Survival Analysis: Lifetables and Cox Proportional Hazard Model

While thinking of survival analysis, potentially the most popular models are lifetables and Cox proportional hazard model, lifetables seem to be the simplest way to analyze survival time, i.e. time that is measured for each subject from the beginning of the observation period till the defined event occurrence (e.g. an adverse event occurrence, death, full recovery) or till the end of observation period if subject did not experience the event. The idea of this method is to estimate the survival function, which is the probability of ‘survival’ (in other words: probability of not experiencing the event) at least to the timepoint t. Such estimation is being performed at each timepoint t at which an event occurred. Between the subsequent event occurrences, survival function is assumed to be constant. On the other hand, the hazard function is being estimated, which is interpreted as the conditional probability of the event occurrence at timepoint t provided that the event had not occurred before. Again, hazard function is being estimated at timepoints at which an event occurred and is constant between subsequent event occurrences.

Lifetables are quite simple to estimate and to interpret – it is very helpful to present survival function on the plot to see how the survival probability or hazard function changes over time. However, this method has its drawbacks as well. The most painful is probably the fact that it is impossible to assess impact of continuous variables (e.g. age) on the hazard level. The only thing we can do is to divide the whole sample into categories, on the basis of categorical variable (e.g. age group) and compare survival functions between such groups.
There are several tests available that make it possible to state whether survival functions for two separate sub-samples are statistically different or not. But still, even if we know that younger subjects have different survival function than older, we cannot say exactly how the survival function changes if a person gets one year older. For such estimations, Cox proportional hazard model becomes very useful. The model takes the hazard level as a dependent variable. Not only is it possible to include in the model continuous variables (and estimate the approximate change in hazard level if, for example, subject gets one year older) but also we can add so-called time-dependent variables.

In general, thanks to the Cox model in clinical programming we can compare two subjects with respect to their hazard level in quantitative manner, i.e. on the basis of parameters estimates we can say that among women, subject who is 40 years old is twice as likely to experience a given adverse event as a subject who is 20 years old. Time-dependent variables are necessary in cases when this relation is not constant, i.e. varies over time. 

Of course, Cox model, as much more complicated than lifetables requires several restrictive assumptions to be satisfied. More on model definition, its assumptions and how to perform the model adequacy assessment process will feature in future posts in our blog.