Abstract

The aim of this paper is to provide a Bayesian formulation of the so-called magnitude-based inference approach to quantifying and interpreting effects, and in a case study example provide accurate probabilistic statements that correspond to the intended magnitude-based inferences. The model is described in the context of a published small-scale athlete study which employed a magnitude-based inference approach to compare the effect of two altitude training regimens (live high-train low (LHTL), and intermittent hypoxic exposure (IHE)) on running performance and blood measurements of elite triathletes. The posterior distributions, and corresponding point and interval estimates, for the parameters and associated effects and comparisons of interest, were estimated using Markov chain Monte Carlo simulations. The Bayesian analysis was shown to provide more direct probabilistic comparisons of treatments and able to identify small effects of interest. The approach avoided asymptotic assumptions and overcame issues such as multiple testing. Bayesian analysis of unscaled effects showed a probability of 0.96 that LHTL yields a substantially greater increase in hemoglobin mass than IHE, a 0.93 probability of a substantially greater improvement in running economy and a greater than 0.96 probability that both IHE and LHTL yield a substantially greater improvement in maximum blood lactate concentration compared to a Placebo. The conclusions are consistent with those obtained using a ‘magnitude-based inference’ approach that has been promoted in the field. The paper demonstrates that a fully Bayesian analysis is a simple and effective way of analysing small effects, providing a rich set of results that are straightforward to interpret in terms of probabilistic statements.

Highlights

  • A key interest in sports science is the estimation and evaluation of small effects, such as the difference in finishing times between world-class athletes, or the impact of exercise training and/ or lifestyle interventions such as dietary changes or sleep behaviors on performance [1]. While such an interest is not confined to this context [2], there are some features of sports science that make accurate and relevant estimation of small effects challenging

  • The aim of this paper is to provide a Bayesian formulation of the method proposed by Batterham and Hopkins (2006) and provide a range of probabilistic statements that parallel their intended magnitude-based inferences

  • We hope that this template will encourage those who deal with small samples and small effects to explore the full Bayesian method, which is well suited to the analysis of small samples

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Summary

Introduction

A key interest in sports science is the estimation and evaluation of small effects, such as the difference in finishing times between world-class athletes, or the impact of exercise training and/ or lifestyle interventions such as dietary changes or sleep behaviors on performance [1]. While such an interest is not confined to this context [2], there are some features of sports science that make accurate and relevant estimation of small effects challenging. The authors translate these proportions to a set of likelihood statements about the magnitude of the true effect

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