Dwarfs on the Shoulders of Giants: Bayesian Analysis With Informative Priors in Elite Sports Research and Decision Making

Abstract

While sample sizes in elite sports are necessarily small, so are the effects that may be relevant. This conundrum is complicated by an understandable reluctance of athletes to comply with extensive study requirements. In Bayesian analyses, pre-existing knowledge (e.g., from sub-elite trials) can be formally included to supplement scarce data. Moreover, some design specifics for small sample research extend to the extreme case of a single subject. This provides the basis for actionable feedback (e.g., about individual responses) thereby incentivising participation. As a proof-of-concept, we conducted a replicated cross-over trial on the effect of cold-water immersion (CWI) on sprint performance recovery in soccer players. Times for 30 m linear sprint and the initial 5 m section, respectively, were measured by light gates before and 24 h after induction of fatigue. Data were analysed by Bayesian and by standard frequentist methods. Informative priors are based on a published metaanalysis. Seven players completed the trial. Sprint performance was 4.156 ± 0.193 s for 30 m linear sprint and 0.978 ± 0.064 s for the initial 5 m section. CWI improved recovery of sprint time for the initial 5 m section (difference to control: -0.060 ± 0.060 s, p = 0.004) but not for the full 30 m sprint (0.002 ± 0.115 s, p = 0.959), with general agreement between Bayesian and frequentist interval estimates. On the individual level, relevant differences between analytical approaches were present for most players. Changes in the two performance measures are correlated (p = 0.009) with a fairly good reproducibility of individual response patterns. Bayesian analyses with informative priors may be a practicable and meaningful option particularly for very small samples and when the analytical aim is decision making (use / don't use in the specific setting) rather than generalizable inference.

Keywords: Bayesian statistics; cold-water immersion; individual response; methodology; replicate crossover.

Figure 1

Boxplot of pre-post changes in sprint time for 30 m sprint and the initial 5 m section depending on the recovery condition p-values are derived from the linear mixed model described in the methods section.

Figure 2

Difference between conditions for 30 m sprint and initial 5 m section. Dashed lines highlight zero difference between conditions.

Figure 3

Efficacy of cold-water immersion. 95% Bayesian credible intervals (highest posterior density intervals) and 95% frequentist confidence intervals, respectively. Dashed vertical line: prior mean, Dotted vertical line: data mean, Solid vertical line: Zero Bayesian credible intervals and frequentist confidence intervals calculated with the same data are displayed in one figure for comparison. Please keep in mind their fundamental disparity.

Figure 4

Efficacy of cold-water immersion on the individual level. 95% Bayesian credible intervals (highest posterior density intervals, Approach = Bayes) and 95% frequentist confidence intervals (Approach = Frequ), respectively. Intervals calculated from individual means and the (group-based) standard error for the fixed effect (Approach = FrequMM) are displayed for comparison. Points are observed differences in pre-post changes between conditions. HPDI have been calculated with informative priors. Subjects are ordered by posterior β_CWI for 5 m acceleration. Dashed vertical line: prior mean, Dotted vertical line: data mean, Solid vertical line: Zero Bayesian credible intervals and frequentist confidence intervals calculated with the same data are displayed in one figure for comparison. Please keep in mind their fundamental disparity.