Appendix I

Proportional Hazards Model

(Sources: Kalbfleisch and Prentice [1], Franses and Paap [2])

A second way to include explanatory variables in a duration model is to scale the hazard function by the function , that is,

(I.1)

where denotes an arbitrary, unspecified, baseline hazard function for continuous . Again, because the hazard function has to be non-negative, one usually specifies as

(I.2)

If the intercept is unequal to 0, the baseline hazard is identified upon a scalar. Here, if one opts for a Weibull or an exponential baseline hazard one again has to restrict to 1 to identify the parameters.

The conditional density function of given covariates is

(I.3)

The conditional survivor function for given covariates is

(I.4)

where

(I.5)

The log-likelihood function for the proportional hazards model

is given by

(I.6)

which allows for various specifications of the baseline hazard. If we assume that the parameters of the baseline hazard are summarized in , the first-order derivatives of the log-likelihood are given by

(I.7)

(I.8

The second-order derivatives are given by

(I.9)

(I.10)

(I.11)

which shows that we need the first- and second-order derivatives of the baseline hazard and the integrated baseline hazard. If we assume a Weibull baseline hazard with , the integrated baseline hazard is . Straightforward differentiation gives

(I.12)

(I.13)

The maximum likelihood estimates are found by iterating over

(I.14)

for properly chosen starting values for and