Partially relaxed strained layers and structures are characterised for composition and strain by high-resolution X-ray diffraction, using rocking curves or reciprocal space maps. Multivariate linear regression analysis provides both a guide to the best reflections to measure and also the optimum extraction of strain and composition values and the errors on them. Strains are often large enough to require non-linear elasticity theory, and we show how multivariate linear regression analysis can also be applied here, to literature data, to obtain better estimates of third-order elastic constants and their errors than those given previously. Finally, the meaning of third-order elastic constants is discussed, leading to a new derivation of thermodynamic effective elastic constants under arbitrary strain. These effective elastic constants enable the correct calculation of the deformation of a strained layer. (C) 2002 Elsevier Science B.V. All rights reserved.