Deterministic Model

Deterministic Model in Health Economic Evaluation

In health economic evaluation, a deterministic model is the spreadsheet or code that takes one value for each parameter, passes it through a fixed structure and returns one set of expected costs and QALYs for each strategy. Decision trees and Markov cohort models are almost always built this way, because they track the expected share of a cohort in each state instead of simulating chance events for individual patients. The word describes how the calculation works and says nothing about whether the inputs are known, so the same deterministic model also serves as the engine of a probabilistic sensitivity analysis. Whether a single run at the mean input values gives the expected result that a decision needs depends on whether the model is linear in its inputs, and most health economic models are not.

What makes a health economic model deterministic

A model is deterministic when its output is a fixed function of its inputs and its structure. Krijkamp and colleagues, in their microsimulation tutorial, state that the result of a deterministic cohort model is precisely determined given a set of initial conditions and parameters.

$$Y = g(\boldsymbol{\theta}; M)$$

where $Y$ is the model output (for example expected costs and QALYs for each strategy), $\boldsymbol{\theta}$ is the vector of input parameter values, $M$ is the model structure and $g$ is the calculation the model performs. Running $g$ twice with the same $\boldsymbol{\theta}$ and $M$ gives the same $Y$.

A Markov cohort model is the standard example. The ISPOR-SMDM report on state-transition modelling by Siebert and colleagues explains that in a cohort simulation the entire cohort is redistributed across states after each cycle, and that each state is homogeneous, with every individual in it facing the same transition probabilities. The Markov trace therefore records expected proportions, and chance plays no part in the arithmetic. A decision tree solved by averaging payoffs over its branch probabilities works in the same way.

Stochastic patient-level models are built differently. In an individual-level microsimulation, the same report notes, whether an individual facing a given transition probability makes the transition depends on a random number. Briggs and colleagues group discrete event simulation with microsimulation as patient-level stochastic simulation. Krijkamp and colleagues show that a microsimulation with no memory and no baseline heterogeneity should converge on the deterministic cohort result as the number of simulated individuals grows.

Which uncertainty a deterministic model removes and which it keeps

The ISPOR-SMDM report on parameter estimation and uncertainty by Briggs and colleagues separates stochastic, or first-order, uncertainty, which is random variability in outcomes between identical patients, from parameter, or second-order, uncertainty in the estimated quantities that govern those outcomes. It treats heterogeneity, the variability between patients that their characteristics explain, as a separate concept, and adds structural uncertainty about the assumptions built into the model.

A deterministic cohort model removes first-order uncertainty by design. The report states that in cohort models parameter uncertainty can be addressed without concern for stochastic uncertainty, whereas patient-level stochastic simulations must first eliminate the stochastic noise, sometimes called Monte Carlo error, before parameter uncertainty can be assessed. Parameter uncertainty and structural uncertainty remain in full. Heterogeneity is handled by running the model separately for each subgroup, which the report links to separate cost-effectiveness analyses.

Running a deterministic model under uncertainty

Briggs and colleagues describe two ways to represent parameter uncertainty. In a deterministic sensitivity analysis parameter values are varied manually to test how sensitive the results are to particular parameters or sets of parameters, while in a probabilistic sensitivity analysis preferably all parameters are varied at once, with sets of values sampled from distributions defined in advance.

Both approaches use the same deterministic engine. In a probabilistic analysis each sampled parameter set is passed through $g$ in the usual way, so the spread of results across many runs reflects uncertainty in the inputs and not chance inside the model. This Monte Carlo simulation over parameters differs from sampling individual patients in a microsimulation. A deterministic analysis is a way of running a model, usually one run at central values or a set of chosen runs, and is a separate idea from a deterministic model.

The glossary of HEMA Report 2 on uncertainty, by Steuten and colleagues, calls deterministic sensitivity analysis useful for checking how a model behaves but unable to say how likely a wrong decision is. One-way sensitivity analysis and threshold analysis belong to this family, and a tornado diagram is the usual way to display a set of one-way results.

Why the run at mean inputs can miss the expected result

Decisions in economic evaluation rest on expected costs and effects. Wilson's methodological note argues that it is reasonable to choose the option with the highest expected net benefit, so model-based evaluations should always present expected outputs. When $g$ is non-linear, a single run with every input at its mean does not give those expected outputs.

$$g\left(E[\boldsymbol{\theta}]; M\right) \neq E\left[g(\boldsymbol{\theta}; M)\right]$$

where $E[\boldsymbol{\theta}]$ is the vector of parameter means, $g(E[\boldsymbol{\theta}]; M)$ is the deterministic run at those means and $E[g(\boldsymbol{\theta}; M)]$ is the expected output averaged over the joint distribution of the inputs, which a probabilistic analysis estimates. The two sides are equal when $g$ is linear in $\boldsymbol{\theta}$ and can differ when it is not.

Markov models are non-linear because the share of the cohort still alive in a given cycle is the product of the survival probabilities of every earlier cycle. Wilson notes that the same principle applies to partitioned survival models and individual sampling models, and concludes that the base case of a non-linear model should be the means from a probabilistic analysis. The HEMA report gives the same reason as the first of its two grounds for making probabilistic analysis the default.

Worked example with an uncertain risk of death

The numbers in this example are illustrative. A two-state Markov cohort model has the states Alive and Dead, annual cycles, no discounting and a lifetime horizon, and counts one life year for everyone alive at the start of each cycle. With a constant annual probability of death $p$, expected life years per person follow a geometric series.

$$L(p) = \sum_{t=0}^{\infty} (1-p)^{t} = \frac{1}{p}$$

where $L(p)$ is expected life years per person, $t$ is the cycle number and $p$ is the annual probability of death. Under standard care $p$ is uncertain and equally likely to be 0.10 or 0.30, so its mean is 0.20. A new treatment halves the probability of death, the quality of life weight in the Alive state is 0.8 under both strategies, and the new treatment costs GBP 120,000 once at the start while standard care costs nothing.

1. Deterministic run at the mean. Standard care gives $L(0.20) = 5.0$ life years and the new treatment gives $L(0.10) = 10.0$, a gain of 5.0 life years or 4.0 QALYs at a weight of 0.8.
2. Expected values over the two equally likely inputs. Standard care gives the average of $L(0.10) = 10.0$ and $L(0.30) = 3.33$, which is 6.67 life years. The new treatment gives the average of $L(0.05) = 20.0$ and $L(0.15) = 6.67$, which is 13.33. The gain is 6.67 life years, or 5.33 QALYs at a weight of 0.8.
3. Incremental cost-effectiveness ratio. The deterministic ICER is GBP 120,000 divided by 4.0, or GBP 30,000 per QALY. The ratio of expected values is GBP 120,000 divided by 5.33 (exactly 16/3), or GBP 22,500 per QALY.
4. Net health benefit at GBP 25,000 per QALY. The cost is worth GBP 120,000 divided by 25,000, or 4.8 QALYs. Net health benefit is 4.0 minus 4.8, or -0.8 QALYs per person, in the deterministic run, and 5.33 minus 4.8, or 0.53 QALYs per person, in expectation.

The deterministic run understates the QALY gain by a quarter, because $L(p) = 1/p$ is convex and the low-mortality value adds more life years than the high-mortality value takes away. At GBP 25,000 per QALY, the lower end of the range used by NICE, the run at mean inputs points to rejection while the expected values point to adoption. A concave relationship would make the deterministic run overstate the benefit instead, and Wilson warns that a deterministic base case can lead to misleading recommendations. In the HEMA report's illustrative cancer example, right-skewed survival gains give a probabilistic mean gain of 0.66 QALYs against a deterministic point estimate of 0.58.

What NICE and the HEMA report expect

Current HTA guidance gives the deterministic run a supporting role behind the probabilistic results. The NICE technology appraisal manual (PMG36, last updated 13 October 2026) states in section 4.7.12 that the committee's preferred cost-effectiveness estimate should be derived from a probabilistic analysis when possible unless the model is linear, and that any use of deterministic results should be clearly justified. Section 4.7.13 adds that scenario analyses should in general also be probabilistic, with deterministic scenarios justified, and notes that this matters less when the base case is shown to be linear or only moderately non-linear. Section 4.7.17 keeps deterministic sensitivity analyses for identifying the parameters to which the decision is most sensitive, but warns that deterministic analysis is less appropriate for decision making when the model is non-linear.

The HEMA report recommends probabilistic analysis as the default and gives deterministic and scenario analysis a narrower role as uncertainty diagnostics. Varying one or two inputs at a time, it notes, helps expose coding and conceptual errors and identify influential inputs, but cannot quantify the uncertainty in a decision. Among its minimum reporting expectations, it asks evidence producers to report the expected incremental costs, health effects and net health benefit from the probabilistic analysis, together with the deterministic result and an explanation of any material difference. Measures of decision uncertainty and the expected value of perfect information require the probabilistic runs, and a single deterministic run supports neither.

Boundaries with nearby concepts

Several neighbouring terms are easily confused with a deterministic model. The differences turn on whether a term describes a type of model or a number that a model produces.

• A deterministic model contrasts with a stochastic model, in which random elements such as sampled patient histories change the output from one run to the next.
• A point estimate is a single output value, and the base case is the preferred set of inputs and assumptions, which for a non-linear model NICE expects to be summarised from a probabilistic run where possible.
• A cohort model is the most common kind of deterministic model in HTA, and a state-transition model can be built either as a deterministic cohort model or as a stochastic microsimulation.

Common misunderstandings about deterministic models

Most errors come from reading the label as a statement about the evidence, or from treating the run at mean inputs as the answer. Four recurring errors stand out.

1. Deterministic means certain. The label refers to the calculation, and the parameters are still estimates with uncertainty. The HEMA report adds that probabilistic analysis is not free from judgement either, because its distributions and correlations rest on the same evidence and modelling assumptions.
2. Repeatable means deterministic. Krijkamp and colleagues note that setting the random number seed allows microsimulation results to be reproduced despite the stochastic process, so a reproducible result does not by itself show that a model is deterministic.
3. The base-case run is the expected result. This holds only for a model that is linear in its inputs, and the worked example above shows how far apart the two can be.
4. A sensitivity range measures uncertainty. The HEMA report notes that varying an input by, for example, plus or minus 20% shows how sensitive the model is to that change, and does not show how uncertain the input itself is.

Sources

• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Krijkamp EM, Alarid-Escudero F, Enns EA, Jalal HJ, Hunink MGM, Pechlivanoglou P. Microsimulation modeling for health decision sciences using R: a tutorial. Medical Decision Making. 2018;38(3):400-422. https://doi.org/10.1177/0272989X18754513
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 4.7.12, 4.7.13, 4.7.15 and 4.7.17. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• 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. https://doi.org/10.1016/j.jval.2012.06.014
• Steuten L, Thavorn K, Cipriano LE, Guertin JR, McQueen B, Schmidler G, Sullivan S, Wailoo A, Sculpher M. Methods to Analyze and Present Uncertainty in Economic Evaluation to Support HTA Decisions. HEMA Report 2. Health Economics Methods Advisory; 2026. https://hemamethods.org/wp-content/uploads/2026/10/HEMA-Report-2_For-Publication_100726.pdf
• Wilson ECF. Methodological note: reporting deterministic versus probabilistic results of Markov, partitioned survival and other non-linear models. Applied Health Economics and Health Policy. 2021;19(6):789-795. https://doi.org/10.1007/s40258-021-00664-2