Modèles joints

class: top, left, title-slide

# Modèles joints
### Henri Panjo
### INSERM
### 29 octobre, 2021

---

# Introduction/Motivation

- La présentation porte sur l'idée d'ajuster simultanément un modèle de survie et un modèle 
longitudinal

- De nombreuses études collectent à la fois des données longitudinales (mesures) et des données sur 
la survie.

- Les données ***longitudinales*** sont des données pour lesquelles une variable réponse `$y$` 
(*biomarker*) est mesurée à différents moments (pression artérielle, le poids, les CD4, etc ...)

- Les données de ***survie*** enregistrent les durées jusqu'à un événement d'intérêt tel que le temps 
jusqu'au décès par cancer par exemple

- En l'absence de corrélation entre la réponse longitudinale et la *survie*, 
chaque *outcome* peut être analysé séparément.

- Lorsque la réponse longitudinale et la *survie* sont liées, les deux outcomes doivent être analysés 
conjointement pour éviter des résultats potentiellement biaisés.

---

# Modèle joints

- Deux procéssus qui sont potentiellement liés :

- Un modèle linéaire (mixte) longitudinal (*biomarker*)  
    
    - Un modèle de survie (survival/time-to-event), survenu d'un évènement (*décès*)
--

- On peut être intéresser par ...

- Estimer la trajectoire du *biomarker* en utilisant la survenu de l'évènement (**informative**)
    
    - Estimer la relation entre la trajectoire du *biomarker* et la survenu de l'évènement

---

# Modèle joints

Les modèles joints sont utiles :

- Pour évaluer l'impact d'une covariable longitudinale (mesurée avec erreur) sur le temps de
survenu d'un événement d'intérêt

- Estimer l'effet des covariables indépendantes du temps sur l'*outcome *longitudinal et la *survie*

- Estimer l'effet des covariables dépendantes du temps sur l'*outcome *longitudinal et la *survie*

- Pour tenir compte des *dropout* informatifs dans l'analyse des données longitudinales

---

# Modèle joints

- On suppose un essai clinique où on observe un *biomarker* mesuré à différents moments et le temps
de survenu d'un évènement d'intérêt

- Soit `$T_i^*$` la vraie durée de vie du patient `$i \in \{ 1, 2 \dots, n \}$` et `$T_i = min(T_i^*,C_i)$` 
la durée de vie observée, avec `$C_i$` le temps de censure

- `$\delta_i,$` l'indicateur de l'évènement (égal à `$1$` si vraie évènement, `$0$` si censure)

- Soit `$y_i(t_{ij}), j = \{1, \dots, m_i\}$` représentant les mesures répétées du *biomarker* pour le patient 
`$i$` au temps `$t_{ij}$`

- On suppose que la censure et les temps de mesure ne sont pas informatifs.

---

# Etudes longitudinales et *drop-out* informatif

- Dans une étude longitudinale, il y a souvent des données manquantes

- Ces données manquantes peuvent être informatives: elles ne doivent pas être ignorées dans les analyses.

- Si la valeur d'un *biomarker* `$y$` est liée au risque de décès :

--
    - les valeurs élevées de `$y$` augmentent le risque de décès
--
    - au fil du temps, les patients disparaissent pour cause de décès
--
    - on se retrouvera avec une population en "meilleur" santé (valeurs plus faible de `$y$`).  
    **L'effet du temps sera donc biaisé**

---

# Données longitudinales .underline[sans] *drop-out*

![](data:image/png;base64,#figure/animate1-1.gif)

---

# Données longitudinales .underline[avec] *drop-out*

![](data:image/png;base64,#figure/animate2-1.gif)

---

# Le (sous) modèle longitudinal

On modélise `$y_i(t_{ij})$`  à l'aide d'un modèle linéaire mixte.

`$$y_i(t_{ij}) = \color{blue}{\mu_i(t_{ij})} + \varepsilon_{ij}, \quad \varepsilon_{ij} \sim N(0, \sigma^2)$$`

avec `$\color{blue}{\mu_i(t_{ij})}$` la fonction de trajectoire "réelle", définit par :

`$$\begin{aligned} 
\color{blue}{\mu_i(t_{ij})} &= (\beta_{0} + b_{i0}) + (\beta_{10} + b_{i1})t_{ij} + \boldsymbol{x}_i^T(t_{ij})\boldsymbol{\beta}, \quad (b_{i0}, b_{i1})^T \sim N(\boldsymbol{0}, \Sigma)
\end{aligned}$$`

- Les effets aléatoires sont indépendants des erreurs de mesures `$\varepsilon_{ij}$`

- Les sujets sont indépendants

---

# Le (sous) modèle de survie

*Proproportional hazards model*, paramétrique (Weibull, autres AFT) ou semi-paramétrique (Cox)

`$$h_i(t) = h_0(t)\exp{\left(\boldsymbol{w}_i^T\boldsymbol{\gamma} + \alpha\color{blue}{\mu_i(t_{ij})} \right)}$$`

- `$h_0(t)$` est la *baseline hazard function*, et `$\boldsymbol{w}_i^T$` est un ensemble de
covariables indépendantes du temps associé et leur coefficient (effet) `$\boldsymbol{\gamma}$`

- `$\alpha$` représente le paramètre d'association.

- Ici on a la paramétrisation "valeur actuelle" (*current value*). C'est l'une des paramétrisations 
la plus utilisée.

- Si des covariables sont dans les deux sous-modèles, on obtient les effets globaux sur la survie en
combinant l'effet direct sur le *biomarker* longitudinal (multiplié `$\alpha$`) + l'effet direct sur la survie.

---

# Autre structure d'association des modèles

## *Random effects association structure*

<br>

- Les sous modèles partagent des effets aléatoires (*intercept* + *slope*).

`$$h_i(t) = h_0(t)\exp{\left(\boldsymbol{w}_i^T\boldsymbol{\gamma} + \alpha_1 \color{blue}{b_{i0}} + \alpha_2 \color{blue}{b_{i1}} \right)}$$`

---

# Autre structure d'association des modèles

## *Gradient or rate of change association structure*

<br>

- On peut lier la pente (taux de variation, *slope*) du *biomarker* directement la survie.

`$$h_i(t) = h_0(t)\exp{\left(\boldsymbol{w}_i^T\boldsymbol{\gamma} + \alpha \color{blue}{\mu_i'(t_{ij})} \right)}$$`
--

avec `$$\color{blue}{\mu_i'(t_{ij})} = \frac{\partial \mu_i(t_{ij})}{\partial t_{ij}}$$`

- C'est la dérivée de la trajectoire par rapport au temps

---

# Autre structure d'association des modèles

## *Cumulative association structure*

<br>

- On peut évaluer l'effet cumulatif du *biomarker* sur la survie

`$$h_i(t) = h_0(t)\exp{\left(\boldsymbol{w}_i^T\boldsymbol{\gamma} + \alpha \color{blue}{\int_0^t\mu_i(u)du} \right)}$$`

---

# Cirrhose biliaire primitive *.font60[(Primary biliary cirrhosis)]*

Pour illustrer on utilise une base de patients avec une cirrhose biliaire primitive (CBP)

- `$312$` patients atteints de CBP recueillis à la Clinique Mayo 1974-1984 (Murtaugh et al., 1994)

- `$158$` sujets randomisées pour recevoir de la D-pénicillamine et `$154$` pour recevoir un placebo 
`$(\texttt{drug})$`

- *Outcome* de survie `$(\texttt{status})$` : `$140$` décès, toute cause

- `$1945$` mesures répétées de bilirubine `$(y = \texttt{serbilir})$`

- Version de la base dans le package `$\texttt{merlin}$` de R ou pour les utilisateurs de 
Stata via ce [lien](https://www.mjcrowther.co.uk/data/jm_example.dta).

---

# Structure de la base R (3 sujets)

<table style="font-size:1em; width:100%; font-size: 20px;  font-family: CMU Serif; margin-left: auto; margin-right: auto;" class="table table table-hover">
 <thead>
  <tr>
   <th style="text-align:center;"> id </th>
   <th style="text-align:center;"> stime </th>
   <th style="text-align:center;"> status </th>
   <th style="text-align:center;"> died </th>
   <th style="text-align:center;"> drug </th>
   <th style="text-align:center;"> trt </th>
   <th style="text-align:center;"> placebo </th>
   <th style="text-align:center;"> sex </th>
   <th style="text-align:center;"> age </th>
   <th style="text-align:center;"> time </th>
   <th style="text-align:center;"> timebis </th>
   <th style="text-align:center;"> serbilir </th>
   <th style="text-align:center;"> logb </th>
   <th style="text-align:center;"> serbilir0 </th>
   <th style="text-align:center;"> logb0 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1.10 </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 58.77 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 14.50 </td>
   <td style="text-align:center;"> 2.67 </td>
   <td style="text-align:center;"> 14.50 </td>
   <td style="text-align:center;"> 2.67 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 58.77 </td>
   <td style="text-align:center;"> 0.53 </td>
   <td style="text-align:center;"> 0.26 </td>
   <td style="text-align:center;"> 21.30 </td>
   <td style="text-align:center;"> 3.06 </td>
   <td style="text-align:center;"> 14.50 </td>
   <td style="text-align:center;"> 2.67 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> 2.77 </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Homme </td>
   <td style="text-align:center;"> 70.07 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 1.40 </td>
   <td style="text-align:center;"> 0.34 </td>
   <td style="text-align:center;"> 1.40 </td>
   <td style="text-align:center;"> 0.34 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Homme </td>
   <td style="text-align:center;"> 70.07 </td>
   <td style="text-align:center;"> 0.48 </td>
   <td style="text-align:center;"> 0.24 </td>
   <td style="text-align:center;"> 1.10 </td>
   <td style="text-align:center;"> 0.10 </td>
   <td style="text-align:center;"> 1.40 </td>
   <td style="text-align:center;"> 0.34 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Homme </td>
   <td style="text-align:center;"> 70.07 </td>
   <td style="text-align:center;"> 1.00 </td>
   <td style="text-align:center;"> 0.50 </td>
   <td style="text-align:center;"> 1.50 </td>
   <td style="text-align:center;"> 0.41 </td>
   <td style="text-align:center;"> 1.40 </td>
   <td style="text-align:center;"> 0.34 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Mort </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> D-penicil </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Homme </td>
   <td style="text-align:center;"> 70.07 </td>
   <td style="text-align:center;"> 2.03 </td>
   <td style="text-align:center;"> 1.02 </td>
   <td style="text-align:center;"> 1.80 </td>
   <td style="text-align:center;"> 0.59 </td>
   <td style="text-align:center;"> 1.40 </td>
   <td style="text-align:center;"> 0.34 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 4.12 </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 0.54 </td>
   <td style="text-align:center;"> 0.27 </td>
   <td style="text-align:center;"> 1.90 </td>
   <td style="text-align:center;"> 0.64 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 1.07 </td>
   <td style="text-align:center;"> 0.54 </td>
   <td style="text-align:center;"> 2.50 </td>
   <td style="text-align:center;"> 0.92 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 2.11 </td>
   <td style="text-align:center;"> 1.05 </td>
   <td style="text-align:center;"> 5.70 </td>
   <td style="text-align:center;"> 1.74 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 3.01 </td>
   <td style="text-align:center;"> 1.50 </td>
   <td style="text-align:center;"> 5.20 </td>
   <td style="text-align:center;"> 1.65 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Censure </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> Placebo </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> Femme </td>
   <td style="text-align:center;"> 38.11 </td>
   <td style="text-align:center;"> 3.98 </td>
   <td style="text-align:center;"> 1.99 </td>
   <td style="text-align:center;"> 19.00 </td>
   <td style="text-align:center;"> 2.94 </td>
   <td style="text-align:center;"> 3.40 </td>
   <td style="text-align:center;"> 1.22 </td>
  </tr>
</tbody>
</table>

---

# Structure de la base Stata (3 sujets)

```stata
set linesize 250
use "data/pbc.dta"
format stime age - logb0 %9.3f
list id stime status died drug trt placebo sex age time timebis logb if inlist(id,1,3,5), noobs sepby(id)
```

```
  | id   stime    status   died        drug   trt   placebo     sex      age    time   timebis    logb |
  |----------------------------------------------------------------------------------------------------|
  |  1   1.095      Mort      1   D-penicil     1         0   Femme   58.767   0.000     0.000   2.674 |
  |  1       .      Mort      .   D-penicil     1         0   Femme   58.767   0.526     0.263   3.059 |
  |----------------------------------------------------------------------------------------------------|
  |  3   2.771      Mort      1   D-penicil     1         0   Homme   70.074   0.000     0.000   0.336 |
  |  3       .      Mort      .   D-penicil     1         0   Homme   70.074   0.482     0.241   0.095 |
  |  3       .      Mort      .   D-penicil     1         0   Homme   70.074   0.997     0.498   0.405 |
  |  3       .      Mort      .   D-penicil     1         0   Homme   70.074   2.034     1.017   0.588 |
  |----------------------------------------------------------------------------------------------------|
  |  5   4.121   Censure      0     Placebo     0         1   Femme   38.106   0.000     0.000   1.224 |
  |  5       .   Censure      .     Placebo     0         1   Femme   38.106   0.545     0.272   0.642 |
  |  5       .   Censure      .     Placebo     0         1   Femme   38.106   1.071     0.535   0.916 |
  |  5       .   Censure      .     Placebo     0         1   Femme   38.106   2.105     1.053   1.740 |
  |  5       .   Censure      .     Placebo     0         1   Femme   38.106   3.006     1.503   1.649 |
  |  5       .   Censure      .     Placebo     0         1   Femme   38.106   3.984     1.992   2.944 |
  +----------------------------------------------------------------------------------------------------+
```

---

# Trajectoire de la réponse (1)

![](data:image/png;base64,#figure/trajplot1-1.png)

---

# Trajectoire de la réponse (2)

![](data:image/png;base64,#figure/trajplot2-1.png)

---

# Trajectoire de la réponse (2) : interprétations

- Il semble y avoir une tendance générale à la hausse

- Beaucoup plus marquée chez les patients décédés que chez ceux qui sont censurés

- Cela indique une association positive entre la réponse longitudinale et la durée avant le décès

-  Un niveau plus élevé du *biomarker* semble être associé à la durée avant le décès

---

# Courbe de survie de Kaplan Meier par valeur de bilirubin

![](data:image/png;base64,#figure/km_graph.svg)

---

# Trajectoire de la réponse (16 sujets)

![](data:image/png;base64,#figure/trajplot3-1.png)

---

# R Modèle Survie : `library(survival)`

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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="2" colspan="1"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="2" colspan="1"><strong>% (n)</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="2" colspan="1"><strong>Event N</strong></th>
      <th class="gt_center gt_columns_top_border gt_column_spanner_outer" rowspan="1" colspan="2">
        <span class="gt_column_spanner"><strong>Cox model</strong></span>
      </th>
      <th class="gt_center gt_columns_top_border gt_column_spanner_outer" rowspan="1" colspan="2">
        <span class="gt_column_spanner"><strong>Weibull model</strong></span>
      </th>
    </tr>
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>HR</strong> <strong>(95% CI)</strong><sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>p-value</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>HR (95% CI)</strong><sup class="gt_footnote_marks">1</sup></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1"><strong>p-value</strong></th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left" style="font-weight: bold;">Traitement</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">D-penicil</td>
<td class="gt_row gt_center">51% (158)</td>
<td class="gt_row gt_center">71</td>
<td class="gt_row gt_center">1.00</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center">1.00</td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">Placebo</td>
<td class="gt_row gt_center">49% (154)</td>
<td class="gt_row gt_center">69</td>
<td class="gt_row gt_center">1.14 (0.80 to 1.61)</td>
<td class="gt_row gt_center">0.469</td>
<td class="gt_row gt_center">1.13 (0.80 to 1.60)</td>
<td class="gt_row gt_center">0.482</td></tr>
    <tr><td class="gt_row gt_left" style="font-weight: bold;">Sexe</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">Femme</td>
<td class="gt_row gt_center">88% (276)</td>
<td class="gt_row gt_center">114</td>
<td class="gt_row gt_center">1.00</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center">1.00</td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">Homme</td>
<td class="gt_row gt_center">12% (36)</td>
<td class="gt_row gt_center">26</td>
<td class="gt_row gt_center">1.00 (0.64 to 1.59)</td>
<td class="gt_row gt_center">0.986</td>
<td class="gt_row gt_center">0.99 (0.62 to 1.57)</td>
<td class="gt_row gt_center">0.963</td></tr>
    <tr><td class="gt_row gt_left" style="font-weight: bold;">Age (années)</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center">140</td>
<td class="gt_row gt_center">1.05 (1.03 to 1.06)</td>
<td class="gt_row gt_center"><0.001</td>
<td class="gt_row gt_center">1.05 (1.03 to 1.06)</td>
<td class="gt_row gt_center"><0.001</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">Mean (SD)</td>
<td class="gt_row gt_center">50 (11)</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_left" style="font-weight: bold;">Bilirubine (base)</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center">140</td>
<td class="gt_row gt_center">2.96 (2.47 to 3.55)</td>
<td class="gt_row gt_center"><0.001</td>
<td class="gt_row gt_center">2.87 (2.41 to 3.43)</td>
<td class="gt_row gt_center"><0.001</td></tr>
    <tr><td class="gt_row gt_left" style="text-align: left; text-indent: 10px;">Mean (SD)</td>
<td class="gt_row gt_center">0.57 (1.03)</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
  </tbody>
  
  <tfoot>
    <tr class="gt_footnotes">
      <td colspan="7">
        <p class="gt_footnote">
          <sup class="gt_footnote_marks">
            <em>1</em>
          </sup>
           
          HR = Hazard Ratio, CI = Confidence Interval, CI = Confidence Interval
          <br />
        </p>
      </td>
    </tr>
  </tfoot>
</table>
</div>

---

# Stata, Modèle Survie

- On déclare à Stata qu'on a des données de survie

```stata
stset stime, failure(died)
```

```
Survival-time data settings

Failure event: died!=0 & died<.
Observed time interval: (0, stime]
     Exit on or before: failure

--------------------------------------------------------------------------
      1,945  total observations
      1,633  event time missing (stime>=.)                  PROBABLE ERROR
--------------------------------------------------------------------------
        312  observations remaining, representing
        140  failures in single-record/single-failure data
  2,000.307  total analysis time at risk and under observation
                                                At risk from t =         0
                                     Earliest observed entry t =         0
                                          Last observed exit t =  14.30566
```

---

# Stata, Cox model : `stcox`

```stata
stcox i.drug i.sex age logb0, nolog cformat(%9.2f)
```

```
        Failure _d: died
  Analysis time _t: stime

Cox regression with Breslow method for ties

No. of subjects =        312                            Number of obs =    312
No. of failures =        140
Time at risk    = 2,000.3066
                                                        LR chi2(4)    = 167.73
Log likelihood = -642.69626                             Prob > chi2   = 0.0000

---

# Stata, Post estimation : Fonction de survie

```stata
stcurve, survival at(drug = (1 2) age = (30 50)) scheme(plotplainblind) xsize(15) ysize(7) ///
legend(pos(12) row(1) size(*1.3) order(1 "D-penicil/30ans" 2 "D-penicil/50ans" 3 "Pacebo/30ans" 4 "Pacebo/30ans")) ///
xlab(,labs(3.5)) ylab(,labs(3.5)) xtitle(,size(4.5)) ytitle(,size(4.5))
```

![](data:image/png;base64,#figure/stcoxpost1.svg)

---

# Stata, Post estimation : Fonction de hazard

```stata
stcurve, hazard at(drug = (1 2) age = (30 50)) scheme(plotplainblind) xsize(15) ysize(7) ///
legend(pos(12) row(1) size(*1.3) order(1 "D-penicil/30ans" 2 "D-penicil/50ans" 3 "Pacebo/30ans" 4 "Pacebo/50ans")) ///
xlab(,labs(3.5)) ylab(,labs(3.5)) xtitle(,size(4.5)) ytitle(,size(4.5))
```

![](data:image/png;base64,#figure/stcoxpost2.svg)

---

# Stata, Weibull model : `streg` (modèle paramétrique)

```stata
streg i.drug i.sex age logb0, dist(weibull) cformat(%9.2f) nolog
```

```
        Failure _d: died
  Analysis time _t: stime

Weibull PH regression

No. of subjects =        312                            Number of obs =    312
No. of failures =        140
Time at risk    = 2,000.3066
                                                        LR chi2(4)    = 166.64
Log likelihood = -271.88609                             Prob > chi2   = 0.0000

------------------------------------------------------------------------------
          _t | Haz. ratio   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        drug |
  D-penicil  |       1.00  (base)
    Placebo  |       1.13       0.20     0.70   0.482         0.80        1.60
             |
         sex |
      Femme  |       1.00  (base)
      Homme  |       0.99       0.23    -0.05   0.963         0.62        1.57
             |
         age |       1.05       0.01     5.65   0.000         1.03        1.06
       logb0 |       2.87       0.26    11.84   0.000         2.41        3.42
       _cons |       0.00       0.00   -12.37   0.000         0.00        0.00
-------------+----------------------------------------------------------------
       /ln_p |       0.36       0.07     5.12   0.000         0.22        0.50
-------------+----------------------------------------------------------------
           p |       1.44       0.10                          1.25        1.65
         1/p |       0.70       0.05                          0.61        0.80
------------------------------------------------------------------------------
Note: _cons estimates baseline hazard.
```

---

# Stata, Weibull model : `gsem` (structural equation model)

```stata
gsem (stime <- i.drug age logb0, family(weibull, fail(died)) cformat(%9.2f) nolog)
```

```
Generalized structural equation model               Number of obs   =      312
Response: stime                                     No. of failures =      140
Family:   Weibull                                   Time at risk    = 2,000.31
Form:     Proportional hazards
Link:     Log                 
Log likelihood = -428.52698

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
stime        |
        drug |
  D-penicil  |       0.00  (base)
    Placebo  |       0.12       0.18     0.70   0.483        -0.22        0.47
             |
         age |       0.05       0.01     6.02   0.000         0.03        0.06
       logb0 |       1.06       0.09    11.95   0.000         0.88        1.23
       _cons |      -6.58       0.51   -12.80   0.000        -7.59       -5.57
-------------+----------------------------------------------------------------
/stime       |
        ln_p |       0.36       0.07                          0.22        0.50
------------------------------------------------------------------------------
```

---

# Stata, Weibull model : `gsem`

```stata
estat eform, cformat(%9.2f) 
```

```
             |     exp(b)   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
stime        |
        drug |
  D-penicil  |       1.00  (base)
    Placebo  |       1.13       0.20     0.70   0.483         0.80        1.60
             |
         age |       1.05       0.01     6.02   0.000         1.03        1.06
       logb0 |       2.87       0.25    11.95   0.000         2.42        3.42
       _cons |       0.00       0.00   -12.80   0.000         0.00        0.00
------------------------------------------------------------------------------
```

---

# Stata, modèle mixte : `mixed`

```stata
mixed logb i.drug time || id: time, mle stddev covariance(unstr) technique(nr) nolog
```

```
Mixed-effects ML regression                     Number of obs     =      1,945
Group variable: id                              Number of groups  =        312
                                                Obs per group:
                                                              min =          1
                                                              avg =        6.2
                                                              max =         16
                                                Wald chi2(2)      =     206.99
Log likelihood = -1525.2746                     Prob > chi2       =     0.0000

------------------------------------------------------------------------------
        logb | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        drug |
  D-penicil  |          0  (base)
    Placebo  |   .1282264   .1120144     1.14   0.252    -.0913179    .3477707
             |
        time |   .1772868   .0123674    14.33   0.000     .1530471    .2015265
       _cons |   .4324002   .0800797     5.40   0.000     .2754469    .5893535
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects parameters  |   Estimate   Std. err.     [95% conf. interval]
-----------------------------+------------------------------------------------
id: Unstructured             |
                    sd(time) |    .170855   .0117884      .1492443    .1955949
                   sd(_cons) |   .9952186   .0424102      .9154727    1.081911
            corr(time,_cons) |   .4183218   .0764693      .2580222    .5562068
-----------------------------+------------------------------------------------
                sd(Residual) |   .3490452   .0067617      .3360409    .3625528
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 2871.90               Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.
```

---

# Stata, modèle mixte : `gsem`

.panelset[

.panel[.panel-name[Stata Code]

- Modèle 1 : *Random Intercept*

```stata
#delimit;

gsem (logb <- i.drug c.time M1[id]@1 c.time#M2[id]@1, fam(gauss) link(id)
covstr(M1[id] M2[id], diagonal) /*covariance(M1[id]*M2[id]@0)*/ nocnsreport nolog) ;

#delimit cr
```

<br>

- Modèle 2 : *Random Intercept, Slope*

```stata
#delimit;

gsem (logb <- i.drug c.time M1[id]@1 c.timebis#M2[id]@1, fam(gauss) link(id)
covstructure(M1[id] M2[id], unstr) /*covariance(M1[id]*M2[id])*/ nolog nocnsreport) ;

#delimit cr
```

]

.panel[.panel-name[Modèle 1]

```
delimiter now ;

Generalized structural equation model                    Number of obs = 1,945
Response: logb    
Family:   Gaussian
Link:     Identity
Log likelihood = -1536.9487

-------------------------------------------------------------------------------
              | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
logb          |
         drug |
   D-penicil  |          0  (base)
     Placebo  |    .134177   .1181044     1.14   0.256    -.0973033    .3656573
              |
         time |   .1617294   .0128887    12.55   0.000      .136468    .1869909
              |
       M1[id] |          1  (constrained)
              |
c.time#M2[id] |          1  (constrained)
              |
        _cons |   .4356243   .0831797     5.24   0.000      .272595    .5986535
--------------+----------------------------------------------------------------
   var(M1[id])|   1.038527   .0877061                      .8800994    1.225473
   var(M2[id])|   .0294898    .004093                      .0224662    .0387091
--------------+----------------------------------------------------------------
   var(e.logb)|   .1209605   .0046763                      .1121339     .130482
-------------------------------------------------------------------------------
```
]

.panel[.panel-name[Modèle 2]

```
delimiter now ;

Generalized structural equation model                    Number of obs = 1,945
Response: logb    
Family:   Gaussian
Link:     Identity
Log likelihood = -1525.2746

------------------------------------------------------------------------------------
                   | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------+----------------------------------------------------------------
logb               |
              drug |
        D-penicil  |          0  (base)
          Placebo  |   .1282264   .1120749     1.14   0.253    -.0914363    .3478891
                   |
              time |   .1772868   .0130372    13.60   0.000     .1517343    .2028393
                   |
            M1[id] |          1  (constrained)
                   |
  c.timebis#M2[id] |          1  (constrained)
                   |
             _cons |   .4324002   .0800887     5.40   0.000     .2754292    .5893712
-------------------+----------------------------------------------------------------
        var(M1[id])|   .9904598   .0844148                      .8380901    1.170531
        var(M2[id])|   .1167657   .0161128                      .0890954    .1530295
-------------------+----------------------------------------------------------------
 cov(M1[id],M2[id])|   .1422613   .0300174     4.74   0.000     .0834282    .2010943
-------------------+----------------------------------------------------------------
        var(e.logb)|   .1218326   .0047203                      .1129235    .1314445
------------------------------------------------------------------------------------
```

]

---

# Stata, modèle joints : `merlin`, *Random Intercept*

.panelset[

.panel[.panel-name[Code]

- *Current value association*

```stata
#delimit;
merlin (logb time M1[id]@1, family(gaussian))
       (stime placebo age EV[logb], fam(weibull, failure(died))),
       intmethod(mvaghermite) intpoints(7) technique(nr) nolog ;
```

<br>

- *Random effect association*

```stata
merlin (logb time M1[id]@1, fam(gaussian)) (stime placebo age M1[id], fam(weibull, failure(died))), nolog
```

<br>

- *Cumulative association*

```stata
merlin (logb time M1[id]@1, fam(gaussian)) (stime placebo age iEV[logb], fam(weibull, failure(died))), nolog
```

]

.panel[.panel-name[Current value association]

```
Fitting fixed effects model:

Fitting full model:

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -2284.7594
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |   .0974722   .0043194    22.57   0.000     .0890063     .105938
      M1[id] |          1          .        .       .            .           .
       _cons |   .5782798   .0649178     8.91   0.000     .4510432    .7055163
  sd(resid.) |   .4913687   .0085918                      .4748144    .5085001
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .0944978   .1792764     0.53   0.598    -.2568775    .4458731
         age |   .0601927    .008457     7.12   0.000     .0436173     .076768
        EV[] |   1.375147    .112922    12.18   0.000     1.153824     1.59647
       _cons |  -8.076551   .6260561   -12.90   0.000    -9.303599   -6.849504
  log(gamma) |    .520452   .0713696     7.29   0.000     .3805702    .6603339
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.106563   .0470289                      1.018123    1.202686
------------------------------------------------------------------------------
```
]

.panel[.panel-name[Random effect association]

```
Fitting fixed effects model:

Fitting full model:

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -2284.7594
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |   .0974722   .0043194    22.57   0.000     .0890063     .105938
      M1[id] |          1          .        .       .            .           .
       _cons |     .57828   .0649178     8.91   0.000     .4510434    .7055165
  sd(resid.) |   .4913687   .0085918                      .4748144    .5085001
-------------+----------------------------------------------------------------
stime:       |            
     placebo |    .094498   .1792764     0.53   0.598    -.2568774    .4458733
         age |   .0601928   .0084569     7.12   0.000     .0436176    .0767681
      M1[id] |   1.375147   .1129219    12.18   0.000     1.153824     1.59647
       _cons |  -7.281341   .5969089   -12.20   0.000    -8.451261   -6.111421
  log(gamma) |   .5204523   .0713696     7.29   0.000     .3805705    .6603341
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.106563   .0470289                      1.018123    1.202686
------------------------------------------------------------------------------
```

]

.panel[.panel-name[Cumulative association]

```
Fitting fixed effects model:

Fitting full model:

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -2356.8201
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |   .0986892   .0043429    22.72   0.000     .0901773    .1072011
      M1[id] |          1          .        .       .            .           .
       _cons |   .5582468   .0644958     8.66   0.000     .4318374    .6846563
  sd(resid.) |   .4913076   .0085898                      .4747572     .508435
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .1967482   .1731024     1.14   0.256    -.1425262    .5360227
         age |   .0549198   .0080979     6.78   0.000     .0390482    .0707914
       iEV[] |   .0935277   .0131188     7.13   0.000     .0678154      .11924
       _cons |  -6.116493   .5267488   -11.61   0.000    -7.148902   -5.084084
  log(gamma) |   .1711225   .0739781     2.31   0.021      .026128    .3161169
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.098356   .0466558                      1.010615    1.193714
------------------------------------------------------------------------------
```

]

---

# Random slope. Current value association

```stata
merlin (logb time M1[id]@1 time#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo age sex EV[logb], fam(weibull, failure(died))),
       cov(unstr) restartvalues(M1 1 M2 0.1) nolog;
```

```
delimiter now ;

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1972.7359
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |    .172882   .0129342    13.37   0.000     .1475315    .1982326
      M1[id] |          1          .        .       .            .           .
 time#M2[id] |          1          .        .       .            .           .
       _cons |   .5046443   .0583123     8.65   0.000     .3903543    .6189343
  sd(resid.) |   .3481897   .0067211                      .3352627    .3616151
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .1376123   .1767057     0.78   0.436    -.2087245    .4839492
         age |   .0578418   .0088909     6.51   0.000      .040416    .0752676
         sex |    .039843   .2385632     0.17   0.867    -.4277324    .5074183
        EV[] |   .7101999   .0806747     8.80   0.000     .5520803    .8683195
       _cons |  -7.369589   .6055144   -12.17   0.000    -8.556375   -6.182803
  log(gamma) |   .2414628   .0722123     3.34   0.001     .0999292    .3829963
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.002638   .0426881                      .9223664    1.089895
      sd(M2) |   .1728658   .0119119                      .1510267    .1978628
 corr(M2,M1) |    .435763   .0734724                      .2814268    .5681226
------------------------------------------------------------------------------
```

---

# Random slope. Random effect association

```stata
merlin (logb time M1[id]@1 time#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo age sex M1[id] M2[id], fam(weibull, failure(died))),
       cov(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

```
delimiter now ;

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1894.1313
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |   .2014861   .0143148    14.08   0.000     .1734295    .2295426
      M1[id] |          1          .        .       .            .           .
 time#M2[id] |          1          .        .       .            .           .
       _cons |   .4929323   .0580162     8.50   0.000     .3792227    .6066419
  sd(resid.) |   .3471596   .0066628                      .3343432    .3604672
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .0490363   .2017467     0.24   0.808    -.3463799    .4444526
         age |   .0668058   .0103209     6.47   0.000     .0465772    .0870344
         sex |  -.2080335   .2737672    -0.76   0.447    -.7446074    .3285405
      M1[id] |   1.204667   .1415654     8.51   0.000     .9272044     1.48213
      M2[id] |   7.508258   .9669993     7.76   0.000     5.612974    9.403542
       _cons |  -8.631326   .8054924   -10.72   0.000    -10.21006    -7.05259
  log(gamma) |   .8468186   .0871465     9.72   0.000     .6760147    1.017623
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   .9991078   .0424512                      .9192752    1.085873
      sd(M2) |   .1963487   .0144576                      .1699621    .2268317
 corr(M2,M1) |   .5132364   .0617975                      .3822387    .6240118
------------------------------------------------------------------------------
```

---

# Random slope. Cumulative association

```stata
merlin (logb time M1[id]@1 time#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo age sex iEV[logb], fam(weibull, failure(died))),
       cov(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

```
delimiter now ;

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -2014.659
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |    .175436   .0130909    13.40   0.000     .1497783    .2010936
      M1[id] |          1          .        .       .            .           .
 time#M2[id] |          1          .        .       .            .           .
       _cons |   .4961729   .0580483     8.55   0.000     .3824003    .6099454
  sd(resid.) |   .3487104   .0067487                      .3357308    .3621917
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .1737923   .1724912     1.01   0.314    -.1642842    .5118688
         age |   .0485811   .0085728     5.67   0.000     .0317788    .0653834
         sex |   .3055313   .2299103     1.33   0.184    -.1450847    .7561472
       iEV[] |   .0413797   .0116459     3.55   0.000     .0185541    .0642052
       _cons |  -6.007028   .5480056   -10.96   0.000    -7.081099   -4.932957
  log(gamma) |   .1086591   .0744668     1.46   0.145    -.0372932    .2546113
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |    .997817   .0425082                       .917886    1.084709
      sd(M2) |   .1722515   .0119054                      .1504288      .19724
 corr(M2,M1) |   .4238052   .0758592                      .2646375    .5604944
------------------------------------------------------------------------------
```

---

# Random slope. Gradiant association, fractional polynomial

```stata
merlin (logb fp(time, pow(1)) M1[id]@1 fp(time, pow(1))#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo age sex dEV[logb], fam(weibull, failure(died))),
       cov(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

```
delimiter now ;
variables created for model 1, component 1: _cmp_1_1_1 to _cmp_1_1_1

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1925.3643
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        fp() |   .2192966   .0143835    15.25   0.000     .1911054    .2474878
      M1[id] |          1          .        .       .            .           .
 fp()#M2[id] |          1          .        .       .            .           .
       _cons |   .4836871   .0559192     8.65   0.000     .3740876    .5932867
  sd(resid.) |   .3554995    .006974                      .3420902    .3694345
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .0706699   .2206293     0.32   0.749    -.3617556    .5030953
         age |   .0564519   .0112882     5.00   0.000     .0343274    .0785763
         sex |   .2171143   .2979299     0.73   0.466    -.3668175    .8010462
       dEV[] |   10.98452   1.231099     8.92   0.000      8.57161    13.39743
       _cons |  -10.85363   1.071445   -10.13   0.000    -12.95362   -8.753637
  log(gamma) |   .8025727   .0976509     8.22   0.000     .6111804    .9939649
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   .9658252   .0409503                      .8888085    1.049516
      sd(M2) |   .2099789   .0146959                      .1830637    .2408514
 corr(M2,M1) |   .6997001   .0394773                      .6138797     .769174
------------------------------------------------------------------------------
```

---

# Modèle joints : traitement dans les 2 modèles

```stata
merlin (logb time placebo M1[id]@1 time#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo EV[logb], family(weibull, failure(died)) timevar(stime)),
       covariance(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

```
delimiter now ;

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1918.5299
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        time |   .1849158   .0133043    13.90   0.000     .1588397    .2109918
     placebo |   .1313651   .1119896     1.17   0.241    -.0881305    .3508608
      M1[id] |          1          .        .       .            .           .
 time#M2[id] |          1          .        .       .            .           .
       _cons |   .4280096   .0802409     5.33   0.000     .2707402    .5852789
  sd(resid.) |   .3471621   .0066736                      .3343254    .3604918
-------------+----------------------------------------------------------------
stime:       |            
     placebo |  -.0392779   .1791411    -0.22   0.826     -.390388    .3118323
        EV[] |   1.241051   .0932787    13.30   0.000     1.058229    1.423874
       _cons |  -4.370797   .2801213   -15.60   0.000    -4.919825    -3.82177
  log(gamma) |   .0194596   .0825675     0.24   0.814    -.1423696    .1812889
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |     1.0003    .042573                      .9202442    1.087321
      sd(M2) |   .1805618   .0123693                      .1578755     .206508
 corr(M2,M1) |   .4246963   .0728335                      .2722046    .5563785
------------------------------------------------------------------------------
```

---

# Overall treatment effect on survival

```stata
matrix list e(b)
```

```
e(b)[1,13]
     _cmp_1_1_1:  _cmp_1_2_1:  _cmp_1_3_1:  _cmp_1_4_1:       cons1:      dap1_1:  _cmp_2_1_1:  _cmp_2_2_1:       cons2:
          _cons        _cons        _cons        _cons        _cons        _cons        _cons        _cons        _cons
y1    .18491577    .13136513            1            1     .4280096   -1.0579634   -.03927789    1.2410515   -4.3707972

dap2_1:      lns1_1:      lns1_2:    art1_1_2:
          _cons        _cons        _cons        _cons
y1    .01945965    .00030031   -1.7116823    .45340794
```

<br>

- Le log hazard-ratio global pour l'effet du placebo sur la survie est

`$$\alpha \beta + \gamma, \mbox{ avec } \alpha = 1.241, \beta = 0.131, \gamma = -0.039$$`

- On peut le calculer avec une combinaison **non linéaire** des paramètres du modèle :

```stata
nlcom (placebo_survival: [_cmp_2_2_1][_cons]*[_cmp_1_2_1][_cons] + [_cmp_2_1_1][_cons])
```

```
placebo_su~l: [_cmp_2_2_1][_cons]*[_cmp_1_2_1][_cons] + [_cmp_2_1_1][_cons]

----------------------------------------------------------------------------------
                 | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
placebo_survival |    .123753   .2293467     0.54   0.589    -.3257582    .5732642
----------------------------------------------------------------------------------
```

---

# Modèle joints : Restricted cubic spline

```stata
merlin (logb rcs(time, df(1)) placebo M1[id]@1 rcs(time, df(1))#M2[id]@1, fam(gaussian) timevar(time))
       (stime placebo age EV[logb], fam(weibull, failure(died))),
       covariance(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

```
delimiter now ;
variables created for model 1, component 1: _cmp_1_1_1 to _cmp_1_1_1

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1972.0731
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
       rcs() |   .1727339   .0129134    13.38   0.000     .1474241    .1980436
     placebo |   .1303464   .1119772     1.16   0.244     -.089125    .3498178
      M1[id] |          1          .        .       .            .           .
rcs()#M2[id] |          1          .        .       .            .           .
       _cons |     .44027   .0802616     5.49   0.000     .2829602    .5975798
  sd(resid.) |   .3482309   .0067222                      .3353017    .3616586
-------------+----------------------------------------------------------------
stime:       |            
     placebo |   .1460657   .1768804     0.83   0.409    -.2006134    .4927449
         age |   .0582723   .0085285     6.83   0.000     .0415568    .0749878
        EV[] |   .7125079   .0798715     8.92   0.000     .5559627    .8690531
       _cons |  -7.354149   .5941311   -12.38   0.000    -8.518624   -6.189673
  log(gamma) |   .2415113   .0721999     3.35   0.001     .1000021    .3830204
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.000521   .0426023                      .9204117    1.087603
      sd(M2) |   .1726132   .0118837                      .1508247    .1975495
 corr(M2,M1) |   .4352359   .0733578                      .2811759    .5674257
------------------------------------------------------------------------------
```

---

# Modèle joints : non-proportional hazards

.panelset[

.panel[.panel-name[Stata Code]

- **Treatment effect** :

`$$h_i(t) = h_0(t)\exp{\left(\gamma_1 \texttt{placebo} + \gamma_2\texttt{placebo}\times\log(t) + \alpha\color{blue}{\mu_i(t_{ij})} \right)}$$`

```stata
merlin (logb fp(time, pow(1)) M1[id]@1 fp(time, pow(1))#M2[id]@1, fam(gauss) timevar(time)) ///
       (stime placebo placebo#fp(stime, pow(0)) EV[logb], fam(weibull, failure(died)) timevar(stime)), ///
       covariance(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

<br>

- **Association effect** :

`$$h_i(t) = h_0(t)\exp{\left(\gamma_1 \texttt{placebo} + \alpha_1\color{blue}{\mu_i(t_{ij})} + \alpha_2\color{blue}{\mu_i(t_{ij})}\times \log(t)\right)}$$`

```stata
merlin (logb fp(time, pow(1)) M1[id]@1 fp(time, pow(1))#M2[id]@1, fam(gauss) timevar(time)) ///
       (stime placebo EV[logb] EV[logb]#fp(stime, pow(0)), fam(weibull, failure(died)) timevar(stime)), ///
       covariance(unstr) restartvalues(M1 1 M2 0.1) nolog ;
```

]

.panel[.panel-name[Treatment effect]

```
delimiter now ;
variables created for model 1, component 1: _cmp_1_1_1 to _cmp_1_1_1
variables created for model 2, component 2: _cmp_2_2_1 to _cmp_2_2_1

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1919.0794
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        fp() |   .1850538   .0133219    13.89   0.000     .1589435    .2111642
      M1[id] |          1          .        .       .            .           .
 fp()#M2[id] |          1          .        .       .            .           .
       _cons |   .4929608   .0582836     8.46   0.000     .3787271    .6071946
  sd(resid.) |   .3471186   .0066719                      .3342852    .3604448
-------------+----------------------------------------------------------------
stime:       |            
     placebo |  -.1401475   .2555729    -0.55   0.583    -.6410611    .3607662
placebo#fp() |   .0884933   .1689263     0.52   0.600    -.2425961    .4195826
        EV[] |   1.247314   .0944558    13.21   0.000     1.062184    1.432444
       _cons |  -4.282816   .3197797   -13.39   0.000    -4.909572   -3.656059
  log(gamma) |  -.0262071   .1217259    -0.22   0.830    -.2647854    .2123712
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |   1.002564    .042663                      .9223385    1.089768
      sd(M2) |   .1808127   .0123957                      .1580791    .2068158
 corr(M2,M1) |   .4248652   .0729237                      .2721674    .5566908
------------------------------------------------------------------------------
```
]

.panel[.panel-name[Association effect]

```
delimiter now ;
variables created for model 1, component 1: _cmp_1_1_1 to _cmp_1_1_1
variables created for model 2, component 3: _cmp_2_3_1 to _cmp_2_3_1

Mixed effects regression model                           Number of obs = 1,945
Log likelihood = -1915.8459
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
logb:        |            
        fp() |   .1864671   .0134396    13.87   0.000      .160126    .2128082
      M1[id] |          1          .        .       .            .           .
 fp()#M2[id] |          1          .        .       .            .           .
       _cons |   .4940006   .0582044     8.49   0.000     .3799222    .6080791
  sd(resid.) |   .3470075   .0066705                      .3341767    .3603309
-------------+----------------------------------------------------------------
stime:       |            
     placebo |  -.1089156   .1811347    -0.60   0.548    -.4639331    .2461019
        EV[] |   1.559629   .1683266     9.27   0.000     1.229715    1.889543
   EV[]#fp() |  -.2507349   .1007199    -2.49   0.013    -.4481423   -.0533274
       _cons |  -5.440698    .540824   -10.06   0.000    -6.500694   -4.380703
  log(gamma) |   .4407332   .1564277     2.82   0.005     .1341405    .7473259
-------------+----------------------------------------------------------------
id:          |            
      sd(M1) |      1.001   .0425932                       .920905    1.088061
      sd(M2) |   .1824288   .0125577                      .1594042    .2087791
 corr(M2,M1) |   .4384078   .0720171                      .2872099    .5682899
------------------------------------------------------------------------------
```

]

---

# Résumé / Conclusion

- L'analyse conjointe des *outcomes* longitudinaux et de survie est nécessaire pour obtenir des
estimations et inférences non biaisées lorsque les deux *outcomes* sont corrélés.

- Les modèles joints peuvent être utilisées :

--
    - pour évaluer les effets des covariables à la *baseline* sur les *outcomes* longitudinaux et de survie
--
    - pour évaluer les effets des covariables dépendantes du temps sur les *outcomes* de survie
--
    - pour tenir compte des *dropouts* informatifs dans l'analyse longitudinale

- On peut utiliser `gsem` de **Stata** pour les modèles conjoints prenant en compte des *outcomes*
longitudinaux non continus. La survie, cependant, est modélisée de manière paramétrique.

- La commande `merlin` (Mixed effects regression for linear, non-linear and user-defined models) de
**Stata**, crée par [Michael J. Crowther](https://www.mjcrowther.co.uk/), permet de "tout" faire 😎🤯😎