Signature
t_abs = (I - Q)^(-1) * c
| 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, with rows for the starting state and columns for the destination | probability per cycle |
c | Vector with one entry equal to 1 for each transient state | no unit |
t_abs | Column vector whose entry i is the expected number of cycles before absorption for a person starting in transient state i | cycles |
|---|
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
Expected cycles before absorption in Excel
With the transient block in the named range TransientQ, Excel 365 returns the column t_abs as a dynamic array. MUNIT builds the identity matrix and SEQUENCE builds the column of ones.
=MMULT(MINVERSE(MUNIT(ROWS(TransientQ))-TransientQ),SEQUENCE(ROWS(TransientQ),1,1,0))
Assumptions
Absorbing chain
At least one state is absorbing and every transient state can reach an absorbing state, directly or through other states. Then Q raised to the power n tends to zero, the whole cohort is eventually absorbed and I minus Q can be inverted.
Time-homogeneous transition matrix
Q is the same in every cycle. A model whose probabilities change with age or cycle, such as one using life-table background mortality, has no single fundamental matrix, and its expected time before absorption comes from summing the cohort trace.
Counting convention
The starting cycle is counted, so the result equals the sum over all cycles of the proportion outside the absorbing states at the start of each cycle. It applies no half-cycle correction and no discounting.
Worked examples
Three-state Well, Sick and Dead model
Using the article's matrix, 85% of Well stay Well, 10% become Sick and 5% die each cycle, and 80% of Sick stay Sick and 20% die. The transient block is Q = [[0.85,0.10],[0,0.80]], and the expected number of cycles before death is 10 from Well and 5 from Sick.
Q = [[0.85,0.10],[0,0.80]]; I = [[1,0],[0,1]]; c = [1,1]; t_abs = [10,5]
Common errors
Including the absorbing state in Q
If the Dead row and column are left in Q, I minus Q has a row of zeros and cannot be inverted; Excel MINVERSE returns #NUM!.
Reading t_abs as a corrected or discounted life expectancy
t_abs counts every cycle begun alive in full. Applying a half-cycle correction to a cohort that starts alive and is fully absorbed reduces it by half a cycle, to 9.5 cycles from Well in the example, and discounting reduces it further.
Using a fundamental matrix with age-dependent mortality
A lifetime model with background mortality rising by age has a different matrix in each cycle. Inverting the matrix of one cycle gives the life expectancy of a cohort whose risks never change.
Sources
Fundamental matrix and time to absorption
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: Definition 11.1, canonical form, Theorem 11.3 (absorption has probability 1), Theorem 11.4 (N equals the inverse of I minus Q) and Theorem 11.5 (t = Nc).
Matrix algebra evaluation of Markov models in medical decision making
Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322-338. A Markov model may be evaluated by matrix algebra, as a cohort simulation or as a Monte Carlo simulation.
Canonical Identity
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