I am designing a within multi-level study in which my data has both nested and multiple-membership structure. The subjects are multimembers of V1 and also are nested in V2.The subjects are all from one year. When applying matching or weighting, should I take the multiple membership into account? if yes, the question would be how? or should I apply matching without taking it into account and then run a multiple membership multi-level regression on the output from matching?
Matching with multi-level multiple membership data
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Without knowing more about your concrete use case, it's hard to recommend specific sources. But check out "cross-classified multilevel analysis". I think that's what you're referring to.