Despite an extensive literature, the "g" construct remains a point of debate. Different models explaining the observed relationships among cognitive tests make distinct assumptions about the role of g in relation to those tests and specific cognitive domains. Surprisingly, these different models and their corresponding assumptions are rarely tested against one another. In addition to the comparison of distinct models, a multivariate application of the twin design offers a unique opportunity to test whether there is support for g as a latent construct with its own genetic and environmental influences, or whether the relationships among cognitive tests are instead driven by independent genetic and environmental factors. Here we tested multiple distinct models of the relationships among cognitive tests utilizing data from the Vietnam Era Twin Study of Aging (VETSA), a study of middle-aged male twins. Results indicated that a hierarchical (higher-order) model with a latent g phenotype, as well as specific cognitive domains, was best supported by the data. The latent g factor was highly heritable (86%), and accounted for most, but not all, of the genetic effects in specific cognitive domains and elementary cognitive tests. By directly testing multiple competing models of the relationships among cognitive tests in a genetically-informative design, we are able to provide stronger support than in prior studies for g being a valid latent construct.