Can scientifically useful hypotheses be tested with correlations?

Historically, interesting psychological theories have been phrased in terms of correlation coefficients, which are standardized covariances, and various statistics derived from them. Methodological practice over the last 40 years, however, has suggested it is necessary to transform such theories into hypotheses on covariances and statistics derived from them. This complication turns out to be unnecessary, because the methodology now exists to test hypotheses on latent structures of correlations directly. Two examples are given. Limitations of correlation structures are also noted.