Here's a hypothetical situation. I want to estimate the proportion of a population who has a certain gene. I am doing this study on region A, and within this region there is a city B. I separately sample $n_A$ and $n_B$ from region A and city B, respectively. However, because city B is within region A, there is a certain probability that an individual in sample $n_A$ is in fact in city B. This probability can be estimated if we have the population of region A and city B.
There are existing methods (e.g., Hartley, Horvitz - Thompson) that allow me to estimate the proportion of population with the gene for the total population, in this case, the whole of region A, as well as the associated variance. It is also possible to estimate the proportion for city B. What is missing, however, is how to calculate the variance associated with the estimate for city B. And also, what happens if I extend this to multi-frames? E.g. what if there is now city C which is also within region A?
Some references:
https://www150.statcan.gc.ca/n1/pub/12-001-x/2011002/article/11608-eng.pdf
https://cran.r-project.org/web/packages/Frames2/Frames2.pdf