12.6 - Model-Based Methods: Time-to-event Outcomes

For a time-to-event outcome variable, proportional hazards regression is available. Let \(\lambda \left(t|X_{1i}, X_{2i}, \dots ,X_{Ki}\right)\) denote the hazard function for the ith patient at time \(t, i = 1, 2, \dots , n\), where the K regressors are denoted as \(X_{1i}, X_{2i}, \dots , X_{Ki}\). The baseline hazard function at time t, i.e., when \(X_{1i} = 0, X_{2i} = 0, \dots , X_{Ki} = 0\), is denoted as \(\lambda_0\left(t\right)\). The baseline hazard function is analogous to the intercept term in the multiple regression model or logistic regression model.

The proportional hazards regression model states that the log of the hazard function to the baseline hazard function at time t is a linear combination of parameters and regressors, i.e.,

\(log\left( \dfrac{\lambda(t|X_{1i},X_{2i}, ... ,X_{Ki})}{\lambda_0(t)} \right)=\beta_1X_{1i},\beta_2X_{2i}, ... , \beta_KX_{Ki}\)

The proportional hazards regression model is nonparametric because we do not specify a specific distribution function for the time-to-event outcome variable. The proportionality assumption, however, is important.

The ratio of hazard functions can be considered a ratio of risk functions, so the proportional hazards regression model is a function of relative risk (unlike the logistic regression models which are a function of the odds ratio). Changes in a covariate have a multiplicative effect on the baseline risk. The model in terms of the hazard function at time t is:

\(\lambda(t|X_{1i},X_{2i}, ... ,X_{Ki})= \lambda_0(t)exp(\beta_1X_{1i},\beta_2X_{2i}, ... , \beta_KX_{Ki})\)