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In a system of intercorrelated variables, the sign and strength of pairwise relationships must conform to a mathematical constraint affecting the entire system. The constraint exists as a property of all correlation matrices, the positive semi-definite (PSD) criterion. We illustrate its implications with a case study of covariation among clutch size, body size, and latitude in the Strigidae. Our analyses indicate that clutch size and body size among strigid owls increase with latitude, while clutch size decreases with body size. The PSD provides a framework to explain the existence of this apparent macroevolutionary paradox. It also shows why the three relationships are never detected concurrently when using simple regressions or independent contrasts. In a system with three intercorrelated variables, the PSD criterion may allow one pairwise relationship to be negative even if the other two relationships are positive. However, the correlation coefficients cannot all be strong. Whenever the PSD criterion may constrain covariation, care should be exercised in inferring evolutionary or ecological processes from patterns of correlation.