Who wants to be a Bayesian? (and why you should…)

# Who wants to be a Bayesian? (and why you should...)

## Gianluca Baio

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

.title-small[
<svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="position:relative;display:inline-block;top:.1em;fill:#00acee;height:0.8em;">  [ comment ]  <path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"></path></svg>  [g.baio@ucl.ac.uk](mailto:g.baio@ucl.ac.uk)
<svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="position:relative;display:inline-block;top:.1em;fill:#EA7600;height:0.8em;">  [ comment ]  <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>  [http://www.statistica.it/gianluca/](http://www.statistica.it/gianluca/)
<svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="position:relative;display:inline-block;top:.1em;fill:#EA7600;height:0.8em;">  [ comment ]  <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>  [https://egon.stats.ucl.ac.uk/research/statistics-health-economics/](https://egon.stats.ucl.ac.uk/research/statistics-health-economics/)
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### Virtual ISPOR US Conference, The Internet

20 May 2021

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Who wants to be a Bayesian?
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Virtual ISPOR US Conference, 20 May 2021
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# What are we talking about?...

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---

## Data

1. We may (or may not!) access **individual level data** for "our" trial, but not for the competitors'    
2. The trial data have a very limited follow up, which implies large amount of censoring
   - This is often OK(-ish!) for "medical stats" analysis. But **HORRIBLE** for economic evaluation! `$\Rightarrow$` Extrapolation    
3. Often the data are manipulated by the stats team within the sponsor and the economic modellers only get summaries/estimates    
   - It is **ALWAYS** good to [leave things to statisticians](https://twitter.com/manuelajoore/status/1329413099678539785). But the modellers can (should?!) be statisticians too, so they could handle the data!...

]

---

## Data

1. Which model is the "best fit" &ndash; how to judge that?

2. Is modelling even enough? (How to make the most of "external data")

3. Should you be Bayesians about this? 
   - (Spoiler alert: the answer is *always* Yes!...)

]

---

# Combining composite data sources

## Increasingly popular
- 13 Technology Assessments (TAs) in immuno-oncology in the period 2019-2021

- 7 formally included external data, of various form

- Sources used to support treatment effect waning (or lack of it) included:   
    - Other non-pivotal clinical trials and published sources with specific % of patients alive at a time point   
    - Flatiron (or other registries such as SEER)   
    - Clinical expert opinion ("soft" vs "hard" data... `$\Rightarrow$` more on this later)   
       - On % of patients surviving at a specific time `$t$`   
       - On clinical implausibility of hazards crossing and becoming higher for intervention vs comparator

## Challenges
- Heterogeneity/representativeness 
   - "Exchangeability"
   
- Afterthought vs plan ahead...

- KOL/Expert opinion/soft evidence: elicitation, formal modelling?...

---

# *To be or not to be (a Bayesian)?...*

.center[
.pull-left[
### Frequentist ("standard")
]
.pull-right[
### Bayesian 
]
]
.center[
<center><img src=./img/unnamed-chunk-2-1.png width='75%' title=''></center>
]

- A Bayesian only speaks one language: probability distributions to describe    
   - Sampling variability (relevant for observ.blue[***ed***] data)
   - Epistemic uncertainty (relevant for .orange[***un***]observ.orange[***able***] parameters + yet .magenta[***un***]observ.magenta[***ed***] future data)
--

- Contextual (="prior") information to be formally included in the construction of the model    
   - Almost irrelevant when evidence is "definitive" (large and consistent data)    
   - Crucial when data are sparse! (... But this isn't preposterous, is it?...)
---

.center[
.pull-left[
### Frequentist ("standard")
<center><img src=./img/two-stage.png width='610px' title=''></center>
]

.pull-right[
### Bayesian     
<center><img src=./img/integrated.png width='610px' title=''></center>
]
]

---

## General structure

`$$\class{myblue}{t \sim f(\mu(\bm{x}),\alpha(\bm{x})), \qquad t\geq 0}$$`

- `$\bm{x}=$` vector of covariates (potentially influencing survival)

- `$\mu(\bm{x})=$` .blue[**location**] parameter
   - Scale or mean &ndash; usually main objective of the (biostats!) analysis   
   - Typically depends on the covariates `$\bm{x}$`
   
- `$\alpha(\bm{x})=$` .olive[**ancillary**] parameters   
   - Shape, variances, etc
   - May depend on `$\bm{x}$`, but often assume they don't (see <svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  [ comment ]  <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> [NICE TSD 14](http://nicedsu.org.uk/technical-support-documents/survival-analysis-tsd/))
   
- **NB**: `$S(t)$` and `$h(t)$` are functions of `$\mu(\bm{x}), \alpha(\bm{x})$`

`$$\class{myblue}{g(\mu_i)=\beta_0 + \sum_{j=1}^J \beta_j x_{ij} [+ \ldots]}$$`
   <span style="display:block; margin-top: -20px ;"></span>
   &ndash; since `$t>0$`, usually, `$g(\cdot) = \log$`

- In a Bayesian setting, complete by putting suitable priors on `$\bm\beta$` and `$\alpha$`

---

# **Bayesian** survival analysis in HTA

.small[.alignright[<svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  [ comment ]  <path d="M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm-28.9 143.6l75.5 72.4H120c-13.3 0-24 10.7-24 24v16c0 13.3 10.7 24 24 24h182.6l-75.5 72.4c-9.7 9.3-9.9 24.8-.4 34.3l11 10.9c9.4 9.4 24.6 9.4 33.9 0L404.3 273c9.4-9.4 9.4-24.6 0-33.9L271.6 106.3c-9.4-9.4-24.6-9.4-33.9 0l-11 10.9c-9.5 9.6-9.3 25.1.4 34.4z"></path></svg> [See here](https://www.jstatsoft.org/article/view/v095i14)]]

---

- We can specify "minimally informative" priors (eg like [`survHE`](http://www.statistica.it/gianluca/software/survhe/) does by default)
   - In many ways, that's the "lazy" option...
   
- Similarly, we can try the various models suggested in the guidelines and see what happens...

- We probably *know* something more about the likely shape of the hazard function
   - Likely to be monotonically increasing? 
   - Definitely unlikely to be constant over time?...
- These considerations should drive the choice of models **over and above** testing all the options!
]

---

- What else do we know?
   - Likely average survival time
   - Chances of surviving after `$t^*$` units of time (eg >75 years old)
   - Population data to "anchor" the extrapolated survival curves
   - `$\ldots$`
]

---

# Example: ICD & Cardiac death

## Basic idea/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> [Benaglia et al (2015)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4847642/)]

Use UK population data (matched by age/sex) to "**anchor**" the ICD population at risk

- Perhaps the easiest way to do this is to relate the hazard between the two populations &ndash; eg **proportional hazard** (PH) model
<span style="display:block; margin-top: -20px ;"></span>
   `$$\class{myblue}{h_{\rm{ICD}}(t) = e^{\beta}h_{\rm{UK}}(t) \qquad \Leftrightarrow \qquad \HR = \frac{h_{\rm{ICD}}(t)}{h_{\rm{UK}}(t)} = e^{\beta} = \style{font-family:inherit;}{\text{Constant}}}$$`
   
<span style="display:block; margin-top: -20px ;"></span>
   
- Relatively easy to model &ndash; but probably very unrealistic!

- ICD patients are at (much?) greater risk of arrhythmia death 
   - If the proportion of deaths caused by arrythmia changes over time, we would induce bias, because we would be extrapolate a constant HR for all causes mortality
   
--

- Formally account for multiple mortality causes (.blue[**Poly-Weibull**] model <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> [Demiris et al, 2015](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4456429/)):
`\begin{align}
\class{myblue}{h_{\rm{ICD}}(t)} &\class{myblue}{= h}_{\rm{\class{red}{ICD}}}^{\rm{\class{myblue}{arr}}}\class{myblue}{(t) + h}_{\rm{\class{red}{ICD}}}^{\rm{\class{myblue}{oth}}}\class{myblue}{(t)} \\
&\class{myblue}{=} \class{orange}{e^\beta} \class{myblue}{h^{\rm{arr}}}_{\rm{\class{blue}{UK}}}\class{myblue}{(t)} + \class{myblue}{h^{\rm{oth}}}_{\rm{\class{blue}{UK}}}\class{myblue}{(t)} \\
&\class{myblue}{=} \class{orange}{e^\beta}\class{myblue}{\alpha_1 \mu_1 t^{\alpha_1-1} + \alpha_2 \mu_2 t^{\alpha_2-1}}
\end{align}`

<span style="display:block; margin-top: -20px ;"></span>
- This assumes that
   -  Arrhythmia hazard is .orange[**proportional**] to matched UK population
   - Other causes hazard is **identical** to matched UK population

---

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# Example: ICD & Cardiac death

## Turning prior *information* into a prior *distribution*

- In the ICD case, age at entry is around 60 &ndash; we **know** that people won't survive more than 60 more years
   - Setting a prior for the scale `$\mu_i \sim \dunif(0,100)$` implies that the prior mean survival of the resulting Weibull distribution is 
   `$$\class{myblue}{\style{font-family:inherit;}{\text{expected survival time}}=\mu_i\Gamma\left(1+\frac{1}{\alpha}\right) < 60}$$`
   
- Can also include some knowledge on the shape `$\alpha$` and the coefficient `$\beta$` to limit their variations in reasonable ranges...

.pull-left-nospace[
<center><img src=./img/friends-gif.gif width='100%' title=''></center>
]
.pull-right-nospace[
- This isn't necessarily easy! 
   - You need to be friends with a statistician...

- Don't be lost in translation...
   - *Elicit* the actual .blue[**information**] and then map it onto a possible and reasonable .red[**distribution**]
   - Mapping changes with the mathematical properties of the underlying sampling distribution selected...
]

---

- Ignoring cause-specific mortality (simple .red[Weibull model]) results in larger bias, especially for females, mostly because the arrhythmia proportion of deaths does vary over time in that subgroup

---

# Example: constraints on `$S(t)$`

### Observed data
<center><img src=./img/surv1-1.png width='72%' title=''></center>

---

### Parametric extrapolation
<center><img src=./img/surv2-1.png width='72%' title=''></center>

---

## What do we see?

- The data are **sparse** and the follow up is limited in comparison to the relevant time horizon

- The **best fitting** model responds by extrapolating a survival curves that implies `$\Pr(\style{font-family:inherit;}{\text{Still alive after 100 months}})>$` 0.5

- This is most likely a ridiculous finding!

<span style="display:block; margin-top: 40px ;"></span>
## What do we know?

- Perhaps we may think a bit more carefully and figure out some kind of "constraint" or upper limit for the survival probability at a given time point in the future...

- Maybe, it's not so controversial to assume that, **before observing any data**, `$\Pr(\style{font-family:inherit;}{\text{Still alive after 70 months}})$` should not exceed, say, 0.20

- We can use this information in our prior specification and let it be modified by the observed data
   - This is a relatively strong prior, so you would need a **really** strong signal to modify it significantly...

<span style="display:block; margin-top: 50px ;"></span>
.alignright[(*Che et al*, work in progress...)]
---

### Constrained semi-parametric
<center><img src=./img/surv3-1.png width='72%' title=''></center>

---

# Mixture "cure" models (MCMs)

- Sustained treatment effect

- Effectively, a fraction of individuals are subject to a different "data generating process"    
   - The overall survival curve is a combination of two components `$\Rightarrow$` **mixture** model
   
   `$$\class{myblue}{S(t, \bm{x}) = S_b(t, \bm{x})[\pi(\bm{x}) + \left(1 − \pi(\bm{x})\right)S_c(t, \bm{x})]},$$`
   - `$\class{myblue}{\bm{x}}=$` a vector of individual level covariates
   - `$\class{myblue}{S_b(t,\bm{x})}=$` (complement of) background mortality for the population with `$\bm{x}$` profile
   - `$\class{myblue}{S_c(t,\bm{x})}=$` (complement of) cancer-specific mortality for the population with `$\bm{x}$` profile
   - `$\class{myblue}{\pi(\bm{x})}=$` "cure fraction" = proportion of individuals with `$\bm{x}$` profile who are "cured"

- Basically, this means that the overall survival in the population with `$\bm{x}$` profile
   - Is the same as the "healthy" population for the proportion who are "cured" 
   - It has an extra multiplicative, independent risk associated with the event (cancer) for those who aren't

<span style="display:block; margin-top: 20px ;"></span>
- We may have some biological/pharmacological insight as to the potential for this phenomenon

- Possibly plausible with immuno-oncological drugs

<span style="display:block; margin-top: 30px ;"></span>
- **BUT**: most likely, we'll have to base our judgement on a limited follow up
   - This may suggest a plateau in one treatment arm, which is likely based on very uncertain evidence

---

count: false
exclude: true
# Mixture "cure" models (MCMs)
## What you see is what you get?...

HERE INCLUDE GRAPH WITH DIFFERENT DATA CUTS

---

# **Bayesian** MCMs

- "Standard" MCMs basically assume no prior knowledge about the "cure fraction" `$\pi$`

- This means that the data are taken at face value
   - Weak evidence of flattening of the survival curve may lead to unrealistically large estimates for the the cure fraction
   
- **BUT**: assuming no prior information on `$\pi$` basically implies that we (implicitly) believe that it can be very large

- Implausible in most cases
   - Can look at other, more established treatments
   
--

- If you're Bayesian about this, you may use a "regularising" prior to avoid "parachute effect"...

.content-box-beamer[
### 
Typically, we can restrict the likely size of some effect, in **real** applications

- If you take 100 people on an airplane, randomise them to either get a parachute or not and measure whether they're still alive after jumping off the plane
   - Almost everybody with the parachute will be OK, almost everybody without will not!
   - You can reasonably expect a large "treatment effect" with OR `$\approx$` 100, or so...

<span style="display:block; margin-top: 20px ;"></span>   
- With pharmaceutical interventions, this is not so common

- Use *skeptical* priors (+ sensitivity analysis!) to restrict the likely range of effects
   - If the data are overwhelming pointing towards a large effect, the model will be able to pick that up
   - **BUT** you want to really see a signal before you call one...
]
---

count: false
# **Bayesian** MCMs
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]
.right-30[
- We may think that the cure fraction is unlikely to exceed 30%
- And that a reasonable expectation would be 20% "success"
- And we may want to allow for `$\pi$` to be actually as low as close to no "cured" at all

<span style="display:block; margin-top: 20px ;"></span>
- We can *encode* this **information** using a `$\dbeta$`(3.97,18.09) distribution
   - The parameters `$\alpha=$` 3.97 and `$\beta=$` 18.09 can be "guessed" by trial and error **OR** working out some algebra
- **Before seeing any data**, this means that you're expecting 
   - mean cure fraction = 0.22
   - 95% range = [0.054; 0.36]
   
- The observed data will modify this &ndash; but unless you have a **very** strong signal in the data, you won't go all crazy suggesting a *definite* plateau...
]

---

# Example: CM017 &ndash; nivolumab

### Early data cut (24 months)

<span style="display:block; margin-top: -20pt ;"></span>
.small[.alignright[<svg viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;fill:currentColor;position:relative;display:inline-block;top:.1em;">  [ comment ]  <path d="M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm-28.9 143.6l75.5 72.4H120c-13.3 0-24 10.7-24 24v16c0 13.3 10.7 24 24 24h182.6l-75.5 72.4c-9.7 9.3-9.9 24.8-.4 34.3l11 10.9c9.4 9.4 24.6 9.4 33.9 0L404.3 273c9.4-9.4 9.4-24.6 0-33.9L271.6 106.3c-9.4-9.4-24.6-9.4-33.9 0l-11 10.9c-9.5 9.6-9.3 25.1.4 34.4z"></path></svg> [Chaudhari et al (2020)](https://tinyurl.com/4f3v3b57)]]

---

### Later data cut (48 months)
<center><img src=./img/plateau2-1.png width='72%' title=''></center>

---

# Conclusions

## Too much, too soon?
- Tension between early introduction in the market and reimbursment decisions on the back of promising, but extremely immature data 
   - Early plateau that doesn't materialise in later data cuts
- Divorce between "medical" and "economic" analysis
   - Lancet papers are OK with estimating median survival time and HRs... Economic evaluations need extrapolation to estimate mean survival time

## All the help you can get
- Long-term data are ideal &ndash; if they're aligned with the population of interest and heterogeneity is manageable (and managed!)
- Often, even defining a comparator is a very complex operation and the market landscape is tricky...
   - Registry data can produce information "in real time". **But**: at the price of confounding/need for confirmation periods (conditional registration/reimbursment?)

## Know what you know
- Some information ***is*** controversial and subjective and could bias the assessment. **But**: other simply isn't and we shouldn't be afraid to use it!

---

NULL

---