The island species–area relationship (ISAR) describes how the number of species increases with increasing size of an island (or island‐like habitat), and is of fundamental importance in island biogeography and conservation. Here, we use a framework based on individual‐based rarefaction to infer whether ISARs result from passive sampling, or whether some processes are acting beyond sampling (e.g., disproportionate effects and/or habitat heterogeneity). Using data on total and relative abundances of four taxa (birds, butterflies, amphibians, and reptiles) from multiple islands in the Andaman and Nicobar archipelago, we examine how different metrics of biodiversity (total species richness, rarefied species richness, and abundance‐weighted effective numbers of species emphasizing common species) vary with island area. Total species richness increased for all taxa, as did rarefied species richness controlling for a given sampling effort. This indicates that the ISAR did not result because of passive sampling, but that instead, some species were disproportionately favored on larger islands. For birds, frogs, and lizards, this disproportionate effect was only associated with species that were rarer in the samples, but for butterflies, both more common and rarer species were affected. Furthermore, for the two taxa for which we had plot‐level data (reptiles and amphibians), within‐island β‐diversity did not increase with island size, suggesting that within‐island compositional effects were unlikely to be driving these ISARs. Overall, our results indicate that the ISARs of these taxa are most likely driven by disproportionate effects, that is, where larger islands are important sources of biodiversity beyond a simple sampling expectation, especially through their influence on rarer species, thus emphasizing their role in the preservation and conservation of species.