Signature
p_ii = exp(-lambda * Delta); D_mean = Delta / (1 - p_ii); T_mean = 1 / lambda; gap_c = (D_mean - T_mean) / Delta
| Inputs | Definition | Unit |
|---|---|---|
lambda | Constant total rate of leaving the state, above zero | per year |
Delta | Length of one model cycle, above zero | years |
p_ii | Probability of staying in the state for another cycle at the constant exit rate | probability per cycle |
|---|---|---|
D_mean | Mean stay of the cycle model in years, counting the cycle of entry and every cycle begun in the state | years |
T_mean | Mean time in the state of the continuous process with the same constant exit rate | years |
gap_c | Excess of the discrete mean stay over the continuous mean, expressed in cycles | cycles |
Function
Memoryless transition property of a Markov state and the sojourn-time distribution it implies
Maps the state a person occupies at cycle t, the cycle itself and the path H_t before it to the probability of each next state, under the restriction that the path adds nothing once the current state and cycle are known. With a constant probability p_ii of staying in state i the restriction makes the number of cycles spent in i on one visit geometric, and in continuous time a constant exit rate makes the time spent in i exponential, so in both cases the chance of leaving does not depend on the time already spent in the state. The package gives the geometric distribution of a stay with its median, the residual-stay identity behind memorylessness with a check of a survival curve for time since entry, the exponential residual stay and median, and the gap between the discrete and continuous mean stays at the same rate. Existing records are cited and not restated. The mean episode length 1 divided by 1 minus p_jj is HE-FM-MKCH-002, the cohort update is HE-FM-MM-001 with the trace totals of HE-FN-CSIM-001 and HE-FM-CSIM-001, rate and probability conversion is HE-FN-TP-001 with HE-FM-TP-001 to HE-FM-TP-004 and HE-CF-TP-001 and HE-CF-TP-002, exponential survival and the mean 1 divided by lambda are HE-FM-BTH-003 and HE-FM-CONT-002, the fundamental matrix is HE-FM-ABS-001 with its discounted form HE-FM-MKCH-001 and HE-CF-MKCH-001, the six-month root of an annual matrix is HE-FM-TMAT-001, and relaxing the assumption with tunnel states is HE-FN-TUN-001, HE-FM-TUN-001 to HE-FM-TUN-003 and HE-CF-TUN-001. Notation follows the Markov Assumption article.
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Implementations
Excel
Discrete and continuous Markov mean stays and their counting gap from named cells
With the total exit rate in ExitRate and the cycle length in years in CycleYears, the four formulas return p_ii, D_mean, T_mean and gap_c, the last held in a cell named GapCycles.
=EXP(-ExitRate*CycleYears); =CycleYears/(1-EXP(-ExitRate*CycleYears)); =1/ExitRate; =(CycleYears/(1-EXP(-ExitRate*CycleYears))-1/ExitRate)/CycleYears
Assumptions
One constant exit rate behind both Markov mean stays
The cycle model and the continuous process share one constant total exit rate, so the stay probability of every cycle is exp(minus lambda Delta) and the comparison isolates the effect of counting whole cycles.
Discrete Markov mean stay counted from the cycle of entry
D_mean counts the cycle of entry, as HE-FM-MKA-001 does. A trace counted at the end of each cycle from model start credits one cycle fewer, Delta times p_ii/(1-p_ii), and gives a gap close to minus one half instead.
Worked examples
Article's memoryless post-event state in annual cycles against continuous time
At a constant death rate of 0.098572 a year in annual cycles the stay probability is 0.90613 and the mean post-event stay is about 10.653 years, the article's 1 divided by q. The continuous mean is about 10.145 years, so the counting gap is about 0.508 of a cycle (computed here for illustration).
lambda = 0.098572; Delta = 1; p_ii = 0.90613; D_mean = 10.6531; T_mean = 10.1449; gap_c = 0.5082
Article's post-event death rate in monthly cycles against continuous time
With the same rate in monthly cycles, Delta of one twelfth of a year, the discrete mean falls to about 10.187 years and the gap to about 0.501 of a cycle, about 0.04 of a year, as the cycle shortens (computed here for illustration).
lambda = 0.098572; Delta = 0.0833333; p_ii = 0.991819; D_mean = 10.1866; T_mean = 10.1449; gap_c = 0.5007
High exit rate of 1 a year in annual cycles against continuous time
At an illustrative exit rate of 1 a year in annual cycles the stay probability is about 0.368, the discrete mean is about 1.582 years against 1 year in continuous time, and the gap rises to about 0.582 of a cycle, which shows that long cycles relative to the stay widen it (computed here for illustration).
lambda = 1; Delta = 1; p_ii = 0.3679; D_mean = 1.582; T_mean = 1; gap_c = 0.582
Common errors
Counting gap read as a structural effect of the Markov assumption
The 0.508-year difference between the article's 10.653 discrete post-event years and the continuous 10.145 comes from whole-cycle counting at one constant rate, whereas the difference between 16 years with the tunnel and 10.653 without it comes from the missing time since entry. A half-cycle correction changes the first and leaves the second untouched, as the article notes.
Sources
Sonnenberg and Beck on counting cohort membership at the start or end of each cycle
Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4) 322-338. Pages 329 to 330, The Half-Cycle Correction, which states that counting at the end of each cycle consistently underestimates the area under the survival curve and counting at the beginning consistently overestimates it, and that the correction assumes transitions occur on average halfway through each cycle.
Grinstead and Snell means of the geometric and exponential distributions
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 6, section 6.1, Example 6.4, page 228, which shows that the time to the first success with P(T = j) = q^(j-1) p has E(T) = 1/p, and section 6.3, Example 6.26, page 273, which shows that an exponential variable with parameter lambda has E(X) = 1/lambda.
Canonical Identity
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