Survival Analysis

<div class = "title">Survival Analysis</div>
<div class = "subtitle">An Overview</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> 
<a href= https://twitter.com/N_Erler><i class="fab fa-twitter"></i> N_Erler</a> 
  <a href= https://github.com/NErler><i class="fab fa-github"></i> NErler</a> 
  <a href= https://nerler.com><i class="fas fa-globe-americas"></i> https://nerler.com</a>
</div>

---

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&emsp;&emsp;&emsp;&emsp;
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</span></div>

---
count: false

## Time-to-event Data

**Endpoint:** Time until an event of interest occurs.
* common baseline time
* event may not be observed within study duration

<br>

**Censoring:**
* end of follow-up
* drop-out &#8680; may be related to "survival" &#8680; **informative censoring**

---

## Events and Censoring

---

## Interval Censoring

---

## Survival Function
Probability of "surviving" until time `$t$`:

`$$S(t) = \Pr(T > t)\quad {\scriptsize\color{var(--nord3)}{\text{with } T: \text{true event time}}}$$`
<img src="index_files/figure-html/unnamed-chunk-3-1.png" width="80%" style="display: block; margin: auto;" />

---

## Hazard function

Instantaneous risk of an event at time `$t$`, given no event has occurred
before `$t$`
`$$h(t) = \lim_{\Delta t\rightarrow 0} = \frac{\Pr(t\leq T < t + \Delta t \mid T\geq t)}{\Delta t}$$`
<img src="index_files/figure-html/unnamed-chunk-4-1.png" width="80%" style="display: block; margin: auto;" />

---

## Cumulative Hazard function
Cumulative risk up to time `$t$` (expected number of events by time `$t$`)
`$$H(t) = \int_0^t h(s) ds\qquad \text{and}\qquad H(t) = -\log S(t)$$`
<img src="index_files/figure-html/unnamed-chunk-5-1.png" width="80%" style="display: block; margin: auto;" />

---

## Hazard & Cumulative Hazard

---

## Hazard & Cumulative Hazard

---

## Hazard & Cumulative Hazard

---

## Hazard & Cumulative Hazard

---

## Median Survival

<div style = "position: fixed; top: 30%; right: 10%; width:450px;">
median survival: &nbsp;
$T_{0.5} = S^{-1}(0.5)$
<br>

The time, by which half of the subjects will experience the event.
</div>

---

## Mean Survival

<div style = "position: fixed; top: 30%; right: 10%; width:450px;">
mean/average survival: 
$\displaystyle \int_0^\infty S(t)dt$

The expected failure time.
</div>

---

## Estimation
**Idea:** Probability of being "alive" at time `$t$`:
`$$\hat S(t) = \Pr(T_i > t) = \frac{\text{number of patients alive at $t$}}{n}$$`
--

**Problem:** We do not observe the true event time for all subjects.
.pull-left[

* observed event time<br>
  `$T_i = \min(T_i^*, C_i)$`
  
* event indicator: 
  - `$\delta_i = 1$` if event,
  - `$\delta_i = 0$` if censored
]

.pull-right[
<br>
<table class=" idtab" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> id </th>
   <th style="text-align:right;"> time </th>
   <th style="text-align:left;"> status </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.00 </td>
   <td style="text-align:left;"> event </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 45.00 </td>
   <td style="text-align:left;"> censored </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 10.12 </td>
   <td style="text-align:left;"> event </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 19.25 </td>
   <td style="text-align:left;"> event </td>
  </tr>
</tbody>
</table>

]

---

## Estimation

Divide the follow-up into intervals and condition on being alive at the
beginning of the time interval:

`\begin{eqnarray*}
\Pr(T_i > 1) &=& \frac{\text{# patients alive at $t = 1$}}{n}\\
\Pr(T_i > 2 \mid T_i > 1) &=& \frac{\text{# patients alive at $t = 2$}}
                                {\text{# patients alive at $t = 1$}}
\end{eqnarray*}`

&#8680; `$\displaystyle \Pr(T_i > 2) = \Pr(T_i > 1) \times \Pr(T_i > 2 \mid T_i > 1)$`

<br>

&#8680; 
`$\displaystyle \Pr(T_i > t) = \Pr(T_i > 1) \times \Pr(T_i > 2 \mid T_i > 1) \times \ldots \times \Pr(T_i > t \mid T_i > t-1)$`

---

## The Kaplan-Meier Estimator

`$$\hat S_{KM}(t) = \prod_{i:t_i\leq t}\frac{r_i - d_i}{r_i}$$`
with
* `$d_i$`: number of events at `$t_i$`
* `$r_i$`: number of patients "at risk" just before `$t_i$`

&#8680; Implies the same survival probability for observed and censored patients.

???

To estimate the survival probability at time `$t$`, take all unique event or 
censoring times `$t_i$` up until `$t$`, and for each of these times, calculate
the ratio of patients who survived the current interval, and the patients at
risk at the beginning of this interval.

---

## The Kaplan-Meier Estimator

---

## The Kaplan-Meier Estimator

---

## Comparing Survival Curves
**Log-Rank Test**
(time stratified Cochran-Mantel-Heanszel test)

* non-parametric
* based on comparison of observed and expected events (&#8680; \$\chi^2\$)

<div>
<table class = "lrtab">
<tr>
<th>&nbsp;</th>
<th>Group 1</th>
<th>Group 2</th>
<th>Total</th>
</tr>
<tr>
<td>event</td>
<td>$d_{1i}$</td>
<td>$d_{2i}$</td>
<td>$d_{i}$</td>
</tr>

<tr>
<td>no event</td>
<td>$r_{1i} - d_{1i}$</td>
<td>$r_{2i} - d_{2i}$</td>
<td>$r_{i} - d_{i}$</td>
</tr>

<br>

* requires **uninformative censoring** & **proportional hazards**
* gives the same weight to all follow-up times

---

## Comparing Survival Curves

**Alternatives:** .sgrey[(weighted versions of the log-rank test with different weights)]
- Peto & Peto modification of the Gehan-Wilcoxon test (aka Breslow)
- Tarone-Ware test
- Flemington-Harrington test
- ...

<br>

**Log-Rank test vs Breslow test:**
- Breslow gives earlier event times more weight
- Log-rank more powerful for proportional hazards
- Breslow more powerful for non-proportional hazards

For crossing survival curves, neither test is optimal!<br>
&#8680; check PH assumption with plot of cumulative hazards

---

## Proportional Hazards Assumption

---

## KM + log-Rank to Identify Risk Factors?

---

## KM + log-Rank to Identify Risk Factors?

* KM estimates **cannot be "adjusted"** for other covariates 
  &#8680; univariable test
  * ok for comparing treatment arms in most **randomized trials**
  * likely **biased in observational studies**
* Continuous covariates have to be categorized <i class = "far fa-frown"></i>

--
<br>

**Also:** Not appropriate for
* time-varying covariates
* competing risk settings

--
<br>

Better: **multivariable analyses** using
* AFT models
* proportional hazards models

---

## Accelerated Failure Time (AFT) models

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

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

--
<br>

&#8680; direct (additive) effect of `$\mathbf x_i$` on `$\log(T_i^*)$`

--
<br>

**But:** Sensitive to specification of the distribution for `$\varepsilon_i$`.<br>
&#8680; May not fit the distribution of the survival times.

???

* 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

---

## Proportional Hazards Model (aka Cox Model)

`$$\underset{\text{baseline hazard}}{h_i(t) = \underset{\uparrow}{h_0(t)} \exp(\beta_1} x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip})$$`
--

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

`$$x_{i'k} = x_{ik} + 1 \quad \Rightarrow \quad \frac{h_{i'}(t)}{h_i(t)} = \exp(\beta_k) \qquad \Rightarrow \quad h_{i'}(t) = h_i(t) \times\exp(\beta_k)$$`
&#8680; **multiplicative effect** of `$x_{ik}$` on `$h_i(t)$`

---

## Proportional Hazards Model (aka Cox Model)

The model can also be written as
`$$\log h_i(t) = \log h_0(t) + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip}$$`
<br>

&#8680; additive effect of `$x_{ik}$` on `$\log h_i(t)$`
`$$x_{i'k} = x_{ik} + 1 \quad \Rightarrow \quad \log h_{i'}(t) = \log h_i(t) + \beta_k$$`

--
<br>

Interpretation of the effect on the log-hazard or hazard (i.e., hazard ratio) is
**not intuitive**.

&#8680; For interpretation: plot of the **expected survival** curves.

---

## Expected Survival

Because `$\displaystyle H(t) = \int_0^t h(s) ds = -\log S(t)$`: 
`$$S(t) = \exp\underset{\;\;\;\;\;\;h_0(t)\exp(\mathbf X\boldsymbol\beta)}{\left\{ -\int_0^t \underbrace{h(s)}ds\right\}}$$`

In the Cox model, `$h_0(t)$` is unspecified<br>
&#8680; we need a separate estimator for the baseline hazard:

$$
\hat S(t) = \exp\bigg\\{
-\underset{\substack{\text{Breslow}\\\\\text{estimator}}}{\underbrace{\hat H_0(t)}} \exp\underset{\substack{\text{from}\\\\\text{Cox model}}}{\mathbf (\mathbf X\underset{\uparrow}{\boldsymbol{\hat \beta}})\bigg\\}}
$$

---

## Expected Survival

1. Fit a model, e.g.,
`$$h_i(t) = h_0(t)\exp(\beta_1 \text{age}_i + \beta_2 \text{sex}_i + 
   \beta_3\text{ascites}_i + \beta_4 \text{hepato}_i + \beta_5 \text{albumin}_i)$$`

2. Create data of hypothetical patients:
<table class=" simpletable" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> age </th>
   <th style="text-align:left;"> sex </th>
   <th style="text-align:right;"> albumin </th>
   <th style="text-align:right;"> hepato </th>
   <th style="text-align:left;"> ascites </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 51 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:left;"> yes </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 51 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:left;"> no </td>
  </tr>
</tbody>
</table>

3. Calculate expected survival for the hypothetical patient(s):<br>

<div style="background-color: var(--nord0); padding: 25px; margin-top: 20px;">
Expected survival probability of a 51 year old female with albumin of
3.5 and no hepatomegaly with or without ascites.
</div>

---

## Expected Survival

---

## Expected Survival `$\neq$` Kaplan Meier

---

## Survival & Cumulative Incidence

Usually: `$\displaystyle \text{cumulative incidence} = 1 - \text{(expected) survival}$`

---

## Competing Risks

Often, multiple things can happen to a patient
* transplant,
* liver-related death
* death from other cause
* ...

When having one event changes the risk of experiencing another event we 
have **competing risks**.

&#8680; Censor at the time of the competing events?

---

## Competing Risks & Kaplan Meier

`$$\color{grey}{\text{informally:}}\qquad\hat S_{KM}(t) = \prod\frac{\text{survivors of interval }i}{\text{at risk in interval }i}$$`

* estimation includes only those still at risk
* implicit extrapolation of the probability to censored subjects

**Example:** Event of interest is LTx<br>
&#8680; KM assumes that patients under follow-up have the same "risk" of LTx as
- patients that were lost to follow up
- patients who died!?!

.warning[
<i class="fas fa-exclamation-triangle" style="color: var(--nord11);"></i>&nbsp;Overestimation of the risk &#8680; overestimation of cumulative incidence&nbsp;<i class="fas fa-exclamation-triangle" style="color: var(--nord11);"></i>
]

---

## Competing Risks & Kaplan Meier

---

## Competing Risks

Two approaches for competing risks:

**Cause specific hazards:**<br>
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

<br>

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

---

## Competing Risk: Example

<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;"> LTx </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;"> 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;"> 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;"> LTx </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;"> LTx </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;"> 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;"> 42 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
</tbody>
</table>
]
]

---

## Competing Risk: Example
<img src="index_files/figure-html/unnamed-chunk-29-1.png" width="100%" style="display: block; margin: auto;" />

---

## Competing Risks: Example
<img src="index_files/figure-html/unnamed-chunk-30-1.png" width="100%" style="display: block; margin: auto;" />

---

## Time-dependent Covariates

Data in long format:
* one row per measurement<br>
  &#8680; multiple rows per subject
* start & stop time of interval
]

.pull-right[
<table class=" idtab" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> id </th>
   <th style="text-align:right;"> tstart </th>
   <th style="text-align:right;"> tstop </th>
   <th style="text-align:right;"> endpt </th>
   <th style="text-align:right;"> age </th>
   <th style="text-align:left;"> sex </th>
   <th style="text-align:right;"> bili </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 192 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 58.77 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 14.5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 192 </td>
   <td style="text-align:right;"> 400 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 58.77 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 21.3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 182 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 1.1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 182 </td>
   <td style="text-align:right;"> 365 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 0.8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 365 </td>
   <td style="text-align:right;"> 768 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 1.0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 768 </td>
   <td style="text-align:right;"> 1790 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 1.9 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1790 </td>
   <td style="text-align:right;"> 2151 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 2.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2151 </td>
   <td style="text-align:right;"> 2515 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 3.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2515 </td>
   <td style="text-align:right;"> 2882 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 4.2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2882 </td>
   <td style="text-align:right;"> 3226 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 3.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3226 </td>
   <td style="text-align:right;"> 4500 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 56.45 </td>
   <td style="text-align:left;"> f </td>
   <td style="text-align:right;"> 4.6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 176 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 70.07 </td>
   <td style="text-align:left;"> m </td>
   <td style="text-align:right;"> 1.4 </td>
  </tr>
</tbody>
</table>

]

---

## Time-dependent Cox Model

The time-dependent Cox model assumes

* **constant** value during the time-interval
* **exogenous** covariates (e.g., value of covariate indep. of whether event has occurred)

<br>

.warning[
<i class="fas fa-exclamation-triangle" style="color: var(--nord11);"></i>&nbsp;Often violated &#8680; bias
]

]

.col[
<img src="index_files/figure-html/unnamed-chunk-32-1.png" width="100%" style="display: block; margin: auto;" />
]
]

---

## Time-dependent Covariates: Joint Model

* model the underlying trajectory of the covariate
* jointly with the survival model

<br>

&#8680; Joint Model for Longitudinal and Survival Data
]

]
]

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## Other Things to Consider

* Clustered Data .sgrey[(e.g., multi-center studies)]
* Discrimination .sgrey[(potential issues with censoring)]
* Stratified Cox .sgrey[(different baseline hazards for different groups)]
* Missing Values .sgrey[(valid imputation in survival setting is not straightforward)]

**Things to watch out for:**
* interval censoring
* competing risks
* time-varying covariates

**and:**
* Consider Kaplan Meier curves to be descriptive, not inferential.
* Expected survival curves to visualize/interpret the model results.