Modelling bivariate relationships when repeated measurements are recorded on more than one subject.

This paper examines the problems of modelling bivariate relationships when repeated observations are recorded for each subject. The statistical methods required to test for a common group model were introduced using an example from exercise physiology, where the oxygen cost of running at four different speeds was recorded for a group of 30 recreationally active males. When data for each subject were studied individually, both the plots and correlations suggested the relationship to be linear. Hence, the homogeneity of the subjects' regression lines was compared using the appropriate ANOVA test. The analysis revealed a significant difference in the slopes and intercepts of the lines, thus precluding the use of a single linear model to represent the group. If the subjects were divided into two groups according to the median maximum oxygen consumption (VO2max), a multivariate analysis of variance of the slope and intercept parameters helped to explain some of this heterogeneity (P less than 0.05). However, for physiological rather than statistical reasons, it was necessary to re-analyse the data without the fourth running speed. The revised analysis suggested that the subjects' lines would be better modelled with a common slope but separate intercepts. As before, by dividing the subjects into two groups according to the median VO2max score, a simple t-test indicated that differences in the subjects' intercept parameters were not significant (P = 0.08). Notwithstanding the relatively homogeneous nature of the 30 subjects in terms of VO2max, the statistical methods showed that differences in running economy are, to some extent, dependent on VO2max.