P 540
Computing the mean of refraction measurements
R. Kusel, U. Oechsner
Introduction: Sometimes it is necessary to compute means and variances of the results of refraction measurements. Examples are the comparison of the results of refraction measurements with different methods, or different surgical methods concerning the optics of the eye. Then direct direct averaging of the components sphere, cylinder and axis is not useful. This results from the fact that these components are not mathematically independent. The spherical refraction value depends on the sign of the cylinder, and the meaning of the error of the angle depends on the cylinder value.
Methods: From the n refraction values si for the sphere, ci for the cylinder (which in the following are assumed to be negative) and f i for the axis the quantities se,i = si + ci / 2, c1,i = - ci / 2 cos 2f i and c2,i = - ci / 2 sin 2f i are computed. These quantities are mathematically independent allowing computation of the mean. From these mean values the means of sphere, cylinder, and axis can be computed using the reverse of the formulas indicated above. Adittionally, applying the error propagation law allows to compute the variances of these values.
Discussion: In general, even a refraction measurement, as every physical measurement, should include an error estimate. But under clinical circumstances repetitions of the measurements are not justifiable. This is particularly true when the accuracy of the method is known, allowing a critical evaluation of the measurement values. In case of scientific studies, and for the evaluation of new methods the assessment of the measurement errors can be useful. Then the described method should be applied. For the judgement of the accuracy of refraction measurements one should consider that the error of the axis itself is meaningless.
University Eye Clinic, Medical Optics Lab., Martinistr. 52, 20246 Hamburg