Signature
n_S_tunnels = n_S + n_tunnels - 1
| Inputs | Definition | Unit |
|---|---|---|
n_S | Number of states in the model before the tunnelled state is split, the original state included | count |
n_tunnels | Number of states that replace the original state, the long-term state included | count |
n_S_tunnels | Number of states after expansion | count |
|---|
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
Expanded state count in one cell
Excel returns the size of the expanded model from named cells holding the number of states before expansion and the number of states the tunnelled state is split into.
=StatesBefore+TunnelSplit-1
Assumptions
One state expanded into a tunnel
The count applies to one tunnelled state. The tutorial discusses a single tunnel and notes that the same principles apply to more; each further tunnelled state adds its own n_tunnels minus one states.
Tunnel length set by the cycle length
n_tunnels is the number of cycles over which the duration effect changes plus one for the long-term state, so it depends on the cycle length: an effect that changes over three years needs four states with annual cycles and 37 with monthly ones.
Worked examples
Post-infarction state split into three annual tunnel states and a long-term state
The article's memoryless model has event-free, post-infarction and Dead. Splitting post-infarction into T1, T2, T3 and L gives six states, as in its staggered-entry example.
n_S = 3; n_tunnels = 4; n_S_tunnels = 6
Sick state of the Sick-Sicker tutorial split over a 75-cycle horizon
In the tutorial the duration effect in the Sick state lasts the whole 75-cycle horizon, so Sick becomes 75 tunnel states and the four-state model grows to 78 states.
n_S = 4; n_tunnels = 75; n_S_tunnels = 78
Three-year duration effect in a monthly model
With monthly cycles the same three-year effect needs 36 tunnel states and a long-term state, so the three-state model grows to 39 states rather than six (computed here for illustration).
n_S = 3; n_tunnels = 37; n_S_tunnels = 39
Common errors
Leaving the long-term state out of the tunnel count
Counting only the three one-cycle tunnel states as n_tunnels gives five states for the article's model in place of six. The missing state is the long-term state, so the survivors of T3 have nowhere to go unless T3 is given a non-zero diagonal, which breaks the tunnel.
Keeping the number of tunnel states when the cycle is shortened
Three tunnel states represent three years with annual cycles but only three months with monthly ones. A model moved to a shorter cycle needs the tunnel recounted, and the growth in states is the state explosion that the ISPOR-SMDM report warns about.
Sources
Alarid-Escudero and colleagues on the size of a model with 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. Section 3: the total number of states for a model with tunnels is n_S plus n_tunnels minus 1, one being subtracted because the original state is replaced; section 4.2: 75 Sick tunnel states plus three other states give 78.
ISPOR-SMDM task force on state explosion
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. Background: a Markov model can handle memory by creating states that include history, but this can greatly increase the number of states, called state explosion; recommendation III-1 on choosing a cohort or an individual-level model.
Canonical Identity
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