Why the Observation Rule Matters
A recorded outcome is not always the outcome itself. A laboratory instrument may report only that a concentration is below its detection limit; a study may retain a person only if their duration exceeds a threshold. These situations can produce datasets that look similar, but the observation mechanisms differ: in one, a unit is recorded with incomplete information about its outcome; in the other, some units never enter the observed sample.
The distinction matters because analysis must reflect how the data came to be observed. A reported value at a measurement limit is not necessarily the unit’s true value. And the outcomes of units excluded by a selection rule are not simply recorded as missing values in the sample. Before choosing an econometric model, describe the rule connecting each latent outcome to the data record.
The words “left” and “right” describe which side of the outcome scale is limited. They do not, by themselves, say whether the mechanism is censoring or truncation. A left-censoring rule might report a floor value for outcomes below a threshold, while a left-truncation rule might exclude those outcomes altogether. Always state both the direction and the observation mechanism.
Censoring: Units Remain, Outcomes Are Partly Known
Under censoring, the dataset contains a record for each unit in the study, including units whose outcomes are not observed exactly. For a simple one-sided rule, let \(c\) be a known reporting limit. Under left censoring, values below the limit are recorded at the limit; under right censoring, values above it are recorded at the limit. These rules describe a recorded value, not a claim that the latent outcome equals the limit.
The max and min expressions summarize the rules case by case. For left censoring, if \(Y<c\), then \(\max(Y,c)=c\); if \(Y\ge c\), then \(\max(Y,c)=Y\). For right censoring, if \(Y>c\), then \(\min(Y,c)=c\); if \(Y\le c\), then \(\min(Y,c)=Y\). The boundary value \(c\) is therefore compatible with more than one latent outcome. A record equal to \(c\) should not automatically be interpreted as an exact outcome at \(c\); its censoring status is essential.
Worked Example: A Left-Censored Laboratory Measurement
Suppose a hypothetical assay records every sample, but its reporting floor is \(2.0\) milligrams per litre (mg/L). Values below \(2.0\) mg/L are recorded as \(2.0\), with a flag indicating that the record is left-censored. Apply the rule to latent concentrations \(1.4\), \(2.0\), and \(3.1\) mg/L.
For left censoring, \(Y^{\mathrm{obs}}=\max(Y,2.0)\). The calculations are \(\max(1.4,2.0)=2.0\), \(\max(2.0,2.0)=2.0\), and \(\max(3.1,2.0)=3.1\) mg/L. The first record is flagged as censored; the third is exact. The second lies exactly at the stated boundary, so its status depends on the assay’s specified boundary convention. If the rule censors only values strictly below \(2.0\), that record is exact. The observed number \(2.0\) alone cannot settle the question.
All three samples remain represented in the data. In particular, the censored first sample is not known to have concentration \(2.0\) mg/L: its record says only that its value is below the reporting floor.
In duration data, a common right-censoring rule arises when follow-up ends before an event occurs. Let \(T\) be the time to the event and \(C\) the time at which observation ends. The recorded time is \(Y^{\mathrm{obs}}=\min(T,C)\). An indicator \(\Delta=1\) if the event is observed by the end of follow-up, and \(\Delta=0\) if follow-up ends first. This pair makes explicit whether the recorded time is an event time or a censoring time.
Worked Example: Right Censoring at the End of Follow-Up
In a hypothetical study, follow-up ends at 30 days for every participant. Consider three latent event times: 12, 30, and 44 days. Use \(Y^{\mathrm{obs}}=\min(T,30)\), with \(\Delta=1\) when \(T\le30\) and \(\Delta=0\) when \(T>30\).
For \(T=12\), the record is \((Y^{\mathrm{obs}},\Delta)=(\min(12,30),1)=(12,1)\) days: the event was observed. For \(T=30\), it is \((30,1)\) under the stated convention \(T\le30\). For \(T=44\), it is \((\min(44,30),0)=(30,0)\) days: follow-up ended at 30 days, so the event time is known only to exceed 30 days. The two records with a recorded time of 30 have different meanings because their indicators differ.
With interval censoring, the outcome is known to lie between two limits, rather than being recorded exactly or replaced by a single censoring limit. If the record states \(L<Y\le U\), the data contain the interval \((L,U]\). For example, periodic assessments may establish that an event occurred after the last negative assessment and by the first positive assessment. The interval endpoints are observation times; they are not an exact event time. Boundary inclusion should follow the measurement or assessment rule, especially for discrete outcomes.
Worked Example: An Outcome Known Only Within an Interval
A hypothetical monitoring program measures a pollutant concentration at scheduled visits. At one visit it is \(4.0\) mg/L, and at the next it is \(5.5\) mg/L; assume the concentration crossed a specified threshold sometime between those visits, with no further timing information. If \(Y\) denotes the crossing time in days and the visits occurred on days 8 and 15, the record is \(8<Y\le15\) days under the convention that the crossing had not occurred at day 8 but had occurred by day 15.
The record is the interval \((8,15]\), not the midpoint \(11.5\) days and not an exact value of 15 days. Its width is \(15-8=7\) days. Any particular crossing time in that interval is consistent with the observations; the monitoring schedule does not identify which one occurred.
Truncation: Inclusion Depends on the Outcome
Under truncation, units outside a specified outcome region are absent from the observed sample. Their outcomes are not recorded as boundary values or censoring intervals in that sample. To describe the rule, define an inclusion indicator \(S\), equal to 1 when a unit is sampled and 0 otherwise. For a fixed left-truncation threshold \(a\), inclusion may be defined by \(S=\mathbf{1}(Y\ge a)\), where \(\mathbf{1}(\cdot)\) equals 1 when its condition holds and 0 otherwise. A right-truncation rule at \(b\) may use \(S=\mathbf{1}(Y\le b)\). Interval truncation to \([a,b]\) uses \(S=\mathbf{1}(a\le Y\le b)\). The outcome is recorded only for units with \(S=1\).
The exact inequalities are part of the observation rule. For a continuous outcome, a single boundary value often has probability zero under a continuous distribution, but for a discrete outcome that boundary may have positive probability. Therefore, replacing \(\ge\) by \(>\), for example, can change which units enter the sample.
Worked Example: Left Truncation at an Eligibility Threshold
A hypothetical registry records participants only when their duration of a condition is at least 6 months. Suppose five eligible-for-review people have durations 2, 6, 9, 5, and 12 months. The inclusion rule is \(S=\mathbf{1}(Y\ge6)\).
Applying the rule gives inclusion indicators \(0,1,1,0,1\), respectively. The observed sample therefore contains durations 6, 9, and 12 months, for a sample size of 3. The 2-month and 5-month outcomes do not appear as records in this sample. They are not recorded as 6 months, and the sample does not include censoring flags for those excluded people.
This is left truncation, not left censoring: the threshold determines who appears, rather than replacing below-threshold values with a recorded floor. Because the rule is \(Y\ge6\), a duration exactly equal to 6 months is included.
A Direct Comparison of Observation Rules
| Mechanism | What happens to the unit? | What the record says about \(Y\) |
|---|---|---|
| Left censoring at \(c\) | The unit remains represented. | If recorded at \(c\) as censored, \(Y<c\); otherwise the value is exact. |
| Right censoring at \(c\) | The unit remains represented. | If recorded at \(c\) as censored, \(Y>c\); otherwise the value is exact. |
| Interval censoring | The unit remains represented. | \(Y\) lies within the reported interval. |
| Left truncation at \(a\) | Only units satisfying the inclusion rule appear. | For observed units, \(Y\ge a\) under that rule; excluded units have no sample record. |
| Right truncation at \(b\) | Only units satisfying the inclusion rule appear. | For observed units, \(Y\le b\) under that rule; excluded units have no sample record. |
The table describes common fixed-threshold rules; practical systems can combine mechanisms. For example, a sample may be selected using a truncation rule and then have some of its recorded outcomes censored. In that case, specify the selection rule and the recording rule separately. A complete description answers two different questions: which units enter the dataset? and what does each included unit’s record reveal about its outcome?
Why the Distinction Matters
Censoring and truncation carry different information. A censoring flag or interval tells the analyst something about an included unit’s outcome, even though it may not give an exact value. Truncation instead means the sample is drawn from units satisfying an outcome-based inclusion rule; units outside that region are not present to be described by individual records. Consequently, the observed outcomes in a truncated sample need not represent the full population’s outcome distribution.
This tutorial specifies observation rules, not an estimation method. The likelihood contributions for censored records and the likelihood for truncated regression require additional model assumptions and are treated separately. At this stage, the essential step is to encode the data-generating observation mechanism accurately before selecting a model.
Common Mistakes
- Calling every value at a limit exact. A value reported at a detection or follow-up limit may represent many possible latent outcomes. Check the censoring indicator and the stated boundary rule.
- Treating truncation as censoring. A unit excluded by truncation has no outcome record in the sample; it is not a censored observation at the threshold.
- Replacing an interval by its midpoint. The midpoint is a summary one could choose to calculate, not an outcome value established by interval-censored data.
- Leaving the inequality unspecified. Whether the threshold itself qualifies for inclusion or is treated as censored can matter, particularly for discrete outcomes. State the boundary convention explicitly.
- Ignoring how the sample was selected. A sample containing only outcomes above a threshold does not, by itself, represent all outcomes in the source population.
Check Your Understanding
For each question, state the observation mechanism and explain what the record establishes.
- A test reports every subject’s result, but replaces values below 1.5 units with 1.5 and marks them as below the reporting limit. Is this left censoring or left truncation?
- Follow-up ends at day 20. A participant has no recorded event by then, and the data record day 20 with an event indicator of 0. What can be concluded about the event time?
- A visit schedule shows that an event had not occurred on day 4 but had occurred by day 10. What interval describes the event time under that rule?
- A registry includes only people whose outcome is greater than 100. What happens to people with outcomes of 100 or less, and why is this not censoring?
- Under a left-censoring rule \(Y^{\mathrm{obs}}=\max(Y,3)\), what are the recorded values for latent outcomes 2, 3, and 5? Which record requires censoring-status information to interpret?