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We establish a connection between Hoeffding\u2019s D statistic and the profile association and derive a permutation test with theoretical guarantees for consistency and power under alternatives to the null hypothesis of independence (equivalently, no association). We extend this framework to the conditional setting, where the independence between random objects given a Euclidean predictor is of interest. In simulations across various metric spaces, the proposed profile independence test is found to outperform existing approaches. The practical utility of this framework is demonstrated with applications to brain connectivity networks derived from magnetic resonance imaging and age-at-death distributions for males and females obtained from human mortality data.<\/jats:p>","DOI":"10.1214\/26-aos2632","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T08:16:00Z","timestamp":1787040960000},"page":"1923-1949","source":"Crossref","is-referenced-by-count":0,"title":["Association and independence test for random objects"],"prefix":"10.1214","volume":"54","author":[{"given":"Hang","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research & School of Data and Information Sciences, University of North Carolina at Chapel Hill"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hans-Georg","family":"M\u00fcller","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California, Davis"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"10","doi-asserted-by":"crossref","unstructured":"<small>Bolstad, B. 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