Why Observation Details Belong in the Likelihood
In interval-censored and panel-observed data, the event history is only partly seen. An event may be known to have happened between two visits without its exact time being known. Likewise, a person observed in the same state at two visits may have changed state and returned between them. These are not small bookkeeping details: the likelihood must represent what was observed, rather than replace it with a more convenient but unobserved history.
Three shortcuts are particularly tempting: assigning an event the midpoint of its censoring interval, assuming no transition occurred unless a visit records one, and using an event-time likelihood that does not account for the actual inspection schedule. Each shortcut can change estimates or uncertainty. The corrections follow from a simple principle: write the probability of the observed data under the stated model, integrating or summing over histories that were not observed.
Error 1: Replacing an Interval by Its Midpoint
If an event is known to occur in \((L,R]\), its time is not observed to be \(m=(L+R)/2\). Under a continuous event-time model with cumulative distribution function \(F\) and density \(f\), the correct probability is \(F(R)-F(L)\). Substituting the midpoint as if it were the exact event time instead uses a density contribution such as \(f(m)\). A midpoint approximation to the interval probability might use \(f(m)(R-L)\), but this is an approximation, not the observed-data likelihood in general.
A narrow interval and a density that changes little across it can make the approximation numerically close. That does not make the midpoint an observed event time, nor does it guarantee that parameter estimates or standard errors will be close. The approximation is especially risky when intervals are wide, the density varies rapidly within them, or interval widths differ systematically across participants.
Worked Example: Exponential Events Observed in One Interval
Suppose, hypothetically, ten independent participants each have an event known to occur in \((2,5]\) time units. Assume event times follow an exponential distribution with rate \(\lambda\) per time unit. For one participant, the correct likelihood contribution is \(P(2<T\le5)=e^{-2\lambda}-e^{-5\lambda}\). Compare the estimate from this interval likelihood with the estimate obtained by imputing every event at the midpoint.
Write the interval width as \(d=5-2=3\). The likelihood for the ten identical interval observations is \(L(\lambda)=(e^{-2\lambda}-e^{-5\lambda})^{10}\). Since the observations are identical, maximizing the likelihood is equivalent to maximizing the one-observation contribution. Its log is \(\ell(\lambda)=-2\lambda+\log(1-e^{-3\lambda})\). Differentiating and setting the score equal to zero gives
The interval-likelihood estimate is therefore \(0.3054\) events per time unit. If instead each event is imputed at \(m=(2+5)/2=3.5\), an exponential exact-time likelihood gives the estimate \(1/\bar{T}=1/3.5=0.2857\) events per time unit. The midpoint procedure gives a different estimate because it treats the unobserved event times as known. Both calculations use the same ten participants; only the assumed data differ.
The exponential model is assumed here to apply to event times, observations are independent, and the intervals are the recorded observation of each event time. These assumptions make the likelihood comparison well defined. They do not establish that the exponential model fits a real process; model adequacy still needs assessment.
For uncensored event times, a density is appropriate because the time is observed exactly. For interval-censored times, the event-time likelihood integrates the density over the interval: \(P(L<T\le R)=\int_L^R f(t)\,dt\). This integral is the direct correction, not a special adjustment to an exact-time likelihood.
Error 2: Treating Unobserved Transitions as Absent
Panel data record states at visits, not necessarily every transition between visits. If the observed state is 0 at both ends of a visit gap, the history could include no transition, or a transition from 0 to 1 followed by a return from 1 to 0. In a multi-state process, several sequences may connect the same observed states. A likelihood based only on the path “no change” discards possible histories and understates the probability of the observed data whenever returns are possible.
For a continuous-time Markov model with transition-intensity matrix \(Q\), the transition-probability matrix over a gap of length \(u\) is \(P(u)=\exp(Qu)\). Its entry \(P_{ab}(u)\) is the probability of being in state \(b\) after \(u\) time units, given state \(a\) at the start. It sums the probabilities of all paths that begin at \(a\) and end at \(b\), including paths with intermediate transitions.
Worked Example: Same State at Both Visits Does Not Mean No Change
Consider a hypothetical two-state continuous-time Markov process. The rate from state 0 to state 1 is \(\alpha=0.3\) per time unit, and the rate from state 1 to state 0 is \(\beta=0.5\) per time unit. A participant is observed in state 0 at times 0 and 1. Assume the process is time-homogeneous and Markov, with the stated rates applying throughout the one-unit gap.
The probability of no transition from state 0 during the gap is \(e^{-0.3}=0.7408\). But this is not the probability of observing state 0 at the next visit. The two-state transition probability is
At \(u=1\), substitution gives
Thus the probability of the observed endpoint state is about 0.7935, not 0.7408. The difference, about 0.05268, is the probability contribution from paths that leave state 0 and return by time 1. The exact number of transitions is not identified by these two observations; the model averages over compatible paths.
This calculation relies on the specified two-state Markov model, constant transition rates, and correct state classification at both visits. If states are misclassified, or the process is not adequately represented by this model, the transition probabilities may not describe the data. Even when the model is suitable, the observed endpoints alone do not reveal which path occurred.
A related shortcut is to force each observed change to be exactly one transition. For example, an observation from state 0 at one visit to state 1 at the next establishes different endpoint states, not necessarily exactly one transition. A path with three transitions could also end in state 1. The transition probability \(P_{01}(u)\), rather than the probability of a single specified jump, is the relevant contribution when intermediate states are unobserved.
Error 3: Ignoring the Actual Inspection Schedule
The length of each observation gap affects the probability of the observed state at the next visit. Using the correct sequence of states but substituting the wrong gap lengths therefore changes the likelihood. For a time-homogeneous Markov model observed at fixed or conditionally ignorable visit times, a sequence \(x_0,\ldots,x_K\) contributes \(\prod_{j=1}^K P_{x_{j-1},x_j}(t_j-t_{j-1})\), conditional on the initial state. Each factor uses its own actual time gap.
Conditioning on the observed schedule is appropriate when the visit process can be treated as fixed by design or is non-informative for the unobserved process, possibly after conditioning on included covariates. If visit timing depends on latent disease status or impending events, conditioning on the schedule alone may not be sufficient: the observation process may need to be modeled jointly or handled with an appropriate method. The schedule is not automatically harmless just because visit dates are recorded.
Worked Example: Unequal Visit Gaps Change the Likelihood
Suppose a hypothetical irreversible process has state 0 before an event and absorbing state 1 after it, with constant event rate \(\lambda=0.2\) per time unit. One participant is observed in state 0 at time 0, state 0 at time 1, and state 1 at time 4. Assume exact state classification, a constant rate, and a visit schedule that is fixed or ignorable conditional on the model.
For a gap of length \(u\), the probability of remaining in state 0 is \(P_{00}(u)=e^{-0.2u}\), while the probability of moving from state 0 to state 1 is \(P_{01}(u)=1-e^{-0.2u}\). The correct likelihood contribution for the two observed gaps is
If the actual gaps are wrongly replaced by two gaps of length 2, the contribution becomes
These are different probabilities for different observation schedules. Replacing the actual gaps changes the likelihood even though the ordered states are unchanged. Here the actual gaps are 1 and 3 time units, not 2 and 2.
The absorbing-state assumption matters: once state 1 is entered, the process cannot return to state 0. Under this model, the two observed 0 states imply no event before time 1, while the state 1 at time 4 implies an event by time 4. The event time remains interval-censored by the observations; it is not established to be exactly time 4.
A Practical Check on the Likelihood
Before fitting a model, write down what is actually observed and identify which parts of the history remain unknown. Then check that the likelihood assigns probability to the observations, rather than to an invented complete history. A useful construction for panel data under a Markov model is:
Keep the actual visit times \(t_0,\ldots,t_K\), and distinguish a known state at a visit from an unknown state between visits.
State the allowed states, transition intensities, time-dependence assumptions, and any covariate effects before calculating transition probabilities.
For each consecutive observed pair, use \(P_{x_{j-1},x_j}(t_j-t_{j-1})\), then multiply the factors under the conditional Markov likelihood.
Assess whether conditioning on visit times is defensible. If visit timing carries information about latent state or transition risk, the assumed likelihood may omit relevant information.
Several diagnostics can expose these errors. Compare the intervals and visit gaps actually present in the data with those used in the likelihood. Check whether a purported exact event time is truly known or only bounded by visits. For panel states, verify that transition probabilities include plausible returns and intermediate states. Inspect fitted transition probabilities against observed state frequencies across comparable gaps, as well as residual or goodness-of-fit diagnostics appropriate to the model.
A warning sign is a large change in estimates when midpoint assignments, interval widths, or visit-gap lengths are handled correctly. That change does not by itself prove which model is right, but it shows that the shortcut was consequential. Also examine whether visit times vary with observed covariates, symptoms, or earlier states; such patterns may challenge an assumption that the observation schedule is ignorable.
Corrections, Alternatives, and Limits
The preferred correction depends on what is observed and what process is being modeled. For a single interval-censored event time, use the probability of the recorded interval under the event-time model. For panel-observed states under a continuous-time Markov model, use transition probabilities over the actual gaps. If rates change over time, a time-homogeneous transition matrix may not suffice; piecewise-constant intensities or another time-varying model can represent that change. Discrete-time transition models can be appropriate when the scientific model is defined at visit intervals, but they are not automatically equivalent to a continuous-time process.
Midpoint imputation can be used as a deliberately rough approximation or for a clearly stated descriptive purpose, but it does not preserve interval uncertainty and should not be confused with the interval likelihood. Treating observed state changes as single transitions may be reasonable only under a defensible design or model that rules out intermediate transitions. Ignoring inspection gaps is acceptable only if the likelihood truly uses the same relevant time scale and gaps; otherwise, the mismatch needs correction.
No likelihood can recover detail that the observation design did not record without relying on modeling assumptions. Sparse visits may leave multiple transition histories compatible with the data, and different models can fit the same observed endpoints while implying different hidden histories. Model-based estimates should therefore be interpreted as conditional on the chosen transition structure, rate assumptions, and treatment of the observation schedule.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An event is known to occur in \((1,4]\). Under a continuous event-time model, what is the likelihood contribution, and why is the midpoint not an exact observation?
- A two-state process has rate 0.2 from state 0 to state 1 and rate 0.4 from state 1 to state 0. In one time unit, what is the probability of observing state 0 at both visits under the two-state Markov model?
- For an irreversible event process with rate 0.1, a participant is observed event-free at time 0 and in the event state at time 2. What is the interval likelihood contribution?
- Give one condition under which it is reasonable to condition on the visit schedule, and one situation that could make this inadequate.
Solutions. (1) The contribution is \(F(4)-F(1)=\int_1^4 f(t)\,dt\). The midpoint 2.5 is not recorded as the event time; it is only a possible summary of the interval. (2) Here \(\alpha=0.2\), \(\beta=0.4\), and \(\alpha+\beta=0.6\). Thus \(P_{00}(1)=0.4/0.6+(0.2/0.6)e^{-0.6}=0.6667+0.3333(0.548812)=0.8496\). This includes paths that leave and return. (3) The event occurred in \((0,2]\), so its contribution is \(1-e^{-0.1(2)}=1-e^{-0.2}=0.1813\). (4) Conditioning can be reasonable when visits are fixed by design or non-informative given modeled covariates. It may be inadequate if visit timing depends on latent disease severity or impending transitions not captured by the model.