Interval-Censored Survival Data Analysis

The contribution is:

Patient 04

Patient 04 remained progression-free through Week 32.

Therefore:

The contribution is:

Patient 05

Patient 05 was already known to have progression by the first post-baseline assessment at Week 8.

Therefore:

The contribution is:

Complete Likelihood for the Example

Ignoring covariates for simplicity, the likelihood for these five patients is:

This expression contains all of the available survival information.

No artificial progression dates are required.

How the Data Should Be Stored

A useful interval-censored dataset typically contains at least:

A common convention is:

• Finite left and finite right = interval censored
• right = Inf = right censored
• left = 0 with finite right = left censored

R: Creating Interval-Censored Data

The following example creates a simple dataset:

dat <- data.frame(
  id = 1:10,

  left = c(
    8, 16, 24, 32, 0,
    8, 16, 24, 32, 40
  ),

  right = c(
    16, 24, 32, Inf, 8,
    16, 24, 32, 40, Inf
  ),

  treatment = c(
    "Control", "Control", "Control", "Control", "Control",
    "Treatment", "Treatment", "Treatment", "Treatment", "Treatment"
  )
)

dat

Using Surv() for Interval-Censored Data

The survival package provides a useful data representation through Surv().

For interval-censored observations represented by a left and right endpoint, the type = "interval2" representation is useful.

library(survival)

y <- with(
  dat,
  Surv(left, right, type = "interval2")
)

y

The important distinction is that the two columns represent an interval rather than an exact event time and an ordinary right-censoring indicator.

R: Parametric Interval-Censored Model

The survival package can fit parametric survival models using interval-censored data.

For example, a Weibull accelerated failure time model can be fitted using survreg():

library(survival)

fit_weibull <- survreg(
  Surv(left, right, type = "interval2") ~ treatment,
  data = dat,
  dist = "weibull"
)

summary(fit_weibull)

The coefficient for treatment is interpreted on the log-time scale.

The corresponding time ratio is:

exp(coef(fit_weibull)["treatmentTreatment"])

A time ratio greater than 1 indicates longer event times for the treatment group under the AFT model.

R: Log-Normal Model

A log-normal interval-censored model can be fitted similarly:

fit_lognormal <- survreg(
  Surv(left, right, type = "interval2") ~ treatment,
  data = dat,
  dist = "lognormal"
)

summary(fit_lognormal)

Comparing several plausible parametric distributions can help assess the sensitivity of conclusions to distributional assumptions.

Turnbull's NPMLE in R

Several R packages provide specialized tools for interval-censored survival analysis.

One commonly used package is icenReg, which provides nonparametric and regression methods for interval-censored data.

install.packages("icenReg")

library(icenReg)

A nonparametric interval-censored model can then be fitted using:

fit_np <- ic_np(
  Surv(left, right, type = "interval2") ~ 1,
  data = dat
)

summary(fit_np)

The estimated survival distribution can be examined using the package's plotting methods.

plot(fit_np)

Interval-Censored Regression with icenReg

Interval-censored regression can incorporate treatment and baseline covariates directly.

For example, a semiparametric proportional hazards model can be fitted with an appropriate icenReg function:

fit_ph <- ic_sp(
  Surv(left, right, type = "interval2") ~ treatment,
  data = dat,
  model = "ph"
)

summary(fit_ph)

A proportional odds formulation is also available:

fit_po <- ic_sp(
  Surv(left, right, type = "interval2") ~ treatment,
  data = dat,
  model = "po"
)

summary(fit_po)

These models provide alternatives to simply imputing event times and fitting ordinary right-censored regression.

What Does the Proportional Hazards Model Assume?

The proportional hazards model assumes:

The hazard ratio does not depend on time under the standard proportional hazards assumption.

The interval-censored estimation procedure must nevertheless account for the fact that the exact failure time is unknown.

Checking the Proportional Hazards Assumption

The usual proportional-hazards diagnostic based on exact event-time residuals is not automatically transferable to interval-censored data.

Instead, investigators should use diagnostics appropriate to the fitted interval-censored model.

Potential approaches include:

• Comparing fitted survival curves across treatment groups
• Evaluating alternative model specifications
• Including treatment-by-time interactions where supported
• Comparing proportional hazards and alternative semiparametric models
• Performing sensitivity analyses using parametric models

Comparing Treatment Groups

When there are two treatment groups, one objective is to compare their survival distributions.

For ordinary right-censored data, the log-rank test is commonly used.

For interval-censored data, specialized tests are preferable because the standard risk sets cannot simply be constructed from artificial event times.

Depending on the analysis method and software, comparisons may be based on likelihood-ratio tests, score tests, Wald tests, or interval-censored nonparametric procedures.

Confidence Intervals

Inference for interval-censored models can use model-based standard errors, likelihood-based confidence intervals, or resampling methods, depending on the estimator.

For a regression coefficient:

a Wald confidence interval can have the familiar form:

For a hazard ratio:

the confidence interval is obtained by exponentiating the corresponding coefficient-scale limits.

Confidence Intervals for Survival Probabilities

Suppose the estimated survival probability at Week 52 is:

The confidence interval should reflect the estimation method used for the interval-censored survival distribution.

The method should be documented rather than simply borrowing the default Greenwood calculation from an ordinary Kaplan-Meier analysis.

Median Survival Under Interval Censoring

The median survival time is the time \(t_{0.5}\) satisfying:

With interval-censored data, however, the data may not identify the median as precisely as in an exact-event analysis.

In some datasets, the estimated survival curve may remain above 0.5 across an interval or may cross 0.5 only within a region determined by the observation intervals.

Restricted Mean Survival Time

Restricted mean survival time (RMST) is defined as:

For interval-censored data, an estimated survival distribution can be used to obtain an RMST estimate.

This can be attractive when proportional hazards is questionable because RMST does not require a proportional-hazards interpretation.

A treatment contrast can be expressed as:

The clinical interpretation is the average event-free time gained or lost over the prespecified horizon \(\tau\).

Parametric vs. Nonparametric Analysis

A strong clinical analysis often considers both approaches rather than assuming that one method is universally superior.

Choosing a Parametric Distribution

If a parametric analysis is planned, candidate distributions should be chosen based on scientific plausibility and evaluated using appropriate diagnostics.

For example:

• Exponential: constant hazard
• Weibull: monotonic hazard that can increase or decrease
• Log-normal: flexible non-monotonic hazard
• Log-logistic: flexible hazard and survival shape

Model selection should not be based solely on whichever distribution produces the smallest information criterion.

Using AIC and BIC

Information criteria can assist in comparing candidate parametric models.

The Akaike Information Criterion is:

where \(\ell\) is the maximized log-likelihood and \(k\) is the number of estimated parameters.

The Bayesian Information Criterion is:

Lower values are preferred within the set of models being compared.

But AIC and BIC Are Not Proof of Model Adequacy

AIC or BIC should not be treated as evidence that a model is correct.

Two misspecified models can have different AIC values.

Model evaluation should also consider:

• Observed versus fitted survival behavior
• Residual or goodness-of-fit diagnostics appropriate to the model
• Clinical plausibility
• Stability of treatment-effect estimates
• Sensitivity to alternative distributions

Interval-Censored Data and Missing Visits

A missed assessment can create a wider interval.

Suppose a patient was last confirmed progression-free at Week 8 but missed the Week 16 scan and is found to have progression at Week 24.

The event is now known only to satisfy:

set.seed(2026)

n <- 400

treatment <- rbinom(
  n,
  size = 1,
  prob = 0.5
)

# Example Weibull event-time generation
shape <- 1.4
scale_control <- 18
scale_treatment <- 24

scale <- ifelse(
  treatment == 1,
  scale_treatment,
  scale_control
)

u <- runif(n)

true_time <- scale *
  (-log(u))^(1 / shape)

# Scheduled assessments
visits <- seq(4, 48, by = 4)

left <- numeric(n)
right <- numeric(n)

for(i in seq_len(n)) {

  positive <- visits[
    visits >= true_time[i]
  ]

  if(length(positive) == 0) {

    left[i] <- max(visits)
    right[i] <- Inf

  } else {

    right[i] <- min(positive)

    previous <- visits[
      visits < right[i]
    ]

    left[i] <- ifelse(
      length(previous) == 0,
      0,
      max(previous)
    )
  }
}

simdat <- data.frame(
  treatment,
  left,
  right
)

head(simdat)