Likelihoods for Interval-Censored Outcomes

Why an Interval Can Be the Observation

A continuous outcome is not always recorded as an exact number. A laboratory report may give a concentration as “between 8 and 12,” a survey may record income in brackets, or a database may round a measurement to the nearest whole unit. In each case the underlying outcome is not observed exactly; what is known is that it lies in a particular interval.

That distinction changes the likelihood contribution. An exact continuous observation contributes a density evaluated at its value. An interval-censored observation contributes the probability that the outcome falls between the recorded bounds. Using a density at an arbitrary point in the interval discards the information actually recorded and does not give the appropriate likelihood.

Definition: An outcome \(Y_i^*\) is interval-censored when the data for observation \(i\) reveal only that \(Y_i^*\) lies in a recorded interval with lower bound \(L_i\) and upper bound \(U_i\), where \(L_i<U_i\). The observed information is the event \(L_i\leq Y_i^*<U_i\), not an exact value of \(Y_i^*\). For a continuous outcome, changing whether a finite endpoint is included does not change the interval probability.

Model and Likelihood Contribution

Suppose the conditional distribution of the latent outcome given covariates \(x_i\) has density \(f_i(y;\theta)\) and cumulative distribution function \(F_i(y;\theta)\). The parameter vector \(\theta\) may include regression coefficients and distributional parameters. For the likelihood below, assume the conditional distribution is correctly specified, observations are conditionally independent, and the recorded interval is the information produced by the observation process. In particular, the interval-reporting process must not introduce additional unmodeled selection that depends on the latent outcome or parameters.

Given an interval \([L_i,U_i)\), the observed event is that \(Y_i^*\) falls within those bounds. Its probability is the integral of the density over the interval. By the definition of the cumulative distribution function, that integral is the difference between the CDF values at the upper and lower bounds.

\begin{aligned} P(L_i\leq Y_i^*<U_i\mid x_i;\theta) &=\int_{L_i}^{U_i} f_i(y;\theta)\,dy\\ &=F_i(U_i;\theta)-F_i(L_i;\theta). \end{aligned}

This probability is the likelihood contribution from observation \(i\). Under conditional independence, the sample likelihood is the product of the contributions:

L(\theta)=\prod_{i=1}^{n} \left[F_i(U_i;\theta)-F_i(L_i;\theta)\right].

It is often more convenient to maximize the log-likelihood, which converts the product into a sum:

\ell(\theta)=\log L(\theta) =\sum_{i=1}^{n}\log\left[F_i(U_i;\theta)-F_i(L_i;\theta)\right].

Infinite bounds cover one-sided information. Define \(F_i(-\infty;\theta)=0\) and \(F_i(\infty;\theta)=1\). Thus, an observation known only to be below \(U_i\) contributes \(F_i(U_i;\theta)\), while one known to exceed \(L_i\) contributes \(1-F_i(L_i;\theta)\). These are interval-probability contributions too, with one infinite endpoint.

Key distinction: For an interval-censored observation, use \(F_i(U_i;\theta)-F_i(L_i;\theta)\). For an exact continuous observation \(y_i\), the likelihood contribution is the density \(f_i(y_i;\theta)\). A zero-width interval has probability zero under a continuous distribution, so it cannot be substituted for an exact observation.

Worked Examples

Worked Example: A Single Normal Interval

Suppose a hypothetical measurement has latent distribution \(Y^*\sim N(10,4)\), so its mean is 10 units and its standard deviation is 2 units. The report says only that the measurement lies between 8 and 12 units. Find its likelihood contribution under this model.

The recorded interval is \([8,12)\). Standardizing its endpoints gives \((8-10)/2=-1\) and \((12-10)/2=1\). Therefore, using the standard normal CDF \(\Phi\),

\begin{aligned} P(8\leq Y^*<12) &=\Phi(1)-\Phi(-1)\\ &\approx 0.68268949\\ &\approx 0.6827. \end{aligned}

The likelihood contribution is approximately 0.6827. It is the probability, under the specified normal model, of obtaining a report in this interval. It is not the normal density evaluated at 10, nor does the report establish that the latent measurement equals 10. The probability is calculated from the unrounded CDF difference; rounding the two CDF values first can give a less accurate final result.

Worked Example: Grouped Counts Under Two Candidate Models

Suppose 20 hypothetical continuous outcomes are reported in three categories: below 0, from 0 to 1, and at least 1. The observed counts are 8, 7, and 5, respectively. Compare the likelihoods under \(Y^*\sim N(0,1)\) and \(Y^*\sim N(0.5,1)\), where the second argument is the variance.

For the first model, the three category probabilities are \(p_1=\Phi(0)=0.5000000\), \(p_2=\Phi(1)-\Phi(0)=0.3413447\), and \(p_3=1-\Phi(1)=0.1586553\). For the second model, the cutpoints standardized relative to mean 0.5 are \(-0.5\) for 0 and \(0.5\) for 1. Its category probabilities are \(p_1=\Phi(-0.5)=0.3085375\), \(p_2=\Phi(0.5)-\Phi(-0.5)=0.3829249\), and \(p_3=1-\Phi(0.5)=0.3085375\). The middle probability uses the difference of the unrounded CDF values.

For category counts \(n_1,n_2,n_3\), the likelihood is proportional to \(p_1^{n_1}p_2^{n_2}p_3^{n_3}\). The multinomial coefficient \(20!/(8!7!5!)\) can be included for the probability of the particular count vector, but it is the same under both models and cancels in their likelihood ratio. Hence,

\frac{L(0.5,1)}{L(0,1)} = \left(\frac{0.3085375}{0.5000000}\right)^8 \left(\frac{0.3829249}{0.3413447}\right)^7 \left(\frac{0.3085375}{0.1586553}\right)^5.

Evaluating with unrounded normal probabilities gives \(\log\{L(0.5,1)/L(0,1)\}\approx 0.2681\), and therefore \(L(0.5,1)/L(0,1)\approx \exp(0.2681)=1.307\). The observed grouped data are about 1.31 times as likely under the second candidate model as under the first. This comparison alone does not establish that either model is correct, or quantify uncertainty about which parameter value generated the data.

Worked Example: Outcomes Rounded to Whole Units

Consider three hypothetical measurements from a model \(Y_i^*\sim N(10,4)\), independent across observations. Each is rounded to the nearest whole unit, and the reported values are 9, 10, and 12. Assume the rounding rule assigns a reported integer \(r\) to values in \([r-0.5,r+0.5)\). Find the likelihood for these three reports under the stated model.

The reports imply the intervals \([8.5,9.5)\), \([9.5,10.5)\), and \([11.5,12.5)\). Standardizing each endpoint using mean 10 and standard deviation 2 gives the following contributions:

\begin{aligned} P(8.5\leq Y_1^*<9.5) &=\Phi(-0.25)-\Phi(-0.75) \approx 0.1746663,\\ P(9.5\leq Y_2^*<10.5) &=\Phi(0.25)-\Phi(-0.25) \approx 0.1974127,\\ P(11.5\leq Y_3^*<12.5) &=\Phi(1.25)-\Phi(0.75) \approx 0.1209776. \end{aligned}

By independence, multiply these interval probabilities:

L =(0.1746663)(0.1974127)(0.1209776) \approx 0.0041715, \qquad \ell=\log L\approx -5.480.

The product is the probability of these three rounding reports under the specified model. Treating 9, 10, and 12 as exact observations and multiplying their normal densities would answer a different question: it would use information the reporting procedure did not provide.

From Individual Intervals to Grouped Data

Grouped data can be expressed either as counts in categories or as interval records for individuals. If category \(j\) corresponds to \([a_j,b_j)\), its probability under a model is \(p_j(\theta)=F(b_j;\theta)-F(a_j;\theta)\). If \(n_j\) observations fall in that category, the parameter-dependent part of the grouped likelihood is

L(\theta)\propto\prod_{j=1}^{J}p_j(\theta)^{n_j}, \qquad \ell(\theta)=\sum_{j=1}^{J}n_j\log p_j(\theta).

This follows by multiplying the category probability once for each observation. When only category counts are retained, there are multiple possible orderings of the same categories; counting those orderings introduces the multinomial coefficient \(n!/\prod_j n_j!\). Because that coefficient does not depend on \(\theta\), it does not affect maximum-likelihood estimation or likelihood ratios between models fitted to the same counts.

The intervals must describe the actual recording scheme. For deterministic rounding, the reporting rule maps an outcome to a known rounding interval. For bracketed survey responses, the reported bracket supplies the interval. If the observation process instead determines who enters the data set based on an unobserved outcome, the likelihood must account for that selection mechanism; simply treating the selected observations as ordinary interval reports is not generally valid.

Common Mistakes and a Practical Check

Using a density at the midpoint. A midpoint is not the observed outcome. For an interval, integrate the density across the interval, equivalently subtract the endpoint CDF values.
Subtracting rounded probabilities too early. CDF values may each be rounded for display, but calculate their difference and the likelihood using unrounded values whenever possible. Small probability errors can become material when counts or sample sizes are large.
Confusing the event probability with a parameter probability. A likelihood contribution evaluates how probable the observed interval is under a specified parameter value. It is not the probability that the parameter value is true.
Ignoring the recording rule. Rounding to the nearest unit produces half-unit intervals, not intervals of width one centered at the rounded value’s lower endpoint. State the rule before assigning bounds.
Applying the independence product without justification. The product likelihood requires independent observations under the model, or an appropriate conditional independence assumption. Dependence requires a joint likelihood rather than a simple product.

Write the information recorded.
For each observation, identify its lower and upper bounds and state the interval implied by the reporting rule.

Specify the outcome distribution.
Write the conditional CDF \(F_i(\cdot;\theta)\) implied by the model before forming a likelihood contribution.

Calculate and combine probabilities.
Use \(F_i(U_i;\theta)-F_i(L_i;\theta)\) for each interval, then multiply contributions or add their logarithms under the stated independence assumption.

Key takeaway: Interval-censored data tell us where an outcome lies, not its exact value. The correct likelihood contribution is the model probability of that recorded interval. This principle applies to finite brackets, one-sided bounds, grouped counts, and rounded continuous measurements.

Check Your Understanding

Answer each question using the interval information actually recorded.

A continuous outcome has CDF \(F\), and an observation is known to lie in \([a,b)\). Write its likelihood contribution.
Why is the density at the midpoint of an interval not generally the likelihood contribution for an interval-censored observation?
Under \(Y^*\sim N(10,4)\), a report states that \(Y^*\) is at least 12. Write the probability contribution using \(\Phi\).
For grouped counts \(n_j\) in categories with probabilities \(p_j(\theta)\), write the parameter-dependent log-likelihood.
A measurement is rounded to the nearest tenth and reported as 3.2. Give the implied interval under the convention that the lower endpoint is included and the upper endpoint excluded.