A hazard function can be approximated using a piecewise constant (PC) function (https://en.wikipedia.org/wiki/Step_function). The parameters of a PC function are jump points, at which the value of the hazard can change, and hazard levels, which remain constant between consecutive jump points.
PC functions have attractive features in building intensity models.
- They can be added and multiplied together, and the resulting function is still a PC function, thus multiplicative and additive hazard models can be easily constructed.
- They can be shifted by adding or subtracting a constant from the jump point values, thus different time scales can be incorporated in the model by selecting the constant appropriately, for example, the time of birth, diagnosis of a disease and onset of a treatment.
- They can also be integrated easily, thus calculation of the Poisson likelihood is computationally simple and efficient.
- If the “true” hazard function has a strict changepoint, at which the hazard function increases/decreases considerably during a short time interval, a PC function can fit to these changes accurately.