Male-biased dimorphism in body size is usually attributed to sexual selection acting on males, through either male competition or female choice. Brown antechinuses (Antechinus stuartii) are sexually dimorphic in size, and heavier males are known to sire more offspring in the wild. We investigated four possible mechanisms that might explain this large-male reproductive advantage. We tested if there is a female preference for large males, a female preference for dominant males, if larger males compete more effectively for mates, and if there is a survival advantage for large males during the mating season. We established nesting groups of males in captivity and conducted mate choice trials in which males from nesting groups either could or could not interact. We assessed male dominance rank and recorded survival times after mating. Females did not prefer larger males directly. The results suggest that the other three mechanisms of sexual selection tested account for the large-male advantage: large males competed more successfully for mates, so were socially dominant; females rejected subordinates (males they saw losing twice in contests to previous mates); and dominant males survived for longer after their first mating. Females judged male rank based on direct observation of male competitive interactions at the time of mating and apparently could not distinguish rank from male scent. Effects of size and dominance on male reproductive success are not confounded by age because male antechinuses are semelparous.