Models for timetoevent data
Here, observations are the "times at which events occur". An event may be oneoff (e.g., death, hardware failure) or repeated (e.g., epileptic seizures, metro strike).
Sommaire 
Single event
To begin with, we will consider a oneoff event. Depending on the application, the length of time to this event may be called the survival time (until death), failure time (until hardware fails), etc. To be general, we can just say event time.
The random variable representing the event time for subject $i$ is typically written $T_i$. Several situations are then possible to define the observations:
 The event time is exactly observed.
 Then, the observation for individual $i$ is $y_i = t_i$, where $t_i$ is a realization of the random variable $T_i$.
 We may know the event has happened in an interval $I_i$ but not know the exact time $t_i$. This is interval censoring. For example, at a routine checkup, cancer recurrence may be detected, and we only know that it has occurred at some point in time since the last checkup.
 The observation for individual $i$ is the event: $y_i = $ "$a_i < t_i \leq b_i$".
 If we assume that the trial ends at time $\tstop$, then the event may happen after the end of the trial period. This is right censoring.
 There are several variations of this for defining what the observations are:
 If events (before $\tstop$) are exactly observed, then for $i=1,2,\ldots, N$,
 If events before $\tstop$ are interval censored, then for $i=1,2,\ldots, N$,
Probability distributions
Several functions play key roles in timetoevent analysis: the survival function, the hazard function and the cumulative hazard function. We are still working under a population approach here and so these functions, detailed below, are therefore individual functions, i.e., each subject has its own. As we are using parametric models, this means that these functions depend on individual parameters $(\psi_i)$.
 The survival function $S(t; \psi_i)$ gives the probability that the event happens to individual $i$ after time $t>t_{start}$:
 The hazard function $\hazard(t;\psi_i)$ is defined for individual $i$ as the instantaneous rate of the event at time $t$, given that the event has not already occurred:
 This is equivalent to:
 Another useful quantity is the cumulative hazard function $\cumhaz(a,b;\psi_i)$, defined for individual $i$ as:
 Note that (1) implies that:
Equation (1) shows that the hazard function $\hazard(t;\psi_i)$ characterizes the problem, because knowing it is the same as knowing the survival function $S(t;\psi_i)$. The probability distribution of survival data is therefore completely defined by the hazard function.
Let $\qcyipsii$ be the conditional distribution of the observation $y_i$ given the vector of individual parameters $\psi_i$. Its pdf can be easily computed for the various censoring situations discussed above:
 If the event is exactly observed with $y_i=t_i$, the density is the derivative of the cumulative density function, i.e., the derivative of $1  S(t_i;\psi_i)$:
 If the event is intervalcensored with $y_i=\,$ "$a_i<t_i\leq b_i$":
 If the event is rightcensored with $y_i= \,$ "$t_i>t_{stop}$":
Repeated events
Sometimes, an event can potentially happen again and again, e.g., epileptic seizures, heart attacks, etc. For any given hazard function $\hazard$, the survival function $S$ for individual $i$ now represents survival since the previous event at $t_{i,j1}$, written here in terms of the cumulative hazard from $t_{i,j1}$ to $t_{i,j}$:
Censoring and probability distributions
Taking into account censoring for repeated events is slightly more complicated than for oneoff events. First, let us assume that a trial starts at time $t_{start}$ and ends at time $t_{stop}$. Let $(T_{i1}, T_{i2}, \ldots )$ be random event times after $t_{start}$. Then, we can distinguish between the two following situations:

1. Exactly observed events: A sequence of $n_i$ event times is precisely observed before $t_{stop}$, i.e., ${\rm y_i = (t_{i,1},t_{i,2},\ldots,t_{i,n_i}, \quad t_{i,n_i+1}>\tstop)}$.
 The conditional pdf of $y_i$ is given by:
 The conditional pdf of $y_i$ is given by:
 where $t_{i0}=\tstart$.
\(
\pcyipsii(y_i  \psi_i) = \left(\prod_{j=1}^{n_i}\hazard(t_{ij};\psi_i)e^{\cumhaz(t_{i,j1},t_{i,j};\psi_i)} \right)e^{\cumhaz(t_{n_i},\tstop;\psi_i)} ,
\)

(1) 

2. Intervalcensored events: Let $(b_{0}, b_1], (b_{1}, b_2], \ldots , (b_{K1}, b_K]$ be a sequence of successive intervals with $\tstart=b_0<b_1<b_2 < \ldots <b_K = \tstop$. We do not know the exact event times, but a sequence $(m_{ik}; \, 1 \leq k \leq K)$ is observed, where $m_{ik}$ is the number of events that occurred for individual $i$ in interval $(b_{k1}, b_k]$.
 We can show that the conditional pdf of $y_i$ is given by:
 In other words, the number of events per interval for individual $i$ is a (possibly nonhomogeneous) Poisson process with intensity $\cumhaz(b_{k1}, b_k;\psi_i)$ in interval $(b_{k1}, b_k]$.
\(
\pcyipsii(y_i  \psi_i) = \prod_{k=1}^{K} e^{\cumhaz(b_{k1}, b_k;\psi_i)} \displaystyle{\frac{\cumhaz^{m_{ik} }(b_{k1}, b_k;\psi_i)}{m_{ik}!} } .
\)

(2) 
Examples of hazard functions
 Constant hazard model:
 The most simple case is that of a constant hazard function: $\hazard(t;\psi_i) = \hazard_i \in \Rset$. Here, $\psi_i=\hazard_i$.
 Proportional hazards model:
 Here, the hazard is decomposed into two terms: a baseline function $\hazard_0$ of $t$, and an "individual" term, function of some individual covariates $c_i$. $ \langle \beta , c_i \rangle$ means a scalar product, i.e., a linear function of $c_i$. In a proportional hazards model, a unit increase in the value of a covariate has a multiplicative effect on the hazard.
 In the usual proportional hazard model, $\alpha_i$ is a population constant ($\alpha_i=\alpha$). Then, $\psi_i$ can be decomposed into a set of population parameters $\alpha$ and an individual parameter $ \langle \beta , c_i \rangle$. A straightforward extension consists in assuming that $\alpha_i$ is also an individual parameter.
 Extended proportional hazards model:
 Another possible extension assumes that the hazard function is a (possibly nonlinear) function $u$ of a regression variable $x_i$:
 Consider for example that $x_i(t)$ is the plasmatic concentration of a drug at time $t$ for individual $i$. Then, $u(\beta_i,x_i(t))$ is the term that represents (i.e., models) the effect of the drug on the hazard, while $\hazard_0(t;\alpha_i)$ might model the effect of disease progression on the hazard.
 In this example, $x_i(t)$ is the "true" plasmatic concentration for subject $i$ at time $t$, and it is a continuous function of time. However, in practice it is only measured at precise times, so a longitudinal model for plasmatic concentration is needed to give a concentration value for each $t$.
 Therefore, in practice we need to develop a joint model in order to simultaneously model timetoevents data and longitudinal data. Such an approach is introduced in the Joint models section.
 Accelerated failure time (AFT) model:
 Unlike proportional hazards models, the AFT model supposes that a change in a covariate has a multiplicative effect not on the hazard but the predicted event time. This can be written as:
 where $\xi_i$ is a zeromean random variable, e.g., a centered normal distribution. Usually, parameters are fixed effects: $\psi_i=\psi$ for each subject $i$.
 To calculate the hazard function, let us first denote $p_{\xi_i}$ the density and $F_{\xi_i}$ the cdf of $\xi_i$, and to simplify, denote $\mu_i = \langle \psi_i , c_i \rangle$ the mean of $\log(T_i)$. We begin by calculating the survival function:
 Calculating (1) then gives the hazard function:
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Bibliography
Aalen, O., Borgan, O., Gjessing, H.  Survival and Event History Analysis.
Duchateau, L., Janssen, P.  The Frailty Model. Statistics for Biology and Health