Meridional profiles of variance-covariance of dioptric power. Part 1. The basic theory

It has recently become possible to calculate and represent variation or spread of dioptric power in a meaningful way. This is important for the proper analysis and interpretation of data on dioptric power in a number of areas of the vision sciences. The representation takes the form of a symmetric matrix of six (usually) variances and covariances. Although the matrix is satisfactory for several formal statistical purposes, such as the testing of hypotheses, it does not give an intuitively satisfactory picture of the extent and nature of the variation, nor is it easy to interpret in a way that could be useful to the researcher or clinician. A useful graphical representation of variation can be constructed from the variance-covariance matrix. It consists of curves that show the meridional dependence of the variation. These meridional profiles of variation give a complete and intuitively satisfactory picture of the nature and extent of the variation of power and are potentially of general use to researcher and clinician. A complete theoretical basis is provided for the construction of meridional profiles of variation of dioptric power. An accompanying paper employs the theory to construct profiles for a number of representative samples of dioptric power.