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Mathematical coupling causes spurious correlation within the conventional acute-to-chronic workload ratio calculations
  1. Lorenzo Lolli1,
  2. Alan M Batterham1,
  3. Richard Hawkins2,
  4. David M Kelly2,3,
  5. Anthony J Strudwick2,
  6. Robin Thorpe2,3,
  7. Warren Gregson3,
  8. Greg Atkinson1
  1. 1 Health and Social Care Institute, Teesside University, Middlesbrough, UK
  2. 2 Medicine and Science Department, Manchester United Football Club, Manchester, UK
  3. 3 Football Exchange, Research Institute for Sport and Exercise Sciences, Liverpool John Moores University, Liverpool, UK
  1. Correspondence to Lorenzo Lolli, School of Health and Social Care, Health and Social Care Institute, Teesside University, Middlesbrough TS1 3BA, UK; L.Lolli{at}

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The monitoring of training loads is now a much-researched topic in team sports.1 Within this topic, researchers and practitioners are particularly interested in the impact of relatively short (acute) periods of higher training loads normalised for the prior and longer term (chronic) loads. In recent years, a well-established approach for normalising this acute ‘spike’ to chronic load has been by calculating the ‘acute:chronic workload ratio’ (ACWR). Importantly, the term "load" was retained given its common use in this research area. Both this index and chronic load itself have been reported to be independent predictors of training-related injuries.2 It has also been reported, particularly in team sports competitors, that there are associations between acute spikes in training loads (relative to chronic loads) and time-loss injuries.1

The ACWR is usually calculated as the simple ratio of recent (ie, 1 week) to longer term (ie, 4 weeks) training loads.1 While it is important for the numerator and denominator of any ratio to be correlated only through biological mechanisms,3 one aspect of the ACWR calculation is that the acute load also constitutes a substantial part of the chronic load.4 This ‘mathematical coupling’ between two variables,5 also referred to as ‘relating a part to the whole’,6 is unusual and raises the possibility that research inferences and athlete monitoring might be compromised by resulting spurious correlations.3 A spurious correlation is one that exists between two variables irrespective of any true biological/physiological association between those variables.3 5

Mathematical coupling in the ACWR calculation

Irrespective of different data smoothing approaches over a 28-day period,7 the conventional calculation of the ACWR is ultimately:

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where A is the 7-day acute load, and hypothetical W1, W2 and W3 are the preceding 7-day loads, respectively.1 4 Given the conceptual definition of acute and chronic load variables,4 we hypothesised that ‘mathematical coupling’ might exist, leading to a spurious correlation between acute and chronic training load estimates.3

To test our hypothesis with adequate statistical precision, we generated data to simulate four 7-day periods of high-speed distance data reported in a recent study involving elite Australian footballers2 for a hypothetical squad of 1000 players (online supplementary file 1). Each of the four sets of data was randomly generated and was completely independent from the other datasets. The most recent 7-day period was designated as the acute period (A), while the 28-day period defining chronic load was calculated as a conventional rolling average. The mean±SD high-speed distance for W1, W2 and W3 and A were 2021±889 m, 1977±880 m, 1968±860 m and 2035±901 m, respectively. None of the preceding 7-day datasets were found to be substantially correlated with A (r<0.06). However, as demonstrated in figure 1, there was a moderate-to-large, positive correlation between the calculated chronic and acute load data; r=0.52 (95% CI 0.47 to 0.56). If A was not included in the calculation of C, then the correlation between A and C was, as expected, close to zero; r=0.01 (95% CI −0.05 to 0.07).

Supplementary file 1

Figure 1

The spurious correlation between the simulated acute phase data and the chronic phase data. Although the 4 weeks of data were uncorrelated with each other, this correlation is explained by the fact that the acute phase data are part of the calculation of the chronic phase data leading to mathematical coupling. This spurious correlation will be present irrespective of any true physiological association between acute and chronic load variables, leading to biased inferences.

The moderate-to-large but spurious (false) correlation between the acute and chronic load variables substantiated the presence of mathematical coupling, since the acute load represents a term in the calculation of the denominator in the ACWR.3 Any functions that are designed to quantify the association between acute and chronic load variables must be mathematically distinct from each other and not naturally associated if any true physiological explanations or likelihood of injury are attempted to be researched.3 Accordingly, the mathematical coupling issues we observed could also affect the chronic load variance and, crucially, its physiological range of measurements.3 In our simulated data, the SD for chronic high-speed distance (with the acute data period included) was ±439 m (data range: 654–3469 m). Nevertheless, following removal of the acute period data from the calculation of the chronic period distance, the SD was a higher ±499 m (data range: 541–3553 m). Furthermore, the formulation of rolling averages might also influence the observed SD.8 Therefore, and as expected, inclusion of the acute data in the calculation artefactually reduced the between-athlete variability in chronic load.

The mathematical coupling issue can also alter the ACWR itself. For example, with an acute high-speed distance of 2375 m, the chronic high-speed distance can be calculated conventionally to be 1639 m yielding an ACWR of 1.45. Rather, the chronic load value and the ACWR without mathematical coupling should really be 1393 m and 1.71, respectively. Therefore, the traditional mathematical definition of the chronic load term in the ACWR protocol also appears to limit a valid and unbiased interpretation of the observed ACWR estimates.


Collectively, our findings have demonstrated that the numerator and denominator in the ACWR are mathematically coupled and, therefore, spuriously correlated. The simplest solution is not to include acute load periods in the calculation of chronic load if the training load–injury aetiological relationship, grounded on the magnitude of the ACWR, is to be examined accurately.



  • Contributors LL and GA developed the article concept. All authors contributed to write, provide feedback and revise the manuscript.

  • Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

  • Competing interests None declared.

  • Provenance and peer review Not commissioned; externally peer reviewed.