Bivariate analysis of surgically induced regular astigmatism. Mathematical analysis and graphical display.

OBJECTIVE: The purpose of the study was to develop methods for simultaneous description of astigmatic direction and magnitude on aggregate data, with special reference to refractive surgery.
DESIGN: Mathematical analysis of astigmatisms employing bivariate statistical methods.
RESULTS: The mean of several astigmatisms is a new astigmatism of specific direction and magnitude, while the confidence region is an area, which may be determined exactly.
CONCLUSIONS: Astigmatisms may conveniently be symbolized as an astigmatic direction and magnitude, but are actually composed of refractive powers in the form of polar values. We are operating with two different entities, a net astigmatism and a power vector in the form of polar values. There is an unequivocal point-to-point correlation between these entities. Mathematical conversions can only be performed with polar values, but never by using net astigmatisms. All net astigmatisms must be converted to their appropriate refractive powers and the relevant calculations performed with these entities. The final result, such as an average of several astigmatisms, variances or confidence areas, may be point-to-point reconverted to and symbolized by a net astigmatism. These principles allow for exact description and comparison of surgical methods, but may be employed to describe and analyze any other population of astigmatisms, such as subjective cylinders and spectacle corrections.