The Cox regression model is used to measure the effects of one or more risk factors (or exposures; x-variables) on the hazard rate. The hazard rate is the effect measure in Cox models: the risk of the occurrence of a health event, given the individual’s survival until that time point. The basic Cox model notation states that the hazard at time t is equal to the baseline hazard at time t multiplied by the exponentiated product of the vector of regression coefficients and the vector of covariates.

Let’s dig in!

As we mentioned in Non-parametric, parametric, and semi-parametric models, health events modelled using parametric models (e.g., exponential, Weibull, Gompertz, and Poisson) are each assumed to have a distinct distribution that is described by one or several parameters. In other words, the baseline hazard for each of these models has a distinct shape and varies in a specific way. Using parametric survival models require that you understand the assumption(s) underlying the shape of the respective distribution and have an idea that the baseline hazard in your data approximately follows this shape.

Cox regression is a semi-parametric approach; the model does contain a parametric component, but also a non-parametric component. In Cox models, the baseline hazard function is non-parametric: it can wander freely with no parameters. This means that the Cox model does not make any assumptions about the shape of the baseline hazard or the distribution of survival times. In fact, estimating the baseline hazard is not needed to make inferences about the relative hazard rates. Unlike parametric models where we need to be very careful about which model we specify, we do not need to specify a distribution for the Cox model. Importantly though, the baseline hazard, no matter the shape, is assumed to be the same for every unit of observation.

Summary so far The baseline hazard at time t is the value of the hazard when all covariates are equal to zero, the baseline hazard does not have a specific shape but is the same for all units of observation, and it is not assumed that survival times follow a specific distribution.

In Cox models, the covariate vector (or covariate function; group of one or more covariates, x-variables) is modelled parametrically. Covariates influence the baseline hazard in a specific way. The hazard function is multiplied by the covariate vector to obtain the effect of the covariates. The covariate vector induces a multiplicative and proportional shift in the baseline hazard, but does not change the shape of the baseline hazard. Furthermore, the multiplicative effect of the covariates is not time dependent; it is the same at any time t during the follow-up period. Please note that this is true for fixed covariates, or covariates that do not depend on time. Cox regression can also be used to model time-dependent covariates, which may vary over time, but we will not discuss time-dependent covariates in this guide. The effect of the covariates underlies a very important assumption in Cox regression: the proportional hazards assumption.

Proportional hazards

Now we know that the effect of any (group of) covariate(s) is the same at any point in time during follow-up. Therefore, the relationship between the covariates and the event (outcome; dependent variable; y-variable) is constant. This means that, for any two units of observation (for example, any two individuals), the ratio of the hazard functions is constant and dependent on the covariate values. In other words, the hazard functions are proportional to one another, at any point in time. The estimated hazard ratio (which compares the hazard functions of one individual to another) does not vary over time, even if the size of the hazard changes (e.g., increases, decreases, increases then decreases, etc.) or remains constant. Therefore, if the proportional hazards assumption holds, the hazard curves for any two individuals should be proportional to one another over time. When graphed, these curves should be parallel to each other, and definitely should not cross. Since this is a very important assumption when using Cox regression, you should always formally test that the proportional hazards assumption is valid – but more on this later.

Example Below is a simplified illustration of what we mean by proportional hazards. We want to estimate the risk of dying among indoor versus outdoor cats diagnosed with feline leukemia virus over a five-year period. If the risk of dying (the hazard) among outdoor cats is 1.4 times higher than the risk of dying among indoor cats at the beginning of the observation period, the proportional hazards assumption implies that the risk of dying among outdoor cats remains 1.4 times higher at all later time points. In other words, the difference between the two hazards should remain 1.4 across the five-year period, no matter the shape of the distribution.

Tied failure times

As noted, tied failure times (ties) occur when two (or more) individuals have the same time to event, or they experience the event at the same time. For example, in the Olympic games, a tie for first place can be problematic: instead of one person each being ranked first, second, and third, suddenly two people are ranked first, no one is ranked second, and one person is ranked third. What does this have to do with Cox models? Cox regression is based on the partial likelihood function: the product of the conditional probabilities. Stay with us.

Imagine that we have a group of individuals who are at risk of the event (failure) at time t. For each event (failure time), we can calculate what is called the conditional probability of the event occurring (failure). To calculate the likelihood function, the numerator should contain only the individual who experiences the event at time t; the denominator contains all the other individuals in the group (risk set) for whom the event has not yet occurred. As such, calculating the likelihood function depends on the order of the failure times, not when the failures occur. Therefore, in theory, only one individual can fail at each failure time. If more than one individual fails at a single failure time, this is a tie, or a tied failure. Suddenly we have more than one individual in the numerator, the ordering of the failures is unclear, and we have no idea who won the gold medal. In conclusion, we want to minimize the number of tied failures.

Other names for Cox regression

Cox regression is sometimes called proportional hazards regression.