MATHEMATICAL CONSTANTS THAT VARY?

Abstract

Dear Editor-in-Chief: Hinckson and Hopkins (1) used mathematical modeling to predict test–retest error of time trials (TT) from variability in time-to-exhaustion tests (TTE). Sample mean estimates of model parameters were considered mathematical constants and used to convert TTE to TT variability; for example, in the critical power model, the value of the term “a/m” was set at 0.506. This sample mean value was then used as a constant factor for converting the sample mean coefficient of variation (CV) in TTE to that in TT. Hinckson and Hopkins' data suggest that the conversion factors employed are not constants at all, but are themselves prone to both between- and within-subject variability. The between- and within-subject CV for the “a/m” conversion factor were 28 and 16%, respectively. It seems inappropriate to use a constant value to convert changes in TTE to changes in TT, when the relationship between these two variables differs substantially between individuals and over time. An analogous situation would be one in which a researcher calculates the work rate corresponding to a certain %V̇O2max for all individuals in all situations using constant values of slope and intercept for the V̇O2–work rate relationship. We request that TT is predicted from TTE using estimates of model parameters specific to a particular subject at a specific time. Once these predicted TT times have been calculated for all subjects at the six test times using specific conversion factors, a CV can be calculated in the normal way. We predict that this CV for predicted TT is larger than those reported in (1), because the between- and within-subject variability in the conversion factors themselves was originally “averaged out.” Aside from the mathematical modeling issues, the exact contribution of variability in pacing strategy to variability in overall TT time is not known at present. Pacing strategy, being an inherent component of real performance, should not be used to deem that something is wrong with TT protocols. Moreover, TTE tests have their own unique components of error. For example, in a relatively long TTE test performed at a lower %V̇O2max, the athlete may be more likely to stop the test, not because of exhaustion, but because of boredom or the need to fulfil other obligations. Physiological variables are conventionally measured at regular periods during longer TTE tests. These measurements would merely serve to provide the athlete with feedback about time completed. Such sources of error in TTE tests should be investigated before it is claimed that any concerns about time-to-exhaustion tests have been laid to rest. Greg Atkinson Research Institute for Sport and Exercise Sciences Liverpool John Moores University Liverpool, United Kingdom Alan Nevill School of Sport, Performing Arts and Leisure University of Wolverhampton Walsall, United Kingdom