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We thank Shrier et al. for a thoughtful expansion on the topic of non-linearity.1 The comments from the authors provide valuable insights and detail to both the handling and the interpretation of fractional polynomials and splines, and may interest readers who seek more information than the short introduction in Bache-Mathiesen, et al. 2.
We are especially grateful for elaborating on the interpretation of restricted cubic splines, and the solution of adding a small constant (i.e. 0.1) to all values to handle the value “0” when using fractional polynomials. These topics could not be sufficiently addressed within the limited wordcount of the original article, and we encourage readers to consider these comments.
Conflict of Interest:
1. Shrier I, Wang C, Stokes T, et al. Important Nuances for Non-Linear Modeling. BMJ Open Sport & Exercise Medicine 2021
2. Bache-Mathiesen LK, Andersen TE, Dalen-Lorentsen T, et al. Not straightforward: modelling non-linearity in training load and injury research. BMJ Open Sport & Exercise Medicine 2021;7(3):e001119. doi: 10.1136/bmjsem-2021-001119
We would like to thank Bache-Matiesen et al.(1) for their thoughtful article on non-linear modelling in sport medicine. Our own study on the non-linear relationship between acute: chronic workload ratio (ACWR) and injury risk in children was published as a preprint (2) and recently accepted by the American Journal of Epidemiology.(3) Below, we highlight some additional underlying principles in non-linear modelling that readers should understand.
Models are based on information, which includes both data and assumptions. Simple linear models are more prone to bias because they assume a data generating process that is likely incorrect. The flexibility of non-linear models leads to less risk of bias, but also less precision. The optimal choice between bias and uncertainty depends on the context.(4)
Bache-Matiesen describe three non-linear modelling options: quadratic modelling, fractional polynomials (FP), and restricted cubic splines (RCS, where knots are determined by either data driven or a priori methods). These all fall under generalized additive models (GAMs), or generalized additive mixed models (GAMMs; if one uses “random effects” to adjust for repeated measures on participants).
FP methods use a single polynomial function over the entire range of exposures to predict the outcome. Quadratic models are special cases of FP (with exponents of 0, 1 and 2) and are too restrictive to be generally recommended. RCS separate data i...
FP methods use a single polynomial function over the entire range of exposures to predict the outcome. Quadratic models are special cases of FP (with exponents of 0, 1 and 2) and are too restrictive to be generally recommended. RCS separate data into sections by “knots”, determine which polynomials best predict the observed data within the sections defined by the knots, and apply a smoothing function to join the polynomials. The placement of the knots is important. In simulations, we know the true data generating process. In real-world observational data, we do not. Although we believe subjective a priori knot placement is sometimes better than data driven methods, researchers should be aware that gross errors may occur if the assumptions are incorrect.
RCS methods are more flexible than FP because they allow the functions between knots to be different from each other. We highlight two technical points. First, the “restriction” in the RCS method used by Bache-Matiesen appears restricted to using linear functions before the first knot and after the last knot. RCS can include other types of restrictions as well. Second, FP and RCS are just two forms of GAM(M)s. Two other popular forms are penalized regression spline and thin plate regression spline (e.g. default in mgcv package in the statistical program R (5)). The major difference between RCS and the penalized/thin plate regression splines is the shape of the polynomial that is used. The optimal choice depends partly on the research question and partly on the observed data.
SPECIFIC COMMENTS ON RECOMMENDATIONS BY BACHE-MATIESEN
1. Based on their simulations, Bache-Matiesen suggests RCS performs better than FP for predictive modeling. We believe the conclusion is too strong. RCS can be used in situations when there is a more complex underlying structure than FP. However, fitting more complex structure requires more data. When there is limited data and less complex structure, FP could outperform RCS.
2. Bache-Matiesen claim that RCS allowed the authors to model the relationship at high training loads where there were few data points. We caution against this due to the limited information. In our own study on children,(3) we restricted our conclusions about the relationship between acute:chronic workload ratio (ACWR) and injury risk to ACWR <3 where there were enough data. We show the full range of the relationship in the supplementary material to be fully transparent and to help generate hypotheses for future studies but did not feel it appropriate to make inferences.
3. We disagree with Bache-Matiesen that FP is more interpretable than RCS for causal effects, and that RCS results “can only be interpreted in the form of p values and visualisation”. First, we hope that most sport medicine researchers have moved beyond making inferences on p-values because of its severe limitations.(6, 7) Second, for causal questions, we generally estimate the difference in outcome when exposure is set to two different levels. When causal inference assumptions are reasonable, one estimates the causal effect with g-computation using predicted data from the model;(8) this is applicable for both linear and non-linear models. In brief, the magnitude of the causal effect in a non-linear model necessarily depends on the two chosen exposure levels. Although FP has only a single function over its entire range, the causal effect between the chosen exposure levels still requires using g-computation and the predicted values from the FP function. The causal effect using other GAM(M) methods is obtained with the same process; one uses the predicted values provided by the statistical software over the chosen exposure range.
4. The authors discuss the need to add a small constant to training load when it can equal 0, in order to allow for analyses on the log scale. The choice should have theoretical justification. For example, if activity (i.e. training load) is a proxy for time at risk (e.g. game injuries and a game was not played), adding a constant is inconsistent with the research question. However, in the ACWR, activity is a proxy for fatigue in the numerator and a proxy for fitness in the denominator. Both fatigue and fitness are affected by activities of daily living (e.g. occupation, transportation) outside of regular sports. In our analysis on children, we added a constant of 0.1 to activity to reflect our belief that activities of daily living might contribute 10% to each of fatigue and fitness. In supplementary analyses, we explored the effects if the contribution were 25% or 50%.
5. Finally, if we believe tripling the activity will triple the injury risk (e.g. 3 games vs 1 game), then activity (or ACWR) should be plotted on the log scale.(3)
1. Bache-Mathiesen LK, Andersen TE, Dalen-Lorentsen T, et al. Not straightforward: modelling non-linearity in training load and injury research. BMJ Open Sport Exerc Med. 2021;7(3):e001119.
2. Wang C, Stokes T, Vargas JT, et al. Injury risk increases minimally over a large range of changes in activity level in children. arXiv. 2021;2010.02952v2 [q-bio.QM].
3. Wang C, Stokes T, Vargas JT, et al. Injury risk increases minimally over a large range of changes in activity level in children (In Press). Am J Epidemiol. 2021.
4. Kaufman JS. Commentary: Why are we biased against bias? Int J Epidemiol. 2008;37(3):624-626.
5. Wood S. mgcv. In: R: A language and environment for statistical computing R Foundation for Statistical Computing Vienna, Austria; 2021.
6. Amrhein V, Greenland S, McShane B. Scientists rise up against statistical significance. Nature. 2019;567(7748):305-307.
7. Wasserstein RL, Lazar NA. The ASA statement on p-values: context, process, and purpose. Am Stat. 2019;70(2):129-133.
8. Hernán MA, Robins JM. Causal inference: What if. Boca Raton: Chapman & Hall/CRC, 2020.