Signature
B = (I - Q)^(-1) * R
| Inputs | Definition | Unit |
|---|---|---|
I | Square matrix with 1 on the diagonal and 0 elsewhere, of the same size as Q | no unit |
Q | Probabilities of moving between transient states in one cycle | probability per cycle |
R | Probabilities of moving from each transient state to each absorbing state in one cycle | probability per cycle |
B | Matrix whose entry i, j is the probability that a person starting in transient state i ends in absorbing state j | probability from 0 to 1 |
|---|
Function
Absorbing Markov chain function
A state k is absorbing when p_kk = 1 and p_kj = 0 for every other state j, so no one who enters it can leave. Ordering the transient states first writes the transition matrix in canonical form, with Q the transitions among transient states, R the transitions from transient to absorbing states, a zero block and an identity block for the absorbing states. From Q and R the expected time before absorption and the probability of ending in each absorbing state follow without running the cohort trace s_(t+1) = s_t P.
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Implementations
Excel
Absorption probabilities in Excel
With the transient block in TransientQ and the transient to absorbing block in AbsorbR, Excel 365 returns B as a dynamic array.
=MMULT(MINVERSE(MUNIT(ROWS(TransientQ))-TransientQ),AbsorbR)
Assumptions
Absorbing chain with a time-homogeneous matrix
As for the expected time before absorption, every transient state can reach an absorbing state and Q and R do not change between cycles.
Worked examples
Two death states in the Well and Sick model
The article's Dead state is split into death from other causes and death from the disease. Well people die only of other causes, at 5% a cycle; Sick people die of other causes at 5% and of the disease at 15% a cycle. Starting in Well, half of the cohort eventually dies of each cause; starting in Sick, 25% die of other causes and 75% of the disease. The split of Sick mortality is illustrative.
Q = [[0.85,0.10],[0,0.80]]; I = [[1,0],[0,1]]; R = [[0.05,0],[0.05,0.15]]; B = [[0.5,0.5],[0.25,0.75]]
Common errors
Reading one-cycle death probabilities as lifetime shares
In the example, other causes account for only a quarter of Sick people's one-cycle death risk, yet they account for half of all deaths among people starting in Well, because a third of people starting in Well die of other causes without ever falling sick.
Sources
Absorption probabilities from the fundamental matrix
Grinstead CM, Snell JL. Grinstead and Snell's Introduction to Probability. The CHANCE Project version of 4 July 2006, based on the 2nd edition published by the American Mathematical Society. Chapter 11 Markov Chains, section 11.2 Absorbing Markov Chains, Theorem 11.6 (B = NR).
Canonical Identity
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