Simultaneous Events In Matrix Product Integrals

Why a Shared Event Time Needs One Update

In observed multi-state data, distinct transitions can have the same recorded time. Some may be different transition types from the same origin state; others may start from different states. If the time records are tied, the product-integral update treats all transitions at that time as one jump in the estimated cumulative transition-hazard matrix. It does not invent an order among events whose order was not observed.

The key operation is therefore to count transitions by origin and destination using the risk set just before the shared time, assemble one matrix increment, and apply one update factor. This is not merely a convenient calculation. Processing tied transitions one at a time can make the result depend on an arbitrary ordering and can introduce paths that were not observed.

This convention uses row probabilities: row \(i\) describes probabilities conditional on starting in state \(i\), and a current transition-probability matrix is updated on the right. The diagonal increment makes every row of \(d\widehat A(t)\) sum to zero. Thus the diagonal entry of the update factor is the estimated fraction still in the origin state through that event-time update, while its off-diagonal entries are the estimated fractions transitioning to each destination.

The risk set must be defined consistently as the set at risk immediately before the shared time. In particular, the data protocol must say how an event tied with censoring is handled. A zero risk set cannot support an increment estimate: if \(Y_i(t)=0\), the ratio is undefined, and no transition increment from state \(i\) should be calculated at that time.

Constructing the Tied-Time Factor

Suppose there are several possible destinations from a particular origin state \(i\). First count every observed transition from \(i\) at \(t\), separately for each destination \(j\). Divide each count by the same pre-time risk set \(Y_i(t)\). The denominator is not reduced after each tied event: all those events belong to the same event-time update, not to a sequence of updates.

Counts from different origin states use their own risk sets. This gives the complete matrix increment, including rows for origins with no observed transition at \(t\), whose increments are zero. The update factor \(I+d\widehat A(t)\) is then applied once to the transition-probability matrix accumulated up to just before \(t\).

This recursion is the relevant product-integral step at a discrete event time. The tied counts determine one matrix factor; they do not supply an ordering for multiplying separate event factors. If there are no events from a state at \(t\), the corresponding row of the factor is the identity row.

Why the Matrix Has This Form

The empirical transition fraction for the move from \(i\) to \(j\) at \(t\) is the number of such moves divided by the number exposed to the possibility of leaving state \(i\) just before \(t\). Those fractions fill the off-diagonal entries. Since a subject who does not make one of the counted outgoing transitions remains in the origin state through this update, the remaining fraction is one minus the sum of outgoing fractions.

This calculation explains both the diagonal and why all tied outgoing counts from an origin state enter together. When each individual can have at most one outgoing transition at that instant, the sum of those counts cannot exceed the risk set: \( \sum_{j\ne i}dN_{ij}(t)\le Y_i(t)\). Consequently, the diagonal of the update factor is nonnegative, and each row sums to one. A row with no outgoing events has diagonal one and zeros elsewhere.

The product-integral recursion then carries the previous transition probabilities forward by distributing each row of the current matrix across the single tied-time factor. This is the event-time counterpart of the general recursion for a product integral; the important tie-specific point is that all counts at \(t\) define one increment before multiplication.

Why Tied Events Must Not Be Serialized

An event-time tie is not evidence that one of the tied transitions happened first. Sequential multiplication treats it as if an order were known. That can create a transition path through two states at one recorded instant and make the result depend on which event was arbitrarily processed first.

Assumptions, Checks, and Limitations

The calculation is well-defined when the state occupied just before each transition time is known, the risk sets and transition counts refer to that same time convention, and the risk set is positive for every origin with an observed outgoing transition. For the empirical fractions to describe the intended transition process, observation and censoring must also support valid risk-set estimation; in particular, censoring should not selectively remove individuals with different transition prospects after conditioning on the information used by the model.

For the update factor to be a probability matrix, each row must have nonnegative entries and sum to one. A useful check is to verify, for every origin \(i\),

If the sum exceeds one, do not repair the row by truncating its diagonal. Investigate whether the risk set is incorrect, whether events were double-counted, or whether the data contain multiple transitions by one individual at the same recorded time. The one-jump update assumes a subject cannot make two successive state changes at one instant. If that is not a reasonable representation of the data, the observation process or model needs reconsideration.

Other warning signs include a transition count from an empty risk set, inconsistent event and censoring tie conventions, or an updated matrix with negative entries or rows that fail to sum to one apart from rounding. These checks assess the arithmetic and data structure; they do not establish that censoring assumptions or the transition model are appropriate.

The main alternative—processing individual tied events in sequence—requires information about within-time ordering that the recorded data do not supply. The matrix-exponential approach is a different model-based calculation for continuous, constant intensities over an interval; it is not a substitute for the tied-time empirical update described here. The single-factor method is tied to the recorded event-time increments and cannot recover a true ordering that was lost through time rounding. If times have been coarsened, ties may combine events that occurred at distinct underlying times.