Statistical specifications and assumptions in relation to the risk interval, risk set, baseline hazard and within-person correlation in the extended Cox proportional hazards (CoxPH) models
| Components | Andersen-Gill (A-G) | Frailty | Wei-Lin-Weissfeld total time (WLW-TT) marginal | Prentice-Williams-Petersen gap time (PWP-GT) conditional |
|---|---|---|---|---|
| Risk interval | Duration since starting observation | Duration since starting observation | Duration since starting observation | Duration since previous injury |
| Risk set for injury k at time t | Independent injuries (any given injury occurrence is not affected by previous injuries) | A random effect (or frailty) term is used to account for the within-player correlation between injuries to enable modelling of the phenomenon by which some players are intrinsically more or less susceptible to experiencing a given injury than others are | All players who have not experienced injury k at time t | All players who have experienced injury k−1, and have not experienced injury k at time t |
| Baseline hazard | Common/same baseline hazard across all injuries | Heterogeneity is directly incorporated via a random effect so that the baseline hazard is allowed to vary with each injury | Common baseline hazard for all injuries within a player | Stratifies the data by injury so that the baseline hazard is allowed to vary with each injury |
| Within-person correlation | The within-person injuries are independent | Captures within-person correlation due to both injury dependence and heterogeneity | The within-person injuries are independent | The current injury is unaffected by earlier injuries that occurred to the player |
| Comment | A-G model is recommended when there is no injury dependence and no covariate/injury effects | The frailty approach accounts for heterogeneity. The random effect (the frailty) has a multiplicative effect on the baseline hazard function and the mixture of individuals with different injury risks | At any time point (matches), WLW-TT describes all players who have not yet experienced k injuries are assumed to be at risk for the kth injury which is not realistic in the sports setting injury data | PWP-GT model takes into account the ordering of events |