AVALOS, M., P. HELLARD, and J.-C. CHATARD. Modeling the Training-Performance Relationship Using a Mixed Model in Elite Swimmers. Med. Sci. Sports Exerc., Vol. 35, No. 5, pp. 838-846, 2003.
Purpose: The aim of this study was to model the relationship between training and performance in 13 competitive swimmers, over three seasons, and to identify individual and group responses to training.
Methods: A linear mixed model was used as an alternative to the Banister model. Training effect on performance was studied over three training periods: short-term, the average of training load accomplished during the 2 wk preceding each performance of the studied period; mid-term, the average of training load accomplished during weeks 3, 4, and 5 before each performance; and long-term, weeks 6, 7, and 8.
Results: Cluster analysis identified four groups of subjects according to their reactions to training. The first group corresponded to the subjects who responded well to the long-term training period, the second group to the long- and mid-term periods, the third to the short- and mid-term periods, and the fourth to the combined periods. In the model, the intersubject differences and the evolution over the three seasons were statistically significant for the identified groups of swimmers. Influence of short-term training was negative on performance in the four groups, whereas mid- and long-term training had, on the average, a positive effect in three groups out of four. Between seasons 1 and 3, the effect of mid-term training declined, whereas the effect of long-term training increased. The fit between real and modeled performances was significant for all swimmers (0.15 ≤ r2 ≤ 0.65;P ≤ 0.01).
Conclusion: The mixed model described a significant relationship between training and performance both for individuals and for groups of swimmers. This relationship was different over the 3 yr. Personalized training schedules could be prescribed on the basis of the model results.
The training-performance relationship is particularly important for elite sports coaches who search for reproducible phenomena useful for organizing the athlete's training program. Many authors have studied the relative influence of training (7,22,23,27) and found that reactions to training depend on volume, intensity, and frequency of the training sessions (7,16,23). Others have reported divergent results (4,9), perhaps related to the fact that delayed effects and interindividual differences were not taken into account.
For individual swimmers, mathematical models have been developed to describe the dynamic aspect of training and the consequences of succession of training loads over time (2). The Banister model (2-4) and its modifications (5,6) are based on two antagonistic functions, both calculated from the training impulse. Studies on cellular adaptability reactions to exercise (3) have demonstrated that the negative function can be assimilated to a fatiguing impulse. The positive function can be compared with a fitness impulse resulting from the organism's adaptation to training. Expressed as an exponential, the functions account for the decreasing impact of the training effect. When iterative training sessions are considered, the time course of performance is described by:MATH
where pt is the known performance at week (or day) t; ws is the known training load per week (or day) from the first week of training to the week (or day) preceding performance pt; ka and kf are the fitness and fatigue multiplying factors, respectively; τa and τf are the fitness and fatigue decay time constants, respectively; and p0 corresponds to an initial basic level of performance.
There is no clear consensus on just how many data points are needed per parameter to ensure a stable solution in a regression analysis. Proposals reported in the literature have ranged from 5 to 50. Stevens (26) recommends a nominal number of 15 observations per parameter (except the intercept parameter) for a multiple linear regression. But as the Banister model is a nonlinear model, inference is based on asymptotic theory (10), which implies more data points per parameter than for a linear regression model. This means a large number of observations would be required to obtain precise results and enable pertinent statistical analysis. The Banister model also assumes the parameters remain constant over time, an assumption that is not consistent with observed time-dependent alterations in response to training (3,4,6,22).
When few repeated measurements are available for several subjects, mixed models provide an attractive solution (29). Instead of constructing a personal model for each subject, a model of popular behavior is constructed, allowing parameters to vary from one individual to another, to take into account the heterogeneity between subjects. Particular care in characterizing random variation in the data is required to recognize two levels of variability: random variation among measurements within a given individual (intraindividual variation) and random variation among individuals (interindividual variation) (10). In addition, mixed models analyze responses corresponding to different dose inputs (10), a common situation in swimming as training loads differ with age, specialty, and/or competition level (7,23).
The aim of the present study was to investigate the effect of training on performance of 13 elite swimmers taking into account a) individual profiles, b) subpopulation profiles, and c) time effect over three seasons.