Bathtub Hazard

A hazard pattern that is high early in follow-up, declines to a stable middle period, then rises again later, resembling a bathtub's shape.

Source: Collett 2015

A bathtub-shaped hazard has three phases across time. Early on the risk is high, as in the period just after surgery or birth when frailty or complications take their toll; it then falls to a low, roughly steady level through a long middle period; finally it rises again as ageing or late effects set in. The curve of risk against time dips in the middle and turns up at both ends, resembling a bathtub. Standard single-shape distributions cannot capture all three phases at once. Klein and Moeschberger (2003) describe this pattern.

Source: Klein & Moeschberger 2003

A bathtub-shaped hazard arises when different processes dominate risk at different times: high early hazard may reflect an initial vulnerable period, such as early mortality after diagnosis or a procedure; the low middle period reflects a stable phase once early risks pass; and rising late hazard reflects the accumulation of risk over time, such as ageing or disease progression. The combination of these distinct sources of risk at the beginning and end, with a safer middle, produces the characteristic bathtub shape.

Source: Collett 2015

The bathtub hazard pattern is seen in human mortality over the lifespan, with high infant mortality, a low-risk middle, and rising mortality in old age, and in some clinical settings, such as after major surgery, where early post-operative risk is high, then falls, then rises later with ageing or disease. It also appears in reliability engineering for equipment failure. Wherever early and late risks are elevated but the middle period is relatively safe, the hazard follows a bathtub shape.

Source: Collett 2015

The bathtub hazard matters for survival modelling because standard parametric distributions often cannot capture a hazard that both falls and rises, so fitting them to bathtub-shaped data or extrapolating may misrepresent the risk over time. Modelling such patterns may require flexible distributions or piecewise approaches that allow the hazard to change shape. Recognising a bathtub hazard is important for choosing an appropriate model, since assuming a monotonic hazard where the true pattern is bathtub-shaped would bias survival estimates and extrapolation.

Source: Kalbfleisch & Prentice 2002

A bathtub hazard is modelled using approaches flexible enough to represent a hazard that decreases then increases, since simple distributions with monotonic hazards cannot. Options include flexible parametric distributions with the necessary shape, spline-based models that let the hazard vary freely, or piecewise models that fit different hazards to early, middle, and late periods. The choice depends on the data and the need to capture the early decline and late rise. Such flexible modelling is needed to represent the bathtub pattern accurately.

Source: Collett 2015