Specific tests for the equality of two dependent interclass correlations, estimated from a multivariate sample, have been examined in some detail by Dunn & Clark (1969, 1971). In this paper I wish to consider the dependencies among correlation coefficients in more generality, whether interclass or intraclass, and hence to suggest appropriate large-sample tests of hypotheses concerning them. This problem is of particular interest to geneticists, who may be concerned with the analysis of correlations between pairs of relatives. Rao, Morton & Yee (1974) have recently presented a general linear model for such an analysis, in which an attempt is made to apportion the total variability of a trait to environmental and genetic causes when independent estimates of the various correlations are available. In practice, however, correlations such as parent-offspring and full-sib are often estimated from a single set of nuclear families, parents and children, and so the estimates are not independent. After deriving asymptotic expressions for the correlations between each of the five different pairs of correlation coefficients possible, I indicate how they can be used to test hypotheses and to obtain pooled correlation estimates from families of different sizes.