Transition-Intensity Matrices And Their Diagonals

Why Put Transition Intensities in a Matrix?

In a multi-state model, a subject can move among a specified set of states. For each possible move, the transition intensity describes the instantaneous rate of moving from one state to another, conditional on being in the origin state just before that time. Placing these intensities in a matrix gives a compact account of the model’s outgoing transitions from every state. The diagonal entries are not additional transition rates: they record the negative total intensity of leaving each state.

This convention matters in calculations with transition probabilities, but its immediate purpose is more basic: it gives a complete, internally consistent representation of the possible transitions at a given time. The row and column positions have a fixed interpretation, and each diagonal entry is determined by the off-diagonal entries in its row.

Definition: Consider \(K\) states numbered \(1,\ldots,K\). Write \(\lambda_{jk}(t)\) for the transition intensity from state \(j\) to state \(k\) at time \(t\), for \(j\ne k\). The transition-intensity matrix \(Q(t)\) has \(Q_{jk}(t)=\lambda_{jk}(t)\) off the diagonal and \(Q_{jj}(t)=-\sum_{k\ne j}\lambda_{jk}(t)\) on the diagonal. Rows represent origin states and columns represent destination states.

An intensity is a rate: if time is measured in years, its units are inverse years. Informally, conditional on occupying state \(j\) immediately before time \(t\), the probability of a transition to \(k\) during a sufficiently short interval of length \(h\) is approximately \(\lambda_{jk}(t)h\), with the approximation error becoming negligible relative to \(h\) as \(h\) approaches zero. Intensities are nonnegative. A transition that the model rules out has intensity zero.

For a finite-state continuous-time model, the matrix representation assumes that the specified transition intensities exist and that transitions occur through the allowed moves in the state space. If the intensities are represented as functions of time and current state alone, that is a Markov specification; in models with additional history or covariate dependence, the intensity matrix must be understood as conditional on that information as well. A state is structurally absorbing if no transition out of it is possible, so all its outgoing intensities are zero.

Why the Diagonal Is the Negative Exit Total

Suppose the process is in state \(j\) at time \(t\), and the allowed destinations are states other than \(j\). In a short interval, the separate outgoing transition probabilities are approximately \(\lambda_{jk}(t)h\). Adding them gives the total probability of leaving \(j\), to first order in \(h\). Therefore the total probability of remaining in \(j\) is approximately one minus that sum. The diagonal is set to the negative of the sum so that each row of the intensity matrix sums to zero.

P(\text{leave }j\text{ during }[t,t+h)\mid X(t)=j) \approx h\sum_{k\ne j}\lambda_{jk}(t), \qquad P(\text{remain in }j) \approx 1-h\sum_{k\ne j}\lambda_{jk}(t).

This argument uses the short-interval interpretation of an intensity: probabilities of multiple transitions within the interval are negligible at first order. The diagonal does not represent a transition from a state to itself. It is a bookkeeping entry that makes the row sum zero, reflecting the balance between remaining in the state and leaving it.

Key result: For every state \(j\), \(\sum_{k=1}^K Q_{jk}(t)=0\). Off-diagonal entries are nonnegative, diagonal entries are nonpositive, and the diagonal is exactly the negative of the row’s off-diagonal total.

Worked Examples

Worked Example: Constructing a Matrix From Instantaneous Rates

Suppose a hypothetical illness-death model has three states: state 1 is alive without the illness, state 2 is alive with the illness, and state 3 is dead. At a particular time, assume the estimated instantaneous rates are \(0.08\) per year for \(1\to2\), \(0.03\) per year for \(1\to3\), and \(0.12\) per year for \(2\to3\). Suppose the other transitions are not allowed.

Place each rate in the row for its origin and the column for its destination. In row 1, the total rate of leaving is \(0.08+0.03=0.11\) per year, so \(Q_{11}=-0.11\) per year. In row 2, the exit total is \(0.12\) per year, so \(Q_{22}=-0.12\) per year. Since death is absorbing in this model, state 3 has no outgoing rates and \(Q_{33}=0\).

Q(t)= \begin{pmatrix} -0.11 & 0.08 & 0.03\\ 0 & -0.12 & 0.12\\ 0 & 0 & 0 \end{pmatrix} \quad\text{per year}.

The first row sums to \(-0.11+0.08+0.03=0\), the second to \(0-0.12+0.12=0\), and the third to zero. Every off-diagonal entry is nonnegative. The entries \(Q_{12}\) and \(Q_{13}\) describe distinct possible destinations from state 1; their sum, not either rate alone, determines the diagonal entry \(Q_{11}\).

Worked Example: Assembling Nelson–Aalen Increments

Consider hypothetical event-time data for three states, with state 3 absorbing. At one event time, suppose 40 individuals are at risk in state 1, with 2 observed transitions \(1\to2\) and 1 observed transition \(1\to3\). Suppose 25 are at risk in state 2, with 3 observed transitions \(2\to1\) and 2 observed transitions \(2\to3\). Assume the risk sets and transition counts are correctly defined for this time.

The Nelson–Aalen increment for an allowed transition \(j\to k\) at this time is \(d\widehat A_{jk}=dN_{jk}/Y_j\), where \(dN_{jk}\) is the number of such transitions and \(Y_j\) is the number at risk in state \(j\) just before the event time. Thus the outgoing increments from state 1 are \(2/40=0.05\) for \(1\to2\) and \(1/40=0.025\) for \(1\to3\). The diagonal increment is their negative sum, \(-0.075\).

From state 2, the increments are \(3/25=0.12\) for \(2\to1\) and \(2/25=0.08\) for \(2\to3\). Their sum is \(0.20\), so the diagonal increment is \(-0.20\). There are no outgoing transitions from state 3.

d\widehat A(t)= \begin{pmatrix} -0.075 & 0.05 & 0.025\\ 0.12 & -0.20 & 0.08\\ 0 & 0 & 0 \end{pmatrix}.

These are estimated cumulative-intensity increments, not instantaneous rates per year: the entries are dimensionless increments at the event time. The row sums are zero. Interpreting the increments as rates would assign them the wrong units. The calculation also relies on correct state-specific risk sets and valid transition counts; if the risk set for an origin state is empty, the ratio cannot be calculated there.

Worked Example: Checking a Four-State Matrix

Suppose a hypothetical model allows transitions \(1\to2\), \(1\to3\), \(2\to1\), \(2\to4\), \(3\to2\), and \(3\to4\); state 4 is structurally absorbing. At one time, let the corresponding intensities, in inverse years, be \(0.04,\ 0.06,\ 0.02,\ 0.03,\ 0.07,\) and \(0.01\).

For state 1, the exit total is \(0.04+0.06=0.10\), giving \(Q_{11}=-0.10\). For state 2, it is \(0.02+0.03=0.05\), giving \(Q_{22}=-0.05\). For state 3, it is \(0.07+0.01=0.08\), giving \(Q_{33}=-0.08\). State 4 has no outgoing transitions, so its entire row is zero.

Q(t)= \begin{pmatrix} -0.10 & 0.04 & 0.06 & 0\\ 0.02 & -0.05 & 0 & 0.03\\ 0 & 0.07 & -0.08 & 0.01\\ 0 & 0 & 0 & 0 \end{pmatrix} \quad\text{per year}.

Each row sums to zero. Now suppose no \(2\to1\) transitions were observed in a particular sample interval, although individuals were at risk in state 2. The estimated increment for that transition may be zero, but that observation alone does not make state 2 structurally absorbing: other exits may be possible, and a zero estimate for one transition is not proof that its underlying intensity is always zero. Structural absorption is a property of the model’s state space, not merely a feature of a finite sample.

Estimated Matrices and Practical Checks

Earlier material on Nelson–Aalen estimation defines the transition-specific cumulative hazard estimate by accumulating event-count-to-risk-set ratios. To form an estimated matrix, place the estimated off-diagonal quantities in their origin and destination positions, then calculate each diagonal as the negative sum of the off-diagonal entries in that row. This rule applies at an event time to the increment matrix, and it applies to cumulative estimates when the cumulative quantities are assembled consistently.

The quantities must be compared like with like. An instantaneous transition intensity is a rate, while a cumulative-intensity increment is dimensionless. Both can be organized using the same row-and-column convention and diagonal rule, but their units and interpretations differ. In particular, do not describe a Nelson–Aalen event-time increment as a per-year rate unless a rate estimate has actually been formed.

Before using an estimated matrix, check the following:

Orientation: rows are origin states and columns are destination states. Transposing the matrix changes the meaning of every off-diagonal entry.
Signs and totals: off-diagonal entries must be nonnegative, and each diagonal must equal the negative row total of the off-diagonal entries.
Allowed transitions: entries for transitions excluded by the state-space design must be zero.
Absorbing states: a structurally absorbing state has a zero row, not a row with a positive exit intensity.
Risk sets and units: verify that each count uses the appropriate origin-state risk set, and label rates and increments correctly.

A failed row-sum check often signals an arithmetic error, a misplaced transition, or a diagonal that was not updated after adding an outgoing rate. A negative off-diagonal entry is not an ordinary sampling fluctuation in a Nelson–Aalen increment; it usually indicates a coding or data-processing problem. Conversely, a zero estimated transition increment can occur simply because no such transition was observed at that event time.

How This Representation Differs From Alternatives

Estimating each transition intensity separately and assembling the results into a matrix is a direct alternative to reporting a list of transition-specific estimates. The list can be useful for a small state space, but the matrix makes the origin and destination indexing explicit and exposes the row-sum constraint. The diagonal is determined by the outgoing estimates; it is not fitted as an independent transition.

A matrix of transition probabilities is a different object. Its entries describe probabilities of occupying destination states at a specified time, conditional on an initial state; its rows sum to one. A transition-intensity matrix instead describes instantaneous rates, has nonnegative off-diagonal entries and nonpositive diagonal entries, and has rows summing to zero. Confusing these two matrices leads to errors in both units and interpretation.

There are also limitations to the intensity matrix itself. It records the modeled transition structure and its rates, but does not by itself give the probability of being in a state at a later time. Estimation depends on appropriate event classification and risk sets, and inference can be uncertain when risk sets are small or transitions are rare. A zero estimated rate should therefore be interpreted with the design, data, and state-space assumptions in view.

Check Your Understanding

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

A model has rates \(0.05\) per year for \(1\to2\) and \(0.02\) per year for \(1\to3\). What is the diagonal entry for state 1, and why?
At an event time, \(Y_2=50\) and \(dN_{21}=1,\ dN_{23}=4\). Find the diagonal increment for state 2.
Why does a transition-probability matrix not use the same row-sum rule as a transition-intensity matrix?
A state has estimated outgoing increments of zero at one event time, but it is not defined as absorbing. What can and cannot be concluded?

Solutions. (1) The total exit rate is \(0.05+0.02=0.07\) per year, so \(Q_{11}=-0.07\) per year; the diagonal is the negative total outgoing intensity. (2) The off-diagonal increments are \(1/50=0.02\) and \(4/50=0.08\), so the diagonal increment is \(-(0.02+0.08)=-0.10\). This is a dimensionless cumulative-intensity increment. (3) A transition-probability row contains probabilities over destination states and sums to one. An intensity-matrix row contains off-diagonal rates and a negative diagonal balancing their total, so it sums to zero. (4) No outgoing transitions were estimated at that event time. This does not establish that the state is structurally absorbing or that its true outgoing intensities are zero at all times.