Traditionally, confirmatory factor analytic models are tested against a null model of total independence. Using randomly generated factors in a matrix of 46 aptitude tests, this approach is shown to be unlikely to reject even random factors. An alternative null model, based on a single general factor, is suggested. In addition, an index of model efficiency is introduced as a useful adjunct to contemporary indices of overall fit. The usefulness of these procedures are demonstrated in a confirmatory factor analysis based on Guilford's Structure of the Intellect model. (Author)