Overview
The Cox proportional hazards model is a model that assumes that the hazard function (and hence the survival function) is dependent on factors other than time, here indicated as a vector, {% \vec{X} %}. The Cox proportional hazards model assumes that the hazard function has the following form:
{% h(t, \vec{X}) = h_0(t) \times exp(\sum_i \beta_i X_i) %}
where {% \vec{X} %} is a vector of factors that affect the observed datapoint. That is, {% \vec{X} %}
can vary between individuals, but not across time.
Hazard Ratio
The hazard ratio measures the ratio of the hazard function for two individuals.
{% HR = \frac{h(t, \vec{X})}{h(t, \vec{X}')} %}
The baseline hazard rate drops out of the equation and the result is.
{% HR = exp(\sum \beta_i(X_i - X'_i)) %}
The hazard ratio shows the relative likelihood of the event ocurring to one individual over another.