A recipe for disaster: Survival modelling and extrapolation in health economic evaluation

# A recipe for disaster: Survival modelling and extrapolation in health economic evaluation

## Gianluca Baio

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

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### Life course analysis: data and methodological challenges, ISER, University of Essex

7 July 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[
A recipe for disaster
]

.footer-right[
Life course analysis &ndash; Uni of Essex,  7 Jul 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... 😉

---

# Outline

1. Health economic evaluation

- What is it?
   - How does it work?

2. Survival data in health economic evaluations

- Extrapolation from clinical trials   
   - Parametric models
   - Why you should always be a Bayesian about this (and everything else...)

3. Example
   - ICD + prior knowledge

4. Conclusions

---

# 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

---

# *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>
]
]

---

Time-to-event data constitute the main outcome in a large number of HTAs (e.g. for cancer drugs)

<center><img src=./img/cake.gif width='100%' title=''></center>
]

## 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! `$\Rightarrow$` .blue[**Extrapolation**] (more on this later...)   
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. But the modellers can (should?!) be statisticians too, so they could handle the data!...

]

---

Time-to-event data constitute the main outcome in a large number of HTAs (e.g. for cancer drugs)

<center><img src=./img/destinyschild.gif width='100%' title=''></center>
]

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!...)

]

---

---

# Survival analysis in HTA

.alignleft[
.ubuntublue[Trial data &ndash; [Kaplan-Meier](https://en.wikipedia.org/wiki/Kaplan–Meier_estimator) curves]
]

---

---

---

# Extrapolation

## A recipe for disaster?...

.pull-left[
<center><img src=./img/ristorante.png width='74%' title=''></center>
]
.pull-right[
<center><img src=./img/pizza.png width='185%' title=''></center>
]

---

## A recipe for disaster?...

---

## A recipe for disaster?...

---

## A recipe for disaster?...

---

## A recipe for disaster?...

- **NB**: Any \*IC can only tell us about model fit **for the observed data**!     
- Extrapolation (like missing data) is based on (virtually) untestable assumptions

---

## A recipe for disaster?...

---

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

- For more complex models, MLE-based estimates may fail to converge

– This may be an issue for multi-parameter models, where limited data (not compounded by relevant prior information) are not enough to fit all the model parameters
   
   – **NB**: you would normally need to fit more complex models for cases where the survival curves are "strange" and so the usual parametric models fail to provide sufficient fit

.lightgray[
- When there is strong correlation among the survival parameters, the results of the uncertainty analysis may be (strongly) biased under a more simplistic frequentist model

– This matters most in health economics, because this bias carries over the economic modelling, optimal decision making and assessment of the impact of parametric uncertainty!
   
   – **A full Bayesian approach propagates directly correlation and uncertainty in the model parameters through to the survival curves and the economic model**
]

---

---

# Example (1): ICD & Cardiac death

.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/)]

## Set up/interventions

- ICD (Implantable Cardioverter Defibrillators) compared to anti-arrhythmic drugs (AAD) for prevention of sudden cardiac death in people with cardiac arrhythmia

## Data

- Individual data from cohort of 535 UK cardiac arrhythmia patients implanted with ICDs between 1991 and 2002

-  Meta-analysis of three (non-UK) RCTs providing published HRs
   – Relatively short-term follow-up: approximately 75% people, followed for less than 5 years, maximum 10 years
   
- UK population mortality statistics by age, sex, cause of death

## Objective

- Estimate the survival curve over the lifetime of ICD and AAD patients in UK

- Extrapolate the output to inform the wider economic model

---

## Basic idea

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

---

## Basic idea

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

---

# <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="M470.38 1.51L150.41 96A32 32 0 0 0 128 126.51v261.41A139 139 0 0 0 96 384c-53 0-96 28.66-96 64s43 64 96 64 96-28.66 96-64V214.32l256-75v184.61a138.4 138.4 0 0 0-32-3.93c-53 0-96 28.66-96 64s43 64 96 64 96-28.65 96-64V32a32 32 0 0 0-41.62-30.49z"></path></svg> You don't know what you're doing...

</style>
.left-column30[
<iframe width="560" height="315" src="https://www.youtube.com/embed/JhTCOVnU0gY" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]
.right-column70[
- To set up a full Bayesian model including a reasonable specification of the priors can be a hard task
- Often people claim that they have "no prior information". 
]

---

count: false
# <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="M470.38 1.51L150.41 96A32 32 0 0 0 128 126.51v261.41A139 139 0 0 0 96 384c-53 0-96 28.66-96 64s43 64 96 64 96-28.66 96-64V214.32l256-75v184.61a138.4 138.4 0 0 0-32-3.93c-53 0-96 28.66-96 64s43 64 96 64 96-28.65 96-64V32a32 32 0 0 0-41.62-30.49z"></path></svg> You don't know what you're doing...

.left-column30[
<iframe width="560" height="315" src="https://www.youtube.com/embed/JhTCOVnU0gY" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]
.right-column70[
- To set up a full Bayesian model including a reasonable specification of the priors can be a hard task
- Often people claim that they have "no prior information". **But: don't they?...**
]

---

<span style="display:block; margin-top: 2em ;"></span>
- 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 
   `$$\class{myblue}{\mu_i\Gamma\left(1+\frac{1}{\alpha}\right) < 60}$$`
   
- Can also include some knowledge on the shape `$\alpha$` and the coefficient `$\beta$` to limit their variations in reasonable ranges...

]

---

# Results

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

---

# Example (2): 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>

---

# 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

---