Aalen-Johansen State-Occupation Estimates After a Landmark

What the Landmark Estimate Describes

A landmark analysis begins by selecting a cohort using information available at a specified time \(L\). The prediction clock then starts at \(L\): elapsed follow-up \(u=0\) corresponds to calendar time \(L\), and elapsed time \(u\) corresponds to calendar time \(L+u\). The Aalen–Johansen estimator uses the observed transitions after the landmark to estimate how probability is distributed across states as follow-up proceeds.

The target is conditional on membership in the landmark cohort and its defining information. For example, if the cohort consists of people who are alive and in state \(r\) at \(L\), the curve for state \(k\) estimates the probability of being in state \(k\) at \(L+u\), conditional on being eligible and in state \(r\) at \(L\). It is not the probability of an event among only those who remain uncensored up to \(L+u\).

Definition: Let \(X_i(u)\) denote subject \(i\)'s state at elapsed time \(u\) after the landmark. For a landmark cohort whose members are all in state \(r\) at \(u=0\), the state-occupation probability is \(p_{rk}(u)=P\{X_i(u)=k\mid X_i(0)=r,\ i\text{ belongs to the landmark cohort}\}\). The Aalen–Johansen estimator estimates these probabilities by estimating transition probabilities from observed transition counts and combining them over time.

A state can be transient, from which a subject may move on, or absorbing, from which no further transitions occur in the model. A competing event is represented as a transition into its own state; it is not treated as censoring. The introductory material on the landmark state-occupation estimand defines the prediction target; here the emphasis is on estimating it from a landmark cohort.

Transition Counts and the Aalen–Johansen Product

Suppose the possible states are \(1,\ldots,K\). For a transition from state \(j\) to a different state \(k\), let \(Y_j(u)\) be the number of cohort members in state \(j\) and still under observation immediately before elapsed time \(u\). Let \(dN_{jk}(u)\) be the number of observed \(j\)-to-\(k\) transitions at \(u\). The estimated transition increment is the observed transition count divided by the risk set:

d\widehat A_{jk}(u)=\frac{dN_{jk}(u)}{Y_j(u)},\quad j\ne k.

The diagonal entries are chosen to make each row sum to zero: \(d\widehat A_{jj}(u)=-\sum_{k\ne j}d\widehat A_{jk}(u)\). Thus \(I+d\widehat A(u)\) is the estimated one-step transition matrix at a jump time, where \(I\) is the identity matrix. With row-vector convention, probabilities multiply these matrices from the left.

Definition: The Aalen–Johansen estimate from elapsed time 0 to \(u\) is the ordered product \(\widehat P(0,u)=\prod_{0<s\le u}\{I+d\widehat A(s)\}\), with factors multiplied in increasing time order. For a cohort starting in state \(r\), the estimated probability of state \(k\) at \(u\) is the \(k\)th entry of \(e_r\widehat P(0,u)\), where \(e_r\) is a row vector with 1 in position \(r\) and 0 elsewhere.

If landmark-eligible subjects start in more than one state, the initial row vector can instead be their empirical distribution of states at the landmark. Multiplying that distribution by the product estimates state occupation for the landmark cohort with that starting-state mix. The initial distribution is part of the target: a curve starting from a particular state is not interchangeable with a curve averaged over several starting states.

The construction has a direct recursion. At each event time, the current estimated state distribution is updated by the estimated transition matrix for that time. If \(\widehat p(u-)\) is the row vector just before a jump, then \(\widehat p(u)=\widehat p(u-)\{I+d\widehat A(u)\}\). Repeated application gives the ordered product above. This explains why transitions at later times are applied to the probability mass that has reached each state by then, rather than being added as though all subjects remained in their initial state.

Worked Example: A Landmark Cohort Starting in One State

Suppose six hypothetical patients are alive and in the well state at a landmark \(L=90\) days. After the landmark, the possible states are well (0), complication (1), and death (2). Death is absorbing. One patient moves from well to complication and one moves from well to death at elapsed time \(u=1\). One of the four patients still well is censored at \(u=1.5\). At \(u=2\), one of the three patients still at risk in the well state moves to complication. Of the two patients then in the complication state, one is censored at \(u=2.5\); the remaining patient moves to death at \(u=3\). Assume the censoring times do not depend on future transitions within the landmark cohort.

At \(u=1\), the well-state risk set is 6. The estimated transition probabilities from well at that time are \(1/6\) to complication, \(1/6\) to death, and \(4/6\) to remain well. At \(u=2\), the well-state risk set is 3, with one transition to complication, giving probabilities \(2/3\) to remain well and \(1/3\) to move to complication. At \(u=3\), the complication-state risk set is 1 and its sole member moves to death.

In the order (well, complication, death), the event-time transition matrices are:

I+d\widehat A(1)= \begin{pmatrix} 2/3&1/6&1/6\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \quad I+d\widehat A(2)= \begin{pmatrix} 2/3&1/3&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \quad I+d\widehat A(3)= \begin{pmatrix} 1&0&0\\ 0&0&1\\ 0&0&1 \end{pmatrix}.

Starting in well gives \(e_0=(1,0,0)\). Immediately after \(u=1\), the estimated distribution is \((2/3,1/6,1/6)\). At \(u=2\), the well probability is \((2/3)(2/3)=4/9\), and the complication probability is \(1/6+(2/3)(1/3)=7/18\); the death probability remains \(1/6\). At \(u=3\), the complication probability moves to death. Therefore the estimated state-occupation probabilities are:

\widehat p(3)=\left(\frac{4}{9},\,0,\,\frac{1}{6}+\frac{7}{18}\right) =\left(\frac{4}{9},\,0,\,\frac{5}{9}\right).

At three elapsed time units after the landmark, the estimated probability of being alive and well is \(4/9\), or about 0.444; the estimated probability of death is \(5/9\), or about 0.556. These are conditional estimates for the landmark cohort, not estimates for all patients at calendar day 93.

Assumptions and Interpretation

The arithmetic of the estimator is simple, but its interpretation as an estimate of state-occupation probabilities depends on assumptions. The process must be observed with enough detail to identify the states and transition times used in the analysis. Subjects should contribute independent follow-up, or the analysis must account for dependence in its design. Most importantly, censoring after the landmark must be independent of future transitions, at least conditional on the information used to define or stratify the cohort. If people at higher unobserved risk are more likely to be censored, the observed transition rates among those still followed need not represent the rates in the target cohort.

For a transition-matrix interpretation, the transition intensity from state \(j\) to state \(k\) is assumed to depend on the current state and elapsed time, rather than on unrepresented earlier history. This is the Markov structure used for state-occupation prediction. Under that structure, the transition probabilities estimated from the risk sets can be combined to describe probabilities from the landmark state. If transition behavior depends on earlier history even after conditioning on current state and elapsed time, the estimator based only on current-state risk sets need not recover the desired conditional curves.

The landmark definition must also be applied consistently. Eligibility, starting state, and any conditioning information should be determined at \(L\), before the post-landmark transitions are counted. A subject who experiences a competing event after \(L\) has moved to another state; that subject is not censored. By contrast, a subject lost to observation has an unknown state thereafter and leaves the relevant risk set at the censoring time.

Worked Example: A Mixed Starting-State Distribution

Consider a hypothetical landmark cohort of 10 people: 6 are in state A and 4 are in state B at the landmark. State C is absorbing. By elapsed time \(u=1\), one of the six people in A has moved to B, and one of the four people in B has moved to C. Suppose there is no censoring before these transitions.

The estimated transition matrix at \(u=1\), in state order (A, B, C), is:

I+d\widehat A(1)= \begin{pmatrix} 5/6&1/6&0\\ 0&3/4&1/4\\ 0&0&1 \end{pmatrix}.

The landmark cohort's empirical starting distribution is \(\widehat q=(6/10,4/10,0)\). Multiplying gives:

\widehat q\{I+d\widehat A(1)\} = \left(\frac{6}{10}\frac{5}{6},\ \frac{6}{10}\frac{1}{6}+\frac{4}{10}\frac{3}{4},\ \frac{4}{10}\frac{1}{4}\right) =(0.500,0.400,0.100).

Thus, at \(u=1\), the estimated probabilities of occupying A, B, and C are 0.500, 0.400, and 0.100. This is a prediction for the cohort's particular mix of starting states. It is not the curve conditional on starting in A alone or in B alone.

As a check on the calculation, each estimated transition matrix has nonnegative entries and row sums equal to one. Consequently, multiplying a probability distribution by it preserves nonnegativity and total probability. The state-occupation estimates should therefore be between 0 and 1 and sum to 1, apart from rounding. A failure of this check often signals an arithmetic error, an incorrectly constructed diagonal, or incompatible state and transition definitions.

Censoring, Alternatives, and Failure Modes

Worked Example: Why Censoring Is Not an Occupied State

Five hypothetical people start in the well state at a landmark; their possible destinations are states A and B, both absorbing. One person is censored at \(u=0.5\). At \(u=1\), one of the four people still at risk moves from well to A. At \(u=2\), one of the three still at risk moves from well to B. Assume independent censoring within this cohort.

At \(u=1\), the estimated probability of moving from well to A is \(1/4\), so the estimated distribution is \((3/4,1/4,0)\). At \(u=2\), one of the three well-state risk-set members moves to B, leaving probability \(2/3\) in well and adding \((3/4)(1/3)=1/4\) to B. The estimate at \(u=2\) is therefore \((1/2,1/4,1/4)\), in the order (well, A, B).

A tempting but incorrect calculation is to count the censored person as still well at \(u=2\). That would give crude proportions \((3/5,1/5,1/5)\), but the censored person's state at \(u=2\) is unknown. Treating censoring as continued well-state occupation changes the estimand and generally biases the result. In this small example, excluding the censored person and dividing observed states by four happens to produce the same proportions as the Aalen–Johansen estimate; that coincidence does not make complete-case proportions a general substitute.

Aalen–Johansen estimation is nonparametric in the sense that it does not require specifying a parametric form for the transition intensities. It differs from a model-based prediction, which estimates transition behavior through a chosen statistical model and may incorporate predictors. The Aalen–Johansen estimate is attractive when a well-defined landmark cohort has adequate risk-set support and a common transition process is plausible. A model may be more useful when predictions must vary across covariate patterns, but then its model assumptions become central.

Diagnostics should focus on the data and assumptions that make each transition increment meaningful:

Inspect risk-set sizes. Small \(Y_j(u)\) makes a transition fraction unstable; a single transition can produce a very large increment.
Check event and censoring patterns. Censoring that is strongly related to future transition risk can undermine the independent-censoring assumption.
Check transition coding. Competing events should be transitions to their own states, while loss to follow-up is censoring. Confusing them changes the estimated probabilities.
Check time and state definitions. Follow-up must use elapsed time from the landmark, and the state just before each event must determine the transition-specific risk set.
Review the implied process. If people with the same current state and elapsed time have systematically different transition behavior because of their histories, a current-state transition estimator may be inadequate.

The estimator also has limitations. It provides cohort-level curves for the landmark group and starting-state distribution used in its construction; it does not automatically provide individualized curves. Its late-follow-up estimates can be imprecise when few subjects remain at risk, and it cannot recover information about states after censoring without assumptions. The curves are estimates, not known probabilities; their uncertainty should be considered when drawing conclusions.

Key takeaway: At each observed transition time, estimate destination probabilities within the risk set for the current state, form a transition matrix, and multiply the matrices in time order. Interpret the result as conditional on the landmark cohort and its starting-state distribution, with censoring and process assumptions made explicit.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A landmark cohort starts entirely in state 0. At a transition time, \(Y_0=8\), with one observed transition from 0 to 1 and one from 0 to 2. What is the estimated probability of remaining in state 0 at that time?
Why is a competing event represented as a transition rather than as censoring in a state-occupation analysis?
A landmark cohort has empirical initial distribution \((0.7,0.3)\). Explain what the estimated curve obtained by multiplying this row vector by an Aalen–Johansen product represents.
Name two checks that can reveal problems with an Aalen–Johansen state-occupation analysis.

Solutions. (1) The estimated probability of remaining in state 0 is \(1-(1/8+1/8)=3/4=0.750\). (2) A competing event is an observed outcome that places the person in a different state; censoring instead means the subsequent state is unknown. (3) It estimates occupation probabilities for the landmark cohort with 70% starting in the first state and 30% in the second, not a curve conditional on only one starting state. (4) Check risk-set sizes and transition coding; also assess whether censoring plausibly is independent of future transitions and whether curves remain in the probability range and sum to one.

Aalen-Johansen State-Occupation Estimates After a Landmark

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