Loghazard

Loghazard is a term used in statistics and survival analysis to denote the natural logarithm of a hazard rate, h(t). The hazard rate represents the instantaneous risk of an event occurring at time t, given survival up to that point. Applying the logarithm yields the log hazard, log h(t).

In models such as the Cox proportional hazards model, the hazard is expressed as h(t|X) = h0(t) exp(βX). Taking logs gives log h(t|X) = log h0(t) + βX, which means covariate effects are additive on the log hazard scale. This transformation facilitates linear modeling and interpretation of covariate effects via hazard ratios.

Interpretation and use: the hazard ratio for a unit change in X is exp(β). The difference on the log hazard scale is β, and the baseline log hazard is log h0(t). Estimation typically uses partial likelihood or related methods, with the log hazard being a fundamental construct in presenting model results.

Limitations: h(t) must be positive, so log h(t) is defined only when the hazard is positive. At times with zero hazard estimates, the log hazard is undefined and may require smoothing or alternative modeling. The term loghazard is not universally standardized and is sometimes used informally to refer to either the log hazard function or to the log of a hazard ratio.

See also: hazard function, survival analysis, Cox proportional hazards model, log transformation, hazard ratio.