Sympercents: symmetric percentage differences on the 100 loge scale simplify the presentation of log transformed data

Abstract

The results of analyses on log transformed data are usually back-transformed and interpreted on the original scale. Yet if natural logs are used this is not necessary – the log scale can be interpreted as it stands. A difference of natural logs corresponds to a fractional difference on the original scale. The agreement is exact if the fractional difference is based on the logarithmic mean. The transform y=100 log_ex leads to differences, standard deviations and regression coefficients of y that are equivalent to symmetric percentage differences, standard deviations and regression coefficients of x. Several simple clinical examples show that the 100 log_e scale is the natural scale on which to express percentage differences. The term sympercent or s% is proposed for them. Sympercents should improve the presentation of log transformed data and lead to a wider understanding of the natural log transformation. Copyright © 2000 John Wiley & Sons, Ltd.