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.