Cohort Markov & cost-effectiveness

Compare long-term costs and quality-adjusted life-years (QALYs) when people move among a small set of health states.

Method lesson · Updated October 6, 2026

Concepts and reasoning

A model is an explicit account of what can happen

A cohort state-transition model represents a population by the fraction occupying each health state over time. For example, Stable, Progressed, and Dead describe three possible states at the start of a cycle. The model does not identify which particular patient progresses; it calculates how much of the population moves along each allowed transition. Those fractions can represent expected proportions in a very large cohort.

States should be mutually exclusive, so a person is not counted twice, and collectively exhaustive, so nobody disappears. Their definitions should correspond to clinically meaningful differences in prognosis, cost, or quality of life. A state called “treated” and a state called “severe disease” would overlap unless the model explicitly separates their combinations. The chosen states determine what history the model remembers.

The Markov assumption is a statement about memory

In a basic Markov model, the distribution of the next state depends on the present state, not the complete path taken to reach it. Two patients currently in Progressed have the same modeled transition probabilities even if one progressed yesterday and the other several years ago. That may be a useful simplification or a serious mismatch. Tunnel states, time-dependent models, and individual simulation can represent additional history, but simply drawing more patients from the same memoryless transition matrix cannot.

The cycle is the time interval over which a transition probability applies. An annual probability cannot be pasted unchanged into a monthly model. If a constant event rate is appropriate, the corresponding probability over duration is . Multiple competing transitions require a coherent joint conversion rather than independently converting each exit probability and hoping the row still represents a valid process. The DARTH introductory tutorial develops the cohort approach.

Mathematical foundations

On a small screen, scroll wide equations and tables horizontally.

Updating the cohort

Let the row vector contain the state proportions at cycle , and let be the probability of moving from state to state during one cycle:

The row-sum condition conserves population mass. An absorbing Dead state has a self-transition probability of 1 and zero probabilities of leaving it. A state with a self-transition of 0.9 is not absorbing: 10% exits each cycle.

Attaching consequences

With an annual state-cost vector , utility vector , and cycle length in years, start-of-cycle rewards are and . These are weighted averages, not separate estimates for each person. Discounting divides future rewards by . End-of-cycle or trapezoidal timing changes which occupancy enters the reward; timing must match the interpretation of costs and events.

For two strategies, define and . Incremental net monetary benefit is

The threshold converts a QALY into the monetary opportunity-cost scale used for the decision. Positive INMB favors strategy 1 under that threshold and the stated model.

Worked example

Hypothetical data for learning; independent of your analysis inputs.

Follow a three-state cohort by hand

Start with everyone Stable, so . Consider the following hypothetical annual matrix:

From / to	Stable	Progressed	Dead
Stable	0.850	0.100	0.050
Progressed	0.000	0.800	0.200
Dead	0.000	0.000	1.000

After one cycle, . After the second, Stable is ; Progressed is ; Dead is . The entries sum to 1. This second step shows why applying only the initial Stable row repeatedly would be wrong: people who have progressed face a different future.

Suppose annual costs are 1000, 4000, and 0, and utilities are 0.85, 0.50, and 0. At the start of the second one-year cycle, expected cost is and expected QALYs are , before discounting. These numbers use start-of-cycle timing.

Finally, suppose an intervention adds 0.20 lifetime QALYs and 5000 in cost. The ICER is 25,000 per QALY. At a threshold of 50,000, INMB is . If the intervention instead costs more and produces fewer QALYs, a negative ICER would obscure the fact that it is dominated; the signs of both increments must be inspected.

Interpreting the results

Connect occupancy to accumulated outcomes

A state-occupancy trace describes the proportion of a modeled cohort in each state at each cycle boundary. It is not a set of individual histories. Costs and quality-adjusted life-years accumulate from time spent in states, so similar final occupancy can conceal different total outcomes earlier in the horizon.

Read incremental outcomes as intervention minus comparator. A positive incremental cost and positive health gain imply a trade-off; a negative cost with positive health gain indicates dominance within the model. An incremental cost-effectiveness ratio is difficult to interpret when the health difference is near zero. Always report its numerator and denominator.

At a willingness-to-pay threshold λ, incremental net monetary benefit is λ times incremental health minus incremental cost. A positive value favors the intervention at that threshold and under the modeled assumptions. It does not demonstrate affordability, clinical effectiveness, or certainty about the choice.

Assumptions and limitations

Decide what a state must remember

States should be mutually exclusive and collectively cover the modeled population. The Markov assumption means the next transition depends on the current state rather than the full previous history. If time since progression changes prognosis, a single progressed state may be inadequate; additional states or another model structure may be needed.

A time-homogeneous model uses the same transition probabilities each cycle. It cannot represent aging, treatment waning, or changing hazards merely by extending the horizon. Each probability must correspond to the chosen cycle length, and competing transitions in a row must form a coherent joint set.

Costs, utilities, perspective, discounting, and horizon also define the question. Annual state costs differ from one-off transition costs. A state utility summarizes health-related quality of life, not the probability of occupying that state. These distinctions affect the reward calculation even when every transition row is numerically valid.

Diagnostics and sensitivity analysis

Check conservation and timing

Every occupancy row should sum to one, and an absorbing death state should never lose population. Check the first transition manually with the initial distribution and transition matrix. Then test limiting cases: identical strategies should produce zero incremental outcomes; zero costs should produce zero accumulated cost.

Vary horizon, cycle length with appropriately converted probabilities, discount rates, and reward timing. Shorter cycles require revised transition probabilities; changing only the number of cycles does not create an equivalent model. Inspect how much survival or cost remains beyond the chosen horizon.

Compare predicted survival or state prevalence with evidence not used solely to set the same parameters. Parameter sensitivity cannot repair an omitted disease stage or an implausible transition. Document structural alternatives as separate scenarios. The DARTH cohort-model tutorial provides reproducible examples of model traces and economic outcomes.

References and other material

Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.

DARTH: introductory cohort modeling tutorial — An author-maintained tutorial with reproducible health economic models.