Modeling between-subject differences and within-subject changes for long distance runners by age

A time-varying predictor in a longitudinal model can be separated into between-and within-subject components. This is important if a researcher is specifically interested in differences between subjects or changes within subjects. If no separation is made, then the effect associated with the time-varying predictor is a combination of the between-and within-subject effects. In this article, running race performances are considered as a nonlinear function of age, for which both absolute and relative difference terms are used to separate between- and within-subject components. Longitudinal data were obtained from 12 consecutive years of the 10-km Bolder Boulder for analysis, for the class of competitive recreational runners. Mixed models were used to fit the data after log transforming the nonlinear model. Results of model fits demonstrated that between-subject differences between ages were somewhat similar to within-subject changes, although the latter tended to change somewhat more slowly, especially after the peak racing ages. For example, the between-subject rate of change for 43 year-old runners was about 1% per year on average (increase in race time), while the within-subject change for a runner of the same age averaged between 0.6 and 0.8% per year.