Linear Combinations of Ordered Symmetric Observations with Applications to Visual Acuity

In vision research, the Snellen chart is commonly used to assess visual acuity and is made up of letters of graduated sizes. By combining letter size and chart distance it is possible to determine the minimum visual angle of retinal resolution. A visual acuity of 20/30 means that at 20 feet the minimum angle of resolution is 30 20 -times the resolution standard (about 5 minutes of arc). The vision of a normal eye is recorded as 20/20 and corresponds to zero in the scale determined by the logarithm of the minimum angle of resolution (Log MAR). Normally, a single measure of visual acuity is made in each eye, say Y1, Y2, together with one or more covariates X , such as the subject’s age, reading performance, physical condition, etc. Because smaller values of Log MAR correspond to better visual acuity, the extremes of visual acuity are defined in terms of the “best” acuity Y(1) = min{Y1, Y2} and the “worst” acuity Y(2) = max{Y1, Y2}.