Statistical methods for analysing discrete and categorical data recorded in performance analysis

In this paper, we identify appropriate statistical methods for analysing categorical differences in discrete variables or 'performance indicators' resulting from performance analysis. The random mechanisms associated with discrete events do not follow a normal distribution; that is, the normal distribution is a continuous not a discrete probability distribution. We propose appropriate statistical methods based on two key discrete probability distributions, the Poisson and binomial distributions. Two approaches are proposed and compared using examples from notational analysis. The first approach is based on the classic chi-square test of significance (both the goodness-of-fit test and the test of independence). The second approach adopts a more contemporary method based on log-linear and logit models fitted using the statistical software GLIM. Provided relatively simple one-way and two-way comparisons in categorical data are required, both of these approaches result in very similar conclusions. However, as soon as more complex models or higher-order comparisons are required, the approach based on log-linear and logit models is shown to be more effective. Indeed, when investigating those factors and categorical differences associated with binomial or binary response variables, such as the proportion of winners when attempting decisive shots in squash or the proportion of goals scored from all shots in association football, logit models become the only realistic method available. By applying log-linear and logit models to discrete events resulting from notational analysis, greater insight into the underlying mechanisms associated with sport performance can be achieved.     

Science and cycling: current knowledge and future directions for research

In this holistic review of cycling science, the objectives are: (1) to identify the various human and environmental factors that influence cycling power output and velocity; (2) to discuss, with the aid of a schematic model, the often complex interrelationships between these factors; and (3) to suggest future directions for research to help clarify how cycling performance can be optimized, given different race disciplines, environments and riders. Most successful cyclists, irrespective of the race discipline, have a high maximal aerobic power output measured from an incremental test, and an ability to work at relatively high power outputs for long periods. The relationship between these characteristics and inherent physiological factors such as muscle capilliarization and muscle fibre type is complicated by inter-individual differences in selecting cadence for different race conditions. More research is needed on high-class professional riders, since they probably represent the pinnacle of natural selection for, and physiological adaptation to, endurance exercise. Recent advances in mathematical modelling and bicycle-mounted strain gauges, which can measure power directly in races, are starting to help unravel the interrelationships between the various resistive forces on the bicycle (e.g. air and rolling resistance, gravity). Interventions on rider position to optimize aerodynamics should also consider the impact on power output of the rider. All-terrain bicycle (ATB) racing is a neglected discipline in terms of the characterization of power outputs in race conditions and the modelling of the effects of the different design of bicycle frame and components on the magnitude of resistive forces. A direct application of mathematical models of cycling velocity has been in identifying optimal pacing strategies for different race conditions. Such data should, nevertheless, be considered alongside physiological optimization of power output in a race. An even distribution of power output is both physiologically and biophysically optimal for longer ( > 4 km) time-trials held in conditions of unvarying wind and gradient. For shorter races (e.g. a 1 km time-trial), an 'all out' effort from the start is advised to 'save' time during the initial phase that contributes most to total race time and to optimize the contribution of kinetic energy to race velocity. From a biophysical standpoint, the optimum pacing strategy for road time-trials may involve increasing power in headwinds and uphill sections and decreasing power in tailwinds and when travelling downhill. More research, using models and direct power measurement, is needed to elucidate fully how much such a pacing strategy might save time in a real race and how much a variable power output can be tolerated by a rider. The cyclist's diet is a multifactorial issue in itself and many researchers have tried to examine aspects of cycling nutrition (e.g. timing, amount, composition) in isolation. Only recently have researchers attempted to analyse interrelationships between dietary factors (e.g. the link between pre-race and in-race dietary effects on performance). The thermal environment is a mediating factor in choice of diet, since there may be competing interests of replacing lost fluid and depleted glycogen during and after a race. Given the prevalence of stage racing in professional cycling, more research into the influence of nutrition on repeated bouts of exercise performance and training is required.     

Describing the plastic deformation of aluminium softball bats

Hollow aluminium bats were introduced over 30 years ago to provide improved durability over wooden bats. Since their introduction, however, interest in hollow bats has focused almost exclusively around their hitting performance. The aim of this study was to take advantage of the progress that has been made in predicting bat performance using finite elements and apply it to describe bat durability. Accordingly, the plastic deformation from a ball impact of a single-wall aluminum bat was numerically modelled. The bat deformation from the finite-element analysis was then compared with experiment using a high-speed bat test machine. The ball was modelled as an isotropic, homogeneous, viscoelastic sphere. The viscoelastic parameters of the ball model were found from instrumented, high-speed, rigid-wall ball impacts. The rigid-wall ball impacts were modelled numerically and showed good agreement with the experimentally obtained response. The strain response of the combined bat-ball model was verified with a strain-gauged bat at intermediate ball impact speeds in the elastic range. The strain response of the bat-ball model exhibited positive correlation with the experimental measurements. High-speed bat-ball impacts were performed experimentally and simulated numerically at increasing impact speeds which induced correspondingly increased dent sizes in the bat. The plastic deformation from the numerical model found good agreement with experiment provided the aluminium work hardening and strain rate effects were appropriately described. The inclusion of strain rate effects was shown to have a significant effect on the bat deformations produced in the finite-element simulations. They also helped explain the existence of high bat stresses found in many performance models.