Esomeprazole and Aspirin in Barrett’s Oesophagus

<div class = "title">Esomeprazole and Aspirin in Barrett’s Oesophagus</div>
<div class = "subtitle">Minimization Randomization, Time-to-Event Data, and Interim Analyses</div>
<div class  = "author">Nicole Erler</div>
<div class = "institute">Department of Biostatistics, Erasmus MC</div>
<div class = "contact">
<i class="fas fa-envelope"></i> <a href="mailto:n.erler@erasmusmc.nl" class="email">n.erler@erasmusmc.nl</a> 
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  <a href= https://nerler.com><i class="fas fa-globe-americas"></i> https://nerler.com</a>
</div>

---

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---
count: false

## Paper

[
<img src="graphics/Lancet_Cover.png" width="100%" style="display: block; margin: auto;" />
](https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(18%2931388-6/fulltext)

???

For this journal club:

- motivating paper 
- deeper look into the methods
  - see what the authors have done
  - explain the related topics
- focus on longitudinal data analysis

---

## Background

<br>

???

- Oesophageal adenocarcinoma (OeAC) is a health problem:
  - incidence has been increasing in Europe & North America in the past 40 years
  - 5-year survival is <10% when detected through symptoms
- preventing progression to OeAC could reduce deaths
- increase may be related to rise in gastro-oesophageal reflux disease, which
  is one of the main risk factors for Barrett’s oesophagus (BE)
  - BE is major risk factor for OeAC but most BE patients die from CVD or chest
    infections &#8680; preventive strategies should affect overall mortality
- Proton pump inhibitors (PPIs) reduce acid reflux (= assumed main driver of BE)
  - Esomeprazole is the most common PPI in the USA
- Aspirin may be associated with reduced risk of OeAC (observational studies
  with mixed results)
- So far there are no randomized trials to evaluate PPIs or aspirin for
  improving outcomes in BE patients

---

## Aim & Design

> We aimed to evaluate the efficacy and safety of [esomeprazole & aspirin],
  especially their ability to reduce all-cause mortality, OeAC, and high-grade
  dysplasia.

- Phase 3 randomized prospective factorial study
- 84 centers across England, Scotland, Wales, Northern Ireland, Canada
- Includes new and existing BE diagnoses

.pull-left[
Inclusion criteria
- &ge; 18 years
- (specific criteria for BE)
- ...
]
.pull-right[
Exclusion criteria
- pre-existing oesophageal AC
- high-grade dysplasia
- taking NSAIDs at baseline
- ...
]

???

From Wikipedia:

In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable.

> Because women with Barrett’s oesophagus have a lower risk of OeAC […] we
limited recruitment of women to approximately 500.

---

## Randomisation

- Randomisation 1 : 1 : 1 : 1 in a 2x2 factorial design

<div style = "position: absolute; right: 60px; top: 120px;">
<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> esomeprazole </th>
   <th style="text-align:center;"> aspirin </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:center;"> low dose </td>
   <td style="text-align:center;"> - </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:center;"> high dose </td>
   <td style="text-align:center;"> - </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:center;"> low dose </td>
   <td style="text-align:center;"> + aspirin </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:center;"> high dose </td>
   <td style="text-align:center;"> + aspirin </td>
  </tr>
</tbody>
</table>
</div>

- some patients couldn't / had to take aspirin<br>
  &#8680; only in the PPI randomization

<br>

- Randomization by **minimisation with random element of 0.8**

- **minimisation factors:**<br>
  risk factors for developing high-grade dysplasia, AC, and death
  - length of BE
  - age
  - intestinal metaplasia (yes / no)
  
- women and men randomized separately

---

## Outcomes

**Co-primary aims**
- efficacy of high-dose PPI vs low-dose PPI
- efficacy of aspirin vs no aspirin
  
<p class = "smallbreak">&nbsp;</p>

**Primary composite endpoint:**<br>
time to all-cause mortality, ESAC or high grade dysplasia (whichever occurred first)

Secondary aims:
- each treatment’s effect on time to the individual components of the composite endpoint,
- cause-specific mortality,
- and the composite endpoint analysed by sex.
    
---

## Statistical Analysis (1)

- ITT analysis for all efficacy analyses
- check significance of the treatment interaction term 
  (<i class="fas fa-bolt red"></i> low power!)
  - marginal analyses

- **AFT models** with adjustment for minimization factors
    - interpreted in terms of the time to an event using the time ratio (TR)<br>
      &#8680; TR > 1 means longer survival
    
    > […] its benefit being that results are reported as a delay to an event 
      over the entire trial period compared with the hazard ratio result, which
      is interpreted as risk of an event at any one given time.

???

> We checked the significance of the treatment interaction term by first adding
  an interaction term to a primary model before using at-the-margins and 
  within-table results to produce an interaction ratio.

---

## Statistical Analysis (2)

- additionally:<br>
  **(competing risk) Cox models** for comparison with other research
  
- **PH assumption** tested with **Schoenfeld** tests and plots of residuals

- **Median follow-up** was calculated using a **reverse Kaplan-Meier method**

---

## Sample Size Calculation

**Assumptions**

<ul class = "ul-wider">
<li>no interaction between aspirin & PPI</li>
<li>exponential time-to-event with constant event rate of 0.76%/year</li>
<li>constant event HR 1.4</li>
<li>recruitment over 2 years, follow-up 8 years</li>
<li>10% loss to -follow-up</li>
<li>20% non-compliance with medication</li>
<li>80% power for 2-sided test at 5% significance</li>
</ul>

&#8680; N = 5000 (1250 per group)

---

## Sample Size Calculation

**Assumptions**, .ins[adjusted based on new, external data]
<ul class = "ul-wider">
<li>no interaction between aspirin & PPI</li>
<li>exponential time-to-event with constant event rate of <strike>0.76%/year</strike> .ins[1%/year]</li>
<li>constant event HR <strike>1.4</strike> .ins[1.5]</li>
<li>recruitment over <strike>2 years</strike> .ins[3 years], follow-up <strike>8 years</strike> .ins[10 years]</li>
<li>10% loss to -follow-up</li>
<li><strike>20% non-compliance with medication</strike></li>
<li>80% power for 2-sided test at 5% significance</li>
</ul>

&#8680; <strike>N = 5000 (1250 per group)</strike> .ins[N = 2224]

(2535 Subjects in total in the analysis)

---

## Interim Analyses

<ul class = "ul-wider">
<li>Interim analyses planned at 2 and 4 years of follow-up.</li>
<li>Interim analyses of the <strong>primary outcome</strong>.</li>
<li>Interim results presented confidentially to the DSMB.</li>
<li>Statistical significance at interim: p-value < 0.001</li>
</ul>

---

## Topics

**Randomization**
* Minimization Randomization

**Survival Analysis**
<ul class = "ul-wider">

<li>AFT models & Cox models: definition & interpretation</li>
<li>Proportional Hazards Assumption
  <ul>
  <li>Schoenfeld Residuals</li>
  </ul></li>
<li>Competing Risks</li>
<li>Median follow-up time / Reverse Kaplan-Meier Method</li>

</ul>

**Sample Size**
* Interim Analyses

---

# Randomization

---

## From the Paper

- **minimisation factors:**<br>
  risk factors for developing high-grade dysplasia, AC, and death
  - length of BE
  - age
  - intestinal metaplasia (yes / no)
]

---

## Minimization Randomization

.reference[[<i class="fas fa-file-alt"></i> Altman & Bland (2005)](https://www.bmj.com/content/bmj/330/7495/843.full.pdf)]

Blocking & stratification is not effective in small trials<br>
&#8680; only widely accepted alternative: **Minimization Randomization**

Assign patients sequentially to the group that **minimises the imbalance** across multiple factors.<br>

* **first patient:**<br>
  assign randomly<br>

* **next patient(s):**<br>
  Which treatment would lead to a better balance in the variables of interest?<br>
  &#8680; Sum per arm over the counts of participants with the same characteristic.

---

## Minimization Randomization

**Example**
new patient: with IM, 55 years old

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;">  </th>
   <th style="text-align:center;"> A<br>(n = 3) </th>
   <th style="text-align:center;"> B<br>(n = 3) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> IM: yes </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> age &gt; 50 </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 2 </td>
  </tr>
</tbody>
</table>
]

<br>

* **A:** 1 .sgrey[(IM: yes)] + 2 .sgrey[(age > 50)] = 3
* **B:** 3 .sgrey[(IM: yes)] + 2 .sgrey[(age > 50)] = 5

&#8680; assignment to group A would improve balance.
]

<br>

- **deterministic assignment**<br>
  assign to group with smaller sum (or random if equal)
- **random assignment**<br>
  assign randomly with high probability to decrease the imbalance,<br>
  e.g., 80% prob. to assign group A ("random element")
  
  
---

## Minimization Randomization

**Minimization**

* works for trials with > 2 arms.
* requires continuous covariates to be categorized.

**Random Minimization**
* decreases the balance slightly
  (but still a lot better than random assignment).
* makes the allocation less predictable.

Balance between patient characteristics is especially desirable when there are
strong prognostic factors and modest treatment effects.

---

## ICH E9 - Section 2.3.2 Randomisation .reference[[<i class="fas fa-external-link-alt"></i> ICH E9](https://www.ema.europa.eu/en/documents/scientific-guideline/ich-e-9-statistical-principles-clinical-trials-step-5_en.pdf)]

- stratification by important prognostic baseline factors may help to promote balance within strata (especially beneficial in small trials)
    - usually &le; 3 stratification factors are sufficient
      <span class = "sgrey">(more is logistically difficult)</span>
      
- Dynamic allocation may help to achieve balance across a number of
  stratification factors simultaneously.<br>
  * **Deterministic** dynamic allocation procedures **should be avoided** and an
    appropriate element of randomisation should be incorporated for each
    treatment allocation.

- **Factors on which randomization has been stratified should be accounted for
  later in the analysis.**

- Details of the randomisation that facilitate predictability (e.g., block
  length) should not be contained in the trial protocol.
  
---

## CONSORT 2010 .reference[[<i class="fas fa-external-link-alt"></i> CONSORT ](https://doi.org/10.1136/bmj.c332)]

<br>

> Minimisation offers the **only acceptable alternative to randomisation**, and some have argued that it is superior. On the other hand, minimisation **lacks the theoretical basis for eliminating bias** on all known and unknown factors. Nevertheless, in general, trials that use minimisation are **considered methodologically equivalent** to randomised trials, even when a random element is not incorporated.

---

# Survival Analysis

---

## From the Paper

**AFT models** with adjustment for minimization factors
  - interpreted in terms of the time to an event using the time ratio (TR)<br>
    &#8680; TR > 1 means longer survival
    
  > […] its benefit being that results are reported as a delay to an event 
    over the entire trial period compared with the hazard ratio result, which
    is interpreted as risk of an event at any one given time.

]

---

## From the Paper

> All analyses used **accelerated failure time (AFT) modelling**, with
adjustment for minimisation factors. The [AFT] models were **interpreted [...]
using the time ratio (TR)**. 
We used AFT because of the **intuitive nature of the TR**, which models survival
time, [...]

]

---

## Accelerated Failure Time (AFT) models

"Direct" **extension** of linear regression **for survival data**.

<br>

**Linear Regression** would be
`$$T_i^* = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip} + \varepsilon_i$$`
with `$\varepsilon_i\sim N(0, \sigma^2).$`
<div class = "sgrey" style = "position: relative; left: 800px; bottom: 50px;">
$ T_i^* $: true event time
</div>

<i class="fas fa-bolt red"></i> Would allow for negative survival times `$T_i^*$`.

---

## Accelerated Failure Time (AFT) models

**AFT-model**:
`$$\color{var(--nord14)}{\log(T_i^*)} = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip} + \varepsilon_i$$`
with `$\varepsilon_i\sim$` Normal, Student's `$t$`, Logistic, Extreme Value, ...

**direct (additive) effect** of `$\mathbf x_i$` on `$\color{var(--nord14)}{\log(T_i^*)}$`:<br>
1-unit increase: `$\qquad x_{\color{var(--nord11)}{i'}1} = x_{i1} + 1$`
&emsp;
.sgrey[(while everything else is kept the same)]

`\begin{align}
\log(T_{\color{var(--nord11)}{i'}}^*) &= \beta_0 + \beta_1 \overset{x_{\color{var(--nord11)}{i'}1}}{(\overbrace{x_{i1} \bbox[#3B4252, 2px]{+ 1}})} + \ldots + \beta_p x_{ip} + \varepsilon_i\\
&= \beta_0 + \beta_1 x_{i1} \bbox[#3B4252, 2px]{+ \beta_1} + \ldots + \beta_p x_{ip} + \varepsilon_i\\[2ex]
\Rightarrow\log(T_{\color{var(--nord11)}{i'}}^*) &= \log(T_i^*) \bbox[#3B4252, 2px, border: 1px solid #BF616A]{+ \beta_1}
\end{align}`

???

* AFT models are direct extension of linear regression models to time-to-event data. 
* Differences:
    - response `$T_i^*$` is always positive &#8680; use `$\log$`
    - censoring &#8680; more sensitive to choice of error distribution

---

## Accelerated Failure Time (AFT) models

<div class = "small grey" style = "position: relative; top: 55px;">
from the previous slide:
</div>
`$$\log(T_{\color{var(--nord11)}{i'}}^*) = \log(T_i^*) \bbox[#3B4252, 2px]{+ \beta_1}$$`

`$$\Rightarrow \bbox[#3B4252, 2px]{\beta_1} = \log(T_{\color{var(--nord11)}{i'}}^*) - \log(T_i^*) = \log\left(\frac{T_{\color{var(--nord11)}{i'}}^*}{T_i^*}\right)$$`

`$$\Rightarrow \text{time ratio:}\quad\frac{T_{\color{var(--nord11)}{i'}}^*}{T_i^*} = \exp\left(\bbox[#3B4252, 2px]{\beta_1}\right)\qquad \text{or:}\quad T_{\color{var(--nord11)}{i'}}^* = T_i^* \times \exp\left(\bbox[#3B4252, 2px]{\beta_1}\right)$$`

&#8680; `$x_i$` has a multiplicative effect on the time-to-event

---

## From the Paper

> **Cox proportional hazards** survival analyses, and where appropriate, 
  **Cox competing risks** survival analyses were also performed on all
  comparisons to allow for comparison with other research.

]

---

## Proportional Hazards Model (aka Cox Model)

`$$\underset{\substack{\uparrow\\\text{hazard}}}{h_i(t)} = \underset{\substack{\uparrow\\\text{baseline}\\\text{hazard}}}{h_0(t)} \exp(\beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip})$$`
**Hazard** (function): .sgrey[instantaneous risk of an event given the event has not yet happened]
`$$h_i(t) = \lim_{\Delta t \rightarrow 0} \frac{\Pr(t\leq T_i < t + \Delta t\mid T_i\geq t)}{\Delta t}$$`

**Hazard Ratio:** &emsp; (patient `$\color{var(--nord11)}{i'}$` vs `$i$`)

`$$\frac{h_{\color{var(--nord11)}{i'}}(t)}{h_i(t)} = \frac{h_0(t) \exp(\beta_1 x_{\color{var(--nord11)}{i'}1} + \beta_2 x_{\color{var(--nord11)}{i'}2} + \ldots + \beta_p x_{\color{var(--nord11)}{i'}p})}{h_0(t) \exp(\beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip})}$$`

---

## Proportional Hazards Model (aka Cox Model)

1-unit increase: `$\qquad x_{\color{var(--nord11)}{i'}1} = x_{i1} + 1$`
&emsp;
.sgrey[(while everything else is kept the same)]

`\begin{align}
\frac{h_{\color{var(--nord11)}{i'}}(t)}{h_i(t)} &= 
\frac{h_0(t) \exp(\beta_1 \overset{x_{\color{var(--nord11)}{i'}1}}{(\overbrace{x_{i1} \bbox[#3B4252, 2px]{+ 1}})} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip})}{h_0(t) \exp(\beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip})\qquad}\\
&= \frac{h_0(t)}{h_0(t)} \; 
\frac{\exp(\beta_1 x_{i1}) \; \bbox[#3B4252, 2px]{\exp(\beta_1)}}{\exp(\beta_1 x_{i1})} \;
\frac{\exp(\beta_2 x_{i2} + \ldots + \beta_p x_{ip})}{\exp(\beta_2 x_{i2} + \ldots + \beta_p x_{ip})}
\end{align}`

`$$\Rightarrow \; \text{Hazard Ratio:}\quad \frac{h_{\color{var(--nord11)}{i'}}(t)}{h_i(t)} = \bbox[#3B4252, 2px]{\exp(\beta_1)} \qquad \Rightarrow \quad h_{\color{var(--nord11)}{i'}}(t) = h_i(t) \bbox[#3B4252, 2px]{\times\exp(\beta_1)}$$`
**&#8680; multiplicative effect** of `$x_{i1}$` on `$h_i(t)$`

---

## Proportional Hazards Model (aka Cox Model)

The model can also be written on the **log hazard** scale:
`$$\log \{h_i(t)\} = \log\{h_0(t)\} + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip}$$`

<br>

1-unit increase: `$\qquad x_{\color{var(--nord11)}{i'}1} = x_{i1} + 1$`
&emsp;
.sgrey[(while everything else is kept the same)]

`$$\Rightarrow \quad \log\{h_{\color{var(--nord11)}{i'}}(t)\} = \log\{h_i(t)\} \bbox[#3B4252, 2px]{+ \beta_1}$$`

**&#8680; additive effect** of `$x_{i1}$` on the log hazard, `$\log\{h_i(t)\}$`

---

## Cox Model vs AFT Model: Interpretation

Say, `$\beta_1 = 0.5$` &emsp; &#8680; `$\exp(\beta_1) = 1.65$`

A 1-unit increase in `$x_1$` is associated with...

**Cox Model:**
* ... an increase in the **hazard** by a **factor of 1.65**.<br>
  &#8680; `$\exp(\beta_1) > 1$` &emsp;&#8680; larger hazard = shorter time-to-event
* ... an **increase of 0.5** in the **log-hazard**.<br>
  &#8680; `$\beta_1 > 0$` &emsp;&#8680; larger log-hazard = shorter time-to-event

**AFT Model:**
* ... an **increase of 0.5** in the **log event time**.<br>
  &#8680; `$\beta_1 > 0$` &emsp;&#8680; longer time-to-event = lower risk
* ... a **1.65 times** longer **time-to-event**.<br>
  &#8680; `$\exp(\beta_1) > 1$` &emsp;&#8680; longer time-to-event = lower risk

---

## From the Paper

> Before the use of both AFT and Cox survival models, the **assumption of
proportional hazards** was tested with **Schoenfeld tests and plots of
residuals**.

]

---

## Proportional Hazards Assumption

From previous slides:

`$$\text{Hazard Ratio:}\quad \frac{h_{\color{var(--nord11)}{i'}}(t)}{h_i(t)} = \bbox[#3B4252, 2px]{\exp(\beta_1)} \qquad \Rightarrow \quad h_{\color{var(--nord11)}{i'}}(t) = h_i(t) \bbox[#3B4252, 2px]{\times\exp(\beta_1)}$$`
<br>

**&#8680; (Implied) Assumptions:**
* The hazard `$h_{\color{var(--nord11)}{i'}}(t)$` is **proportional** to the 
  hazard `$h_i(t)$`, i.e., multiplied by a "constant".
* The **hazard ratio**
  `$\displaystyle \frac{h_{\color{var(--nord11)}{i'}}(t)}{h_i(t)}$`
  **is independent of time**: `$\exp(\beta_1)$`.
  
--

How can we check this assumption?

---

## Proportional Hazards Assumption

**Idea:** Investigate / plot the "observed hazards" in the data.

&#8680; Problem: `$h_i(t)$` is not observed.

**But** 
* We know a non-parametric estimator of the survival:<br>
the **Kaplan-Meier** estimator.

* We know: `$\displaystyle \underset{\substack{\text{cumulative}\\\text{hazard}\\\text{ function}}}{H(t)} = \int_0^t h(s) ds = -\log\underset{\substack{\uparrow\\\text{survival}\\\text{ function}}}{S(t)}$`

---

## Proportional Hazards Assumption

`\begin{align}
h_{\color{var(--nord11)}{i'}}(t) &= h_i(t) \times \underset{\text{const.}}{\underbrace{\exp(\beta_1)}}\\
\underset{H_{\color{var(--nord11)}{i'}}(t)}{\underbrace{\int_0^t h_{\color{var(--nord11)}{i'}}(s)\;ds}} &= \int_0^t h_i(s) \times \text{const.} \;ds
= \underset{H_i(t)}{\underbrace{\int_0^t h_i(s)\;ds}} \times \text{const.}
\end{align}`

`$$H_{\color{var(--nord11)}{i'}}(t) = H_i(t) \times \text{const.}$$`

<br>

&#8680; Plot the cumulative hazards derived from the Kaplan-Meier estimator.<br>
&#8680; Should be proportional (as well).

---

## Proportional Hazards Assumption

**What about continuous covariates?**

---

## Schoenfeld Residuals

.reference[[<i class="fas fa-file-alt"></i> Schoenfeld (1982)](https://doi.org/10.1093/biomet/69.1.239)]

* only defined for subjects with an event
* one residual per subject **per covariate**

**Schoenfeld residual** for subject `$i$` and covariate `$k$` with event at time `$t_i$`:

`$$\text{residual}_{ik} = \underset{\substack{\uparrow\\\text{observed}\\\text{covariate}\\\text{value}}}{x_{ik}} - \underset{\text{expected value of }x_{ik}\\\text{based on the risk set}}{\underbrace{\overset{\color{grey}{\substack{\text{risk set}\\\text{at }t_i}}}{\sum_{i = 1}^{j\in \color{grey}{\overbrace{\color{white}{R(t_i)}}}}} \overset{\color{grey}{\substack{\text{likelihood of}\\\text{patient }j \text{ to have}\\\text{an event at } t_i}}}{\underset{\color{grey}{\substack{\text{covariate value}\\\text{of patient }j}}}{\underset{\color{grey}{\uparrow}}{x_{kj}}\;\overset{\color{grey}{\downarrow}}{p_j}\qquad}\quad}}}$$`

---

## Schoenfeld Residuals

---

## Proportional Hazards Assumption

<div style = "position: absolute; left: 120px; top: 300px; width: 350px;">
What, if the proportional hazards assumption is <strong>violated</strong>?
</div>

---

## Proportional Hazards Assumption

---

## From the Paper

We **used AFT** [...], its benefit being that results are reported as a **delay
to an event over the entire trial period** compared with the **hazard ratio
result, which is interpreted as risk of an event at any one given time**.

]

---

## From the Paper

> Secondary aims [...] included each treatment’s effect on time to the
individual components of the composite endpoint [all-cause mortality,
oesophageal adenocarcinoma, or high-grade dysplasia], cause-specific mortality,
and [...].

]

---

## Competing Risks: Example .reference[[<i class="fas fa-file-alt"></i> Putter et al. (2007)](https://doi.org/10.1002/sim.2712)]

<table class=" risktab" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>Risk of an event.</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> gender </th>
   <th style="text-align:left;"> dead </th>
   <th style="text-align:left;"> AC </th>
   <th style="text-align:left;"> alive </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> female </td>
   <td style="text-align:left;"> 10% </td>
   <td style="text-align:left;background-color: #3B4252 !important;"> 27% </td>
   <td style="text-align:left;"> 63% </td>
  </tr>
  <tr>
   <td style="text-align:left;"> male </td>
   <td style="text-align:left;"> 30% </td>
   <td style="text-align:left;background-color: #3B4252 !important;"> 30% </td>
   <td style="text-align:left;"> 40% </td>
  </tr>
</tbody>
</table>

<br>

.flex-grid[
.col[
<table class=" risktab" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>baseline</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> gender </th>
   <th style="text-align:right;"> alive </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> female </td>
   <td style="text-align:right;"> 100 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> male </td>
   <td style="text-align:right;"> 100 </td>
  </tr>
</tbody>
</table>
]
.col[
<table class=" risktab" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>after 1 year</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> gender </th>
   <th style="text-align:right;"> dead </th>
   <th style="text-align:right;"> AC </th>
   <th style="text-align:right;"> alive </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> female </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 27 </td>
   <td style="text-align:right;"> 63 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> male </td>
   <td style="text-align:right;"> 30 </td>
   <td style="text-align:right;"> 30 </td>
   <td style="text-align:right;"> 40 </td>
  </tr>
</tbody>
</table>
]
.col[
<table class=" risktab" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>after 2 years</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> gender </th>
   <th style="text-align:right;"> dead </th>
   <th style="text-align:right;"> AC </th>
   <th style="text-align:right;"> alive </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> female </td>
   <td style="text-align:right;"> 16 </td>
   <td style="text-align:right;background-color: #3B4252 !important;"> 45 </td>
   <td style="text-align:right;"> 39 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> male </td>
   <td style="text-align:right;"> 42 </td>
   <td style="text-align:right;background-color: #3B4252 !important;"> 42 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
</tbody>
</table>
]
]

---

## Competing Risks: Example
<img src="index_files/figure-html/unnamed-chunk-16-1.png" width="100%" style="display: block; margin: auto;" />

---

## Competing Risks: Example

---

## Competing Risks

<span style = "font-weight: bold; color: var(--nord11);">
The Kaplan-Meier Estimator is biased!!!</span>

**Cause specific hazards:**
* Cox model in which alternative events are censored
* HR remains valid!
* cumulative incidence calculation changes (cause specific!)
* survival = overall event free survival (not cause specific)
* interaction of covariates with cause-specific cumulative incidences

**Fine-Grey model:**
* directly models the cumulative incidence
* can't handle time-varying covariates

---

## From the Paper

> Median follow-up was calculated using a **reverse Kaplan-Meier method**.

]

---

## (Median) Follow-up Time

**How should the (median) follow-up time be defined?** .reference[[<i class="fas fa-file-alt"></i> Schemper (1996) ](https://doi.org/10.1016/0197-2456(96%2900075-x)]

* **observation time**, i.e., time to event or censoring<br>
  &#8680; with increasing risk events happen earlier<br>
  &#8680; shorter FU time (but same information content)

* **censoring time**, i.e., time to censoring (exclude subjects with event)<br>
  &#8680; low probability to observe long time-to-censoring because of events<br>
  &#8680; underestimation of FU time

* **time to end of study**<br>
  &#8680; does not penalize for loss-to-follow-up &#8680; overestimation of FU time

* **known function time**: time-to-censoring or time to EOS (if event)<br>
   &#8680; tends to overestimate potential FU of patients with increasing loss
   to FU

---

## (Median) Follow-up Time

**Reverse Kaplan-Meier Method**<br>
Kaplan-Meier estimate for censoring (i.e., reverse the role of event and censoring)
--

<div style="position: fixed; top: 35%; left: 25%; width:250px; font-size: 0.5rem;">
median survival: &nbsp;
$T_{0.5} = \hat S_T^{-1}(0.5)$
<br>

The time, by which half of the subjects have experienced the event.
</div>
]
--
.pull-right[
<img src="index_files/figure-html/unnamed-chunk-21-1.png" width="100%" style="display: block; margin: auto;" />

<div style="position: fixed; top: 35%; right: 5%; width:240px; font-size: 0.5rem;">
median follow-up: &nbsp;
$C_{0.5} = \hat S_C^{-1}(0.5)$
<br>

The time, by which half of the subjects are censored.
</div>

]

---

# Interim Analyses

---

## From the Paper

>The **primary outcome** was analysed and presented confidentially to the
trial's data safety monitoring committee as specified in the protocol **after 2
and 4 years of follow-up** as interim analyses, with a **p-value &le; 0.001**
regarded as significant.
The **committee recommended trial continuation** and neither interim analysis
was disseminated further.

]

---

## Reasons to Stop a Trial Early

.reference[[<i class="fas fa-external-link-alt"></i> STAT 509: Design and Analysis of Clinical Trials](https://online.stat.psu.edu/stat509/lesson/9)]

* Treatments are convincingly different.
* Treatments are convincingly not different.
* Side effects, toxicity.
* Poor data quality, poor adherence .sgrey[(preventing an answer to the primary question)].
* Accrual is too slow.

* The scientific questions are no longer important.
* Resources to perform the study are no longer available.
* The study integrity has been undermined by fraud or misconduct.

**Dangers of Interim Analyses**
- Knowledge of interim results may affect objectivity.
- Repeated statistical testing increases the Type I error rate.

---

## Interim Analyses

**Trade-off:**
.pull-left[
Stopping early will
- save costs and labour,
- expose as few patients as possible to inferior treatments, and
- allow disseminating information about the treatments quickly.
]

.pull-right[
Continuation of the trial
- increases precision,
- reduces errors of inference,
- is necessary for sufficient statistical power <span class = "sgrey">(account for prognostic factors, examine subgroups, ...)</span>,
- gather information on secondary endpoints.
]

Statistical criteria can **provide guidelines** for terminating a trial but
no statistical method is a substitute for judgement.

---

## Interim Analyses: Group Sequential Approach

Repeated hypothesis testing **increases the type-I error**!<br>
&#8680; Control for this by adjusting the p-value cut-off

<br>

.flex-grid[
.col[
<div style = "width: 350px; margin-top: 50px; margin-bottom: 20px;">
Most common methods to select adjusted cut-offs for statistical significance:
</div>

* O'Brian-Flemming (OF)
* Haybittle-Peto (HP)
* Pocock
]

.col[
<table class=" pval-tab" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="3"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">p-value cut-off</div></th>
</tr>
  <tr>
   <th style="text-align:right;">   </th>
   <th style="text-align:right;"> O'Brian-<br>Flemming </th>
   <th style="text-align:right;"> Haybittle-<br>Peto </th>
   <th style="text-align:right;"> Pocock </th>
  </tr>
 </thead>
<tbody>
  <tr grouplength="2"><td colspan="4" style="border-bottom: 1px solid;"><strong>Two interim analyses</strong></td></tr>
<tr>
   <td style="text-align:right;padding-left: 2em;" indentlevel="1"> 1 </td>
   <td style="text-align:right;"> 0.0054 </td>
   <td style="text-align:right;"> 0.002 </td>
   <td style="text-align:right;"> 0.0294 </td>
  </tr>
  <tr>
   <td style="text-align:right;padding-left: 2em;" indentlevel="1"> 2 </td>
   <td style="text-align:right;"> 0.0492 </td>
   <td style="text-align:right;"> 0.050 </td>
   <td style="text-align:right;"> 0.0294 </td>
  </tr>
  <tr grouplength="3"><td colspan="4" style="border-bottom: 1px solid;"><strong>Three interim analyses</strong></td></tr>
<tr>
   <td style="text-align:right;padding-left: 2em;" indentlevel="1"> 1 </td>
   <td style="text-align:right;"> 0.0006 </td>
   <td style="text-align:right;"> 0.001 </td>
   <td style="text-align:right;"> 0.0221 </td>
  </tr>
  <tr>
   <td style="text-align:right;padding-left: 2em;" indentlevel="1"> 2 </td>
   <td style="text-align:right;"> 0.0151 </td>
   <td style="text-align:right;"> 0.001 </td>
   <td style="text-align:right;"> 0.0221 </td>
  </tr>
  <tr>
   <td style="text-align:right;padding-left: 2em;" indentlevel="1"> 3 </td>
   <td style="text-align:right;"> 0.0471 </td>
   <td style="text-align:right;"> 0.050 </td>
   <td style="text-align:right;"> 0.0221 </td>
  </tr>
</tbody>
</table>
]
]

---

## Interim Analyses: Group Sequential Approach

.flex-grid[
.col[
<div style = "width: 550px; margin-top: 0px; margin-bottom: 20px;">
The approaches differ in how they "spent" the $\alpha$ over the repeated analyses.
</div>

<ul>
<li><strong>Pocock</strong> spends it equally
<ul class = "arrowlist">
<li>highest chance of early stopping</li>
<li>low chance of significance at final analysis</li></ul><br></li>
<li><strong>HP</strong> (based on intuitive reasoning) &<br>
    <strong>OF</strong> (based on statistical reasoning)
<ul class = "arrowlist">
<li>early stopping is unlikely</li>
<li>less impact on final analysis</li></ul></li>
</ul>
]

---

## Interim Analyses: Alpha Spending

**Drawbacks** of the group sequential approach:
- number of interim analyses has to be pre-determined
- equal spacing between scheduled analyses with respect to accrual
    
--

Alternative: **Alpha Spending Function** approach
    
&#8680; Use a function of the **information fraction available** at interim:
`$\alpha(\tau)$`
* `$\tau = \frac{n}{N}$`: ratio of participants (or number of events) included
  in the interim
* `$\alpha(\tau)$` is monotonically increasing from 0 to  `$\alpha$` at the end of
  the trial

Regardless approach to deflate type-I error:
* The estimates of a **treatment effect** will be **biased** when a trial is
  terminated at an early stage.
* The earlier the decision, the larger the bias.

---

## Futility Assessment with Conditional Power

Other reason to stop a trial:<br>
(Negative) results at the interim are unlikely to change with more patients.

<br>

- **Unconditional power:**<br>
probability of a significant result as calculated at the beginning of the trial<br>

- **Conditional power:**<br>
probability of rejecting `$H_0$` once some data are available<br>
&#8680; If this prob. is very small, it would be futile to continue to trial.

---

# Thank you for your attention!

<div class="contact">
<i class="fas fa-envelope"></i> <a href="mailto:n.erler@erasmusmc.nl" class="email">n.erler@erasmusmc.nl</a>&emsp;
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  <a href="https://github.com/NErler" target="_blank"><i class="fab fa-github"></i> NErler</a>&emsp;
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</div>

---
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