Why the Recorded Time May Be an Interval
In many clinical and omics studies, the event time is not observed continuously. A patient may have a negative assessment at one visit and a positive assessment at the next. The event is known to have occurred between those visits, but its exact time is not known. This is not the same observation as an exact event time, and it is not the same as right censoring. The statistical model must respect what was actually observed.
For example, if a participant is known to be event-free at 2 years and the event is detected at the 5-year visit, the event time lies in an interval. A biological assay may be measured once at baseline, while the clinical outcome is monitored only at scheduled visits. In that setting, treating the visit date as the event date can misstate the timing information available for estimating an association.
The endpoint convention matters: an event at \(R\) is included, and an event at \(L\) is excluded. With a continuous event-time distribution, the probability of an event at one specified time is zero, but stating the convention still makes the data representation unambiguous. A visit schedule with no recorded event by the last assessment gives right censoring; a positive assessment following a negative one gives interval censoring.
Likelihood Contributions
Write \(S(t\mid x)=P(T>t\mid x)\) for the survival function conditional on covariates \(x\), and \(F(t\mid x)=P(T\le t\mid x)=1-S(t\mid x)\) for the event-time distribution function. The likelihood contribution is the probability of the observation that was actually recorded, conditional on the covariates.
The subtraction works because \(\{T>R\}\) is contained in \(\{T>L\}\): the difference between those events is exactly \(\{L<T\le R\}\). Equivalently, the contribution is \(F(R\mid x)-F(L\mid x)\). For a left-censored observation known only to satisfy \(T\le R\), the contribution is \(F(R\mid x)=1-S(R\mid x)\). For right censoring at \(L\), it is \(S(L\mid x)\).
If subjects are independent conditional on their covariates and each subject's inspection process is noninformative for the event time, the likelihood for the event-time parameters is the product of these contributions. Noninformative inspection means that, conditional on modeled covariates and the design, assessment timing or missing visits do not carry additional information about the unobserved event time. If visits are more likely to be missed or delayed because an event has occurred or is imminent, the inspection process may be informative and cannot simply be ignored.
Worked Example: Building Contributions Under an Exponential Model
Suppose, hypothetically, event times follow an exponential distribution with a constant hazard \(\lambda=0.20\) per year. Then \(S(t)=e^{-\lambda t}\). Consider three independent participants: one with an event in \((2,5]\) years, one known to be event-free through 4 years, and one whose event is known to have occurred by 3 years.
For the interval-censored event, the contribution is \(S(2)-S(5)=e^{-0.20(2)}-e^{-0.20(5)}=e^{-0.4}-e^{-1}\). Using unrounded exponential values, this is \(0.670320-0.367879=0.302441\), or \(0.3024\) to four significant figures.
For the right-censored participant, the contribution is \(S(4)=e^{-0.80}=0.449329\), or \(0.4493\). The observation says the event time exceeds 4 years; it does not say that an event occurred at year 4.
For the left-censored participant, the contribution is \(F(3)=1-S(3)=1-e^{-0.60}=1-0.548812=0.451188\), or \(0.4512\).
The joint likelihood contribution for these three observations is the product \(0.302441\times0.449329\times0.451188\), approximately \(0.0613\). This product is a likelihood value for the specified model parameter, not a probability that the parameter is correct.
Adding Omics Covariates
An omics measurement can enter an event-time model through its effect on the survival distribution. To make the likelihood concrete, suppose the conditional hazard is exponential: \(\lambda_i=\lambda_0\exp(\beta x_i)\), where \(x_i\) is a baseline omics feature, \(\lambda_0>0\) is the baseline hazard per unit time, and \(\beta\) is the feature coefficient. This model assumes a constant hazard over time for each subject, conditional on \(x_i\), and a log-linear relationship between the feature and hazard. It also assumes the chosen covariate scale and functional form are suitable.
Under this model, \(S_i(t)=\exp\{-\lambda_0\exp(\beta x_i)t\}\). An interval observation therefore contributes \(\exp\{-\lambda_0\exp(\beta x_i)L_i\}-\exp\{-\lambda_0\exp(\beta x_i)R_i\}\). The feature coefficient is estimated by combining these interval probabilities across subjects, rather than by pretending that each interval has a known event time.
Worked Example: An Omics Feature Changes the Interval Probability
Consider a hypothetical one-feature model with \(\lambda_0=0.10\) per year and \(\beta=\log(2)\). Compare a participant with feature value \(x=0\) and one with \(x=1\), each observed to have an event in \((1,4]\) years. The model implies hazards of \(0.10\) and \(0.10\exp\{\log(2)\}=0.20\) per year, respectively.
For \(x=1\), the probability of the recorded interval is \(e^{-0.20(1)}-e^{-0.20(4)}=e^{-0.2}-e^{-0.8}\). Substitution gives \(0.818731-0.449329=0.369402\), or \(0.3694\).
For \(x=0\), the interval probability is \(e^{-0.10(1)}-e^{-0.10(4)}=e^{-0.1}-e^{-0.4}=0.904837-0.670320=0.234517\), or \(0.2345\).
These are probabilities of falling in the specified interval under the two model settings. The positive \(\beta\) makes the modeled hazard twice as large at \(x=1\), but this example alone does not establish a population association or a causal effect. Estimation and uncertainty assessment require data from many participants and the model assumptions described above.
Why Ordinary Cox Regression May Not Fit
The ordinary Cox partial likelihood uses observed event times to form risk sets: at an event time, it compares the covariates of the subject who experienced the event with those of subjects still at risk. With interval-censored outcomes, the event time and often the ordering of events within overlapping intervals are unknown. The risk sets at the unobserved event times therefore cannot be constructed from the recorded data as if those times were exact.
Replacing each interval by its midpoint does not solve this problem. The midpoint is an imputed value, not an observed event time. It can change event ordering, distort risk sets, and understate uncertainty, especially when assessment intervals are wide or vary across participants. Ordinary right-censoring methods are also inadequate if they treat a detected event as occurring at the first positive visit without accounting for the preceding negative assessment.
This does not mean that all proportional-hazards modeling is impossible. A suitable interval-censored method can use a likelihood or other estimation procedure designed for interval observations. The key distinction is between a method that explicitly accommodates the uncertainty about event times and ordinary Cox partial likelihood applied as though exact event times were recorded. Exact-time survival methods are appropriate when the event times really are observed with sufficient precision; ordinary right-censoring methods are appropriate when event times are exact for observed events and otherwise only known to exceed a censoring time.
Record the last known event-free time \(L_i\), the first known event time bound \(R_i\), or the appropriate one-sided censoring information.
Define \(S_i(t)\) or \(F_i(t)\) conditional on covariates and state assumptions such as the hazard form and covariate effect.
For each interval, use \(S_i(L_i)-S_i(R_i)\); for one-sided censoring, use the corresponding \(F_i\) or \(S_i\) contribution.
Assess whether assessment timing, visit attendance, and censoring can reasonably be treated as noninformative given the model and study design.
Likelihood Comparison and Diagnostics
Worked Example: Comparing Two Candidate Hazard Values
Suppose a hypothetical sample of three independent participants has event observations \((0,2]\), \((1,3]\), and right censoring at 2 years. Under an exponential model with common hazard \(\lambda\), the likelihood is \((1-e^{-2\lambda})(e^{-\lambda}-e^{-3\lambda})e^{-2\lambda}\). The first factor is the probability of an event by 2 years, the second is the probability of an event in \((1,3]\), and the last is survival beyond 2 years.
At \(\lambda=0.20\) per year, the three factors are \(1-e^{-0.4}=0.329680\), \(e^{-0.2}-e^{-0.6}=0.269919\), and \(e^{-0.4}=0.670320\). Thus the likelihood is approximately \(0.329680\times0.269919\times0.670320=0.05965\).
At \(\lambda=0.40\) per year, the factors are \(1-e^{-0.8}=0.550671\), \(e^{-0.4}-e^{-1.2}=0.369126\), and \(e^{-0.8}=0.449329\). Their product is approximately \(0.09133\). Among these two candidate values, \(0.40\) per year gives the larger likelihood. This comparison does not prove that \(0.40\) is the maximum-likelihood estimate: the likelihood must be optimized over the full allowed range of \(\lambda\), and uncertainty must also be assessed.
Likelihood calculations are only as credible as their assumptions and data representation. Useful checks include confirming that each interval matches the actual assessment record; inspecting interval widths and visit schedules by cohort, treatment, and omics-feature groups; and checking whether missed visits or study withdrawal are associated with disease status or predictors of the event. A wide interval contains less timing information than a narrow one, even if both are coded as events.
The exponential example is deliberately restrictive: it imposes a constant hazard. A poor fit to the observed event proportions across follow-up, or major discrepancies between predicted and observed interval frequencies, can signal that the event-time distribution is misspecified. Other limitations include measurement error in the omics feature, changes in the feature after baseline, unmodeled confounding, and insufficient information when many subjects have very broad or one-sided intervals. An interval-censored likelihood accounts for coarsened timing; it does not by itself correct these other problems.
The practical choice depends on the measurement process. When exact event times are recorded, exact-time methods preserve that information. When participants are only known to remain event-free until a censoring time, right-censoring likelihoods use survival probabilities. When events are detected only between assessments, interval-censored likelihoods use probabilities over the recorded ranges. Choosing the correct representation is essential before interpreting any estimated omics association.
Check Your Understanding
Answer each question, then compare with the solutions.
- A participant is event-free at year 3 and has an event detected at year 6. State the censoring type and likelihood contribution.
- Under an exponential model with \(\lambda=0.10\) per year, calculate the contribution for an event known to occur in \((2,5]\).
- Why is assigning the midpoint of each interval not equivalent to observing exact event times?
- State one condition under which the inspection schedule may be ignored in the event-time likelihood, and one situation that could violate it.
Solutions. (1) This is interval censoring, with \(L=3\) and \(R=6\); the contribution is \(S(3)-S(6)\). (2) The contribution is \(e^{-0.10(2)}-e^{-0.10(5)}=e^{-0.2}-e^{-0.5}=0.818731-0.606531=0.212200\), or \(0.2122\). (3) The midpoint is an imputed time, not an observed time; it can impose an unsupported event ordering and produce incorrect risk sets. (4) The schedule may be ignored if it is noninformative for event time conditional on the modeled covariates and design. It may be informative if participants with worsening disease are more likely to miss or receive unscheduled visits.