Analysis of relative change and initial value in biological studies

In analyzing longitudinal data, the relationship between initial value (x1) and change in response to a maneuver or over time is often of interest. This relationship is often determined by the Pearson correlation between x1 and subsequent change (x2 - x1), but a correlation will often be obtained by mathematical necessity even when x1 and x2 are uncorrelated. It has been suggested that use of relative change, or (x2 - x1)/x1, avoids these mathematical artifacts. By reference to early work by Pearson (Proc. R. Soc. London 60: 489-498, 1897) on an approximation to the correlation between ratios and by computer simulations, this paper shows that this solution is frequently not valid. An alternative approach to the analysis of the relationship between x1 and relative change is presented and is illustrated by a study of renal blood flow.