Transforming scales of measurement of visual acuity at the group level

To provide formulae that may be used to transform sample-based estimates of group-level mean and standard deviation of visual acuity (VA) across different scales of measurement.

We focused on 3 transformations: (1) ETDRS letters - LogMAR (2) Decimal - LogMAR and (3) Snellen - LogMAR. We assumed that logMAR follows a normal distribution in the underlying population and used the empirical asymptotic normal approximation of the joint distribution of average and standard deviation in order to derive formulae for transformation of group-level estimates. We considered that the true population parameters are not known and are to be estimated using data from a sample of patients (which is essentially always the case). We compared estimates obtained with the proposed sample-based approach with those based on a "naïve" approach in which individual-level formulae are used directly for transformation of means and standard deviations at the group-level.

Applying formulae that are appropriate for transformations of scales of measurement for data at the individual- (or patient-) level, to transform VA at the group level, can lead to biased estimates of means and standard deviations. In particular, it could lead to underestimation of the average logMAR VA in studies that use decimal VA. Such bias will be greater in magnitude when disease strongly affects VA.

This paper provides formulae that can be easily implemented in standard spreadsheet software programs, and which allow appropriate transformations of group-level estimates of mean and standard deviation of VA across different scales of measurement. These transformations are helpful for performing meta-analyses or for comparisons of results across studies when VA is expressed in different units.