11.12 Regression models

• The hazard function can be used to specify a likelihood (for maximum likelihood methods)

$$L = \prod_{i=1}^{n}h(y_i)^{\delta_i}S(y_i)$$

• For a non-censored data point, the factor is $h(y_i)S(y_i) = f(y_i)$ , the probability of dying in an tiny interval around $y_i$

• For a censored data point, the factor is just $S(y_i)$, the probability of surviving at least until $y_i$.

• This could be used for some parameterized model of $h$, Exercise 9 looks at this for a simple (constant hazard) example.

• But we really want to do regression, and one approach is to assume functional form like $h(t|x_i) = exp \left(\beta_0 + \sum_{j=1}^{p}\beta_j x_{ij}\right)$. This could be used in the likelihood to estimate the parameters, but the lack of time dependence is very restrictive.