We’ve mostly focused on a population mean E(Y) as our quantity of interest. We saw how methods extend to estimating a subgroup mean E(Y | V=1), e.g. voters.
What about estimating a general conditional mean E(Y | X) ? We talked a lot (4 posts) about calibrating this to a known population mean E(Y), e.g. via the “logit shift”. But first we start with an estimate of E(Y | X) from survey data.
Lumley 2010 Section 5.2 says:
The polar bear has been going thru the pile of papers he was sitting on last week and found this:
Replace R (whether you respond to a survey) with T (whether you are treated) and you can see that my drawing is heavily inspired by Johansson et al. (2022) Figure 3:
We’ve talked about connections between survey random sampling and randomized experiments. There are also connections between nonprobability surveys and observational studies. We will explore more analogies between survey statistics and causal inference. Favorite references ?
https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1294076
“Using Standard Tools From Finite Population Sampling to Improve Causal Inference for Complex Experiments”
Peng:
Thanks for sharing. This reminds me a bit of the unification of sampling and causal inference in chapter 8 of Bayesian Data Analysis (originally in chapter 7 of the first edition of that book). Which makes sense, as Rubin is a coauthor of both documents.
Thanks, Andrew ! That’s a great chapter, I’ll pull from that for sure.
Thanks, Peng ! Excited to read. Does this also cover nonprobability surveys and observational studies ?
Peng, I also found some connections in your excellent book (Ding, Peng. A first course in causal inference. Chapman and Hall/CRC, 2024 https://www.routledge.com/A-First-Course-in-Causal-Inference/Ding/p/book/9781032758626):
p.173 “Using a working model to improve efficiency is an old idea from survey sampling. Little and An (2004) and Lumley et al. (2011) pointed out its connection with the doubly robust estimator.”
https://arxiv.org/abs/2308.11458
This seems to overview some relevant material.
Thanks, Anon ! Excited to read.
Just to be sure, be free to disagree with me! If you replace R with T, as you proposed in the text, you are working with two qualitatively different concepts, they are no interchangeable. To truly have “T = R” relation, the R has to be “whether you are asked to participate in a survey” or T needs to change to “whether you follow the treatment level as assigned, by protocol”.
Thanks, Roman ! Great catch, I edited the text to say “whether you are treated” for T.