Overview
The maximum likelihood function starts with a dataset,
{% \{ (\vec{X}_1, Y_1), (\vec{X}_2, Y_2), ... (\vec{X}_n, Y_n), \} %}
Here {% \vec{X}_i %} represents a vector of factor realizations for observation i, and {% Y_i %} is the time of the event for
realization i.
This model makes some simplifying assumptions, which can be relaxed.
- Right censored data - the maximum likelihood
- Factor values for a given observation do not change over time
Partial Likelihood
For the purposes of stating the Likelihood function, we assume that the dataset given above has been sorted so that the observation that represents the earliest occurence of the event is given first.
The Cox likelihood is the product of n different likelihoods.
{% L = L_1 \times L_2 \times ... \times L_n %}
where
{% L_i = \frac{h_0(t_i) \times exp(\vec{\beta} \cdot \vec{X}_i)}{ \sum_{j=i}^n h_0(t_i) \times exp(\vec{\beta} \cdot \vec{X}_j) } %}
(see kleinbaum)
the likelihood funciton is constructed by asking what is the likelihood of the ordering of the data points. That is, the algorithm successively asks
how likely the next point in the series was to have ocurred as the next point.
As can be seen, the baseline hazard function, {% h_0(t) %} cancels out in the likelihood function given. That means that the likelihood can be fit to the data, and a set of {% \beta %} parameters estimated without having to determing the baseline hazard function.
Log Partial Likelihood
{% Log_i = Log(exp(\vec{\beta} \cdot \vec{X}_i)) - Log(\sum_{j=i}^n \times exp(\vec{\beta} \cdot \vec{X}_j)) %}
{% = \vec{\beta} \cdot \vec{X}_i - Log(\sum_{j=i}^n exp(\vec{\beta} \cdot \vec{X}_j)) %}
Maximum Log Partial Likelihood
The next step is to optimize the log likelihood. The analytic method to optimize the log-likelihood is to find the values of {% \beta %} such that the partial derivatives are zero.
{% \frac{\partial LogLikelihood}{\partial \beta_i} = 0 %}
Other numeric methods can also be applied.
see Optimization