Bayesian methods in health technology assessment

Gianluca Baio

Department of Statistical Science | University College London

g.baio@ucl.ac.uk

https://gianluca.statistica.it

https://egon.stats.ucl.ac.uk/research/statistics-health-economics

https://github.com/giabaio https://github.com/StatisticsHealthEconomics

@gianlubaio@mas.to gianluca-baio

Karolinska Institute, Stockholm (Sweden)

SFO Epidemiology/Biostatistics and the KI Health Economics Network Seminar

4 December 2025

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Summary

Health economic evaluation
- What is and why do we need health economics?
A framework for health economic evaluation
- Statistical modelling
- Economic modelling
- Decision analysis
- Uncertainty analysis
Standard vs Bayesian HTA
- Two-stage vs integrated approach
Decision-making
- Cost-effectiveness plane; ICER; EIB

References

Bayesian Methods in Health Economics, chapters 1, 3
Bayesian cost effectiveness with the R package BCEA
Evidence Synthesis for Decision Making in Healthcare

Health technology assessment (HTA)

Objective

Combine costs and benefits of a given intervention into a rational scheme for allocating resources

Health technology assessment (HTA) is a method of evidence synthesis that considers evidence regarding clinical effectiveness, safety, cost-effectiveness and, when broadly applied, includes social, ethical, and legal aspects of the use of health technologies. The precise balance of these inputs depends on the purpose of each individual HTA. A major use of HTAs is in informing reimbursement and coverage decisions, in which case HTAs should include benefit-harm assessment and economic evaluation. Luce et al, 2010

(Quote stolen from a brilliant presentation by Cynthia Iglesias)

A relatively new discipline

Basically becomes “a thing” in the 1970s
Arguably, a historical accident…
- Economists take the lead in developing the main theory \(\Rightarrow\) Health Economics
- But there’s so much more to it (more on this later…)

(Truly…) World-beating Britain

Since its establishment, the National Institute for Health and Care Excellence (originally: National Institute for Clinical Excellence, NICE), has gained prominence as the global powerhouse for HTA

Health technology assessment (HTA)

NICE

Established in 1999, during the first New Labour government
Health Secretary Frank Dobson on whether it will work:

Probably not, but it’s worth a bloody try! Rawlins, 2009

Main driver: tackle the inequalities and inefficiencies generated by the “postcode lottery”
- Decisions about which drugs to fund through the NHS had historically been taken at a local level
- Concerns over the fact that patients in some areas of the country could access treatments that people elsewhere, sometimes in neighbouring streets, could not \(\Rightarrow\) large inequalities in access to resources!

Ancillary objectives
- Set quality standard nationally (although NICE is technically responsible for England only…)
- De-politicise reimbursement/coverage decisions
- Align with growing body of literature and experience in other countries (PBAC in Australia, CADTH in Canada, NZHTA in New Zealand,…)

Health technology assessment (HTA)

Objective

Combine costs and benefits of a given intervention into a rational scheme for allocating resources

Quality-Adjusted Life Years

Typically, experimental clinical studies set their primary outcome at some hard measure of relative effectiveness
- Physical measurements (e.g. blood pressure)
- Incidence of a specific clinical outcome (e.g. cardiovascular failure)
- Time until a clinically meaningful event occurs (e.g. time to death)

From a wider decision-making perspective, these are not necessarily ideal
- Complex to compare e.g. the economic performance of a highly specialised cancer drug (whose main outcome is reduction in cancer mortality) against an innovative form of physiotherapy to relieve from back pain
Hard clinical outcomes may fail to account for both the quantity and the quality of life for a given individual under a particular intervention. So: “health related quality of life” (HRQL) data
Assign a utility score to each specific health state and then compute the overall Quality-Adjusted Life Years

where:

\({\color{red} u_{ij}} =\) utility score for individual \(i\) at time \(j\)
\({\color{blue} \delta_j}=\) duration of the interval between \(j-1\) and \(j\)
\(\qalys = \displaystyle\sum_{j=1}^{J} \frac{\left(u_{j}+u_{j-1}\right)}{2} \delta_{j}\)

Health technology assessment (HTA)

Objective

Combine costs and benefits of a given intervention into a rational scheme for allocating resources

Economic modelling+Decision analysis

Base-case scenario

\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\hat\theta}_2]}}_{\class{red}{\hat\mu_{e2}}}} \class{myblue}{-} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\hat\theta}_1]}}_{\class{red}{\hat\mu_{e1}}}}\]

\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\hat\theta}_2]}}_{\class{red}{\hat\mu_{c2}}}} \class{myblue}{-} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\hat\theta}_1]}}_{\class{red}{\hat\mu_{c1}}}}\]

Limitations of ICER

The ICER is not an ordered statistic

–200/200 better than –100/200 better than –100/100 in terms of decision, but ratios are \(-1\), \(-1/2\), \(-1\)
ICERs in the NW quadrant indicate an intervention that is dominated (\(+\) costs / \(-\) effectiveness)

Decision-theoretic approach to HTA

Analytic framework for decision-making in the face of uncertainty

Considers a set of prescriptive axioms to ensure rationality in decision-making
Identifies the best course of action given:
- Model specification
- Current evidence

Basic principle: maximise the expected utility

Typical utility function in HTA: Monetary Net Benefit \(\class{myblue}{u(e, c; t) = nb_t = ke_t − c_t}\)
- \(k\) is the “willingness to pay”, i.e. the cost per extra unit of effectiveness gained – sometimes referred to as “cost-effectiveness threshold”
- Fixed, linear form, which simplifies computations
- Assumes decision-maker is risk neutral. Not necessarily true!

Decision-making

For each intervention, we can compute the expected net benefit \(\class{myblue}{\mathcal{NB}_t = \E\left[nb_t \right]}\)
The intervention associated with the highest expected net benefit \(\class{myblue}{\mathcal{NB}^*=\max_t \mathcal{NB}_t}\) is then selected as the most cost-effective

Expected Incremental Benefit

Assuming we are considering only two interventions \(t=(1,2)\), decision-making can be effected by looking at the Expected Incremental Benefit

\[ \begin{aligned}[b] \eib & = \mathcal{NB}_2 - \mathcal{NB}_1 \\ & = \E\left[k e_2 - c_2\right] - \E\left[k e_1 - c_1 \right] \\ & = k\E\left[e_2 - e_1\right] - \E\left[c_2 - c_1 \right] \\ & = k\E\left[\Delta_e \right]- \E\left[\Delta_c \right] \end{aligned} \]

The reference treatment \(t=2\) is more cost-effective than the comparator \(t=1\) if \[\begin{equation} \class{myblue}{\style{font-family:inherit;}{\text{EIB}}>0 \Rightarrow k>(<)\frac{\E[\Delta_c]}{\E[\Delta_e]} = \style{font-family:inherit;}{\text{ICER}} \quad \style{font-family:inherit;}{\text{ if }} \E[\Delta_e]>(<)0} \end{equation}\]

Cost-effectiveness plane

A QALY is a QALY is a QALY(?)…

The cost-effectiveness threshold around the world

https://twitter.com/mikepaulden/status/1333467554317156353

A QALY is a QALY is a QALY(?)…

Changes to the NICE threshold

NICE website, December 2025

In a statement, the [UK] Department of Health and Social Care said: “It means NICE will be able to approve medicines that deliver significant health improvements but might have previously been declined purely on cost-effectiveness grounds – this could include breakthrough cancer treatments, therapies for rare diseases and innovative approaches to conditions that have long been difficult to treat.”

BUT…

Commenting on the announcements, Sally Gainsbury, senior policy analyst at health think tank The Nuffield Trust said: “A big increase in the price the NHS pays by raising the NICE threshold will not bring additional benefits for the population as a whole, it will just make healthcare more expensive. The NHS budget is already under intense pressure and so the reported £3bn extra cost will need to be fully funded by the Treasury. However, even if it is not to come from day-to-day NHS budgets, that will not stop this being a deal that undermines the NHS’s ability to get the most health benefits for patients out of its resources”, she added.

Karl Claxton, professor of health economics at the centre for health economics at the University of York, said: “We urgently need to see an impact assessment, which takes account of the full weight of robust research evidence with a comprehensive valuation of all the impacts. Only then can NHS patients and the general public understand the consequences of this decision made on their behalf and appropriate parliamentary scrutiny can then be applied to this deal to examine whether it constitutes a good use of scarce public funds.”

Health technology assessment (HTA)

The straight path from the “Statistical model” to the “Decision analysis” represents the decision-making process given current evidence and conditionally on the assumptions made in the statistical analysis
This does not fully account for the inherent uncertainty in the estimates for the model parameters, which in turn determine the economic summaries and therefore have a potentially substantial impact on the uncertainty in the decision-making process

More importantly, the decision is generally not binary, because we could always consider a third option…
- What if we are not confident in making a decision just yet, because the data are not definitive and thus there remains too much uncertainty?
- In that case, the optimal decision is in fact to postpone it until we can obtain more data to reduce the current uncertainty (VoI)

Health technology assessment (HTA)

Objective

Combine costs and benefits of a given intervention into a rational scheme for allocating resources

Uncertainty analysis

Frequentist/“standard” approach

Uncertainty analysis

Uncertainty induced by \(g(\boldsymbol{\hat\theta}_1),g(\boldsymbol{\hat\theta}_2)\) — typically independent simulations

Statistical model

Economic model

Status quo

New drug

Decision analysis

Status quo
Benefits	Costs
741	670382.1
699	871273.3
...	...
726	425822.2
716.2	790381.2

New drug
Benefits	Costs
732	1131978
664	1325654
...	...
811	766411.4
774.5	1066849.8

Uncertainty analysis

Uncertainty induced by \(g(\boldsymbol{\hat\theta}_1),g(\boldsymbol{\hat\theta}_2)\) — typically independent simulations

Statistical model

Economic model

Status quo

New drug

Decision analysis

Status quo
Benefits	Costs
741	670382.1
699	871273.3
...	...
726	425822.2
716.2	790381.2

New drug
Benefits	Costs
732	1131978
664	1325654
...	...
811	766411.4
774.5	1066849.8

Uncertainty analysis

Uncertainty induced by \(g(\boldsymbol{\hat\theta}_1),g(\boldsymbol{\hat\theta}_2)\) — typically independent simulations

Statistical model

Economic model

Status quo

New drug

Decision analysis

Status quo
Benefits	Costs
741	670382.1
699	871273.3
...	...
726	425822.2
716.2	790381.2

New drug
Benefits	Costs
732	1131978
664	1325654
...	...
811	766411.4
774.5	1066849.8

Uncertainty analysis

Uncertainty induced by \(g(\boldsymbol{\hat\theta}_1),g(\boldsymbol{\hat\theta}_2)\) — typically independent simulations

Statistical model

Economic model

Status quo

New drug

Decision analysis

Status quo
Benefits	Costs
741	670382.1
699	871273.3
...	...
726	425822.2
716.2	790381.2

New drug
Benefits	Costs
732	1131978
664	1325654
...	...
811	766411.4
774.5	1066849.8

\[ \begin{align} \class{myblue}{\style{font-family:inherit;}{\text{ICER}}} & \class{myblue}{=} \frac{\class{myblue}{\style{font-family:inherit;}{\text{276468.6}}}}{\class{myblue}{\style{font-family:inherit;}{\text{58.3}}}}\\ & \class{myblue}{= \style{font-family:inherit;}{\text{6497.1}}} \end{align} \]

Uncertainty analysis*

\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_2]}}_{\class{red}{\mu_{e2}}}} \class{myblue}{-} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{e1}}}}\]

\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_2]}}_{\class{red}{\mu_{c2}}}} \class{myblue}{-} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{c1}}}}\]

*Induced by \(\class{myblue}{g(\boldsymbol{\hat\theta}_1),g(\boldsymbol{\hat\theta}_2)}\)

What’s wrong with this?…

Potential correlation between costs & clinical benefits (Individual level + Aggregated level Data)
- Strong positive correlation – effective treatments are innovative and result from intensive and lengthy research \(\Rightarrow\) are associated with higher unit costs
- Negative correlation - more effective treatments may reduce total care pathway costs e.g. by reducing hospitalisations, side effects, etc.
- Because of the way in which standard models are set up, bootstrapping generally only approximates the underlying level of correlation – (MCMC does a better job!)

Joint/marginal normality not realistic (Mainly ILD)
- Costs usually skewed and benefits may be bounded in \([0; 1]\)
- Can use transformation (e.g. logs) – but care is needed when back transforming to the natural scale
- Should use more suitable models (e.g. Beta, Gamma or log-Normal) – (generally easier under a Bayesian framework)
- Particularly relevant in presence of partially observed data – more on this later!

Particularly as the focus is on decision-making (rather than just inference), we need to use all available evidence to fully characterise current uncertainty on the model parameters and outcomes (Mainly ALD)
- A Bayesian approach is helpful in combining different sources of information
- (Propagating uncertainty is a fundamentally Bayesian operation!)

Decision-theoretic approach to HTA

Process of (Bayesian) rational decision-making

Describe uncertainty on all unknown quantities by means of a (possibly subjective) probability distribution \(\class{myblue}{p(\boldsymbol\omega) = p(e, c \mid \boldsymbol\theta)p(\boldsymbol\theta)}\)
For each intervention \(t\), outcomes \(o = (e, c)\) are valued by means of a pre-specified measure of utility \(\class{myblue}{u(e, c; t)}\)
Select as the most “cost-effective” the intervention that is associated with the maximum expected utility \(\class{myblue}{\mathcal{U}^t = \E_\boldsymbol\omega [u(e, c; t)]}\)

Under the MNB, the expected utility is

\[\begin{align} \class{myblue}{\mathcal{U}^t = \mathcal{NB}_t} & \class{myblue}{= \E_{\class{blue}{\boldsymbol\omega}}[u(e,c;t)]} \\ & \class{myblue}{= k\E_{\class{blue}{\boldsymbol\omega}}[e_t] - \E_{\class{blue}{\boldsymbol\omega}}[c_t]} \\ & \class{myblue}{= k\E_{\class{red}{\boldsymbol\theta}}[e\mid \boldsymbol\theta_t] - \E_{\class{red}{\boldsymbol\theta}}[c\mid \boldsymbol\theta_t]} \\ & \class{myblue}{= k\E[\mu_{et}] - \E[\mu_{ct}]} \end{align}\]

The expectation is taken with respect to \(p(\boldsymbol\omega)\) so \(\mathcal{NB}_t\) is a pure number!

Uncertainty analysis

Bayesian approach

Uncertainty analysis

Uncertainty induced by \(\class{myblue}{p(\boldsymbol\theta\mid \txt{data})}\) — uses the joint posterior of all the parameters!

Statistical model

Economic model

Status quo

New drug

Decision analysis

Status quo
Benefits	Costs
741	670382.1
699	871273.3
...	...
726	425822.2
716.2	790381.2

New drug
Benefits	Costs
732	1131978
664	1325654
...	...
811	766411.4
774.5	1066849.8

Economic model + Uncertainty analysis*

*Induced by \(\class{myblue}{p(\boldsymbol{\theta} \mid \style{font-family:inherit;}{\text{data}})}\)

Decision analysis

Summarising PSA

	Parameter simulations				Expected utility
Iteration	\(\lambda_1\)	\(\lambda_2\)	\(\lambda_3\)	\(\ldots\)	\(\nb_1(\boldsymbol\theta)\)	\(\nb_2(\boldsymbol\theta)\)	\(\ib(\boldsymbol\theta)\)
1	0.585	0.3814	0.4194	\(\ldots\)	77480	67795	-9685
2	0.515	0.0166	0.0768	\(\ldots\)	87165	106535	19370
3	0.611	0.1373	0.0592	\(\ldots\)	58110	38740	-19370
4	0.195	0.7282	0.7314	\(\ldots\)	77480	87165	9685
\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)
1000	0.0305	0.204	0.558	\(\ldots\)	48425	87165	38740
				Average	\(\mathcal{NB}_1=\)72365.35	\(\mathcal{NB}_2=\)77403.49	\(\eib=\)5038.14

\(\color{blue}\nb_t(\boldsymbol\theta)=k\mu_{et}-\mu_{ct}\) is the “known distribution” utility
\(\color{blue}\ib(\boldsymbol\theta)=\nb_2(\boldsymbol\theta)-\nb_1 (\boldsymbol\theta)\) is the incremental benefit (as a function of \(\boldsymbol\theta\))
Can summarise uncertainty in the decision-making process using the cost-effectiveness acceptability curve

\[\color{blue}\ceac=\Pr(\ib(\boldsymbol\theta) >0 \mid \style{font-family:inherit;}{\text{data}})\]

Upon varying \(k\), this is the probability that the optimal decision would not be reversed by reducing uncertainty

CE plane vs CEAC