Introduction
A continuous-time state model often presents transition intensities in a matrix. The numbers alone are not enough to identify a transition: the state ordering and the matrix convention determine what each position means. Under the convention used here, rows identify the source state and columns identify the destination state.
This is a reading rule, not a calculation rule. Given a state ordering and an entry in the matrix, the task is to identify which ordered transition that entry represents. Reversing the row and column reverses the direction of the transition, and can therefore give a different answer.
How to Read a Matrix Coordinate
The state ordering is part of the model information. If a problem lists the states as “Active, Disabled, Dead,” then row 1 and column 1 both refer to Active, row 2 and column 2 both refer to Disabled, and row 3 and column 3 both refer to Dead. The coordinate \((2,3)\) therefore refers to a transition from Disabled to Dead, not from Dead to Disabled.
For a transition from state \(i\) to a different state \(j\), locate the row for \(i\), then move across to the column for \(j\). Keep the direction in the same order as the words “from” and “to.” In particular, the entry for \(i\) to \(j\) need not equal the entry for \(j\) to \(i\).
Use the ordering explicitly supplied with the matrix. Do not assume a familiar ordering from another problem.
Find the source state's position in the ordering. That position selects the row.
Find the destination state's position. That position selects the column.
The matrix entry at that row and column is the quantity associated with the directed transition. Preserve the time unit stated for the model.
For distinct states \(i\) and \(j\), the relationship between the coordinate convention and the transition-intensity definition can be seen directly from conditional transition probabilities. Let \(p_{ij}(t,h)\) be the probability of being in state \(j\) at time \(t+h\), conditional on being in state \(i\) at time \(t\). The intensity from \(i\) to \(j\) is defined by the small-interval limit:
The conditioning state \(i\) is the starting information, so it indexes the row of the conditional transition table. The state \(j\) whose probability is being considered is the outcome, so it indexes the column. Taking the limit changes the quantity in the cell from a transition probability over a short interval to an intensity, but it does not change which state labels the row and which labels the column. This justifies the row-source, column-destination convention.
The matrix coordinate also does not remove the need to check units. If time is measured in years, an intensity is expressed per year; if time is measured in months, the model's stated time unit must be used consistently. An intensity is not itself the probability of making the transition over an arbitrary finite interval.
Worked Example: Reading Transitions in a Three-State Model
A hypothetical income-protection model has states ordered as Active, Disabled, Dead. Its intensity matrix is given in units per year:
Identify the matrix entries corresponding to (a) Active to Disabled, (b) Disabled to Active, (c) Disabled to Dead, and (d) Active to Dead. Also state the direction represented by row 3, column 1.
Method. Number the listed states in order: Active is 1, Disabled is 2, and Dead is 3. For each directed transition, use the source number as the row and the destination number as the column.
(a) Active to Disabled has source 1 and destination 2. Read row 1, column 2: \(q_{12}=0.11\) per year.
(b) Disabled to Active has source 2 and destination 1. Read row 2, column 1: \(q_{21}=0.06\) per year.
(c) Disabled to Dead has source 2 and destination 3. Read row 2, column 3: \(q_{23}=0.15\) per year.
(d) Active to Dead has source 1 and destination 3. Read row 1, column 3: \(q_{13}=0.03\) per year.
Row 3, column 1 means source Dead and destination Active. Its entry is 0. The coordinate therefore represents Dead to Active; the zero is the value shown for that coordinate. It is not the Active-to-Dead entry, which was found at row 1, column 3.
Check. The source and destination numbers in each coordinate match the verbal direction: \(1\to2\), \(2\to1\), \(2\to3\), and \(1\to3\). The rates retain the stated unit, per year. The common mistake avoided is swapping the row and column when reversing a transition.
State Ordering Determines What an Entry Means
A matrix position has no fixed actuarial meaning independent of its state ordering. The same coordinate can refer to different transitions in two matrices if the states are listed in different orders. Conversely, the same transition can occupy different coordinates after a reordering. Always read the order attached to the particular matrix rather than carrying over row numbers from another representation.
This matters when a problem gives a state diagram, a verbal description, and a matrix together. The diagram may make direction visually clear, but the matrix still has to be read using its declared ordering. When in doubt, write the state number beside each state before selecting an entry.
Worked Example: Mapping a Four-State Pension Model
Consider a hypothetical pension model whose states, in order, are Not Yet Eligible (N), Deferred (E), Pensioner (P), and Dead (D). The entries below are stated per year:
Find (a) the entry for Deferred to Pensioner, (b) the entry for Pensioner to Dead, (c) the entry in row 3, column 2 and its transition direction, and (d) the entry for Deferred to Not Yet Eligible.
Method. Use the order N = 1, E = 2, P = 3, and D = 4. Each requested transition is located by writing its source first and destination second.
(a) Deferred to Pensioner is \(E\to P\), or \(2\to3\). The coordinate is row 2, column 3, so \(q_{23}=0.12\) per year.
(b) Pensioner to Dead is \(P\to D\), or \(3\to4\). The coordinate is row 3, column 4, so \(q_{34}=0.09\) per year.
(c) Row 3, column 2 has source P and destination E. It represents Pensioner to Deferred, and the displayed entry is \(q_{32}=0\).
(d) Deferred to Not Yet Eligible is \(E\to N\), or \(2\to1\). The coordinate is row 2, column 1, so \(q_{21}=0.02\) per year.
Check. The two transitions involving states E and P occupy opposite coordinates: \(E\to P\) is row 2, column 3, while \(P\to E\) is row 3, column 2. Their entries are 0.12 and 0, respectively. The common mistake avoided is treating a pair of states as an unordered pair and overlooking the direction.
Reordering States Without Changing the Convention
A state list may be rearranged for presentation, but the row-source, column-destination rule remains the same. What changes is the mapping between state names and row or column numbers. If a matrix is copied or compared after reordering, a reader must update both mappings before interpreting any coordinate.
For example, moving Dead from the third position to the first position changes the coordinate for an Active-to-Dead transition. The transition itself is still Active to Dead; only its row and column positions change. This is a common reason that a matrix may appear to disagree with a diagram when the two displays use different state orders.
Worked Example: Comparing Two State Orderings
A hypothetical three-state model is first listed as Active (A), Disabled (D), Dead (X), with matrix
The same model is then displayed with the ordering Dead (X), Active (A), Disabled (D):
Using the second matrix, find (a) Active to Dead, (b) Disabled to Active, and (c) the transition represented by row 2, column 3. Then compare that last coordinate with row 2, column 3 of the first matrix.
Method. For the second matrix, assign positions from its own ordering: X is 1, A is 2, and D is 3.
(a) Active to Dead is \(A\to X\), or \(2\to1\). Row 2, column 1 of the second matrix is 0.03.
(b) Disabled to Active is \(D\to A\), or \(3\to2\). Row 3, column 2 is 0.06.
(c) Row 2, column 3 in the second matrix means source A and destination D, so it represents Active to Disabled and has value 0.11.
In the first matrix, the ordering is A, D, X. Row 2, column 3 therefore means Disabled to Dead, and its entry is 0.15. The same coordinate, \((2,3)\), represents different transitions in the two displays because the state ordering changed.
Check. Active to Dead is read as row 2, column 1 under the second ordering, while its value 0.03 agrees with row 1, column 3 under the first ordering. The common mistake avoided is comparing equal coordinates across differently ordered matrices as if the coordinates had unchanged meanings.
Common Reading Errors and Why the Convention Matters
The most frequent error is transposition: using the destination to select the row and the source to select the column. This silently changes a transition \(i\to j\) into \(j\to i\). In a model with different intensities in the two directions, the result is not merely a different label; it is a different numerical quantity.
A second error is ignoring the state ordering. Row 2 does not inherently mean Disabled, or any other particular state. It means whichever state appears second in the ordering attached to that matrix. A third error is reading an intensity as a finite-interval probability. The entry is an intensity with a time unit; it is not automatically the probability of transition over a year, month, or other specified period.
Diagonal coordinates have the same state as both row and column. They are not transitions between two distinct state labels. The interpretation of diagonal entries is a separate issue from identifying directed transitions between different states; do not use a diagonal coordinate as though it identified a change from one named state to another.
| Question | Reading rule |
|---|---|
| What selects the row? | The source state, the state occupied at the start of the transition. |
| What selects the column? | The destination state, the state being transitioned to. |
| What does changing the state order do? | It changes the state labels assigned to row and column positions. |
| What do the units describe? | The time scale used for the intensity, such as per year. |
| Does a matrix entry alone give a finite-interval probability? | No. An intensity is not itself a finite-interval transition probability. |
Check Your Understanding
For each problem, identify the relevant state order before interpreting matrix entries. All rates are in the units stated in the question.
- A hypothetical care model uses the order Independent (I), Needs Assistance (A), Deceased (D), with \(q_{13}=0.025\) per year and \(q_{31}=0\). State the transition and value represented by each entry. (Answer: \(q_{13}\) is Independent to Deceased at 0.025 per year; \(q_{31}\) is Deceased to Independent at 0 per year.)
- A hypothetical annuity model has states Waiting (W), In Payment (P), and Dead (D), in that order. Its matrix has \(q_{23}=0.08\) per year and \(q_{32}=0\). Identify both directed transitions and explain why they are not the same entry. (Answer: \(q_{23}\) is In Payment to Dead at 0.08 per year; \(q_{32}\) is Dead to In Payment at 0 per year. The source and destination are reversed.)
- A hypothetical health model is ordered Recovered (R), Ill (I), Dead (D). The entry \(q_{21}\) is 0.18 per year. Give its direction, then identify the coordinate that would represent the reverse direction. (Answer: \(q_{21}\) represents Ill to Recovered at 0.18 per year; Recovered to Ill is \(q_{12}\).)
- A hypothetical pension model is ordered Deferred (D), Active (A), Retired (R), Dead (X). The matrix entry in row 3, column 4 is 0.04 per year. Identify the transition and give the coordinate for Active to Retired. (Answer: Row 3, column 4 is Retired to Dead at 0.04 per year; Active to Retired is row 2, column 3, or \(q_{23}\).)
- A three-state model is first ordered Active (A), Disabled (D), Dead (X), then rearranged as Dead (X), Disabled (D), Active (A). In the first ordering, Active to Disabled is \(q_{12}=0.09\) per year. In the rearranged ordering, give its new coordinate and explain what row 1, column 2 means there. (Answer: Active to Disabled is row 3, column 2, or \(q_{32}\), in the rearranged ordering. Row 1, column 2 means Dead to Disabled.)