Across group-living animals, linear dominance hierarchies lead to disparities in access to resources, health outcomes, and reproductive performance. Studies of how dominance rank affects these outcomes typically employ one of several dominance rank metrics without examining the assumptions each metric makes about its underlying competitive processes. Here we compare the ability of two dominance rank metrics - ordinal rank and proportional or ‘standardized’ rank - to predict 20 distinct traits in a well-studied wild baboon population in Amboseli, Kenya. We propose that ordinal rank best predicts outcomes when competition is density-dependent, while proportional rank best predicts outcomes when competition is density-independent. We found that for 75% (15/20) of the traits, one of the two rank metrics performed better than the other. Strikingly, all male traits were better predicted by ordinal than by proportional rank, while female traits were evenly split between being better predicted by proportional or ordinal rank. Hence, male and female traits are shaped by different competitive regimes: males’ competitive environments are largely driven by density-dependent resource access (e.g., access to estrus females), while females’ competitive environments are shaped by both density-independent resource access (e.g. distributed food resources) and density-dependent resource access. However, traits related to competition for social and mating partners are an exception to this sex-biased pattern: these traits were better predicted by ordinal rank than by proportional rank for both sexes. We argue that this method of comparing how different rank metrics predict traits of interest can be used as a way to distinguish between different competitive processes operating in animal societies.