The straightforward procedure to analyze censored data is to assume a parametric model for the failure times. It is possible to fit Accelerated failure time (AFT) models for a variety of distributions to interval censored data. In the statistical area of survival analysis, an accelerated failure time model is a parametric model that provides an alternative to the commonly-used proportional hazards models. A proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant; an AFT model assumes that the effect of a covariate is to multiply the predicted event time by some constant. AFT models can therefore be framed as linear models for the logarithm of the survival time.
The results of AFT models are easily interpreted. For example, the results of a clinical trial with mortality as the endpoint could be interpreted as a certain percentage increase in future life expectancy on the new treatment compared to the control. So a patient could be informed that he would be expected to live (say) 15% longer if he took the new treatment. Hazard ratios are harder to explain in layman’s terms. More probability distributions can be used in AFT models than parametric proportional hazards models. A distribution must have a parameterization that includes a scale parameter to be used in an AFT model. The logarithm of the scale parameter is then modeled as a linear function of the covariates.
The Weibull distribution (including the exponential distribution as a special case) can be parameterized as either a proportional hazards model or an AFT model, and is the only family of distributions to have this property. The results of fitting a Weibull model can therefore be interpreted in either framework. Unlike the Weibull distribution, log-logistic distribution can exhibit a non-monotonic hazard function which increases at early times and will decrease at later times.
Other distributions suitable for AFT models include the log-normal and log-gamma distributions, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form.
The SAS procedure LIFEREG provides a way of fitting accelerated failure time models for a variety of distributions to interval censored data. The AFT model is defined by the transformation
where 
is the failure time random variable for an individual with covariate z and 
is the failure time that the individual would have if they had covariate value 0. The effect of changing covariates is to shrink or stretch the time to event. If 
is negative, then the covariate has the effect of “speeding up time” so that individuals with larger values of z have higher failure rates and hence shorter survival times. The survival function can be written as
where 
is the survival function for an individual with covariate value 0. Taking natural logarithm, the AFT model can be expressed as
If we assume that 
can be expressed as
, where W is a random variable, then the model can be written in a linear model-like form:
The PROC LIFEREG module of SAS fits this model, except that the sign is changed on the regression coefficients. That is, SAS fits
It is possible to include a variety of distributions to be placed on the error term W with SAS, including the log of the exponential, log-normal and log-gamma distributions. The intercept parameter 
and the scale parameter 
are usually not of direct interest, although for some distributions, there is a relationship between the AFT model and a proportional hazards model through the scale parameter. For example, if W is an extreme value distribution (log of a unit exponential), then T has a Weibull distribution. Note that because of the change in sign implicit in the AFT formulation, the direction of covariate effects will be opposite to those fit with a Cox proportional hazards model.
The Kaplan-Meier estimator estimates the survival function for life-time data. In medical research, it might be used to measure the fraction of patients living for a certain amount of time after treatment. An economist might measure the length of time people remain unemployed after a job loss. An engineer might measure the time until failure of machine parts.
A plot of the Kaplan-Meier estimate of the survival function is a series of horizontal steps of declining magnitude which, when a large enough sample is taken, approaches the true survival function for that population. The value of the survival function between successive distinct sampled observations is assumed to be constant.
An important advantage of the Kaplan-Meier curve is that the method can take censored data into account, for instance, if a patient withdraws from a study. When no truncation or censoring occurs, the Kaplan-Meier curve is equivalent to the empirical distribution.
Let 
be the probability that an item from a given population will have a lifetime exceeding t. For a sample from this population of size N let the observed times until death of N sample members be
Corresponding to each 
is
, the number “at risk” just prior to time
, and
, the number of deaths at time
.
Note that the intervals between each time typically will not be uniform. For example, a small data set might begin with 10 cases, have a death at Day 3, a loss (censored case) at Day 9, and another death at Day 11. Then we have
, 
, and
.
The Kaplan-Meier estimator is the nonparametric maximum likelihood estimate of
. It is a product of the form
When there is no censoring, 
is just the number of survivors just prior to time
. With censoring, 
is the number of survivors less the number of losses (censored cases). It is only those surviving cases that are still being observed (have not yet been censored) that are “at risk” of an (observed) death.
Let T be the random variable that measures the time of failure and let 
be its cumulative distribution function. Note that
Consequently, the right-continuous definition of 
may be preferred in order to make the estimate compatible with a right-continuous estimate of
.
With right-censored data, Kaplan and Meier [
In most applications, the data may be interval-censored. By interval-censored data, a random variable of interest is known only to lie in an interval, instead of being observed exactly. In such cases, the only information available for each individual is that their event time falls in an interval, but the exact time is unknown. A nonparametric estimate of the survival function can also be found in such interval censored situations. The survival function is perhaps the most important function in medical and health studies. In this section, the iterative procedure proposed by Turnbull [
Situations where the observed response for each individual under study is either an exact survival time or a censoring time are common in practice. Other situations, however, can occur, and amongst them we find the longitudinal studies, where the individuals are followed for a pre-fixed time period or visited periodically for a fixed number of times. In this context, the time 
until the occurrence of the event of interest for each individual is only known (whenever it occurs) to be within the interval between visits, i.e., between the visit in time 
and the visit in time
. Note that in such studies, the survival times 
are no longer known exactly. It is only known that the event of interest has occurred within the interval 
with
. Furthermore, note that if the event occurs exactly at the moment of a visit, which is very improbable but can happen, then we have an exact survival time. In this case it is assumed that
.
On the other hand, it is known for the individuals with right censoring that the event of interest did not occur until the last visit but it can happen at any time from that moment on. We therefore assumed in this case that 
can occur within the interval 
with 
being equal to the period of time from the beginning of the study until the last visit and
.
Similarly, it is known for the individuals that are left censored, that the event of interest has occurred before the first visit and, hence, we assume that 
falls in the interval 
with 
representing the beginning of the study and 
is the period of time from the beginning of the study until the first visit.
Note from what we have presented so far that exact survival times as well as right and left censored data, are all special cases of interval survival data with 
for exact times, 
for right censoring and 
for left censoring. We can therefore state that interval survival data generalize any situation with combinations of survival times (exact or interval) and right and left censoring that can occur in survival studies.
As usual in the analysis of non-interval survival data, it is also of interest to estimate the survival function 
and to assess the importance of potential prognostic factors. Few statistical software allow for such data, and for this reason a common practice amongst data analysts is to assume that the event occurring within the interval 
has occurred either at the upper/lower limit of the interval or, at the middle point of each interval. Rucker and Messerer [
Peto [
, the survival times for n patients, are independent random variables with right continuous survival function
. If 
are not observed directly, but instead are known to lie in the interval
, then the likelihood for the n observations is,
By
, we mean
which may be different from
, since 
is left continuous. It is important to note that different authors vary in their conventions regarding definition of the censoring interval. The Convention of Peto [
, was followed. This definition facilitates the accommodation of observations that are known exactly, that is, 
, but necessitates the use of the 
notation in above Equation (2.9) to allow a non-zero contribution to the likelihood for these observations. Finkelstein [
, is
. Good arguments against almost any convention can be made for defining the censoring intervals. In practice, the choice will have little impact and all reasonable conventions can be adopted.
Turnbull [
Since the observed event times are known to occur only within potentially overlapping intervals, the survival curve can only jump within so-called equivalence sets
, 
, where
. The curve between 
and 
is flat. The estimate of 
is unique only up to these equivalence classes; any function that jumps the appropriate amount within the equivalence class will yield the same likelihood.
An analog of the Product-Limit estimator of the survival function for interval-censored data is presented in this section. This estimator, which has no closed form, is based on an iterative procedure and has been suggested by Turnbull [
To construct the estimator, let 
be a grid of time which includes all the points 
and 
for
. For the 
observation, define a weight 
to be 1 if the interval 
is contained in the interval 
and 0, otherwise. The weight 
indicates whether the event which occurs in the interval 
could have occurred at
. An initial guess at 
is made and Turnbull’s algorithm is as follows:
Step 1: Compute the probability of an event occurring at time 
by
Step 2: Estimate the number of events which occurred at 
by
Step 3: Compute the estimated number at risk at time
by 
Step 4: Compute the updated Product-Limit estimator using the pseudo data found in Steps 2 and 3. If the updated estimate of S is close to the old version of S for all
’s, stop the iterative process, otherwise repeat Steps 1- 3, using the updated estimate of S.
Kooperberg and Stone [
Kooperberg and Stone [
is assumed. Examples are: models based on the exponential, Weibull, gamma and generalized F distributions. A second approach might be called a flexible or semi-parametric strategy, where mild assumptions are made about the baseline hazard
. Specifically, time is subdivided into reasonably small intervals and it is assumed that the baseline hazard is constant in each interval leading to a piecewise exponential model. The third approach is a non-parametric strategy that focuses on estimation of the parameters leaving the baseline hazard 
completely unspecified. This approach relies on a partial likelihood function proposed by Cox [
, if resistance was seen to have occurred, and a high proportion of right-censored observations. Because of the sparseness of these data, this is a challenging data set to analyze. The variables of interest were the effects of stage of disease,
