The same question could be asked for estimates of sensitivity and specificity for medical diagnostic tests, when the data used to estimate them are often generated neither from random sampling nor random allocation (ie, they used a convenience sample). Of course, as discussed on this forum previously, that may be the least of the problems with Se/Sp: Sensitivity, specificity, and ROC curves are not needed for good medical decision making
I have seen versions of this question raised before, and the heartburn comes from the obvious alternative: would I rather report point estimates without any confidence intervals at all? Sometimes I wonder if the right answer is Yes, along with showing the 2x2 table with margins. Simply report what the design of the study was, and give the results in summary tables, with no pretense of doing “statistical inference”. A descriptive summary is all the data deserves. I’ve taken this approach many times, but such tables do not allow readily exploring trends with covariates, for example. Other times I have just given the cop-out caveat, “if we were to pretend that the subjects were randomly sampled, the confidence intervals would be…”
Actually many years ago I remember Bert Gunter asking Frank about the statistical inferences in RMS Ch. 12, where the survival data for the Titanic passengers are modeled. This is a more difficult case because the model is used to explore trends in the data; it’s like an example of what Frank has said - the model may be the best way to summarize the complex structure of the data. This structure would not be evident by just making tables, I suspect. While there are no confidence intervals in the chapter, there are likelihood ratio tests; Wald statistics and p-values for different terms in the model; predicted survival probabilities for various settings of covariates, etc. Bert asked something along the lines of: to what population do these inferences apply? Sadly I do not remember Frank’s response, but the question has been stuck in my mind for 2 decades.