Discrete-time survival models
The methods we have discussed so far are only applicable to continuous survival times. In the discrete time frame, we assume that the available data are the same, but we define the hazard function and the link between the hazard and survival functions differently. We divide the continuous survival time into a sequence of J contiguous time intervals (t0,t1](t1,t2],…,(tJ-1,tJ]where t0=0. In this framework, the hazard, or instantaneous risk of the event, in a given interval is the probability that an individual will experience the event during that interval, given that they have survived to the beginning of that interval. Therefore, in discrete time, the hazard is a conditional probability rather than a rate, and as such its value lies between zero and one. Therefore, for an individual with reference covariates Xidanger in interval AJ=(tJ-1,tJ]can be expressed as a conditional probability
$$ \begin{aligned} \lambda_{ij}(X_{i})&=\text{Pr}(T_{i}\in A_{j}|T_{i}>t_{j-1},X_ {i})\\ &= \text{Pr}(t_{j-1}
and the discrete likelihood function is given by
$$f_{ij}=\text{Pr}(T_{i}\in A_{j}|X_{i}) = S(t_{j-1}|X_{i})-S(t_{j }|X_{i}) $$
The probability of surviving beyond a certain time t can be obtained as the product of the conditional survival probabilities for all time intervals up to and including (tJ-1,tJ], so that tJ≤t. This is analogous to specifying the continuous-time survival function as the integrated hazard over all previous times. Thus, the discrete-time survival probability is given by
$$ S_{i}(t|X_{i}) = \text{Pr}(T_{i}>t|X_{i})=\prod_{j:t_{j}\leq t}(1- \lambda_{ij}(X_{i})) $$
(1)
Note that the relationship lij(Xi)=fij/Si(tJ−1|Xi) still holds under these definitions. Build likelihood, subject i yields the product of the conditional survival probabilities for the time intervals in which the event is observed but not experienced. People who are observed to have a failure (i.e. di=1) also contribute to the probability of conditional failure in the interval \(A_{j_{i}}=(t_{j_{i}}1},t_{j_{i}}]\) in which they experience the event of interest.We use Ji to indicate the last interval during which we have information on the subject i, so that \(\phantom {\dot {i}\!}T_{i}\in A_{j_{i}}\). subject i it does not contribute any information to the probability of intervals beyond \(\phantom {\dot {i}\!}A_{j_{i}}\). Here, we only consider right censoring, so the probability is given by
$$\begin{array}{*{20}l} {}L&=\!\prod_{i=1}^{n} \left[\text{Pr}(T_{i}=t_{j_{i}})\right]^{\delta_{i}}\left[\text{Pr}(T_{i}>t_{j_{i}})\right]^{1-\delta_{i}}\\ {}&=\!\prod_{i=1}^{n}\left[\!\lambda_{ij_{i}}(X_{i})\prod_{j=1}^{j_{i}-1}(1-\lambda_{ij}(X_{i}))\!\right]^{\delta_{i}}\left[\prod_{j=1}^{j_{i}}(1-\lambda_{ij}(X_{i}))\right]^{1-\delta_{i}} \end{array} $$
where \(t_{j_{i}}\) indicates this subject i has a survival time in the interval \(\phantom {\dot {i}\!}(t_{j_{i}-1}, t_{j_{i}}]\). We can introduce an indicator of event history \( \phantom {\dot {i}\!}d_{ij}=I(T_{i}\in A_{j_{i}})=I(t_{j-1}
$$ L=\prod_{i=1}^{n}\prod_{j=1}^{j_{i}}\lambda_{ij}(X_{i})^{d_{ij}}(1- \lambda_{ij}(X_{i}))^{1-d_{ij}} $$
(2)
which is equivalent to the likelihood of a binomial model with independent observations dijsubject-specific probabilities lij(Xi) for the matter i experiencing the event in the interval (tJ-1,tJ], and time-fixed covariates Xi. Note that we do not assume that event indicators within a subject are independent and have a binomial distribution. Instead, we note that the likelihood function for the discrete-time survival model under uninformative censoring can be represented by a binomial likelihood that assumes independent event indicators. [27].
To construct this probability from our data, we need to convert it to a person-period dataset, as shown in Figure 1. Subjects contribute one row for each time interval in which they are still in risk at first, that is, all of them J so that T>tJ−1. Each record contains the subject’s failure indicator to experience the event during that interval (i.e., its event history indicator dij), a copy of its reference covariate vector Xiand a factor variable that identifies the interval AJ to which the record corresponds.
Fig. 1
Example of a person-period dataset (right) created from continuous-time survival data (left). In the timeline plot, circles indicate censoring and diamonds indicate events. The horizon of interest is w=5 and there are J=5 specified intervals set to 1: A1=(t0,t1]2: A2=(t1,t2]3: A3=(t2,t3]4: A4=(t3,t4]5: A5=(t4,t5], whose ends are given by t0=0, t1=1, t2=2, t3=3, t4=4, t5=5. ID 1 experiences an event in interval 3 and therefore in the period dataset person has rows corresponding to the first three intervals and for the 3rd interval its event status is 1. The ID 2 is censored in the interval 4 and in the person. -Periods dataset have row corresponding to the first 4 intervals and have an event status of 0 for all of them. ID 3 experiences the event at a time beyond the prediction horizon of interest, so we administratively censor them at the prediction horizon and have a row in the person period for all intervals and have a event status of 0 for all of them.
Due to the binomial structure of the likelihood function in Eq. (2) the discrete survival time formulation is general and any algorithm that can optimize a binomial log-likelihood can be used to obtain parameter estimates. Thus, within this approach we can apply any method to calculate the probability of a binary event and we can choose from various binary classification methods, from traditional regression methods to more complex machine learning approaches. Estimation can be carried out by applying these methods to the data set of the described person period. The resulting estimates can then be used in Eq. (1) to calculate the predicted survival probability for a particular follow-up time.
The advantage of a discrete-time survival approach is that it does not require a proportional hazards assumption for the survival time distribution. Furthermore, it provides a more intuitive interpretation, as the hazard function represents the probability of experiencing the event in a given interval given that the person is alive at the beginning of the interval. Discrete-time models are also able to handle bound failure times without adjustments [26]as required in Cox PH modeling due to its assumption of a continuous hazard in which links are not possible [3].
Parametric classification models
Cox (1972) proposed that since in discrete time hazards, lijare probabilities, they can be parameterized to have a logistic dependence with the predictors and time intervals [4, 27]. That is, we assume that the predictors are linearly associated with the logistic transformation of the hazard (logit-hazard) instead of the hazard probabilities themselves. Specifically, the conditional probabilities of experiencing a failure in each time interval (tJ-1,tJ](given that it has not yet occurred) is assumed to be a linear function of the predictor and interval effects. This model is known as a continuation relationship modeland is specified as
$$ \log\left(\frac{\lambda_{ij}}{1-\lambda_{ij}}|X_{i}\right) = \alpha_{j} + \beta X_{i} $$
(3)
where aJ is the logit of the reference risk for the interval (tJ-1,tJ]i b describes the effect of the other covariates on the baseline hazard on the logit scale, as in a logistic regression. That is, for a binary predictor Xipa positive (negative) coefficient estimate for a bp indicates an upward (downward) change in the logit hazard for those with Xip=1 of the logit-hazard for those with Xip=0. Thus, taking the exponential on both sides of Eq. (3), we see that the probability is proportional for those with Xip=1 vs. 0. This property restricts the discrete hazards of both crossover groups. This rules out the scenario where the predictors have short-term protective effects on survival, but are inferior at later time points. Note also that this effect is not a function of time and is therefore assumed to be constant for all periods.
In the probability given in Eq. (2), the continuation relation model assumes that the ranking model for lij(Xi) is a logistic regression. Estimation is performed by obtaining maximum likelihood estimates of aJ i b using standard logistic regression software applied to the person-period dataset. With budgets for aJ i bthen we can calculate lkj(Xk) for a new topic k of the same population for all time intervals J, and then calculate the survival prediction given in Eq. (1).
With the logistic model, when the magnitude of the hazards is small, we find that the failure probabilities approach the probability of failure (i.e. lij/(1−lij)≈lij) and that the hazards of the proportional odds model and the proportional hazards model are close. The continuation relation model is shown to converge to the Cox model as the length of the discrete time intervals goes to zero (ie, the number of time intervals in a fixed period increases) [45].
In the parameterization g(lij|Xi)=aJ+bXiwe can also consider models with alternative link functions for g which are commonly used for binary results. The Gompertz or pooled proportional hazards model that uses a complementary log-log link, log(− log(lij|Xi)), is a discrete-time equivalent to a Cox PH model. Other parameterizations include the probit model (probit link), Phi-1(lij|Xi), the…