Diagnostics for Censored and Truncated Regression

Why Limited-Outcome Models Need Specialized Diagnostics

A model for a censored or truncated outcome describes more than the numerical values that happen to be observed. It also makes claims about which values are recorded and, for censored data, how often observations should accumulate at a censoring limit. Diagnostics must therefore check the observation pattern as well as the observed numerical outcomes.

A routine residual plot can be misleading here. If an outcome is censored below a known limit, the residuals among uncensored observations are selected from one part of the latent error distribution; they need not have mean zero. Under truncation, observations outside the selection range are absent altogether. The appropriate reference for either case depends on the model and the observation rule.

Model and Observation Rules

Let \(Y_i^*\) be a latent outcome for unit \(i\), with covariate vector \(x_i\), and suppose

Write \(\mu_i=x_i^\top\beta\). In a homoskedastic model, \(\sigma_i=\sigma\); in a heteroskedastic model, the scale may depend on \(x_i\). For a lower-censored outcome with limit \(L\), the recorded value is \(Y_i=\max(L,Y_i^*)\). A unit with \(Y_i=L\) is known to have a latent outcome at or below the limit, but its exact value is not observed. A truncated outcome differs: a unit is recorded only if it meets an observation rule, such as \(Y_i^*>L\), and units outside that rule do not appear in the outcome sample.

The formulas below use a normal latent error model to make the diagnostic reference distributions explicit. A fitted model should be checked against the distribution and variance assumptions it actually uses. If a different error distribution is assumed, the corresponding conditional probabilities and moments should replace the normal formulas.

Check Censoring Frequencies

For lower censoring at \(L\), the fitted model implies a probability that unit \(i\) is censored. With estimated parameters, this probability is

where \(\Phi\) is the standard normal cumulative distribution function. For upper censoring at \(U\), the implied probability of being censored above the limit is \(1-\Phi((U-x_i^\top\hat\beta)/\hat\sigma_i)\). With two limits, each limit has its own implied probability. Comparing these probabilities with observed censoring frequencies can reveal a model that predicts too many or too few censored outcomes.

The comparison should be made overall and, where useful, across meaningful covariate groups or ranges of fitted values. An overall rate can look reasonable even when the model systematically underpredicts censoring for one group and overpredicts it for another. For units with different fitted probabilities, the model’s expected number censored is the sum of their probabilities, not the sample size times a single arbitrary probability.

Residual Patterns Among Uncensored Outcomes

Let \(a_i=(L-\mu_i)/\sigma_i\). Among observations that are uncensored below \(L\), the latent error is known to exceed \(-\mu_i+L\), or, after standardizing, \(Z_i>a_i\), where \(Z_i=\varepsilon_i/\sigma_i\) is standard normal. Define the inverse Mills ratio for this lower-tail selection as \(\lambda(a)=\phi(a)/[1-\Phi(a)]\), with \(\phi\) the standard normal density. The conditional mean and variance of the error are

Thus, the raw residual \(Y_i-\hat\mu_i\) for an uncensored observation does not generally have conditional mean zero. In a truncated sample with the same lower selection rule, the observed outcomes are also drawn from this upper-tail conditional distribution. For two-sided observation over \(L<Y_i^*<U\), put \(a_i=(L-\mu_i)/\sigma_i\), \(b_i=(U-\mu_i)/\sigma_i\), and \(D_i=\Phi(b_i)-\Phi(a_i)\). The conditional mean residual is \(\sigma_i[\phi(a_i)-\phi(b_i)]/D_i\); diagnostics should use the relevant interval rather than an unselected normal reference.

Conditional Distribution Checks

Residual plots can expose systematic patterns, such as residuals that become more dispersed as fitted values increase, or conditional residual means that vary with a covariate. A complementary check uses the probability-integral transform (PIT): under a correctly specified continuous conditional distribution with known parameters, evaluating its cumulative distribution function at an observation produces a uniform value on \([0,1]\). For limited outcomes, the distribution used for this transformation must respect the censoring or selection rule.

For example, with lower censoring at \(L\), uncensored values can be transformed using their distribution conditional on exceeding \(L\). For a normal model, that conditional CDF at \(y>L\) is

Under the fitted conditional model, these transformed values should not show systematic clustering or trends when examined across fitted values or covariates. For a full check of a censored outcome, the censoring mass must also be represented. The general randomized PIT for any distribution with jumps is \(F(Y_i^-\mid x_i)+V_i\{F(Y_i\mid x_i)-F(Y_i^-\mid x_i)\}\), where \(V_i\) is an independent uniform random variable on \([0,1]\) and \(F(Y_i^-\mid x_i)\) is the left limit of the CDF. At a censoring point, this randomization spreads the probability mass across its appropriate interval. Applying an ordinary continuous PIT to the censored value as if it were an exact latent outcome is not valid.

What Can Be Checked Under Truncation?

A truncated outcome sample contains no outcome records for units excluded by the observation rule. Consequently, its observed outcome distribution cannot reveal how many units were excluded unless information on the source population or screening process is also available. If that denominator and the relevant covariates are recorded, predicted selection probabilities can be compared with observed selection rates. Without them, diagnostics should focus on the distribution conditional on selection rather than treating the selected sample as an unselected random sample.

This distinction also affects residual checks. For lower truncation at \(L\), the model’s outcome distribution in the observed sample is the distribution conditional on \(Y_i^*>L\). Residuals should therefore be compared with the conditional moments or conditional CDF, just as for the uncensored portion of a lower-censored outcome. The data do not contain residuals for excluded units, so no plot of observed residuals can directly assess their unobserved outcomes.

Sensitivity Checks and Common Pitfalls

A diagnostic conclusion can depend on assumptions that are difficult to verify from limited outcomes alone. Repeat important checks under plausible alternatives: for example, a different error distribution, a heteroskedastic scale specification, or a substantively justified change to the covariate specification. Compare predicted censoring probabilities and conditional residual patterns across these fits. If the apparent fit changes substantially, report that sensitivity rather than presenting one specification’s diagnostics as decisive.

Common mistake: “The uncensored residuals have a positive mean, so the model must be biased.” This reasoning treats uncensored observations as if they were an unselected sample. Under lower censoring, the uncensored residuals are drawn from the upper tail, whose conditional mean is generally positive; truncation creates the same issue. The correction is to compare residuals with the conditional distribution implied by the observation rule, and then look for departures from that conditional reference.

A second pitfall is relying on one aggregate censoring rate. Matching the overall rate does not guarantee that censoring probabilities are calibrated across covariates, and it says nothing by itself about the distribution of uncensored outcomes. Conversely, a handful of unusual residuals or PIT values is not conclusive evidence against a model. Diagnostics are most useful as a set of complementary checks, interpreted with the sample size, parameter estimation, and observation process in view.