Signature
q_bar = (m_T1 * q_1 + m_T2 * q_2 + m_T3 * q_3 + m_L * q_L) / (m_T1 + m_T2 + m_T3 + m_L)
| Inputs | Definition | Unit |
|---|---|---|
m_T1 | People or proportion of the cohort in T1 at the end of the cycle | people or proportion of cohort |
q_1 | Probability of death during a cycle spent in T1 | probability per cycle |
m_T2 | People or proportion of the cohort in T2 at the end of the cycle | people or proportion of cohort |
q_2 | Probability of death during a cycle spent in T2 | probability per cycle |
m_T3 | People or proportion of the cohort in T3 at the end of the cycle | people or proportion of cohort |
q_3 | Probability of death during a cycle spent in T3 | probability per cycle |
m_L | People or proportion of the cohort in L at the end of the cycle | people or proportion of cohort |
q_L | Probability of death during a cycle spent in L | probability per cycle |
q_bar | Share of the people in T1 to L at the end of a cycle who die in the next cycle | probability per cycle |
|---|
Function
Tunnel-state expansion of a cohort state-transition model
Maps a clinical state whose risks, costs or utilities depend on time since entry into a chain of K one-cycle tunnel states followed by a long-term state, so that a cohort model carries time in the state as well as the state itself. The expanded matrix P_tunnels is run with the cohort update s_(t+1) = s_t P of the Markov Model page (HE-FM-MM-001), which is not restated. Three existing records complete the method: tunnel probabilities from a survival curve for time since entry come from the rise in cumulative hazard across each cycle (HE-FM-CUMH-002), with the clock restarting on entry as HE-AS-CUMH-004 notes; the constant probability of a memoryless comparator calibrated to the same three-year survival is the rescaling formula HE-FM-TP-003 (p_old = 0.29344, Delta_old = 3 and Delta_new = 1 give 0.10933, as in the article); and life years from the expanded trace under a counting rule are HE-FM-CSIM-001 and HE-CF-CSIM-001. Notation follows the Tunnel State article.
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Implementations
Excel
Average death probability of tunnel states in one cell
With the occupancies of T1 to L in a row named TunnelOccRow and their death probabilities in a row of the same length named ExitProbRow, Excel returns the weighted average.
=SUMPRODUCT(TunnelOccRow,ExitProbRow)/SUM(TunnelOccRow)
Assumptions
Weights from the occupancy at the start of the cycle
The weights are the numbers in each state at the end of cycle t, which are the people at risk during cycle t+1. At least one of them is above zero.
Average specific to one cycle and one entry history
The result describes one cycle of one strategy. It changes from cycle to cycle and with the incidence of the entering event, so it summarises a tunnel model rather than replacing it.
Worked examples
Post-infarction average after four years at 5 per cent incidence
In the article's staggered-entry example, 41.53 × 0.20 + 35.34 × 0.08 + 34.59 × 0.04 + 35.33 × 0.03 gives 13.58 expected deaths among 146.79 post-infarction people, an average of 0.0925.
m_T1 = 41.5292; m_T2 = 35.344; m_T3 = 34.592; m_L = 35.328; q_1 = 0.20; q_2 = 0.08; q_3 = 0.04; q_L = 0.03; q_bar = 0.0925
Post-infarction average after four years at 10 per cent incidence
At an incidence of 10 per cent the post-infarction group holds more recent entrants, 70.50 in T1, 63.37 in T2, 65.50 in T3 and 70.66 in L, and the average is 0.0885, as in the article (occupancies computed here from the same model).
m_T1 = 70.4969; m_T2 = 63.368; m_T3 = 65.504; m_L = 70.656; q_1 = 0.20; q_2 = 0.08; q_3 = 0.04; q_L = 0.03; q_bar = 0.0885
Everyone in the first tunnel state after one year
After the first year of the staggered-entry example all 50 post-infarction people are in T1, so the average equals the first-year probability of 0.20, as in the article.
m_T1 = 50; m_T2 = 0; m_T3 = 0; m_L = 0; q_1 = 0.20; q_2 = 0.08; q_3 = 0.04; q_L = 0.03; q_bar = 0.20
Common errors
Averaging the tunnel death probabilities without weights
The simple mean of 0.20, 0.08, 0.04 and 0.03 is 0.0875, against the weighted 0.0925 in the article's year 4 example, and the unweighted figure does not move as the mix of durations changes (computed here for illustration).
Carrying one scenario's average into a single memoryless state
Fixing 0.0925 as the probability of a single post-infarction state fits only that cycle at 5 per cent incidence. The same tunnel gives 0.20 after one year, and 0.0885 or 0.0940 after four years at incidences of 10 or 3 per cent, so a strategy that changes incidence changes the right value.
Sources
Alarid-Escudero and colleagues on when cycle-specific probabilities can replace tunnels
Alarid-Escudero F, Krijkamp E, Enns EA, Yang A, Hunink MGM, Pechlivanoglou P, Jalal H. A tutorial on time-dependent cohort state-transition models in R using a cost-effectiveness analysis example. Medical Decision Making. 2023;43(1):21-41. Discussion: any time-varying feature in a discrete-time model can be implemented as tunnel states, and time-varying transition probabilities are a shortcut only when the cohort experiences the change simultaneously as a function of time from the simulation start.
ISPOR-SMDM task force on carrying time in state in the state definitions
Siebert U, Alagoz O, Bayoumi AM, Jahn B, Owens DK, Cohen DJ, Kuntz KM. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812-820. Defining states: if history, including time spent in a current state, is important in determining transition probabilities, the relevant states should carry that history in their definition.
Canonical Identity
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