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While the principal average treatment effects are the standard target of inference, they may be insufficient when interest lies in the relative ordering of potential outcomes within a principal stratum. We introduce the principal generalized causal effect estimands to accommodate nonlinear contrast functions, providing robust, probability-scale summaries suitable for ordinal outcomes and win\u2013loss comparisons with composite endpoints. Under principal ignorability, we expand the theoretical results in Jiang et al. (J R Stat Soc Series B., 2022, 84(4), 1423\u20131445) to a broader class of causal estimands in the presence of a binary intermediate variable. We develop nonparametric identification results and derive efficient influence functions for the generalized causal estimands in principal stratification analyses. These efficient influence functions motivate multiply robust estimators and lay the ground for obtaining efficient debiased machine learning estimators via cross-fitting based on U-statistics. The proposed methods are illustrated through simulations and the analysis of a data example.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag044","type":"journal-article","created":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T12:38:29Z","timestamp":1768999109000},"page":"1345-1365","source":"Crossref","is-referenced-by-count":0,"title":["Principal stratification with U-statistics under principal ignorability"],"prefix":"10.1093","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6127-9602","authenticated-orcid":false,"given":"Xinyuan","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Mississippi State University , Mississippi State, MS ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6183-1893","authenticated-orcid":false,"given":"Fan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Yale School of Public Health , New 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