The importance of addressing heteroscedasticity in the reliability analysis of ratio‐scaled variables: an example based on walking energy‐cost measurements

Abstract

When analysing the reliability of ratio-scaled variables, such as walking energy cost, variability of the error term often increases with increasing mean values. This phenomenon is called heteroscedasticity, and it makes the analysis of reliability more complicated. This study presents an examination of heteroscedasticity for walking energy cost before analysing the reliability. Walking energy cost was collected from 33 children with cerebral palsy (CP), with varying Gross Motor Function Classification System (GMFCS) levels (19 males; 14 females; mean age: 7y 6mo [SD 2y 6mo]; GMFCS levels I [n=16], II [n=7], and III [n=10]). It was assessed by measuring oxygen uptake during 10 minutes of resting and 5 minutes of walking at comfortable speed. Measurements were performed twice, within 4 to 6 weeks. Primary outcomes included gross energy cost, gross non-dimensional energy cost, net energy cost, net non-dimensional energy cost, speed, and non-dimensional speed. Heteroscedasticity was analysed with Bland-Altman plots and Kendall's tau. Visual inspection of the Bland-Altman plots showed heteroscedasticity for gross energy cost, gross non-energy cost, and net energy cost. This was confirmed by Kendall's tau coefficients. Accordingly, data were logarithmically transformed, and reliability was assessed with ratio statistics. For speed, heteroscedasticity was not observed. Variability of gross energy cost, gross non-energy cost, and net energy cost, assessed across different GMFCS levels in children with CP, was proportional to the mean, indicating the presence of heteroscedasticity. This finding emphasizes the importance of always performing a heteroscedasticity examination in reliability studies on energy cost and reporting the reliability statistics accordingly.