I got some questions about the notation I use for various functions in survival models and the interpretation of some of these functions.
\(T\) is a random variable, defined on the interval \((0, \infty)\). It represents the time at which an event occurs (typically death, but it could be other events).
\(t_1, t_2, ..., t_n\) are the times at which an event occurs or a patient is lost to follow-up (censored).
\(c_1, c_2, ..., c_n\) are indicator (0-1) variables that indicate whether an event occured or the observation was censored.
\(f(\cdot)\) is the probability density function for T. We use \(f(t)\) to compute probabilities.
\(P[a < T < b] = \int_a^b f(t) dt\)
Sometimes f depends on some covariates and parameters. In that case, we write
\(f(\cdot, X_i, \beta)\)
Every density function has a cumulative density associated with it.
\(F(a) = \int_{-\infty}^a f(t) dt\)
\(S(\cdot)\) is the survival function.
\(S(a) = P[T > a] = 1- F(a) = = \int_a^{\infty} f(t) dt\)
Again, the survival function may depend on some covariates and parameters.
\(S(\cdot, X_i, \beta)\)
\(h(\cdot)\) is the hazard function. It is an instantaneous rate, adjusted for the number of events that occur prior to the time of interest.
\(h(t) = \frac{P[t < T < t + \Delta t] / \Delta t}{P[T > t]}\)
If you let \(\Delta t\) approach zero, the hazard function becomes
\(h(\cdot) = \frac{f(\cdot)}{S(\cdot)}\)
Like before, you can show the dependence on some covariates and parameters with the notation
\(h(\cdot, X_i, \beta)\)
The proportional hazards assumption is written in mathematical notation as
\(h(\cdot, X_i, \beta) = e^{X_i \beta}h_0(\cdot)\)
where \(h_0(\cdot)\) is a baseline hazard rate.
An earlier version is here.