Introduction
When an outcome is censored, the recorded value is not necessarily the value of interest. A study may record every value below a threshold as that threshold, leaving the exact outcomes hidden. The recorded data still contain information: an uncensored value reveals the latent outcome directly, while a censored observation tells us that the latent outcome lies on one side of the threshold. The central question is how far that information goes.
The answer depends on assumptions. Censoring rates across covariate values can inform regression parameters when the error distribution is specified. Uncensored values provide information about the part of the outcome distribution above the threshold. But without assumptions on the distribution, the observations do not reveal how censored values are spread below the threshold.
What the Data Record
Consider left censoring at a known threshold \(c\). Let \(Y_i^*\) be the latent outcome for observation \(i\), and let \(Y_i\) be the recorded outcome. The censoring indicator \(D_i\) equals one when the outcome is uncensored and zero when it is censored. Assume there is no probability mass exactly at \(c\).
A common latent regression specification is \(Y_i^*=x_i'\beta+u_i\), where \(x_i\) is a vector of observed covariates, \(\beta\) is a vector of regression parameters, and \(u_i\) is an error term. To express the information in the data precisely, suppose initially that, conditional on \(x_i\), the errors are independent across observations and have a specified continuous distribution with cumulative distribution function \(F\) and density \(f\), location zero, and scale \(\sigma>0\). The threshold is known and the censoring rule is correctly recorded. These assumptions define a model for the latent outcomes; they are not implied by censoring itself.
Two Sources of Information
An uncensored observation contributes its outcome value as well as the fact that it exceeds \(c\). A censored observation contributes an inequality, \(Y_i^*\le c\), rather than a latent value. Conditional on \(x_i\), the probability of censoring is
Thus, the censoring indicator contains information about the latent location \(x_i'\beta\), relative to the threshold and the error scale. With a specified \(F\), covariate patterns associated with different censoring probabilities can help identify regression parameters. The uncensored values also inform those parameters through their distribution above \(c\). Under the assumptions just stated, a single observation's likelihood contribution is
For an uncensored observation, the density term applies because \(Y_i=Y_i^*>c\). For a censored observation, the probability term applies because the recorded event is \(Y_i^*\le c\). In particular, the censored value should not be treated as an exact latent outcome equal to \(c\).
A Short Derivation: How the Censoring Rate Enters
Write \(\mu_i=x_i'\beta\). Since \(Y_i^*=\mu_i+u_i\), censoring occurs when \(u_i\le c-\mu_i\). Dividing by the scale \(\sigma\) places this event on the standardized error scale. Its probability is therefore \(F((c-\mu_i)/\sigma)\). This establishes the censoring-probability formula and shows why the rate depends on both latent location and error scale, not on location alone.
The formula also makes the role of distributional assumptions clear. If \(F\) is known, a censoring probability can be mapped to a standardized threshold index. If \(F\) is unspecified, the rate tells us the probability below the threshold, but does not by itself translate that probability into a value of \(\mu_i\) or describe the outcomes within the censored region.
Worked Example: Using Censoring Rates Under a Gaussian Model
Suppose the threshold is \(c=0\), and the latent regression is \(Y_i^*=\beta_0+\beta_1x_i+u_i\), with \(u_i\sim N(0,2^2)\). In a hypothetical sample, 20 of 100 observations with \(x=0\) are censored, and 50 of 100 with \(x=1\) are censored. To illustrate the information in those rates, treat the observed rates as plug-in probabilities and use the Gaussian model.
At \(x=0\), the censoring probability is \(0.20\). Since \(\Phi^{-1}(0.20)\approx-0.8416\),
At \(x=1\), the censoring probability is \(0.50\), and \(\Phi^{-1}(0.50)=0\). Therefore,
Combining the two equations gives \(\beta_1\approx-1.683\). Under this assumed Gaussian model, the higher censoring rate at \(x=1\) is consistent with a lower latent mean at that covariate value. These calculations illustrate how rates map to a latent regression index; they are not a substitute for estimating the full model. In an actual finite sample, the observed rates are noisy, and likelihood estimation uses both the censoring indicators and the uncensored outcome values.
Why Uncensored Values Are Not the Whole Latent Distribution
Uncensored observations are drawn from the part of the latent distribution above \(c\). Consequently, their average or distribution generally differs from the corresponding latent quantities for the entire population. Even if the uncensored values are measured without error, they do not represent a random sample of all latent outcomes: inclusion as an observed value depends on exceeding the threshold.
Worked Example: The Mean Among Uncensored Outcomes
Let \(Y^*\sim N(0,1)\) and suppose outcomes are left-censored at \(c=0\). The latent mean is \(0\), and half of the outcomes are uncensored. The mean among uncensored outcomes is the mean of a standard normal variable conditional on exceeding zero:
The integral equals \(\phi(0)\) because \(\phi'(y)=-y\phi(y)\), so \(\int_0^\infty y\phi(y)\,dy=\phi(0)\). The uncensored mean is about \(0.7979\) outcome units, not the latent mean of zero. The difference arises from censoring and remains even though the latent model is correctly specified.
Under a fully specified distribution, the model can use the censoring probabilities and the uncensored distribution together to estimate latent regression parameters. That does not mean the unobserved values have been directly measured. Instead, the model uses its assumptions about the distribution to relate the observed portion to the unseen portion.
What Remains Hidden Without Distributional Assumptions
A censoring rate tells us how much probability lies below the threshold, and uncensored values reveal the distribution above it. Those facts do not determine how the probability below the threshold is distributed. The following example holds the censoring rate and all uncensored outcomes fixed while changing the latent mean.
Worked Example: Same Observations, Different Hidden Tails
Consider two hypothetical latent distributions censored at \(c=0\). In both, 40% of outcomes lie below zero; conditional on being above zero, outcomes are uniformly distributed from 0 to 2. Thus, both distributions produce the same censoring rate and the same distribution of uncensored values.
In Distribution A, outcomes below zero are uniformly distributed from \(-1.5\) to \(-0.5\), with mean \(-1\). The latent mean is
In Distribution B, half of the outcomes below zero are uniformly distributed from \(-1\) to \(0\), with mean \(-0.5\), and half are uniformly distributed from \(-5.5\) to \(-4.5\), with mean \(-5\). The mean among censored outcomes is \((-0.5-5)/2=-2.75\), so
The two latent means differ, even though the recorded censoring rate and uncensored distribution are identical. The observed mean after replacing every censored value by zero is \(0.4(0)+0.6(1)=0.6\) in both cases. This example shows that the unseen tail, including its contribution to the latent mean, is not generally identified by the censored data alone.
What the Model Adds—and What It Does Not
A parametric assumption such as Gaussian errors restricts the possible latent distributions. Under that restriction, estimates of location and scale imply a particular distribution below the threshold. The resulting extrapolation can be useful, but it is conditional on the assumed shape. If the true error distribution has a different tail, the inferred latent outcomes may change even when the observed censoring rate and uncensored values are similar.
The distinction is between what the data reveal directly and what a model infers. For values above \(c\), the observed outcomes identify the latent values and, together with the censoring rate, the latent distribution over that region. Below \(c\), the data identify the total probability \(P(Y^*\le c\mid x)\), not the detailed distribution within that region. A model may connect these pieces, but that connection comes from its assumptions.
Censoring rates can be especially informative when they vary across covariates and the error distribution is specified. If nearly all observations are censored for some covariate patterns, however, those patterns provide little direct information about the distribution of uncensored outcomes. If nearly none are censored, the data offer little information about the lower-tail probability. In either case, the amount and type of information depend on the observed covariate support and the model.
Common Mistake
Wrong reasoning: “The uncensored observations are the observed outcomes, so their average estimates the latent mean; censored observations can be treated as equal to the threshold.” Neither claim follows. The uncensored group is selected by exceeding \(c\), while a censored observation only establishes \(Y^*\le c\). Treating the threshold as the latent value replaces an unknown outcome with a known bound.
Correction: Separate the observed information from the model-based inference. Use uncensored outcomes as exact latent values above the threshold and censored observations as inequality events. A latent mean or lower-tail shape requires additional assumptions or information; it should not be presented as directly observed.
Check Your Understanding
Answer each question by distinguishing information in the observed data from information supplied by a model.
- Under left censoring at \(c\), what does a recorded value \(Y_i=c\) establish about \(Y_i^*\)?
- Why can a censoring rate inform the latent regression index when the error distribution is specified?
- In the standard normal example censored at zero, why is the mean among uncensored outcomes greater than the latent mean?
- Can a censoring rate and the distribution of uncensored outcomes determine the shape of the latent distribution below the threshold? Explain.
- What additional role does a parametric error-distribution assumption play when estimating latent outcomes in the censored region?