Proportional hazards fit

A semi-parametric model that describes the relationship between a time to event response variable and one or more explanatory variables using a hazard function.

The Cox proportional hazard model is expressed as a hazard function h₀(t) + exp(∑β_iX_i). The formula expresses the hazard at time t for an individual with a given set of explanatory variables as the product of two quantities. The first quantity is h₀(t), the baseline hazard function. The second quantity is the exponent of the linear sum β_iX_i where the sum is over the p explanatory variables. An important feature of the proportional hazards model is that the baseline hazard function is a function of t but does not include the Xs. Therefore, it is unnecessary to specify the form of the baseline function in estimating the parameters. Also, the parameters in the second quantity are dependent only on the Xs and are independent of time. The unknown model parameters are estimated using maximum-likelihood estimation.

• Hazard ratio estimates
  Hazard ratios are the increase or decrease in the hazard associated with a change of the predictor, all other predictors been held constant.
• Baseline survival function
  A baseline survivual function is an estimate of h₀(t) in a proportional hazards model.
• Effect of model hypothesis test
  A likelihood ratio or Wald X² test formally tests the hypothesis of whether the model fits the data better than no model.
• Effect of term hypothesis test
  A likelihood ratio or Wald X² test formally tests the hypothesis of whether a term contributes to the model.