Rejoinder to Next Generation Models for Subsequent Sports Injuries by Wu et al.

We greatly appreciate the commentary and positive feedback of discussants Prof. Jialiang Li and Dr. Rhythm Grover to enrich our paper and its context.

As noted by Prof. Li, survival models are highly applicable to the subsequent sports injury problem given the temporal dimension of injury data. In the sporting context, censoring can arise, for example, from finite surveillance windows associated with a sporting season, athletes joining and leaving a team, or even extended absence due to injury [1, 2]. However, given the complex systems nature of individual athletes and potentially changing dynamics and susceptibility to injury over time, it is also important to capture the changing state of the athlete explicitly [3]. For example, increasing strength with training over a season could reduce injury risk; however, a serious injury such as an ACL injury could lead to increased susceptibility to subsequent injuries.

Our paper presented a pragmatic approach, as noted by Dr. Grover, to tackle the challenges of modeling subsequent injury, reducing dimensionality through a time-varying Cox Proportional Hazards (PH) model, and using a discrete-time HMM to capture changes in susceptibility and covariate effects over time. Both Prof. Li and Dr. Grover note the potential computational challenge associated with Hidden Markov Models (HMMs) especially in the presence of large-scale and high-dimensional datasets. Hence, the need for dimension reduction, which was undertaken using survival modeling to explicitly cater for the time-to-event nature of injury data and censoring. The appropriateness of using the survival model was supported by checks of the assumptions of the PH model (e.g., proportional hazards, Schoenfeld residuals) and validation results (concordance index) as reported in our paper.

In addition to computational complexity, however, is the somewhat associated challenge of model convergence. Greater model complexity, such as more HMM states or more model covariates, can lead to challenges with model identifiability, estimation, computation, and thus model convergence [4]. This is a current research challenge when faced with limited data as in our subsequent injury application, which is limited to 33 players and 2523 training and competition sessions over one season. Computationally, the proposed discrete-time HMM fitted with Expectation Maximization (EM) took approximately 155 s to converge for the entire team of players over one season, compared to less than a second for the Cox PH model. However, model convergence with more than two states could not be achieved with this limited dataset. Therefore, although the computational cost is feasible in this case study, the data available can limit the level of model complexity that can be achieved. Hence, it highlights the utility of the proposed combination of dimension reduction and state space modelling as a more generalizable approach, and the need for more research in this area.

Along these lines, both Prof. Li and Dr. Grover discuss the challenge of computationally efficient inference and future research in this area, including bootstrapping, additive hazards model, and model-free dimension reduction with censored data. In addition, or in combination with bootstrapping, Bayesian inference of HMMs [5, 6] could be another avenue for investigation for inference with limited data and to help overcome potential challenges of local maxima with EM. Furthermore, a combined approach to variable selection and inference could potentially capture influential variables in a HMM context that are not marginally impactful in a survival model, as noted by Dr. Grover. This is challenging due to the computational and inferential complexity noted above; however, methods for learning Dynamic Bayesian Networks from high dimensional data, which are a generalized form of HMMs, could potentially be adapted to subsequent injury [7, 8]. Another avenue for investigation could be the Continuous Time HMM (CTHMM) as a way to potentially better capture non-uniform time intervals between observations. However, the CTHMM incurs additional model complexity compared to the discrete HMM, as both state transition times and the number of state transitions between observations need to be estimated [4].

In this study, the data was limited to one Australian Football League (AFL) club over the course of one season. Potentially, with additional data over more seasons and/or clubs, we could better assess and study model transferability and generalizability as noted by Dr. Grover. The proposed HMM, however, was able to assess the injury risk of individual players of differing playing positions, exposures, and loads, suggesting some level of generalizability. Larger datasets could also enable the application of modern methods for machine learning including recurrent neural networks and temporal convolutional networks, which to date have been understudied in the subsequent injury domain. Generally, neural networks require large datasets and are challenging to interpret, but can produce very high predictive performance [9]. One potential approach for future research to help address the practical challenge of limited data in elite sports could be to train a model on a larger injury dataset and re-train it for a specific sports club [10]. This is important because, with the competitive nature of sport and the movement of players between clubs, obtaining complete data for individual athletes over time is challenging.

The authors declare no conflicts of interest.