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Is it all for naught? What does mathematical coupling mean for acute:chronic workload ratios?
1. Johann Windt1,2,3,
2. Tim J Gabbett4,5
1. 1 Experimental Medicine Program, University of British Columbia, Vancouver, British Columbia, Canada
2. 2 Sports Medicine Division, United States Olympic Committee, Colorado Springs, Colorado, USA
3. 3 US Coalition for the Prevention of Illness and Injury in Sport, Colorado Springs, Colorado, USA
4. 4 Gabbett Performance Solutions, Brisbane, Queensland, Australia
5. 5 Institute for Resilient Regions, University of Southern Queensland, Ipswich, Queensland, Australia
1. Correspondence to Johann Windt, Experimental Medicine Program, University of British Columbia, Vancouver BC V6T 1Z4, Canada; johannwindt{at}gmail.com

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Traditional calculations of the acute:chronic workload ratio (ACWR) are ‘mathematically coupled’, as the most recent week is included in estimates of both the acute and chronic workloads. As Lolli and colleagues rightly point out, this induces a spurious correlation between the acute and chronic loads of ~0.50 (r=0.52 in their simulated data of 1000 athletes).1 They suggest that the simplest solution is to use uncoupled ACWRs (where the acute load is not part of the chronic load) instead (figure 1).

Figure 1

The distinction between traditional ‘coupled’ and ‘uncoupled’ estimates of the acute:chronic workload ratio. The inclusion or exclusion of the most recent week in ‘chronic load’ calculations is the key distinction.

Notably, at least two studies have already used uncoupled ACWR calculations, both demonstrating that rapid workload increases are associated with higher injury risks.2 3 To this end, Lolli and colleagues’ suggestion warrants consideration—should we use ‘uncoupled’ ACWRs instead of ‘coupled’? We have two aims in this editorial: (1) to further comment on how mathematical coupling affects ACWR estimates and (2) to encourage researchers and practitioners to use a critical approach to load management, wherein ACWRs may play a part.

## Comments on mathematical coupling and ACWRs

### Defining coupled and uncoupled ACWRs

We define mathematical coupling in the same manner as Lolli et al, where a number is represented in both the numerator and denominator of a ratio, contributing to a spurious correlation. In the case of the ACWR, both coupled and uncoupled calculations convey whether recent workloads are increasing or decreasing compared with prior workloads (ACWR>1 increasing, ACWR<1 decreasing). However, including the most recent week in the chronic loading window changes the interpretation/definition of the ACWR.

 Coupled Uncoupled Formula Definition The ratio between the most recent week of work with the average of the most recent 4 weeks. The ratio of the most recent week of work with the average of the three preceding weeks.
• *Where ‘A’ is the acute workload and the last 4 weeks are represented by A, W2, W3 and W4, respectively.

Another way to think of the traditional, coupled ACWR is ‘what proportion of the total 4-week load is provided by the acute load?’ At constant loading conditions, the acute load makes up one-fourth of the total 4-week load. As workloads spike, the acute workload constitutes a greater proportion (table 1). The ACWR equals how many times greater this proportion is than 0.25. This number can approach, but never reach four, since the acute load can constitute virtually all of the chronic load, but never all of it.

Table 1

Coupled and uncoupled (rolling averages over 4 weeks) acute:chronic workload ratio (ACWR) calculations with various hypothetical 4-week workload distributions

### The coupled correlation is expected

The ‘spurious’ correlation of approximately r=0.5 between acute and chronic loads is exactly what we would expect based on the ACWR calculation. Squaring this correlation (r= 0.52 = 0.25) demonstrates that, on average, ~25% of the total variation in our outcome (chronic loads) is explained by its linear relationship with the acute load. This aligns with the fact that the acute load makes up one-fourth of the chronic load calculation. Extending this logic, if a coupled ACWR was calculated over a 5-week window, the correlation would approximate r=0.45 (r2=0.20).

### One can convert between coupled and uncoupled ACWRs

Since both ratios discussed by Lolli et al are calculated from the same four training weeks, one can convert between the two via the following formula (full proof available on request).

### Effects of mathematical coupling on ACWRs

Lolli et al noted that a ‘coupled’ ACWR of 1.45 equals an ‘uncoupled’ ACWR of 1.71. Thus, mathematical coupling ‘suppresses’ traditional ACWRs compared with uncoupled ACWRs. The degree to which the ratio is suppressed increases as the ACWR increases. The ratios are equal at 1.0, vary slightly at first (1.5 vs 1.8) and progressively diverge (2.0 vs 3.0, 3.0 vs 9.0). The coupled ACWR never exceeds four, regardless of how high the uncoupled ACWR gets (figure 2) so a coupled ACWR should not be considered a continuous measure. The way that these scaling properties affect research estimates and the appropriateness of including them in different analyses (eg, regression modelling) deserves further consideration.

Figure 2

The association between coupled and uncoupled estimates of the acute:chronic workload ratio with increasing workload progression. While the coupled ACWR asymptotes at four, uncoupled estimates continue to increase in a linear fashion.

## What now? A call for critical, thoughtful monitoring

Mathematical coupling may support uncoupled ACWRs in the future. However, rather than suggesting one particular approach, we propose that researchers and practitioners should focus on a thoughtful, critical approach to load monitoring, which may or may not include ACWRs. We highlight three practical applications below.

### Do not use any ACWR in isolation

Athlete monitoring is a multifaceted process with information gleaned from external and internal loads, athlete wellness, readiness and other information.4 One factor which may be of interest to practitioners and researchers is the rate of workload change. An ACWR (coupled or uncoupled, rolling average or exponentially weighted) provides a single measure that indicates how recent loads are changing relative to previous loads. Whatever method (coupled or uncoupled) is chosen, it must not be interpreted in isolation. Instead, it must be interpreted in conjunction with other information, for example, training modalities, absolute workload changes, how the athlete is tolerating training, the athlete’s chronic load and the athlete’s internal risk profile (eg, age, aerobic fitness level, strength, etc).5 6

### Understand the implications of how any given ACWR was calculated

Different acute and chronic time windows,7 exponentially weighted moving averages8 and week-to-week changes9 have been proposed in the literature to monitor training progression. Each of these calculations will result in different numbers given the same workload sequence. Notable ‘thresholds’ will also differ depending on how an ACWR is calculated. The proverbial ‘sweet spot’ of 0.8–1.3 for the ACWR has been largely based on coupled, rolling average estimates. If uncoupled (1 week:4 week) ACWR estimates were used, these same thresholds would correspond to different ratios (0.75–1.45). It is up to the user to critically select which variation, if any, they will use and how they will interpret the values.

### Explicitly detail how the ACWRs were calculated

Finally, if researchers choose to use ACWRs, they must explicitly detail how they were calculated, facilitating interpretation and replication. These details should include whether calculations were rolling averages or exponentially weighted,8 the length of the acute:chronic windows, and whether acute and chronic workloads were coupled or not.

Ultimately, developing robust, high-performing athletes depends on building high chronic workloads, but increasing these workloads too quickly will likely increase athletes’ risks of injury.2 10 In the context of a thoughtful, critical athlete monitoring approach, ACWRs may help by estimating workload progression; however they are quantified.

## Acknowledgments

The authors would like to thank Carson Grose for his assistance with the mathematical proof and Professor Bruno Zumbo for his helpful comments on the manuscript.

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## Footnotes

• Contributors Both authors were responsible for the concept, writing and critical revision of the manuscript.

• Funding At the time of initially writing this manuscript, JW was a Vanier Scholar funded by the Canadian Institutes of Health Research.

• Competing interests TJG works as a consultant to several high-performance organisations, including sporting teams, industry, military and higher education institutions. He serves in a voluntary capacity as Senior Associate Editor of BJSM.

• Patient consent Not required.

• Ethics approval No ethics approval was sought for this editorial.

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

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