Updating A Transition Matrix At One Event Time

From Transition Counts to One Matrix Update

A transition matrix update at an event time translates observed transitions into changes in the probability of occupying each state. Its ingredients are local: for each origin state, count the observed transitions out of that state and divide by the number of individuals at risk in it immediately before the event time. The resulting off-diagonal entries describe transitions to other states; the diagonal entry accounts for remaining in the origin state at that update.

This is a single-time calculation, not a complete estimate of transition probabilities over an interval. It is also distinct from an instantaneous transition intensity: the update uses the estimated cumulative transition-hazard increments at that event time. The earlier material on transition-intensity matrices establishes the convention used here: rows index origin states, columns index destination states, and diagonal increments are negative sums of outgoing increments.

Thus, for \(i\ne j\), \(B_{ij}(t)=dN_{ij}(t)/Y_i(t)\), while \(B_{ii}(t)=1-\sum_{j\ne i}dN_{ij}(t)/Y_i(t)\). A row of \(B(t)\) is computed using the risk set for that row's origin state. It is not computed using the total study sample or the risk set in a destination state.

For a row-vector convention, a current transition-probability matrix \(P(s,t-)\), whose entries describe transition probabilities up to just before \(t\), is updated at this event time by \(P(s,t)=P(s,t-)B(t)\). The focus here is how to calculate \(B(t)\), not how to combine all event-time factors across a follow-up period.

Why the Diagonal Is Determined by the Off-Diagonals

The observed transition counts directly determine the estimated increments for transitions between distinct states. The diagonal increment is not a separate count of transitions from a state back to itself. Instead, it is defined to make the total increment in each row zero. Adding the identity matrix then makes each row of the event-time factor sum to one.

The diagonal factor entry is therefore the fraction of the origin-state risk set not assigned to an observed outgoing transition at that time. Under the usual single-transition-per-individual-at-a-time setup, the total number of outgoing transitions from state \(i\) at \(t\) cannot exceed \(Y_i(t)\). Consequently, \(B_{ii}(t)\ge 0\). Together with nonnegative off-diagonal entries, this makes \(B(t)\) a row-stochastic matrix: its entries are nonnegative and each row sums to one.

This check is useful, but it does not validate the data or assumptions by itself. A negative diagonal entry can signal a wrong denominator, duplicated counts, or an incompatible treatment of tied transitions. A row sum different from one usually signals an arithmetic or indexing error.

Multiple Destinations and Multiple Origin States

When several transition types leave the same origin state at \(t\), calculate each off-diagonal entry separately with the same origin-state risk set. Then add those entries to obtain the total outgoing increment and subtract that total from one for the diagonal. Counts from different origin states use their respective risk sets, even when the transitions share the same event time.

The update can be justified directly from the estimated transition-hazard increments. For each allowed transition \(i\to j\), the Nelson–Aalen increment at \(t\) is the observed count divided by the origin-state risk set. The transition-intensity matrix has that increment off the diagonal and the negative sum of outgoing increments on the diagonal. Adding the identity gives the event-time factor. This construction both retains the observed transition increments and produces rows that sum to one.

The formula is defined only for an origin state with a positive risk set if an outgoing count needs to be converted to an increment. If \(Y_i(t)=0\), there should be no observed transition out of state \(i\) at \(t\) under the stated risk-set definition. The event-time factor for that row is conventionally left as the identity row, since the data provide no at-risk individuals from which to estimate an outgoing increment. This is a boundary convention, not evidence that the true transition intensity is zero.

Assumptions, Diagnostics, and Common Errors

The arithmetic requires a correctly specified state system, counts classified by both origin and destination, and risk sets measured immediately before the event time. For interpreting the increments as estimates of transition hazards in a population, the observation process must also be handled appropriately: in particular, censoring must be non-informative for the transition process, at least conditional on variables used in the analysis. The data must support the transition-counting setup, including the stated handling of event times and the rule that an individual cannot contribute multiple incompatible outgoing transitions from one origin at the same instant.

• Check the denominator. \(Y_i(t)\) counts individuals at risk in origin state \(i\), not all participants and not those in the destination state.
• Check the row total. Each row should sum to one, up to rounding. Its diagonal is one minus the sum of the outgoing off-diagonals.
• Check nonnegativity. A negative diagonal under the usual setup suggests that the outgoing counts exceed the risk set or have been mishandled.
• Check zero risk sets. Do not divide by zero. Investigate any claimed transition from an origin state with no one at risk.
• Check the meaning of ties. Multiple individuals can have transitions recorded at the same time. If event-time ties create a more complicated within-time ordering or simultaneous-transition issue, the simple single-factor calculation may not resolve it.

A common mistake is to calculate a separate diagonal as \(1-dN_{ij}(t)/Y_i(t)\) for each destination \(j\), then place all of those values on the same diagonal. That ignores the other outgoing transitions from the origin state. With two destinations, the correct diagonal is \(1-\{dN_{ij}(t)+dN_{ik}(t)\}/Y_i(t)\), so the entire row sums to one. Another error is to use a different denominator for each destination. All outgoing transitions from the same origin at \(t\) use the same \(Y_i(t)\).

This event-time factor is not the only way to represent transition behavior. Transition intensities describe instantaneous rates and have units of inverse time; their increments are accumulated over time to build event-time factors. A constant-intensity model instead represents behavior over an interval through its assumed rate structure. Those approaches answer related but distinct questions and require their own assumptions. The count-over-risk-set factor is a nonparametric, event-time update; it does not smooth across time, repair informative censoring, or determine what happens inside a complex tied event time.