To determine whether the relationship between training load and injury risk is non-linear and investigate ways of handling non-linearity.

We analysed daily training load and injury data from three cohorts: Norwegian elite U-19 football (n=81, 55% male, mean age 17 years (SD 1)), Norwegian Premier League football (n=36, 100% male, mean age 26 years (SD 4)) and elite youth handball (n=205, 36% male, mean age 17 years (SD 1)). The relationship between session rating of perceived exertion (sRPE) and probability of injury was estimated with restricted cubic splines in mixed-effects logistic regression models. Simulations were carried out to compare the ability of seven methods to model non-linear relationships, using visualisations, root-mean-squared error and coverage of prediction intervals as performance metrics.

No relationships were identified in the football cohorts; however, a J-shaped relationship was found between sRPE and the probability of injury on the same day for elite youth handball players (p<0.001). In the simulations, the only methods capable of non-linear modelling relationships were the quadratic model, fractional polynomials and restricted cubic splines.

The relationship between training load and injury risk should be assumed to be non-linear. Future research should apply appropriate methods to account for non-linearity, such as fractional polynomials or restricted cubic splines. We propose a guide for which method(s) to use in a range of different situations.

Hypotheses suggest that the relationship between training load and injury risk is non-linear.

Methods used in previous training load and injury research often assume linearity.

Categorisation has been proven a suboptimal alternative for handling non-linearity.

A non-linear relationship (p<0.001) between session rating of perceived exertion and the probability of injury in elite youth handball players would not have been discovered if linearity had been assumed (p=0.24).

Acceptable Brier scores and C-statistics from a linear model do not mean that the relationship is linear.

Categorising training load by quartiles could not model a linear relationship under skewed data conditions.

Fractional polynomials and restricted cubic splines were the only methods capable of exploring non-linear shapes.

Clinical researchers will have the tools available to perform causal and predictive research on training load and injury risk more accurately.

More consistent methodology between training load and injury risk studies will improve comparability, reproducibility and facilitate meta-analyses.

Injuries can hamper athlete and team performance in a variety of sporting disciplines.

In 2013, Gamble theorised a U-shaped relationship between training load and injury risk. Too little and too much load increases risk,

Despite these hypotheses and calls, methods that assume a linear relationship between training load and injury risk, such as Pearson correlations and logistic regression, are commonly used in the field.

The ideal method to handle non-linearity should be able to: (1) explore non-linear shapes and thus may confirm or reject previously outlined hypotheses; (2) model the non-linear relationship accurately; and (3) offer interpretable results.

The overall aim of this study was to identify the best methods for handling non-linearity in training load and injury research. First, we ascertained the relationship in three sports populations to reveal any potential evidence of non-linearity, to illustrate the problems and to present solutions. Second, we compared different methods in their ability to explore and accurately model potential non-linear shapes. Finally, we used the comparisons to develop a guide for which method(s) to use in different situations.

We obtained training load and injury data collected from three cohorts: Norwegian elite U-19 football players (n=81, 55% male, mean age: 17 years, SD: 1 year),

All participants provided informed consent. Ethical principles were followed in accordance with the Declaration of Helsinki.

In all three cohorts, players reported the number of training sessions and matches daily. They also reported the duration of each activity and their rating of perceived exertion (RPE)

Missing sRPE values are reported in

All load measures were based on players’ daily ratings of perceived exertion (sRPE). We calculated an acute:chronic workload ratio (ACWR) in two different ways:

The mean sRPE across 7 days divided by the exponentially weighted moving average (EWMA) of the previous 21 days, uncoupled (

Illustration of time periods for calculating (A) daily ACWR 7:21-period and (B) micro-cycle ACWR 1:3-period. The first day that ACWR is calculated from is denoted day 0. The space between two tick marks represent 1 day (24 hours). For B, a microcycle period consists of all activity before a new match (M). That is, recovery days after the previous match as well as the training days before the next match. Days denoted with negative numbers are training days before the next match (M-1: being the day before the match; M-2: 2 days before a match and so on). Days with positive numbers are recovery and training days after a match (M+1: being the day after a match, M+2: 2 days after a match). The number of days between matches varies by the match schedule. How a team plan their training and recovery activities varies and is dependent on the teams’ philosophy. For A, injury on the same day is defined as an injury on day 0, and future injury is defined as an injury occurring during the next 4 days excluding day 0. For B, future injury was defined as an injury occurring during the next microcycle excluding day 0. ACWR, acute:chronic workload ratio.

The mean sRPE for each microcycle divided by the EWMA of the previous three microcycles uncoupled (

When computing a ratio, one assumes that there is no relationship between the ratio and the denominator after controlling for the denominator; a ratio is only effective when the relationship between the numerator and the denominator is a straight line that intersects the origin.

The same online questionnaire was used to collect daily health status and training information from all three sports cohorts. The elite U-19 football data and elite youth handball data were collected via the Briteback AB online survey platform, while the Norwegian Premier League football data were collected with Athlete Monitoring, Moncton. The players daily reported whether they had experienced ‘no health problem’, ‘a new health problem’ or ‘an exacerbation of an existing health problem’. In the youth elite handball study, if players reported any new health problems, they were immediately prompted to specify whether it was an injury or illness in the questionnaire. In the football studies, if players reported any new health problems, a clinician contacted them by telephone the following day for a structured interview and classified the health problem as an injury or illness with the UEFA guidelines.

To estimate the relationship between training load and injury risk, mixed effects logistic regression was used.

We considered two outcomes: (1) occurrence of an injury on the same day as the observed training load (day 0) and (2) occurrence of injury in the future, where the current observation day (day 0) was not included. For unmodified training load values and daily ACWR 7:21-period, the future injury was defined as an injury occurring during the next 4 days excluding day 0. For microcycle ACWR 1:3-period, the future injury was any injury occurring during the next microcycle excluding day 0 (see

We adjusted for player age in all analyses. In addition, we adjusted for sex in the U-19 elite football and the elite youth handball models. In all models, the relationship between sRPE and injury risk was modelled with restricted cubic splines (RCSs).

More details about data preparation and calculations are available in a supplementary file in .pdf format (

In addition to analysing real data, we performed (stochastic) simulations to compare different methods for ascertaining non-linear and linear relationships between training load and injury risk. The simulations were based on the elite U-19 football dataset since it had the least missing data (24%). The methodology here is focused on a causal research setting; however, the methods may also be applied in predictive research.

Two datasets were created. The first kept the original 8495 sRPE and 6308 ACWR values. In the second, sRPE and ACWR were sampled with replacement to generate 22 500 training load values.

Artificial injuries were simulated under different assumed scenarios for the relationship between training load and injury risk:

A U shape.

A J shape.

A linear shape.

A U shape between training load and injury risk indicates that the injury risk at lower levels of training load is equal to the injury risk at higher levels of training load. In contrast, moderate levels of training load have the lowest risk. In a J shape, moderate levels of training load have the lowest injury risk, followed by low levels of training load having intermediate risk. Finally, high levels of training load have the highest injury risk. For the U and linear relationship shapes, the simulated probability of an injury was based on the sRPE, while for the J shape, it was based on the ACWR. Any reference to the ‘true’ probability refers to the simulated probability we have created for a given scenario and which we aim to model.

We used mixed effects logistic regression models to estimate the relationship between training load and predefined injury risk, and we compared seven different methods to model the relationship:

Linear model.

Categorising by quartiles (data driven).

Categorising by subjective cut-offs (subjective).

Quadratic model.

Fractional polynomials.

RCSs with automated knots (data driven).

RCSs with subjectively placed knots (subjective).

The root-mean-squared error (RMSE), coverage of prediction intervals, Brier score for model fit and C-statistics for predictive ability were calculated as performance measures. RMSE is a combined measure of accuracy and precision, where the lower the RMSE, the better the method. RMSE is only interpretable by comparing values in the same analysis – the values are meaningless in isolation.

In summary, the four steps of the simulations were:

Sample training load values from the elite U-19 football data.

Simulate injuries with three different shapes for the relationship between injury risk and training load.

Fit seven different models with injury as the outcome and training load as the explanatory variable.

Calculate performance measures.

Steps 1–4 were repeated 1900 times.

For the U-shaped relationship, predicted values were visualised alongside the predefined shape to determine each method’s ability to capture the true relationship. RMSE was also visually compared for the non-linear shapes.

All statistical analyses and simulations were performed using R V.4.0.2.

A strong J-shaped relationship was found between sRPE and the probability of injury on the same day for elite youth handball players (p<0.001,

Probability of injury in elite youth handball on (A) the sameday and (B) the next 4 days, for each level of session rating of perceived exertion (sRPE) measured in arbitrary units (AU), as predicted by mixed effects logistic regression models with restricted cubic splines. The predictions pertain to a 17-year-old female. The yellow area represents 95% cluster-robust CIs around predicted values. The straight line shows the same predictions from an equivalent model without splines (ie, assuming linearity). For figure part B, modelling the response of injury in the next 4 days, multiple injuries on the same day were considered one event and an injury event would pertain to four load values and are therefore included four times.

The quadratic model, fractional polynomials (FPs) and RCSs with subjectively placed knots were the only methods capable of modelling the non-linear U-shaped relationship (

Probability of injury for each level of session rating of perceived exertion (sRPE) as predicted by seven different methods of modelling load. The yellow line represents the ability of the method to capture the U-shaped relationship (shown by the black line). The yellow area corresponds to the prediction interval. The predictions are based on 8494 sRPE values sampled from a highly skewed distribution in a Norwegian elite U-19 football dataset.

The mean root-mean-squared error (RMSE) of 1900 permutations for seven different methods modelling a non-linear (A) U-shaped relationship between session rating of perceived exertion (sRPE) and probability of injury, and (B) J-shaped relationship between acute:chronic workload ratio (ACWR) and probability of injury. The methods are arranged from top-to-bottom by the method with highest RMSE (most error) to the method with lowest RMSE. Thus, the best methods (those with lowest RMSE) are arranged towards the bottom. For figure part A, fractional polynomials and restricted cubic splines (subjectively) were the best methods, while for figure part B, fractional polynomials and the quadratic model were the best methods. The calculations are based on a Norwegian elite U-19 football dataset with 8494 sRPE values for (A) U shape and 6308 ACWR values for (B) J shape. RMSE cannot be compared between the two shapes, only within each shape.

A comparison of mean root-mean-squared error, Brier score, C-statistic and coverage of prediction intervals for 1900 permutations of modelling the relationship between training load and risk of injury in seven different ways, with predetermined relationship shapes

Relationship | Sample size | Method | RMSE | Brier score | C-statistic | Coverage (%) |

U shape | 22 500 | Linear model | 2.344 | 0.097 | 0.827 | 100.000 |

Categorised (quartiles) | 0.995 | 0.101 | 0.809 | 99.678 | ||

Categorised (subjectively) | 0.996 | 0.102 | 0.758 | 94.600 | ||

Quadratic model | 0.993 | 0.097 | 0.826 | 100.000 | ||

Fractional polynomials | 0.994 | 0.096 | 0.829 | 100.000 | ||

Restricted cubic splines (data driven) | 1.065 | 0.097 | 0.826 | 100.000 | ||

Restricted cubic splines (subjectively) | 0.981 | 0.097 | 0.827 | 100.000 | ||

8494 | Linear model | 2.935 | 0.093 | 0.851 | 98.048 | |

Categorised (quartiles) | 0.958 | 0.096 | 0.838 | 98.769 | ||

Categorised (subjectively) | 0.965 | 0.098 | 0.809 | 84.600 | ||

Quadratic model | 0.956 | 0.092 | 0.850 | 98.937 | ||

Fractional polynomials | 0.956 | 0.092 | 0.852 | 98.942 | ||

Restricted cubic splines (data driven) | 1.079 | 0.092 | 0.849 | 98.686 | ||

Restricted cubic splines (subjectively) | 0.936 | 0.092 | 0.851 | 98.687 | ||

J shape | 22 500 | Linear model | 1.044 | 0.063 | 0.618 | 77.694 |

Categorised (quartiles) | 0.993 | 0.064 | 0.689 | 88.652 | ||

Categorised (subjectively) | 0.993 | 0.063 | 0.690 | 96.404 | ||

Quadratic model | 0.984 | 0.061 | 0.732 | 99.997 | ||

Fractional polynomials | 0.986 | 0.061 | 0.740 | 100.000 | ||

Restricted cubic splines (data driven) | 0.992 | 0.061 | 0.735 | 99.999 | ||

Restricted cubic splines (subjectively) | 0.993 | 0.061 | 0.721 | 99.869 | ||

6308 | Linear model | 0.942 | 0.060 | 0.774 | 54.493 | |

Categorised (quartiles) | 0.919 | 0.060 | 0.791 | 79.120 | ||

Categorised (subjectively) | 0.917 | 0.059 | 0.795 | 89.393 | ||

Quadratic model | 0.912 | 0.057 | 0.817 | 93.272 | ||

Fractional polynomials | 0.915 | 0.057 | 0.821 | 95.517 | ||

Restricted cubic splines (data driven) | 0.918 | 0.057 | 0.818 | 94.281 | ||

Restricted cubic splines (subjectively) | 0.919 | 0.057 | 0.812 | 89.959 | ||

Linear | 22 500 | Linear model | 0.999 | 0.239 | 0.591 | 100.000 |

Categorised (quartiles) | 0.999 | 0.240 | 0.588 | 25.000 | ||

Categorised (subjectively) | 0.999 | 0.241 | 0.579 | 99.995 | ||

Quadratic model | 0.999 | 0.239 | 0.591 | 99.999 | ||

Fractional polynomials | 0.999 | 0.239 | 0.592 | 100.000 | ||

Restricted cubic splines (data driven) | 0.999 | 0.239 | 0.591 | 100.000 | ||

Restricted cubic splines (subjectively) | 0.999 | 0.239 | 0.591 | 99.997 | ||

8494 | Linear model | 0.991 | 0.228 | 0.655 | 99.795 | |

Categorised (quartiles) | 0.991 | 0.228 | 0.653 | 24.957 | ||

Categorised (subjectively) | 0.991 | 0.229 | 0.649 | 99.678 | ||

Quadratic model | 0.991 | 0.228 | 0.656 | 99.786 | ||

Fractional polynomials | 0.991 | 0.228 | 0.656 | 99.788 | ||

Restricted cubic splines (data driven) | 0.991 | 0.228 | 0.656 | 99.789 | ||

Restricted cubic splines (subjectively) | 0.991 | 0.228 | 0.656 | 99.791 |

RMSE, root-mean-squared error.

All methods had a similar degree of error, predictive ability and model fit for the linear relationship (

The categorisation methods had the lowest coverage for the U and linear shapes, and categorising by quartiles had particularly poor coverage for the linear shape (25% vs >99% for other methods,

The differences in evaluation metrics between the two different sample sizes, n=22 500 and n=8494 for sRPE, and n=22 500 and n=6308 for ACWR, were negligible (

This is the first study exploring the potential for non-linearity in the relationship between training load and injury risk for football and handball. We found a J-shaped relationship between training load measured as the sRPE and probability of an injury on the same day in an elite youth handball cohort (

We also found that three methods were able to model the non-linear relationships between training load and injury explored in this paper: the quadratic model, FPs and RCSs, which managed to accurately recreate all simulated risk shapes (

All modelled relationships between training load and injury risk were either flat (no relationship) or non-linear. The results showed that the strength and direction of the relationship varied between training load—and injury—definitions in the handball population, while no relationships were found in the two football populations.

If we had assumed linearity and modelled the data accordingly, we would not have discovered these relationships. More grievously, we would have concluded there was no relationship between training load and injury risk for elite youth handball players for injury on the same day (linear model, p=0.24, type II error), when it was, in fact, a strong U-shaped parabola (RCS model, p<0.001,

In 2013, Gamble

As expected, standard logistic regression could not model the U and J shapes, as it assumes linearity. For the U shape, the RMSE was threefold higher for the linear model than all other models (RMSE=2.9 vs RMSE≈0.95,

Categorisation has previously been explored thoroughly in Carey

Recently, some studies have added a quadratic term to the training load and injury model to test for linearity: if the term was non-significant, it was discarded for a linear model; if significant, they categorised the training load variable to handle non-linearity.

Blanch and Gabbett

FPs modelled all risk shapes accurately (

RCSs performance depended on how knot locations (the points where the polynomials that make up cubic splines are joined, see

RCS produces effect sizes that are difficult to use in a practical setting, and results can only be interpreted in the form of p values and visualisation (such as in

When the main objective is causal research, FP is preferred. When the training load measure does not include negative numbers or 0. This includes:

Studies that use the Acute-Chronic Workload Ratio or other metrics that cannot be the value 0 or a negative value.

Studies that model the relationship between training load and injury risk on the same day, or other scenarios where the researchers may wish to remove the days where the athletes were not exposed to any training load from the dataset.

Studies that can justify applying a small constant (such as 0.001, or whatever is considered small in the context of the measuring scale) to all training load values.

When the main objective is predictive research, RCS is preferred.

When the training load measures must have the value 0. This includes studies that wish to capture a change in the effect, regardless how small, going from no training load at all to any amount of training load.

When training load is included in the study merely to adjust for it as a potential confounder and is not the main variable of interest.

To model non-linear relationships, either Fractional Polynomials (FP) or Restricted Cubic Splines (RCS) can be used.

Fractional polynomials are easier to interpret. We recommend FP under the following conditions:

We recommend restricted cubic splines under the following conditions:

We do not recommend changing the study aims or the chosen measure to use FP, nor do we recommend using FP under certain conditions and RCS for other conditions in the same study.

A step-by-step guide to performing FP and RCS in R can be accessed on the primary author’s GitHub page.

A limitation of this study was the sample size, the number of injuries and consequently statistical power. Neither of the two football cohorts satisfied the recommendation of >200 injuries to detect a small to moderate effect.

We used statistical methods commonly used and recommended in the field to demonstrate how non-linear relationships can be ascertained with existing methods. We were consequently limited in the choice of methods. The ACWR model is under debate, and the pros and cons of the method have been explored extensively in recent publications.

Exploratory analyses showed evidence of a non-linear relationship between training load and risk of injury in a sports population. Researchers should assume that the relationship between training load and injury risk is non-linear and use appropriate methods that explore relationships rather than constrain them. Linear methods should only be used when the relationship is first proven to be linear. We promote FPs or RCSs to model non-linear relationships, depending on the scenario.

We would like to thank Christian Thue Bjørndal for access to the elite youth handball data. We would also like to thank Garth Theron for high-quality programming of the Norwegian Premier League football database. This research would not be made possible without the collaboration of coaches and athletes, and we would like to thank the participants who contributed.

@lena_kbm, @DocThorAndersen, @torsteindalen, @benclarsen, @FagerlandWang

LKB-M designed the study and performed statistical analyses in collaboration with and under supervision from MWF and TEA. TDL constructed the novel idea of using microcycles instead of calendar weeks. All authors contributed with notable critical appraisal of the text and approved the final version.

The Oslo Sports Trauma Research Centre provided all funding for performing this study.

None declared.

Patients and/or the public were not involved in the design, or conduct, or reporting, or dissemination plans of this research.

Not commissioned; externally peer reviewed.

Data are available in a public, open access repository. Data are available on reasonable request. Data used for simulations are available in a public, open access repository (

Not required.

Study protocol for all three studies were approved by the Norwegian Centre for Research Data: Norwegian elite U-19 football (5487), Norwegian Premier League football (722773) and Norwegian elite youth handball (407930). They were also approved by the Ethical Review Board of the Norwegian School of Sport Sciences. The Norwegian elite U-19 football study was also approved by the South-Eastern Norway Regional Committee for Medical and Health Research Ethics (2017/1015).