Analysis of Multi-level Data

<div class = "title">Analysis of Multi-level Data</div>
<div class = "subtitle">(and more)</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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---
count: false

## Example

[
<img src="JHep.png" width="100%" style="display: block; margin: auto;" />
](https://doi.org/10.1016/j.jhep.2020.03.029)

???

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

---

## Data

* n = 1093 patients who
  - had cirrhosis,
  - were listed or eligible for listing for liver transplantation,
  - were seen as an outpatient for clinical care.
* 8 liver transplant centers

**Outcome:** Waitlist mortality (death / de-listing because too sick for LTx)

**Covariate of interest:** Frailty &emsp; .sgrey[&#8680; Liver Frailty Index]<br>
* measured at enrolment and every subsequent ambulatory clinic visit
* 3 tests combined into one continuous score (LFI)
* higher LFI &#8680; more frail

---

## Statistical Analysis (I)

**Multilevel mixed-effects linear regression**<br>
&#8680; 2 levels (participant & visit) to take into account correlation

???

<blockquote>
We used multilevel mixed-effects linear regression to model longitudinal data
from each clinic visit and predict ΔLFI per 3 months.<br><br>

Each participant could contribute multiple observations. <br><br>

We structured the model with 2 levels (participant and visit), nesting the
longitudinal visit-level observations within the participant to address
correlation.
</blockquote>

- - -
--

**Fixed effects:**
* baseline age, gender, MELD Na, ascites, hepatic encephalopathy, time of measurement
* selected *a priori*, irrespective of significance
  &emsp; <i class="fas fa-thumbs-up" style = "color: var(--nord8)"></i>

???

<blockquote>
Baseline age, gender, MELD-Na, ascites, and hepatic encephalopathy were modeled
as fixed effects.<br><br>

These characteristics were selected a priori and included in the multivariable
model, regardless of statistical significance, given their biologically
plausible association with frailty.
</blockquote>

- - -

**Random effects:**
* participant
* follow-up time
&#8680; correlated

???

<blockquote>
Participant and follow-up time were modeled as random effects to allow for
random intercepts and random slopes, respectively.<br><br>

The random intercepts and slopes were allowed to be correlated to accommodate,
for example, those with greater baseline frailty worsening more quickly.<br><br>

We addressed varying time between LFI observations by including the actual time
of the measurement (from baseline to each LFI assessment) as a predictor in the mixed-effects model. This allowed the model to account for differences in timing
of LFI measurements.
</blockquote>

- - -

<ul class = "arrowlist">
<li>Calculate patient specific predicted/estimated baseline LFI & $\Delta$LFI.</li>
<li>Categorize $\Delta$LFI into "improved", "stable", "moderate worsening", "worsening".</li>
</ul>

???

<blockquote>
From the multivariable mixed-effects model, we calculated the predicted
intercept (baseline LFI) and slope (ΔLFI) per 3 months for each participant, to
facilitate clinical interpretation of the data and for the purposes of visual
presentation.<br><br>

Higher positive ΔLFI values reflected more rapid frailty progression, positive
ΔLFI values closer to 0 reflected slower frailty progression, and negative ΔLFI
values reflected improving frailty.<br><br>

We categorized ΔLFI into 4 categories: improved, stable, moderate worsening, and
severe worsening. These categories were initially based on quartiles of ΔLFI
slopes in this cohort.<br><br>

[...]
</blockquote>

---

## Statistical Analysis (II)

**Cumulative Incidence** of waitlist mortality per `$\Delta$`LFI category

???

<blockquote>
To evaluate the impact of baseline LFI and predicted ΔLFI on risk of waitlist
mortality, we estimated the cumulative incidence of waitlist mortality within
3-, 6-, and 12-months of study entry by ΔLFI category.
</blockquote>

**Fine & Gray competing risk model** to estimate association  waitlist mortality with covariates
* **event of interest**: death or de-listing because too sick
* **competing risk**: deceased donor liver transplant
* **censoring** at time of living donor transplant or de-listing for other reasons

???

<blockquote>
Then we modeled the cumulative incidence function with Fine and Gray competing
risk regressions to estimate the risk of waitlist mortality associated with each
variable.<br><br>

Risk estimates were described as subhazard ratios (sHR) with 95%
CIs.<br><br>

In these competing risk models, waitlist mortality was defined as a death or
delisting; deceased donor liver transplant was treated as the competing risk.

[...]

</blockquote>

<br>

* Selection of covariates other than MELD-Na based on univariable models
  (p&nbsp;<&nbsp;0.1) and backward elimination
  &emsp; <i class="fas fa-frown" style = "color: var(--nord8)"></i>

???

<blockquote>
For the multivariable regression, all variables associated with waitlist
mortality with a p value of <0.1 in univariable analysis were evaluated for
inclusion in the final multivariable model.<br><br>

Backwards elimination was then performed to derive the final multivariable model
which included only variables associated with a p value <0.05 and MELD-Na due to
the biologically plausible association with waitlist mortality.<br><br>

[...]

</blockquote>
--

Differences in MELD-Na regression coefficients were compared between
models with and without LFI and `$\Delta$`LFI using **bootstrap** methods.

???

<blockquote>
To determine if the MELD-Na sHR changed significantly after adjusting for LFI,
differences in MELD-Na regression coefficients were compared between models with
and without LFI and ΔLFI using bootstrap methods (100 replications).
</blockquote>

---

## Data (Simulated)
<img src="index_files/figure-html/unnamed-chunk-3-1.png" width="100%" style="display: block; margin: auto;" />

???

* Simulated data based on the description of the data in the results section

* repeated measurements over time
* exact time of measurement, not visit number
* **unbalanced** data &#8680; different number of measurements at different times

---

## Data (Simulated)

---

## Mixed (Effects) Model

* contains both **fixed** and **random** effects

**Linear Mixed Model**
`$$y_{ij} = \underset{\text{fixed effects}}{\underbrace{\beta_0 + \beta_1 \text{time}_{ij} + \beta_2 \text{age}_i +  \ldots}} +  
\underset{\text{random effects}}{\underbrace{b_{0i} + b_{1i} \text{time}_{ij}}} +
\varepsilon_{ij}, \quad \substack{i = 1, \ldots, n\\j = 1,\ldots, n_i}$$`

???

Mixed model = Mixed effects model = random effects model

Extension of (generalized) linear regression model that additionally includes
random effects to take into account that observations are not independent of
each other.

`$$\begin{pmatrix}b_{0i}\\b_{1i}\end{pmatrix} \sim N(\mathbf 0, \mathbf D) \qquad
\varepsilon_{ij} \sim N(0, \sigma^2)$$`
* `$b_{0i}$`: random intercept
* `$b_{1i}$`: random slope
* `$\mathbf D$`: variance-covariance matrix

???

* random intercept and slope (random effect of time) are unobserved
* assumed to be jointly normally distributed with mean zero

---

## Fixed & Random Effects

???

* Fixed effects describe population average fit. (same in each panel)
* Random effects deviation of subject-specific fit from population average.

We ignore other covariates here for now.

---

## Fixed & Random Effects

???

Here: zoom in, look at different elements

* fixed effects intercept, `$\beta_0$`, point where population regression line
  crosses the `$y$`-axis

---

## Fixed & Random Effects

???

random intercept: deviation of subject-specific regression line at baseline
to population line

* &#8680; subject-specific line crosses `$y$`-axis at `$\beta_0 + b_{0i}$`
* subject with lower values than the population average: negative random intercept

---

## Fixed & Random Effects

???

* effect of time ($\beta_1$) is slope of population regression line ("difference in `$y$` for a 1-unit change in time")
* corresponding random effect in this special case called random slope
* deviation of subject-specific slope from population slope
* slope of subject-specific regression line is `$\beta_1 + b_{1i}$`

<div style = "position: absolute; bottom: 450px; left: 200px; font-size: 10pt;">
$\color{var(--nord10)}{\hat y = \beta_0 + \beta_1 \text{time}_{ij}} \color{var(--nord3)}{+ \ldots \text{(covariates)}\ldots}$
</div>

<div style = "position: absolute; bottom: 250px; left: 200px; font-size: 10pt;">
$\color{var(--nord15)}{\hat y = (\beta_0 + b_0) + (\beta_1 + b_1)\ \text{time}_{ij}} \color{var(--nord3)}{+ \ldots \text{(covariates)}\ldots}$
</div>

---

## Random Effects

???

What are the implied assumptions of the random intercept and slope with regards
to the outcome trajectories?

* random intercept only: subject-specific trajectories are randomly shifted on
  y-axis
  
* random slope: slope randomly varies

How much variation? &$8680; described by a normal distribution with mean 0

---

## Distribution of the Random Effects
Random effects are unobserved and assumed to be normally distributed
`$$\begin{pmatrix}b_{0i}\\b_{1i}\end{pmatrix} \sim N(\mathbf 0, \mathbf D)$$`

with **variance-covariance matrix**

`$$\mathbf D = \begin{pmatrix}d_{1}^2 & d_{12}\\d_{12} & d_{2}^2\end{pmatrix}$$`

<br>

&#8680; Correlation between intercept and slope: `$\rho = \frac{d_{12}}{d_1 d_2}$`

???

* Bivariate normal distribution
* not variance but variance-covariance matrix
* variance-covariance matrix has the variances on the diagonal and the covariances
on the off-diagonal

Can be expressed in terms of correlation.

Different correlation will result in different patterns of the trajectories.

---

## Random Effects Correlation

---

## Effects of Covariates

`$$y_{ij} = \underset{\text{fixed effects}}{\underbrace{\beta_0 + \beta_1 \text{time}_{ij} + \bbox[#3B4252, 2pt]{\beta_2 \text{sex}_i}}} +  
\underset{\text{random effects}}{\underbrace{b_{0i} + b_{1i} \text{time}_{ij}}} +
\varepsilon_{ij}$$`

---

## Effects of Covariates

`$$y_{ij} = \underset{\text{fixed effects}}{\underbrace{\beta_0 + \beta_1 \text{time}_{ij} + \bbox[#3B4252, 2pt]{\beta_2 \text{sex}_i + \beta_3 \text{sex}_i \text{time}_{ij}}}} +  
\underset{\text{random effects}}{\underbrace{b_{0i} + b_{1i} \text{time}_{ij}}} +
\varepsilon_{ij}$$`

---

## Effects of Time-varying Covariates

`$$y_{ij} = \underset{\text{fixed effects}}{\underbrace{\beta_0 + \beta_1 \text{time}_{ij} + \bbox[#3B4252, 2pt]{\beta_2 \text{BMI}_{\color{var(--nord8)}{ij}}}}} +  
\underset{\text{random effects}}{\underbrace{b_{0i} + b_{1i} \text{time}_{ij}}} +
\varepsilon_{ij}$$`

---

## Effects of Time-varying Covariates

`$$y_{ij} = \underset{\text{fixed effects}}{\underbrace{\beta_0 + \beta_1 \text{time}_{ij} + \bbox[#3B4252, 2pt]{\beta_2 \text{BMI}_{\color{var(--nord8)}{ij}} + \beta_3 \text{BMI}_{\color{var(--nord8)}{ij}}\text{time}_{ij}}}} +  
\underset{\text{random effects}}{\underbrace{b_{0i} + b_{1i} \text{time}_{ij}}} +
\varepsilon_{ij}$$`

---

## Predicted Intercept & Slope

.pull-left[
<img src="index_files/figure-html/unnamed-chunk-17-1.png" width="100%" style="display: block; margin: auto;" />

]
.pull-right[

<img src="index_files/figure-html/unnamed-chunk-18-1.png" width="100%" style="display: block; margin: auto;" />
]

---

## Predicted Intercept & Slope

---

## Results from the Paper

<table class=" simpletable" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="2"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">median (IQR)</div></th>
</tr>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> n </th>
   <th style="text-align:left;"> # visits </th>
   <th style="text-align:left;"> days
between visits </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> severe </td>
   <td style="text-align:left;"> 284 </td>
   <td style="text-align:left;"> 2 (1–4) </td>
   <td style="text-align:left;"> 112 days (91–179) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> moderate </td>
   <td style="text-align:left;"> 252 </td>
   <td style="text-align:left;"> 1 (1–2) </td>
   <td style="text-align:left;"> 112 days (91–161) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> stable </td>
   <td style="text-align:left;"> 38 </td>
   <td style="text-align:left;"> 2 (1–2) </td>
   <td style="text-align:left;"> 119 days (97–175) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> improved </td>
   <td style="text-align:left;"> 170 </td>
   <td style="text-align:left;"> 4 (3–6) </td>
   <td style="text-align:left;"> 168 days (116–210) </td>
  </tr>
</tbody>
</table>
]
.col[
<table class=" simpletable" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="2"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">median (IQR)</div></th>
</tr>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> n </th>
   <th style="text-align:left;"> # visits </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> died/delisted </td>
   <td style="text-align:left;"> 223 </td>
   <td style="text-align:left;"> 2 (1–3) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> other </td>
   <td style="text-align:left;"> 870 </td>
   <td style="text-align:left;"> 2 (1–4) </td>
  </tr>
</tbody>
</table>

]
]

???

median (IQR) number of visits differed significantly by ΔLFI group: 
* 2 (1–4) for the severe worsening 
* 1 (1–2) for the slight worsening
* 2 (1–2) for the stable 
* 4 (3–6) for the improved

median (IQR) number of days between visits (among those with >1 LFI assessment)

* 112 days (91–179) for steep worsening, 
* 112 days (91–161) for moderate worsening, 
* 119 days (97–175) for stable and 
* 168 days (116–210) for improved (p <0.001).

Participants who died/delisted vs. other waitlist outcomes had a similar median number of visits (2 (1–3) vs. 2 (1–4), p = 0.55).

<br>

.warning[
<i class="fas fa-exclamation-triangle" style = "color: var(--nord11)"></i> Beware of Missing Not At Random! <i class="fas fa-exclamation-triangle" style = "color: var(--nord11)"></i>
]

---

---

## Statistical Analysis (II)

**Cumulative Incidence** of waitlist mortality per `$\Delta$`LFI category
<span style = "color:var(--nord11); font-weight: 700;">&#8680; How???</span>

**Remember...**

<div style = "position: relative; bottom: 60px; left: 120px;">
<div class="shareagain" style="min-width:300px;margin:1em auto;max-width:75%;">
<iframe src="https://nerler.github.io/Presentation_MDL2021_Survival/#47" width="1600" height="900" style="border:2px solid var(--nord8);" loading="lazy" allowfullscreen></iframe>
<script>fitvids('.shareagain', {players: 'iframe'});</script>
</div>
</div>

<div style = "position: absolute; left: 60px; bottom: 200px;">
<strong>&#8680; Aalen-Johansen<br>estimator</strong>
</div>

---

.box[
**Fine-Grey Model:**
* model for the cumulative incidence function
* *sub-distribution* hazard instead of *cause-specific* hazard
* person failing from other cause remain in the risk set
* PH assumption for the sub-distribution hazard
]

---

## Bootstrap

.pull-left[
<table class=" bstab" 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="1"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">original</div></th>
<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="6"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">bootstrap samples</div></th>
</tr></thead>
<tbody>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;background-color: var(--nord0) !important;"> 1 </td>
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   <td style="text-align:center;"> ... </td>
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   <td style="text-align:center;background-color: var(--nord0) !important;"> 7 </td>
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   <td style="text-align:left;">  </td>
   <td style="text-align:center;background-color: var(--nord0) !important;"> 9 </td>
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   <td style="text-align:center;"> ... </td>
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   <td style="text-align:left;">  </td>
   <td style="text-align:center;background-color: var(--nord0) !important;"> 10 </td>
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   <td style="text-align:center;"> ... </td>
  </tr>
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   <td style="text-align:left;">  </td>
   <td style="text-align:center;background-color: var(--nord0) !important;">   </td>
   <td style="text-align:center;">   </td>
   <td style="text-align:center;">   </td>
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   <td style="text-align:center;">   </td>
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   <td style="text-align:center;">   </td>
   <td style="text-align:center;">   </td>
  </tr>
  <tr>
   <td style="text-align:left;"> mean </td>
   <td style="text-align:center;background-color: var(--nord0) !important;"> 5.5 </td>
   <td style="text-align:center;">   </td>
   <td style="text-align:center;"> 5.1 </td>
   <td style="text-align:center;"> 5.8 </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 5.2 </td>
   <td style="text-align:center;"> 4.4 </td>
   <td style="text-align:center;"> ... </td>
  </tr>
</tbody>
</table>

]

.pull-right[
<br><br><br>
&#8680; 95% bootstrap CI:<br>
2.5% and 97.5% quantile of the means from the bootstrap samples

]

---

# 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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---