Bayesian approaches for addressing missing data in Cost-Effectiveness Analysis (alongside Randomised Controlled Trials)

# Bayesian approaches for addressing missing data in Cost-Effectiveness Analysis (alongside Randomised Controlled Trials)

## Gianluca Baio

### [Department of Statistical Science](https://www.ucl.ac.uk/statistics/) | University College London

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### RSS South West Local Group, Internettown

16 June 2021

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  <path d="M503.52,241.48c-.12-1.56-.24-3.12-.24-4.68v-.12l-.36-4.68v-.12a245.86,245.86,0,0,0-7.32-41.15c0-.12,0-.12-.12-.24l-1.08-4c-.12-.24-.12-.48-.24-.6-.36-1.2-.72-2.52-1.08-3.72-.12-.24-.12-.6-.24-.84-.36-1.2-.72-2.4-1.08-3.48-.12-.36-.24-.6-.36-1-.36-1.2-.72-2.28-1.2-3.48l-.36-1.08c-.36-1.08-.84-2.28-1.2-3.36a8.27,8.27,0,0,0-.36-1c-.48-1.08-.84-2.28-1.32-3.36-.12-.24-.24-.6-.36-.84-.48-1.2-1-2.28-1.44-3.48,0-.12-.12-.24-.12-.36-1.56-3.84-3.24-7.68-5-11.4l-.36-.72c-.48-1-.84-1.8-1.32-2.64-.24-.48-.48-1.08-.72-1.56-.36-.84-.84-1.56-1.2-2.4-.36-.6-.6-1.2-1-1.8s-.84-1.44-1.2-2.28c-.36-.6-.72-1.32-1.08-1.92s-.84-1.44-1.2-2.16a18.07,18.07,0,0,0-1.2-2c-.36-.72-.84-1.32-1.2-2s-.84-1.32-1.2-2-.84-1.32-1.2-1.92-.84-1.44-1.32-2.16a15.63,15.63,0,0,0-1.2-1.8L463.2,119a15.63,15.63,0,0,0-1.2-1.8c-.48-.72-1.08-1.56-1.56-2.28-.36-.48-.72-1.08-1.08-1.56l-1.8-2.52c-.36-.48-.6-.84-1-1.32-1-1.32-1.8-2.52-2.76-3.72a248.76,248.76,0,0,0-23.51-26.64A186.82,186.82,0,0,0,412,62.46c-4-3.48-8.16-6.72-12.48-9.84a162.49,162.49,0,0,0-24.6-15.12c-2.4-1.32-4.8-2.52-7.2-3.72a254,254,0,0,0-55.43-19.56c-1.92-.36-3.84-.84-5.64-1.2h-.12c-1-.12-1.8-.36-2.76-.48a236.35,236.35,0,0,0-38-4H255.14a234.62,234.62,0,0,0-45.48,5c-33.59,7.08-63.23,21.24-82.91,39-1.08,1-1.92,1.68-2.4,2.16l-.48.48H124l-.12.12.12-.12a.12.12,0,0,0,.12-.12l-.12.12a.42.42,0,0,1,.24-.12c14.64-8.76,34.92-16,49.44-19.56l5.88-1.44c.36-.12.84-.12,1.2-.24,1.68-.36,3.36-.72,5.16-1.08.24,0,.6-.12.84-.12C250.94,20.94,319.34,40.14,367,85.61a171.49,171.49,0,0,1,26.88,32.76c30.36,49.2,27.48,111.11,3.84,147.59-34.44,53-111.35,71.27-159,24.84a84.19,84.19,0,0,1-25.56-59,74.05,74.05,0,0,1,6.24-31c1.68-3.84,13.08-25.67,18.24-24.59-13.08-2.76-37.55,2.64-54.71,28.19-15.36,22.92-14.52,58.2-5,83.28a132.85,132.85,0,0,1-12.12-39.24c-12.24-82.55,43.31-153,94.31-170.51-27.48-24-96.47-22.31-147.71,15.36-29.88,22-51.23,53.16-62.51,90.36,1.68-20.88,9.6-52.08,25.8-83.88-17.16,8.88-39,37-49.8,62.88-15.6,37.43-21,82.19-16.08,124.79.36,3.24.72,6.36,1.08,9.6,19.92,117.11,122,206.38,244.78,206.38C392.77,503.42,504,392.19,504,255,503.88,250.48,503.76,245.92,503.52,241.48Z"></path>
</svg></a>&nbsp;
]

.footer-center[
Missing data in HTA
]

.footer-right[
RSS South West Local Group, 16 Jun 2021
]

---

# Disclaimer...

<center>
<blockquote class="twitter-tweet"><p lang="en" dir="ltr">Best opening sentence <a href="https://twitter.com/hashtag/ISPOREurope?src=hash&amp;ref_src=twsrc%5Etfw">#ISPOREurope</a> from Gianluca Baio: “statisticians should rule the world and Bayesian statisticians should rule all statisticians” <a href="https://t.co/GN2w7liAcR">https://t.co/GN2w7liAcR</a></p>&mdash; Manuela Joore (@ManuelaJoore) <a href="https://twitter.com/ManuelaJoore/status/1191397718930939904?ref_src=twsrc%5Etfw">November 4, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> 
</center>

<span style="display:block; margin-top: 10px ;"></span>
...Just so you know what you're about to get into... 😉

---

# Health technology assessment (HTA)

**Objective**: Combine .red[costs] and .blue[benefits] of a given intervention into a rational scheme for allocating resources

---

**Objective**: Combine .red[costs] and .blue[benefits] of a given intervention into a rational scheme for allocating resources

---

**Objective**: Combine .red[costs] and .blue[benefits] of a given intervention into a rational scheme for allocating resources

---

**Objective**: Combine .red[costs] and .blue[benefits] of a given intervention into a rational scheme for allocating resources

---

<span style="display:block; margin-top: -14.6cm ;"></span>
**Probabilistic Sensitivity Analysis (PSA)**

---

.small["***Two-stage* approach**" ([Spiegelhalter et al, 2004](https://onlinelibrary.wiley.com/doi/book/10.1002/0470092602))]
---

.small["***Integrated* approach**" ([Spiegelhalter et al, 2004](https://onlinelibrary.wiley.com/doi/book/10.1002/0470092602); [Baio et al, 2017](http://www.statistica.it/gianluca/book/bcea/))]
---

# **Individual**-level data

## HTA alongside RCTs

<table class=" lightable-classic table" style="font-family: Ubuntu; font-size: 14px; margin-left: auto; margin-right: auto; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;" colspan="2"></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; color: orange !important;" colspan="3"><div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">Demographics</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; color: olive !important;" colspan="4"><div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">Clinical outcomes</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; color: blue !important;" colspan="4"><div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">HRQL data</div></th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; color: red !important;" colspan="4"><div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">Resource use data</div></th>
</tr>
  <tr>
   <th style="text-align:center;color: black !important; font-weight: bold;"> ID </th>
   <th style="text-align:center;color: black !important; font-weight: bold"> Trt </th>
   <th style="text-align:center;color: orange !important;"> Sex </th>
   <th style="text-align:center;color: orange !important;"> Age </th>
   <th style="text-align:center;color: orange !important;"> $\ldots$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{olive}{y_0}$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{olive}{y_1}$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{olive}{\ldots}$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{olive}{y_J}$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{blue}{u_0}$ </th>
   <th style="text-align:center;color: blue !important;"> $\color{blue}{u_1}$ </th>
   <th style="text-align:center;color: blue !important;"> $\ldots$ </th>
   <th style="text-align:center;color: red !important;"> $\color{blue}{u_J}$ </th>
   <th style="text-align:center;color: red !important;"> $\color{red}{c_0}$ </th>
   <th style="text-align:center;color: red !important;"> $\color{red}{c_1}$ </th>
   <th style="text-align:center;color: red !important;"> $\ldots$ </th>
   <th style="text-align:center;color: red !important;"> $\color{red}{c_J}$ </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;color: orange !important;"> M </td>
   <td style="text-align:center;color: orange !important;"> 23 </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{10}$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{11}$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{1J}$ </td>
   <td style="text-align:center;color: blue !important;"> 0.32 </td>
   <td style="text-align:center;color: blue !important;"> 0.66 </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> 0.44 </td>
   <td style="text-align:center;color: red !important;"> 103 </td>
   <td style="text-align:center;color: red !important;"> 241 </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> 80 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;color: orange !important;"> M </td>
   <td style="text-align:center;color: orange !important;"> 23 </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{20}$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{21}$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{2J}$ </td>
   <td style="text-align:center;color: blue !important;"> 0.12 </td>
   <td style="text-align:center;color: blue !important;"> 0.16 </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> 0.38 </td>
   <td style="text-align:center;color: red !important;"> 1204 </td>
   <td style="text-align:center;color: red !important;"> 1808 </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> 877 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;color: orange !important;"> F </td>
   <td style="text-align:center;color: orange !important;"> 19 </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{30}$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{31}$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $y_{3J}$ </td>
   <td style="text-align:center;color: blue !important;"> 0.49 </td>
   <td style="text-align:center;color: blue !important;"> 0.55 </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> 0.88 </td>
   <td style="text-align:center;color: red !important;"> 16 </td>
   <td style="text-align:center;color: red !important;"> 23 </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> 22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> $\ldots$ </td>
   <td style="text-align:center;"> $\ldots$ </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: orange !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: olive !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: blue !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
   <td style="text-align:center;color: red !important;"> $\ldots$ </td>
  </tr>
  <th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; color: black !important;" colspan="17"><div style="border-bottom: 1px solid #111111; margin-bottom: -1px; "></div></th>
</tbody>
</table>

<p style="margin-left: 2em; margin-top: 10px">– $\class{olive}{y_{ij}}=$ Survival time, event indicator (eg CVD), number of events, continuous measurement (eg blood pressure), ...</p>
<p style="margin-left: 2em;">– $\class{blue}{u_{ij}}=$ Utility-based score to value health (eg <a href="https://euroqol.org/">EQ-5D</a>, <a href="https://www.rand.org/health-care/surveys_tools/mos/36-item-short-form/scoring.html">SF-36</a>, <a href="https://en.wikipedia.org/wiki/Hospital_Anxiety_and_Depression_Scale">Hospital & Anxiety Depression Scale)</a>, ...</p>
<p style="margin-left: 2em;">– $\class{red}{c_{ij}}=$ Use of resources (drugs, hospital, GP appointments), ...</p>

- Usually aggregate longitudinal measurements into a cross-sectional summary and for each individual consider the pair `$\class{myblue}{(e_i,c_i)}$`

- HTA preferably based on .red[**utility-based**] measures of effectiveness

- Quality Adjusted Life Years (QALYs) are a measure of disease burden combining
   - .blue90red60[**Quantity**] of life (the amount of time spent in a given health state)
   - .blue60red90[**Quality**] of life (the utility attached to that state)
   
---

# ("Standard") Statistical modelling

1. Compute individual QALYs and total costs as 
`$$\class{myblue}{e_i = \displaystyle\sum_{j=1}^{J} \left(u_{ij}+u_{i\hspace{.5pt}j-1}\right) \frac{\delta_{j}}{2}} \qquad \style{font-family:inherit;}{\text{ and }} \class{myblue}{\qquad c_i = \sum_{j=0}^J c_{ij}} \qquad \left[\style{font-family:inherit;}{\text{with: }} \class{myblue}{\delta_j = \frac{\style{font-family:inherit;}{\text{Time}}_j - \style{font-family:inherit;}{\text{Time}}_{j-1}}{\style{font-family:inherit;}{\text{Unit of time}}}}\right]$$`

2. (Often implicitly) assume normality and linearity and model .olive[**independently**] individual QALYs and total costs by controlling for (**centered**) baseline values, eg
`$\class{myblue}{u^∗  = (u − \bar{u} )}$` and  `$\class{myblue}{c^∗  = (c − \bar{c} )}$`
`\begin{align}
e_i & = \alpha_{e0} + \alpha_{e1} u^*_{0i} + \alpha_{e2} \style{font-family:inherit;}{\text{Trt}}_i + \varepsilon_{ei}\, [+ \ldots], \qquad \varepsilon_{ei} \sim \dnorm(0,\sigma_e) \\
c_i & = \alpha_{c0} + \alpha_{c1} c^*_{0i} + \alpha_{c2} \style{font-family:inherit;}{\text{Trt}}_i + \varepsilon_{ci}\, [+ \ldots], \qquad\hspace{2pt} \varepsilon_{ci} \sim \dnorm(0,\sigma_c) 
\end{align}`

3. Estimate population average cost and effectiveness differentials 
   - **Under this model specification**, these are `$\class{myblue}{\Delta_e=\alpha_{e2}}$` and `$\class{myblue}{\Delta_c=\alpha_{c2}}$`

4. Quantify impact of uncertainty in model parameters on the decision making process
    - In a fully frequentist analysis, this is done using resampling methods (eg bootstrap)

---

# ("Standard") Statistical modelling

1. Build a population level model (eg decision tree/Markov model)

<img src="./img/unnamed-chunk-3-1.png" style="display: block; margin: auto;" width="75%" title="INSERT TEXT HERE">
    **NB**: in this case, the "data" are typically represented by summary statistics for the parameters of interest `$\bm\theta= (p_0 , p_1 , l, \ldots)$`, but may also have access to a combination of ILD and summaries

<ol style="counter-reset: my-awesome-counter 1;">
<li> Use point estimates for the parameters to build the “base-case” (average) evaluation</li>

<li>Use resampling methods (eg bootstrap) to propage uncertainty in the point estimates and perform uncertainty analysis</li>
</ol>

---

---

# What's wrong with that?...

- Potential correlation between costs & clinical benefits .alignright[.orange[[**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 &ndash; .olive[**MCMC does a better job!**]

- Joint/marginal normality not realistic .alignright[.orange[[**Mainly ILD**]]]
   - Costs usually skewed and benefits may be bounded in `$[0; 1]$` 
   - Can use transformation (e.g. logs) &ndash; but care is needed when back transforming to the natural scale
   - Should use more suitable models (e.g. Beta, Gamma or log-Normal) &ndash; .olive[**generally easier under a Bayesian framework**]
   - Particularly relevant in presence of partially observed data &ndash; 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 .alignright[.orange[[**ILD + ALD**]]]
   - A Bayesian approach is helpful in combining different sources of information 
   - .olive[**Propagating uncertainty is a fundamentally Bayesian operation!**]

---

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = p(e)p(c\mid e) = p(c)p(e\mid c)}$`

---

# Bayesian HTA in action

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = }\class{blue}{p(e)}\class{myblue}{p(c\mid e) = p(c)p(e\mid c)}$`

---

# Bayesian HTA in action

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = }\class{blue}{p(e)}\class{red}{p(c\mid e)}\class{myblue}{ = p(c)p(e\mid c)}$`

---

# Bayesian HTA in action

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = p(e)p(c\mid e) = p(c)p(e\mid c)}$`

- For example:
`\begin{align}
\class{blue}{e_i \sim \style{font-family:inherit;}{\text{Normal}}(\phi_{ei},\tau_e)} && \class{blue}{\phi_{ei}} &\class{blue}{= \alpha_0 [+ \ldots]} && \class{blue}{\mu_e = \alpha_0} \\
\class{red}{c_i\mid e_i \sim \style{font-family:inherit;}{\text{Normal}}(\phi_{ci},\tau_c)} && \class{red}{\phi_{ci}} &\class{red}{= \beta_0 + \beta_1(e_i -\mu_e)[+\ldots]} && \class{red}{\mu_ c = \beta_0}
\end{align}`

---

# Bayesian HTA in action

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = p(e)p(c\mid e) = p(c)p(e\mid c)}$`

- For example:
<span style="display:block; margin-top: -30px ;"></span>
`\begin{align}
\class{blue}{e_{i} \sim \style{font-family:inherit;}{\text{Beta}}(\phi_{ei}\tau_e,(1-\phi_{ei})\tau_e)} && \class{blue}{\style{font-family:inherit;}{\text{logit}}(\phi_{ei})} &\class{blue}{= \alpha_0 [+ \ldots]} && \class{blue}{\mu_e = \frac{\style{font-family:inherit;}{\text{exp}}(\alpha_0)}{1+\style{font-family:inherit;}{\text{exp}}(\alpha_0)}} \\
\class{red}{c_i\mid e_i \sim \style{font-family:inherit;}{\text{Gamma}}(\tau_c,\tau_c/\phi_{ci})} && \class{red}{\style{font-family:inherit;}{\text{log}}(\phi_{ci})} &\class{red}{= \beta_0 + \beta_1(e_i -\mu_e)[+\ldots]} && \class{red}{\mu_ c = \style{font-family:inherit;}{\text{exp}}(\beta_0)}
\end{align}`

---

# Bayesian HTA in action

- In general, can represent the joint distribution as `$\class{myblue}{p(e,c) = p(e)p(c\mid e) = p(c)p(e\mid c)}$`

<span style="display:block; margin-top: -20px ;"></span>
- For example:
<span style="display:block; margin-top: -30px ;"></span>
`\begin{align}
\class{blue}{e_{i} \sim \style{font-family:inherit;}{\text{Beta}}(\phi_{ei}\tau_e,(1-\phi_{ei})\tau_e)} && \class{blue}{\style{font-family:inherit;}{\text{logit}}(\phi_{ei})} &\class{blue}{= \alpha_0 [+ \ldots]} && \class{blue}{\mu_e = \frac{\style{font-family:inherit;}{\text{exp}}(\alpha_0)}{1+\style{font-family:inherit;}{\text{exp}}(\alpha_0)}} \\
\class{red}{c_i\mid e_i \sim \style{font-family:inherit;}{\text{Gamma}}(\tau_c,\tau_c/\phi_{ci})} && \class{red}{\style{font-family:inherit;}{\text{log}}(\phi_{ci})} &\class{red}{= \beta_0 + \beta_1(e_i -\mu_e)[+\ldots]} && \class{red}{\mu_ c = \style{font-family:inherit;}{\text{exp}}(\beta_0)}
\end{align}`

-  Combining "modules" and fully characterising uncertainty about deterministic functions of random quantities is relatively straightforward using MCMC

- Prior information can help stabilise inference (especially with sparse data!), eg
   - Cancer patients are unlikely to survive as long as the general population
   - ORs are unlikely to be greater than `$\pm \style{font-family:inherit;}{\text{5}}$`

---

# Missing data **in HTA**

- Missing data are complicated in any context

- But are fairly established in medical/bio-statistical research
  
  
-  In HTA it's even more complicated...

- Bivariate outcome, usually correlated
   - Normality not reasonable (skewness)
   - Other features of the data ("spikes")
   - Main objective: decision-making, not inference!
   
---

### Selection models: MCAR `$(e,c)$`

<span style="display:block; margin-top: 8px ;"></span>
- `$\class{olive}{m_{ei}\sim\dbern(\pi_{ei}); \qquad \logit(\pi_{ei})=\gamma_{e0}}$`

- `$\class{orange}{m_{ci}\sim\dbern(\pi_{ci}); \qquad \logit(\pi_{ci})=\gamma_{c0}}$`

---

### Selection models: MAR `$(e,c)$`

<span style="display:block; margin-top: 11px ;"></span>
- `$\class{olive}{m_{ei}\sim\dbern(\pi_{ei}); \qquad \logit(\pi_{ei})=\gamma_{e0} + \sum_{k=1}^K \gamma_{ek}x_{eik}}$`

- `$\class{orange}{m_{ci}\sim\dbern(\pi_{ci}); \qquad \logit(\pi_{ci})=\gamma_{c0} + \sum_{h=1}^H \gamma_{ch}x_{cih}}$`

---

### Selection models: MNAR `$(e,c)$`

<span style="display:block; margin-top: 11px ;"></span>
- `$\class{olive}{m_{ei}\sim\dbern(\pi_{ei}); \qquad \logit(\pi_{ei})=\gamma_{e0} + \sum_{k=1}^K \gamma_{ek}x_{eik} + \gamma_{eK+1}e_i;} \class{blue}{\quad \gamma_{eK+1}\sim\style{font-family:inherit;}{\text{Informative Prior}}}$`

- `$\class{orange}{m_{ci}\sim\dbern(\pi_{ci}); \qquad \logit(\pi_{ci})=\gamma_{c0} + \sum_{h=1}^H \gamma_{ch}x_{cih} + \gamma_{cH+1}c_i;} \class{blue}{\quad \gamma_{cK+1}\sim\style{font-family:inherit;}{\text{Informative Prior}}}$`

---

# Motivating example: MenSS trial

- The MenSS pilot RCT evaluates the cost-effectiveness of a new digital intervention to reduce the incidence of STI in young men with respect to the SOC
   - QALYs calculated from utilities (EQ-5D 3L)
   - Total costs calculated from different components (no baseline)
  
 
--

### Partially observed data

<center><img src=./img/tab_menss.png width='75%' title='INCLUDE TEXT HERE'></center>
   
---

### Skewness and "structural values"

---

# Modelling

---

.alignright[<svg viewBox="0 0 384 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  <g label="icon" id="layer6" groupmode="layer">    <path id="path2" d="M 120.19265,177.73779 C 123.18778,77.35076 64.277527,63.999998 64.277527,63.999998 v 31.26245 C 40.834519,83.611374 18.32863,81.929634 18.32863,81.929634 V 337.10903 c 0,0 98.10414,-11.41744 98.10414,84.40952 0,0 36.58424,-153.37442 248.86103,26.48145 0,-61.59342 0.37757,-216.93925 0.37757,-268.28471 C 169.9561,37.131382 120.1931,177.73779 120.1931,177.73779 Z m 187.20631,173.82056 -12.37599,-97.65441 h -0.448 l -40.72819,97.65441 h -17.55994 l -38.9362,-97.65441 h -0.448 l -14.17589,97.65441 h -43.87514 l 28.8015,-169.61925 h 43.42716 l 34.43518,90.6496 36.46566,-90.6496 h 43.87513 l 25.6817,169.61925 h -44.13938 z" style="stroke-width:0.0675239"></path>  </g></svg> [Gabrio et al (2018)](https://pubmed.ncbi.nlm.nih.gov/30565727/)]
1. **Bivariate Normal**     
   - Simpler and closer to "standard" frequentist models     
   - Accounts for .red65blue[correlation between QALYs and costs]     
<span style="display:block; margin-top: -10px ;"></span>
2. **Beta-Gamma**     
   - Account for .red65blue[correlation between outcomes] **and** model the relevant ranges: QALYs `$\in (0,1)$` and costs `$\in (0,\infty)$`    
   - **But**: needs to rescale the observed data `$\class{myblue}{e^*_{it}=(e_{it}-\epsilon)}$` to avoid spikes at 1

---

.alignright[<svg viewBox="0 0 384 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  <g label="icon" id="layer6" groupmode="layer">    <path id="path2" d="M 120.19265,177.73779 C 123.18778,77.35076 64.277527,63.999998 64.277527,63.999998 v 31.26245 C 40.834519,83.611374 18.32863,81.929634 18.32863,81.929634 V 337.10903 c 0,0 98.10414,-11.41744 98.10414,84.40952 0,0 36.58424,-153.37442 248.86103,26.48145 0,-61.59342 0.37757,-216.93925 0.37757,-268.28471 C 169.9561,37.131382 120.1931,177.73779 120.1931,177.73779 Z m 187.20631,173.82056 -12.37599,-97.65441 h -0.448 l -40.72819,97.65441 h -17.55994 l -38.9362,-97.65441 h -0.448 l -14.17589,97.65441 h -43.87514 l 28.8015,-169.61925 h 43.42716 l 34.43518,90.6496 36.46566,-90.6496 h 43.87513 l 25.6817,169.61925 h -44.13938 z" style="stroke-width:0.0675239"></path>  </g></svg> [Gabrio et al (2018)](https://pubmed.ncbi.nlm.nih.gov/30565727/)]
1. **Bivariate Normal**     
   - Simpler and closer to "standard" frequentist models     
   - Accounts for .red65blue[correlation between QALYs and costs]    
<span style="display:block; margin-top: -10px ;"></span>
2. **Beta-Gamma**     
   - Account for .red65blue[correlation between outcomes] **and** model the relevant ranges: QALYs `$\in (0,1)$` and costs `$\in (0,\infty)$`    
   - **But**: needs to rescale the observed data `$\class{myblue}{e^*_{it}=(e_{it}-\epsilon)}$` to avoid spikes at 1    
<span style="display:block; margin-top: -10px ;"></span>
3. **Hurdle model**    
   - Model `$e_{it}$` as a .blue[**mixture**] to account for .red65blue[correlation between outcomes] + relevant ranges + .olive[structural values]
   - May expand further to account for partially observed baseline utilities `$u_{0it}$` (needs untestable assumptions!)

---

.alignright[<svg viewBox="0 0 384 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  <g label="icon" id="layer6" groupmode="layer">    <path id="path2" d="M 120.19265,177.73779 C 123.18778,77.35076 64.277527,63.999998 64.277527,63.999998 v 31.26245 C 40.834519,83.611374 18.32863,81.929634 18.32863,81.929634 V 337.10903 c 0,0 98.10414,-11.41744 98.10414,84.40952 0,0 36.58424,-153.37442 248.86103,26.48145 0,-61.59342 0.37757,-216.93925 0.37757,-268.28471 C 169.9561,37.131382 120.1931,177.73779 120.1931,177.73779 Z m 187.20631,173.82056 -12.37599,-97.65441 h -0.448 l -40.72819,97.65441 h -17.55994 l -38.9362,-97.65441 h -0.448 l -14.17589,97.65441 h -43.87514 l 28.8015,-169.61925 h 43.42716 l 34.43518,90.6496 36.46566,-90.6496 h 43.87513 l 25.6817,169.61925 h -44.13938 z" style="stroke-width:0.0675239"></path>  </g></svg> [Gabrio et al (2018)](https://pubmed.ncbi.nlm.nih.gov/30565727/)]
1. **Bivariate Normal**     
   - Simpler and closer to "standard" frequentist models     
   - Accounts for .red65blue[correlation between QALYs and costs]    
<span style="display:block; margin-top: -10px ;"></span>
2. **Beta-Gamma**     
   - Account for .red65blue[correlation between outcomes] **and** model the relevant ranges: QALYs `$\in (0,1)$` and costs `$\in (0,\infty)$`    
   - **But**: needs to rescale the observed data `$\class{myblue}{e^*_{it}=(e_{it}-\epsilon)}$` to avoid spikes at 1    
<span style="display:block; margin-top: -10px ;"></span>
3. **Hurdle model**    
   - Model `$e_{it}$` as a .blue[**mixture**] to account for .red65blue[correlation between outcomes] + relevant ranges + .olive[structural values]
   - May expand further to account for partially observed baseline utilities `$u_{0it}$` (needs untestable assumptions!)

---

.alignright[<svg viewBox="0 0 384 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  <g label="icon" id="layer6" groupmode="layer">    <path id="path2" d="M 120.19265,177.73779 C 123.18778,77.35076 64.277527,63.999998 64.277527,63.999998 v 31.26245 C 40.834519,83.611374 18.32863,81.929634 18.32863,81.929634 V 337.10903 c 0,0 98.10414,-11.41744 98.10414,84.40952 0,0 36.58424,-153.37442 248.86103,26.48145 0,-61.59342 0.37757,-216.93925 0.37757,-268.28471 C 169.9561,37.131382 120.1931,177.73779 120.1931,177.73779 Z m 187.20631,173.82056 -12.37599,-97.65441 h -0.448 l -40.72819,97.65441 h -17.55994 l -38.9362,-97.65441 h -0.448 l -14.17589,97.65441 h -43.87514 l 28.8015,-169.61925 h 43.42716 l 34.43518,90.6496 36.46566,-90.6496 h 43.87513 l 25.6817,169.61925 h -44.13938 z" style="stroke-width:0.0675239"></path>  </g></svg> [Gabrio et al (2018)](https://pubmed.ncbi.nlm.nih.gov/30565727/)]
1. **Bivariate Normal**     
   - Simpler and closer to "standard" frequentist models     
   - Accounts for .red65blue[correlation between QALYs and costs]    
<span style="display:block; margin-top: -10px ;"></span>
2. **Beta-Gamma**     
   - Account for .red65blue[correlation between outcomes] **and** model the relevant ranges: QALYs `$\in (0,1)$` and costs `$\in (0,\infty)$`    
   - **But**: needs to rescale the observed data `$\class{myblue}{e^*_{it}=(e_{it}-\epsilon)}$` to avoid spikes at 1    
<span style="display:block; margin-top: -10px ;"></span>
3. **Hurdle model**    
   - Model `$e_{it}$` as a .blue[**mixture**] to account for .red65blue[correlation between outcomes] + relevant ranges + .olive[structural values]
   - May expand further to account for partially observed baseline utilities `$u_{0it}$` (needs untestable assumptions!)

---

# Bayesian multiple imputation (MAR)
<span style="display:block; margin-top: -5px ;"></span>

---

# `missingHE`
## A `R` package to deal with missing data in HTA

**Objectives**: Run a set of complex models to account for different level of complexity and missingness

<span style="display:block; margin-top: 30px ;"></span>
<svg viewBox="0 0 496 512" xmlns="http://www.w3.org/2000/svg" style="position:relative;display:inline-block;top:.1em;fill:black;height:0.8em;">  [ comment ]  <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"></path></svg> [GitHub repository](https://github.com/giabaio/missingHE)

---

# Conclusions

- A full Bayesian approach to handling missing data extends standard
"imputation methods"

- Can consider MAR and MNAR with relatively little expansion to the basic model
   
--

- Particularly helpful in cost-effectiveness analysis, to account for
 
   - Asymmetrical distributions for the main outcomes
   - Correlation between costs & benefits 
   - Structural values (eg spikes at 1 for utilities or spikes at 0 for costs)

- Need specialised software + coding skills

- `R` package `missingHE` to implement a set of general models
   
   ```r
   > priormodel<-list ("mu.prior.e"=mu.prior.e,"delta.prior.e"=delta.prior.e)
   >    run_model(data=data, model.eff=e~1, model.cost=c~1,
   +    dist_e="norm",dist_c="norm",type="MNAR_eff",stand=FALSE,
   +    program="JAGS",forward=FALSE,prob=c(0.05,0.95),n.chains=2,n.iter =20000,
   +    n.burnin=floor(20000/2),inits=NULL,n.thin=1,save_model=FALSE,prior=prior
   + )
   ```

- Link up with the "`R-HTA`-verse"

- Other packages that can be used to analyse/post-process the output of economic models
   - [`BCEA`](http://www.statistica.it/gianluca/software/bcea), [`survHE`](http://www.statistica.it/gianluca/software/survhe), [`heemod`](https://cran.r-project.org/web/packages/heemod/vignettes/a_introduction.html), ...
   
---

# Where can I find more, you ask?...

[https://r-hta.org/](https://r-hta.org/)

---

Baio, G. (2012). _Bayesian Methods in Health Economics_. Boca Raton, FL: Chapman Hall, CRC. ISBN: 1439895554. DOI:
[10.1201/b13099](https://doi.org/10.1201%2Fb13099). URL:
[http://www.statistica.it/gianluca/book/bmhe](http://www.statistica.it/gianluca/book/bmhe).

Baio, G. (2020b). "survHE: Survival analysis for health economic evaluation and cost-effectiveness modelling". In: _Journal of
Statistical Software_ 95.14, pp. 1-47. DOI: [10.18637/jss.v095.i14](https://doi.org/10.18637%2Fjss.v095.i14).

---