Refraction and keratometry: departures from and transformations toward multivariate normality.

The multivariate distributional properties of refraction and keratometric data were investigated across eyes with power represented in the coordinate system introduced by Deal and Toop. Normality and departure from normality were assessed with the aid of X 2 and normal probability plots and by the comparison of multivariate sample skewness and kurtosis with critical values. Two of the three data sets show significant departure from normality in each of the marginal distributions and, therefore, the joint distribution too. The keratometric data were normally distributed along the line of spherical powers but departed from normality in the astigmatic plane. Marginal transformations are used to reduce the departure from normality where necessary. The transformation that was found to be successful is essentially an example of a Box/Cox transformation involving a shift, c i , and an exponent, λ i , where i = 1,2,3. For two of the data sets, the values of the exponent, λ i , result in a transformation that is similar to a modified square root transformation.