Glossary

Proportional Hazard Model

Proportional Hazard Model:

Proportional hazard model is a generic term for models (particularly survival models in medicine) that have the form

L(t | x₁, x₂, Ã‚Â¼, x_n) = h(t) exp(b₁ x₁ + Ã‚Â¼+ b_n x_n),

where L is the hazard function or hazard rate, {x_i} are covariates, {b_i} are coefficients of the model – effects of the corresponding covariates, and h(t) gives the effect of duration on the hazard rate.

In a proportional hazard model, the effect of an independent variable on the hazard rate is assumed to be multiplicative. For example, the variable “smoking” in a model might have the effect of increasing the hazard rate 30%.

Examples of proportional hazard model are exponential, Weibull, and Gompertz models given respectively by

h(t) = 1;

h(t) = exp(g ln(t));

h(t) = exp(gt).

Cox proposed an ingenious principle for estimating all proportional hazard models without knowing the function h(t) or even the base hazard rate h₀(t). Using this principle one estimates the effects {b_i} of the covariates {x_i}, but not the effect of duration h(t).

This is known as the Cox Proportional Hazard Model.

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