Gianluca Baio
Department of Statistical Science  University College London
https://gianluca.statistica.it
https://egon.stats.ucl.ac.uk/research/statisticshealtheconomics
https://github.com/giabaio https://github.com/StatisticsHealthEconomics
@gianlubaio@mas.to @gianlubaio
Bayesian modelling for economic evaluation of healthcare interventions
València International Bayesian Analysis Summer School, 7th edition, University of Valencia
10  11 July 2024
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References
Health technology assessment (HTA) is a method of evidence synthesis that considers evidence regarding clinical effectiveness, safety, costeffectiveness 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 benefitharm assessment and economic evaluation. Luce et al, 2010
(Quote stolen from a brilliant presentation by Cynthia Iglesias)
\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\hat\theta}_1]}}_{\class{red}{\hat\mu_{e1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\hat\theta}_0]}}_{\class{red}{\hat\mu_{e0}}}}\]
\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\hat\theta}_1]}}_{\class{red}{\hat\mu_{c1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\hat\theta}_0]}}_{\class{red}{\hat\mu_{c0}}}}\]
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 
\[ \begin{align} \class{myblue}{\style{fontfamily:inherit;}{\text{ICER}}} & \class{myblue}{=} \frac{\class{myblue}{\style{fontfamily:inherit;}{\text{276468.6}}}}{\class{myblue}{\style{fontfamily:inherit;}{\text{58.3}}}}\\ & \class{myblue}{= \style{fontfamily:inherit;}{\text{6497.1}}} \end{align} \]
\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{e1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{e0}}}}\]
\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{c1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{c0}}}}\]
*Induced by \(\class{myblue}{g(\boldsymbol{\hat\theta}_0),g(\boldsymbol{\hat\theta}_1)}\)
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 
\[ \begin{align} \class{myblue}{\style{fontfamily:inherit;}{\text{ICER}}} & \class{myblue}{=} \frac{\class{myblue}{\style{fontfamily:inherit;}{\text{276468.6}}}}{\class{myblue}{\style{fontfamily:inherit;}{\text{58.3}}}}\\ & \class{myblue}{= \style{fontfamily:inherit;}{\text{6497.1}}} \end{align} \]
\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{e1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{e0}}}}\]
\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{c1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{c0}}}}\]
*Induced by \(\class{myblue}{p(\boldsymbol{\theta} \mid \style{fontfamily:inherit;}{\text{data}})}\)
\[\class{myblue}{\Delta_e=}\class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{e1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[e \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{e0}}}}\]
\[\class{myblue}{\Delta_c=}\class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_1]}}_{\class{red}{\mu_{c1}}}} \class{myblue}{} \class{red}{\underbrace{\class{myblue}{\E[c \mid \boldsymbol{\theta}_0]}}_{\class{red}{\mu_{c0}}}}\]
Analytic framework for decisionmaking in the face of uncertainty
Considers a set of prescriptive axioms to ensure rationality in decisionmaking
Identifies the best course of action given:
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 prespecified measure of utility \(\class{myblue}{u(e, c; t)}\)
Select as the most “costeffective” the intervention that is associated with the maximum expected utility \(\class{myblue}{\mathcal{U}^t = \E_\boldsymbol\omega [u(e, c; t)]}\)
\[\begin{align} \class{myblue}{\mathcal{U}^t = \mathcal{NB}_t} & \class{myblue}{= \E_{\class{blue}{\boldsymbol\omega}}[u(e,c;t)]} \\ & \class{myblue}{= k\E_{\class{blue}{\boldsymbol\omega}}[e_t]  \E_{\class{blue}{\boldsymbol\omega}}[c_t]} \\ & \class{myblue}{= k\E_{\class{red}{\boldsymbol\theta}}[e\mid \boldsymbol\theta_t]  \E_{\class{red}{\boldsymbol\theta}}[c\mid \boldsymbol\theta_t] = k\E[\mu_{et}]  \E[\mu_{ct}]} \end{align}\]
The expectation is taken with respect to \(p(\boldsymbol\omega)\) so \(\mathcal{NB}_t\) is a pure number!
\[\begin{align} \class{myblue}{\style{fontfamily:inherit;}{\text{EIB}}} & \class{myblue}{= \mathcal{NB}_1\mathcal{NB}_0} \\ & \class{myblue}{= \left(k\E[\mu_{e1}]\E[\mu_{c1}]\right)  \left(k\E[\mu_{e0}]\E[\mu_{c0}]\right)} \\ & \class{myblue}{= k\E[\Delta_e]  \E[\Delta_c]} \end{align}\]
Parameter simulations

Expected utility



Iteration  \(\lambda_1\)  \(\lambda_2\)  \(\lambda_3\)  \(\ldots\)  \(\nb_0(\boldsymbol\theta)\)  \(\nb_1(\boldsymbol\theta)\)  \(\ib(\boldsymbol\theta)\) 
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  \(\mathcal{NB}_0=\)72365.35  \(\mathcal{NB}_1=\)77403.49  \(\eib=\)5038.14 
\(\color{blue}\nb_t(\boldsymbol\theta)=k\mu_{et}\mu_{ct}\) is the “known distribution” utility
\(\color{blue}\ib(\boldsymbol\theta)=\nb_1(\boldsymbol\theta)\nb_0(\boldsymbol\theta)\) is the incremental benefit (as a function of \(\boldsymbol\theta\))
Can summarise uncertainty in the decisionmaking process using the costeffectiveness acceptability curve
\[\color{blue}\ceac=\Pr(\ib(\boldsymbol\theta)\mid \style{fontfamily:inherit;}{\text{data}})>0\]
1. Intro HTA 2. Bayesian computation 3. ILD 4. ALD 5. NMA © Gianluca Baio (UCL)   Bayesian models in HTA  VIBASS7 2024  10  11 July 2024 