Group Diabetes Program: A Further Statistical Analysis of the Mortality Findings’, JAMA 1971. His is a purely verbal description, there being no established notation back then. In modern terms what he was doing there is arguing against there being much confounding in the trial because the distribution of the fitted E(y;0,z,u) - his proxy for the f(y_0;z,u) distribution - was similar in the treated and untreated groups.
More recent examples in modern notation can be found under the topics of g-estimation and, especially, finding “optimal” treatment regimes. Once the fitted function f(y_x;z,u) is in hand, focus then turns to external validity: whether z is sufficient for transporting the function to a new target population beyond those studied. This problem is more general and difficult than the internal validity issue of whether the fitted potential outcomes can be transported across the treatment or exposure groups within the study - see p. 46 of the Hernan-Robins book for a discussion.
There are now many papers on finding functions for making treatment choices (“optimizing” treatment regimes) and transporting those functions. I haven’t even begun to read them all and would not claim to be able to recommend one best for your purposes. Nonetheless, most I see are illustrated with real examples. One clearly centered around individual patient choices is Msaouel et al. “A Causal Framework for Making Individualized Treatment Decisions in Oncology” in Cancers 2022, which I believe was posted earlier in this thread. One could object to its use of additive risk models and risk differences, but the same general framework can be used with other models.
Regardless of the chosen smoothing model, one can display the risk estimates directly instead of their differences. Survival times and their differences might however be more relevant than risks (probabilities). In either case, the use of differences is defensible to the extent the chosen differences are proportional to loss differences, which I am pretty sure is far more often for risk and survival-time differences than for odds ratios (I’ve yet to see a real medical example in which odds ratios are proportional to actual loss differences).