Interval Censoring and Panel Data · Tutorial 519 of 699

Uncertainty Quantification for Interval-Censored Estimates

Compare likelihood and subject-level bootstrap uncertainty for Turnbull masses and distribution values, and learn how support changes and identifiability affect interpretation.

Advanced 11 min read

What You'll Learn

Write the interval-censored likelihood for masses on Turnbull support cells.
Use information-based standard errors when the support is fixed and the estimate is regular.
Recognize when a distribution value is not identified by the observed intervals.
Resample subjects and recompute Turnbull estimates to assess sampling variability.
Diagnose bootstrap replicates in which cells disappear or change meaning.
Choose uncertainty summaries cautiously near boundaries or under weak identification.

Why uncertainty for Turnbull estimates needs care

A Turnbull estimate summarizes the distribution of event times using probability masses on support cells compatible with the observed intervals. An estimated mass is not automatically accompanied by a trustworthy standard error: the likelihood can be flat in some directions, fitted masses can lie on the boundary, and the support itself can look different in a resampled data set. These features affect both likelihood-based intervals and bootstrap uncertainty estimates.

The goal here is to quantify uncertainty for a specified, identifiable feature of the fitted distribution. It is not enough to ask for an interval around a number produced by software. First check that the number is determined by the observed-data likelihood; then assess whether the regularity conditions behind the chosen uncertainty calculation are plausible.

Definition: Suppose the interval-censored observations generate $m$ Turnbull support cells. Let $p_j$ be the probability mass assigned to cell $j$, where $p_j\geq 0$ and $\sum_{j=1}^m p_j=1$. If observation $i$ is compatible with cell $j$, set $a_{ij}=1$, and otherwise set $a_{ij}=0$. Its likelihood contribution is $\sum_{j=1}^m a_{ij}p_j$, so the observed-data likelihood is $$ L(\mathbf p)=\prod_{i=1}^n\left(\sum_{j=1}^m a_{ij}p_j\right), \qquad \ell(\mathbf p)=\sum_{i=1}^n\log\left(\sum_{j=1}^m a_{ij}p_j\right). $$ The Turnbull estimate maximizes this likelihood over the probability simplex.

This form assumes independent subjects, correctly recorded observation intervals, and a censoring or inspection mechanism that is ignorable for the likelihood being used. It also presumes the support representation is appropriate for the observed intervals. As established in the earlier material on identifiability limits in panel-observed models, distinct mass allocations can sometimes yield the same observed-data likelihood. In that case, uncertainty calculations must respect what is and is not identified.

Likelihood-based uncertainty on a fixed support

When the support cells are treated as fixed, the maximum is unique, all relevant masses are interior, and the information matrix is nonsingular after accounting for the sum-to-one constraint, a local likelihood approximation can quantify uncertainty. One may parameterize the simplex using $m-1$ free coordinates, calculate the observed information from the second derivatives of $\ell$, and use its inverse as an approximate covariance matrix. For a smooth target $g(\mathbf p)$, the delta method gives $$ \widehat{\operatorname{Var}}\{g(\widehat{\mathbf p})\} \approx \nabla g(\widehat{\mathbf p})^\mathsf{T} \widehat{\operatorname{Var}}(\widehat{\mathbf p}) \nabla g(\widehat{\mathbf p}). $$

The approximation is local: it describes curvature near the fitted value. A Wald interval takes the estimate plus or minus a normal critical value times its estimated standard error. A profile-likelihood interval instead retains target values whose constrained log likelihood is not too far below its maximum. The usual chi-square calibration for a one-dimensional profile likelihood relies on regularity, including an identifiable interior target; it is not automatically valid at a boundary or under nonidentifiability.

Worked Example: Information and a Wald Interval for a Turnbull Mass

Suppose 12 hypothetical independent subjects each have an interval that identifies which of two disjoint Turnbull cells contains the event. Eight intervals identify cell 1 and four identify cell 2. Let $p$ denote the mass on cell 1; cell 2 has mass $1-p$. The likelihood is $L(p)=p^8(1-p)^4$, and the maximum-likelihood estimate is $\widehat p=8/12=0.6667$.

Check the conditions. The two cells are fixed and distinguishable in this constructed example, the observations are independent, and $0<\widehat p<1$. The target is identifiable because the intervals distinguish the cells. Thus the ordinary interior information calculation is appropriate as a large-sample approximation, although 12 subjects is small enough that the interval should be viewed cautiously.

The log likelihood is $\ell(p)=8\log(p)+4\log(1-p)$. Its negative second derivative is $8/p^2+4/(1-p)^2$, which at the maximum is $12/\{\widehat p(1-\widehat p)\}$. Therefore the estimated variance and standard error are $$ \widehat{\operatorname{Var}}(\widehat p) =\frac{\widehat p(1-\widehat p)}{12} =\frac{0.6667(0.3333)}{12} =0.01852, \qquad SE(\widehat p)=\sqrt{0.01852}=0.1361. $$

A 95% Wald interval is $0.6667\pm1.96(0.1361)$, or approximately $(0.400,\ 0.933)$. In context, this is an approximate confidence interval for the population probability mass on cell 1 under the stated model and regularity conditions. It does not assign a 95% probability to the fixed parameter being inside this particular interval. The broad interval also signals that 12 subjects provide limited precision.

Check the target before calculating its uncertainty

An estimated mass on a support cell and a distribution value at a time point are not always the same kind of target. A support cell can represent a whole interval of possible event times. If the cell straddles a time $t$, the observed intervals may establish its total mass without establishing how much of that mass lies at or before $t$. A point estimate of $F(t)$ can then depend on an arbitrary placement convention rather than on information in the likelihood.

Worked Example: A Distribution Value That Is Not Identified

Consider a hypothetical fitted support with three cells: one lying entirely at or before time 2 with estimated mass 0.30, one spanning times 2 to 5 with mass 0.40, and one lying entirely after time 5 with mass 0.30. Take the target to be $F(3)$, the probability of an event by time 3.

The first cell contributes 0.30 to $F(3)$, and the last contributes zero. The middle cell is compatible with event times on either side of 3. The observed-data likelihood determines its total mass, 0.40, but does not determine how much of that mass is at or before 3. Consequently, allocations consistent with the same likelihood can give values of $F(3)$ from 0.30 up to 0.70. This is an identification range, not a confidence interval for a uniquely identified point value.

A routine standard error around one chosen value would conceal the problem. Before reporting uncertainty for $F(3)$, use a target and support representation for which the functional is identified, or report the identified range and explain what additional assumptions would be needed to narrow it.

Subject-level bootstrap and changing support

A subject-level nonparametric bootstrap resamples the $n$ observed subjects with replacement. Each resampled subject brings along that subject’s full interval-observation record. For every replicate, reconstruct the Turnbull support from the resampled records and refit the estimator. The resulting estimates approximate sampling variation under repeated sampling from the empirical distribution of subjects. Resampling rows, rather than individual visits or interval endpoints separately, preserves the subject as the independent sampling unit.

Support reconstruction matters. A cell that appears in the original sample may be absent from a replicate if the observations that define it are not selected. Conversely, a replicate’s maximal intersections can have different boundaries or labels. Comparing “mass in cell 2” across fits by cell number is therefore unsafe. A bootstrap analysis needs a consistent target: for example, a fixed-time distribution value that is identifiable in every replicate, or a mass mapped to a prespecified region. If that target is not defined or identified for a replicate, it cannot simply be treated as an ordinary numeric estimate.

Worked Example: Bootstrap Masses When a Cell Disappears

Suppose four hypothetical independent subjects have interval records that identify one of two disjoint cells: three identify cell A and one identifies cell B. The fitted mass on B is $\widehat p_B=1/4=0.25$. In each bootstrap replicate, draw four subjects with replacement from these four records and refit.

Let $K$ be the number of times the one observed B record is selected. Then $K\sim\operatorname{Binomial}(4,0.25)$, and $\widehat p_B^*=K/4$. The bootstrap probabilities are $P(K=0)=0.75^4=0.3164$, $P(K=1)=4(0.25)(0.75)^3=0.4219$, $P(K=2)=6(0.25)^2(0.75)^2=0.2109$, $P(K=3)=4(0.25)^3(0.75)=0.04688$, and $P(K=4)=0.25^4=0.003906$.

Thus 31.64% of the resamples contain no B record. If the support is rebuilt, cell B may vanish; equivalently, its estimated mass is zero in those replicates when the target is retained as mass on the original B region. The bootstrap mean is $E(\widehat p_B^*)=0.25$, and its standard error is $$ SE_{\mathrm{boot}} =\sqrt{\frac{0.25(0.75)}{4}} =\sqrt{0.046875} =0.2165. $$

The empirical 2.5th percentile of these exact bootstrap values is 0, and the 97.5th percentile is 0.75 under the usual nearest-quantile interpretation. This highly discrete interval reflects the small sample and frequent loss of the B pattern; it is not evidence that support changes are harmless. It is a warning that a smooth, symmetric approximation may be poor and that the target mapping across replicates must be stated.

What the two approaches can and cannot tell you

Approach	Useful when	Main caution
Information-based or profile likelihood	The target is identifiable, the support is fixed, and the estimate is regular and away from boundaries.	Curvature and chi-square approximations can fail with boundary masses, flat likelihoods, or changing effective support.
Subject-level bootstrap	The subjects are independent sampling units and refitting captures relevant sampling variability.	Replicates can omit patterns, change support, or leave the target unidentified; percentile intervals can be unstable in small samples.
Identification range	The likelihood permits multiple values for a target, such as a distribution value inside a support cell.	It describes lack of identification, not sampling uncertainty around a single parameter value.

Likelihood methods use the shape of the observed-data likelihood; the bootstrap uses variation among resampled subject records. Agreement between them can be reassuring in a regular, well-identified setting, but neither method repairs an unidentified target. Bootstrap standard errors also do not make a point estimate meaningful when replicate cell labels lack a stable interpretation.

Diagnostics and common failure modes

Before reporting uncertainty, inspect the fitted masses, likelihood surface, and resampled fits. Warning signs include masses at zero or one, nearly flat likelihood directions, very large standard errors, frequent nonconvergence, and many replicates in which the target is undefined or lands on a boundary. For bootstrap output, record how often each relevant observation pattern is absent and how often the support changes. A histogram of estimates can reveal skewness or a point mass at zero that a standard error alone hides.

Misreading a cell-specific estimate. A mass for an interval-valued support cell does not identify where within that cell the event occurred. Do not interpret it as an exact-time probability without justification.
Matching cells by index. “Cell 2” in one replicate need not be the same region as “cell 2” in another. Define and map the target using time regions or a prespecified functional.
Treating a boundary as regular. A zero fitted mass is not an interior estimate. Normal and standard likelihood-ratio approximations may have nonstandard behavior there.
Resampling visits instead of subjects. This changes the sampling unit and can break the observed history. Resample complete independent subject records.
Calling an identification range a confidence interval. Identification concerns which parameter values are compatible with the observed-data distribution; a confidence interval concerns sampling uncertainty. They answer different questions.

A practical analysis sequence

A defensible report makes the target, assumptions, and uncertainty procedure explicit. The following sequence helps avoid producing a precise-looking interval for a quantity the data do not determine.

Define the target.
State whether the estimand is a support-cell mass, a cumulative distribution value, or another functional, and specify its time region or cell mapping.

Check identification and fit.
Confirm that the target is determined by the observed-data likelihood and examine zero masses, flat directions, and convergence.

Choose an uncertainty method.
Use information or profile likelihood only when their fixed-support regularity conditions are plausible; otherwise investigate subject-level resampling and its support behavior.

Report the limitations.
Give the interval or standard error with units and context, describe support changes and failed replicates, and distinguish sampling uncertainty from nonidentification.

These estimates are sensitive to the observation patterns actually represented in the sample. A bootstrap with few subjects may repeatedly omit uncommon patterns, while a likelihood approximation can understate uncertainty if it treats a weakly identified direction as well determined. Report the number of subjects, the target definition, the method, and relevant diagnostics rather than presenting a confidence interval without its conditions.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In the interval-censored likelihood, what does $a_{ij}=1$ mean?
Why can an information-based standard error be unreliable when a fitted mass is zero?
A support cell spans time 4, and its total mass is identified. What extra issue arises when estimating $F(4)$?
In a subject-level bootstrap, why should support cells not be matched solely by their numerical indices?
Four subjects include one record for cell B. What is the probability that a size-four bootstrap sample contains no B record?

Solutions. (1) Subject $i$'s observed interval is compatible with support cell $j$. (2) Zero is on the boundary of the probability simplex, so the usual interior curvature and normal approximations may not apply. (3) The cell’s mass may be split on either side of time 4 in ways the observed likelihood cannot distinguish, so $F(4)$ may not be identified. (4) Resampling can change the support and its boundaries, so equal indices need not denote equal time regions or targets. (5) Each draw misses the sole B record with probability $3/4$, so all four draws miss it with probability $(3/4)^4=0.3164$, or 31.64%.

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