Interval Censoring and Panel Data · Tutorial 521 of 699

Accelerated Failure Time Regression for Interval-Censored Data

Build the interval-censored likelihood for an accelerated failure time model, then interpret its fitted coefficients as multiplicative changes in event-time quantiles.

Advanced 12 min read

What You'll Learn

Write likelihood contributions for interval-, left-, and right-censored observations under an accelerated failure time model.
State the assumptions needed to use the interval likelihood for regression.
Interpret a regression coefficient as a time ratio for event-time quantiles.
Calculate a likelihood from hypothetical inspection intervals.
Use likelihood maximization to fit the model and assess basic signs of poor fit.
Distinguish accelerated failure time effects from proportional hazards effects.

Why model event time on a logarithmic scale?

With interval-censored data, the event time is not known exactly: inspections show only that it occurred after one visit and by a later visit. Regression can relate event timing to covariates without pretending that an inspection time is the event time. An accelerated failure time (AFT) model does this by expressing log event time as a covariate-dependent location plus random variation.

The key interpretation is a time scale: a covariate changes event-time quantiles by a multiplicative factor. To fit the model with interval observations, each subject contributes the probability that their event time fell inside the observed interval. The likelihood must therefore use interval probabilities, rather than a density evaluated at an invented event time.

The AFT model and interval likelihood

Definition: For subject $i$, let $T_i>0$ be the event time, $\mathbf{x}_i$ a covariate vector, and $\eta_i=\beta_0+\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}$. A log-location-scale AFT model is $$ \log T_i=\eta_i+\sigma\epsilon_i,\qquad \sigma>0, $$ where the errors $\epsilon_i$ are independent with a specified continuous distribution function $F_0$. Thus $P(T_i\le t\mid\mathbf{x}_i)=F_0((\log t-\eta_i)/\sigma)$ for $t>0$. In the lognormal AFT model used in the examples, $F_0=\Phi$, the standard normal distribution function.

For an observed interval $(L_i,R_i]$, where $0\le L_i<R_i\le\infty$, the event is known to have occurred after $L_i$ and no later than $R_i$. Define $F_i(t)=P(T_i\le t\mid\mathbf{x}_i)$. The subject's likelihood contribution is $F_i(R_i)-F_i(L_i)$, with $F_i(0)=0$ and $F_i(\infty)=1$. In the lognormal model, when both endpoints are finite and positive, this is $$ P(L_i<T_i\le R_i\mid\mathbf{x}_i) = \Phi\left(\frac{\log R_i-\eta_i}{\sigma}\right) - \Phi\left(\frac{\log L_i-\eta_i}{\sigma}\right). $$

The endpoint conventions give the special cases directly. For left censoring, $T_i\le R_i$, the contribution is $F_i(R_i)$. For right censoring, $T_i>L_i$, it is $1-F_i(L_i)$. An exact event time, if observed, contributes a density rather than an interval probability; interval-censored analyses should not replace a nonzero-width interval with its midpoint and treat that midpoint as exact.

Assuming subjects' event times are independent conditional on their covariates, the event-time likelihood is the product of these contributions. If subject $i$ has observed interval $(L_i,R_i]$, then

L(\boldsymbol{\theta})=\prod_{i=1}^n\{F_i(R_i)-F_i(L_i)\}, \qquad \ell(\boldsymbol{\theta})=\sum_{i=1}^n \log\{F_i(R_i)-F_i(L_i)\},

where $\boldsymbol{\theta}$ includes the regression coefficients and error-scale parameter. The maximum likelihood estimate maximizes $\ell$. In practice, numerical optimization evaluates the interval probabilities repeatedly as the parameters change; log likelihoods are used to avoid underflow from multiplying many small probabilities.

Assumptions and a likelihood justification

The AFT model and its likelihood are useful only when their assumptions are plausible. The model specifies a common error distribution and scale structure, with covariate effects entering through $\eta_i$. For the lognormal model, this means log event times have normal errors with a common standard deviation $\sigma$, conditional on the covariates. The conditional independence assumption means that subjects' event times are independent given their covariates. Dependence, such as clustering by clinic, requires an appropriate extension or variance treatment.

The observation process must also be handled correctly. The usual likelihood above is appropriate when inspection and censoring are noninformative for the event time, conditional on modeled covariates: the visit pattern must not provide additional information about an unobserved event time after conditioning on those covariates. If inspection timing depends on symptoms that also predict the event, ignoring that dependence can bias estimation. The interval endpoints and the meaning of the event must be recorded consistently.

The central likelihood identity follows from the observed information. Given $L_i<T_i\le R_i$, the probability of the observation is the conditional probability that $T_i\le R_i$ minus the probability that $T_i\le L_i$. Since the events $\{T_i\le L_i\}$ and $\{T_i\le R_i\}$ are nested,

P(L_i<T_i\le R_i\mid\mathbf{x}_i) = P(T_i\le R_i\mid\mathbf{x}_i) - P(T_i\le L_i\mid\mathbf{x}_i) = F_i(R_i)-F_i(L_i).

This derivation explains why neither endpoint alone is an event-time observation. The probability mass between the endpoints is the information the likelihood must use.

Worked Example: A Lognormal Interval Contribution

Suppose a hypothetical subject has an event interval $(2,4]$ months, and consider the candidate model $\eta=\log 4$, $\sigma=0.5$. The event time is interval-censored, so the contribution is the probability between two bounds, not a density at month 3.

The standardized bounds are $$ z_L=\frac{\log 2-\log 4}{0.5}=-1.3863, \qquad z_R=\frac{\log 4-\log 4}{0.5}=0. $$

Therefore the interval probability is $$ \Phi(0)-\Phi(-1.3863)\approx 0.5000-0.0828=0.4172. $$

At these candidate parameter values, this subject contributes approximately 0.4172 to the likelihood. This calculation checks the contribution for a given parameter value; it does not by itself estimate the parameters.

From fitted coefficients to time quantiles

Let $q_p(\mathbf{x})$ denote the conditional $p$th event-time quantile, for $0<p<1$. Solving $P(T_i\le q_p\mid\mathbf{x}_i)=p$ gives $$ q_p(\mathbf{x})=\exp\{\eta(\mathbf{x})+\sigma F_0^{-1}(p)\}. $$

For two covariate vectors $\mathbf{x}_a$ and $\mathbf{x}_b$, assuming the same error distribution and scale, the quantile ratio is $$ \frac{q_p(\mathbf{x}_b)}{q_p(\mathbf{x}_a)} = \exp\{(\mathbf{x}_b-\mathbf{x}_a)^\mathsf{T}\boldsymbol{\beta}\}. $$ Thus the ratio is the same for every $p$: the model shifts event times proportionally on the original time scale. A positive coefficient for a binary covariate means longer event times for the group coded 1 than for the reference group, conditional on other covariates. A negative coefficient means shorter times. These are model-based associations, not proof that changing the covariate would cause event times to change.

Worked Example: Interpreting a Binary Covariate

Suppose a hypothetical lognormal AFT fit for time to an event, measured in months, gives $\hat\beta_0=\log 6$, $\hat\beta_1=\log 1.5=0.4055$, and $\hat\sigma=0.4$. The covariate $x$ is 0 for the reference group and 1 for the comparison group. The estimated time ratio is $$ \exp(\hat\beta_1)=\exp(0.4055)=1.500. $$

At every fixed event-time quantile, the comparison group's estimated time is 1.5 times the reference group's, conditional on the model. For example, the median is $q_{0.5}(0)=\exp(\log 6)=6$ months and $q_{0.5}(1)=\exp(\log 6+\log 1.5)=9$ months.

For the 90th percentile, using $F_0^{-1}(0.90)=1.2816$, the estimates are $$ q_{0.90}(0)=\exp(\log 6+0.4(1.2816))\approx 10.02\text{ months}, \quad q_{0.90}(1)=\exp(\log 9+0.4(1.2816))\approx 15.03\text{ months}. $$ Their ratio is approximately $15.03/10.02=1.500$, as required by the model.

If the estimated standard error of $\hat\beta_1$ is 0.18, a large-sample 95% Wald interval on the log time-ratio scale is $0.4055\pm1.96(0.18)=[0.0527,0.7583]$. Exponentiating the endpoints gives an approximate 95% confidence interval of $[1.054,2.135]$ for the time ratio. Under the fitted model and the usual large-sample approximation, the data are compatible with a comparison-group event-time quantile about 1.054 to 2.135 times the reference-group quantile. This interval does not assign a 95% probability to the fixed unknown ratio.

Worked likelihood and what fitting means

Worked Example: Multiplying Contributions From Four Records

Consider four hypothetical independent subjects with no covariates, under the candidate lognormal model $\eta=\log 4$ and $\sigma=0.5$. Their observed records are $(2,4]$ months, $(4,8]$ months, right-censored after 4 months, and left-censored by 2 months. These records use all three contribution forms.

For $(2,4]$, the contribution from the preceding calculation is $0.4172$. For $(4,8]$, the standardized bounds are 0 and $1.3863$, so the contribution is $\Phi(1.3863)-\Phi(0)\approx0.9172-0.5000=0.4172$. For right censoring after 4 months, the contribution is $1-\Phi(0)=0.5000$. For left censoring by 2 months, it is $\Phi(-1.3863)\approx0.0828$.

The likelihood at these candidate parameter values is therefore $$ L\approx(0.4172)(0.4172)(0.5000)(0.0828)\approx0.00721, \qquad \ell=\log L\approx-4.933. $$

This is a likelihood value, not a probability that the parameter values are correct. To fit the model, an optimizer varies $\beta_0$ and $\sigma$ and selects values that maximize the log likelihood. With covariates, it also varies the regression coefficients. A reported estimate should be accompanied by its uncertainty and by checks of whether the distributional and observation-process assumptions are credible.

Diagnostics, alternatives, and limitations

Basic checks should address both the numerical fit and the modeling assumptions. Confirm that intervals are ordered and that left- and right-censoring records have been encoded with the correct endpoint conventions. Check optimizer convergence from more than one starting value, inspect whether the scale estimate is positive and finite, and look for unusually large standard errors or estimates drifting toward parameter boundaries. These can indicate weak information, a flat likelihood, or numerical trouble.

For model checking, compare fitted probabilities of observations or grouped interval frequencies with those seen in the data, and examine whether the fitted distribution gives plausible event-time quantiles across covariate patterns. If visits are sparse, broad intervals may leave many different event-time distributions similarly plausible; apparently precise regression estimates then deserve scrutiny. A mismatch between observed and fitted interval probabilities may signal an unsuitable lognormal error distribution, an inadequate covariate specification, or informative inspection timing. Such checks do not prove the model correct.

Approach	What it provides	Important distinction
Parametric AFT regression	Covariate effects on event-time quantiles through a time ratio; a specified event-time distribution	Interpretation and estimates depend on the chosen error distribution and AFT structure
Proportional hazards regression	Covariate effects on the hazard under a proportional hazards structure	Its primary effect is a hazard ratio, not the AFT time ratio
Turnbull estimation	A nonparametric estimate of the event-time distribution for interval-censored data	It does not by itself provide the same regression-based covariate interpretation

The proportional hazards model is an alternative regression structure, not a synonym for AFT regression. Some parametric distributions can connect the two structures, but the relationship is not universal. Turnbull estimation avoids specifying a particular parametric event-time distribution, but it does not substitute for a covariate model when adjusted regression effects are the target.

Common mistake

Common mistake: Replacing every interval $(L_i,R_i]$ with its midpoint and analyzing that midpoint as an exact event time. This treats information the study did not observe as if it were known, and it changes the likelihood from an interval probability to an exact-time density. The correction is to use $F_i(R_i)-F_i(L_i)$ for the interval, with the appropriate left- or right-censoring contribution at an open endpoint.

A second interpretive trap is to call $\exp(\beta_j)$ a hazard ratio. In an AFT model it is a time ratio for corresponding conditional event-time quantiles when the covariate changes by one unit, holding other covariates fixed. Its units are a ratio of times, so it is dimensionless; the quantiles themselves retain the time units used in the data.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

Under a lognormal AFT model with $\eta=\log 5$ and $\sigma=0.5$, write the likelihood contribution for an observation known to occur in $(5,10]$ weeks.
A fitted binary covariate coefficient is $-0.2231$. What is the estimated event-time quantile ratio for covariate 1 versus 0, and how should it be interpreted?
For a subject right-censored after time $L$, what is the likelihood contribution and why is it not $F_i(L)$?
Give one reason that replacing interval-censored observations by interval midpoints can lead to misleading estimates.

Solutions. (1) The standardized bounds are $0$ and $\log(2)/0.5=1.3863$, so the contribution is $\Phi(1.3863)-\Phi(0)\approx0.9172-0.5000=0.4172$. (2) The ratio is $\exp(-0.2231)\approx0.8000$. The model estimates event-time quantiles about 0.8000 times as large for covariate 1 as for covariate 0, holding other covariates fixed. (3) The contribution is $1-F_i(L)$, the probability $T_i>L$; $F_i(L)$ is instead the probability the event has already occurred by $L$. (4) A midpoint is not an observed event time. Treating it as exact discards the interval uncertainty and uses the wrong likelihood contribution.

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