Context and motivation

• Time-to-event and longitudinal data are fundamental in health-related studies
• A significant feature is its temporal dimension, allowing for dynamic predictions
• These predictions have demonstrated utility in precision medicine

Context and motivation

• Joint models (JMs) provide a suitable framework for analyzing time-to-event and longitudinal data together
• JMs allow dynamic predictions
• Applying JMs to multivariate longitudinal data poses challenges due to the increased number of random effects and parameters

The PBC dataset

• The PBC dataset is a widely used, freely available, dataset that will be used to apply the methods
• By using the PBC dataset, all methods are fully reproducible
• Primary biliary cirrhosis (PBC) is a rare disease that could eventually lead to liver cirrhosis

The PBC dataset

Notation

• \(\require{color}\textcolor{#2E8B57}{T_1^*,\dots, T_n^*}\) true time until event
• \(\require{color}\textcolor{#2E8B57}{C_1,\dots, C_n}\) censoring times
• \(\require{color}\textcolor{#2E8B57}{T_i=\min(T_i^*, C_i)}\) observed times, \(i=1,\dotsc,n\)
• \(\require{color}\textcolor{#2E8B57}{\delta_i=\mathbf{1}\{T_i^*\leq C_i\}}\) censoring indicator, \(i=1,\dotsc,n\)
• Longitudinal data \(\require{color}\textcolor{#D2691E}{\{\boldsymbol{y}_{li}; i=1,\dots,n, l=1,\dots,L\}}\)
  • \(\require{color}\textcolor{#D2691E}{y_{li}(t_{ij})}\) value of the longitudinal outcome at \(t_{ij}\), \(j=1,\dots,n_{li}\)
• \(\require{color}\textcolor{#2E8B57}{w_i}\) vector of baseline covarites

Multivariate longitudinal data

\[\huge{\cdots}\]

\[\require{color}\color{#2E8B57}\begin{split} (T_1 &,\delta_1) \\ &w_1 \end{split}\]

\[\require{color}\color{#2E8B57}\scriptsize{\begin{split} (1.09 , & 1) \\ (\text{female}, \text{ D} & \text{-peni} , 57) \end{split}}\]

\[\require{color}\color{#2E8B57}\begin{split} (T_2 &,\delta_2) \\ &w_2 \end{split}\]

\[\require{color}\color{#2E8B57}\scriptsize{\begin{split} (14.15 , & 0) \\ (\text{female}, \text{ D} & \text{-peni} , 56) \end{split}}\]

\[\require{color}\color{#2E8B57}\begin{split} (T_n &,\delta_n) \\ &w_n \end{split}\]

\[\require{color}\color{#2E8B57}\scriptsize{\begin{split} (3.98 , & 0) \\ (\text{female}, \text{ plac} & \text{ebo} , 33) \end{split}}\]

Multivariate longitudinal data

\[\huge{\cdots}\]

\[\require{color}\scriptsize{\begin{split} \color{#2E8B57}(T_1 &,\color{#2E8B57}\delta_1) \\ & \color{#2E8B57}w_1 \\ \color{#D2691E}(y_{11}(t_{ij}), & \color{#D2691E}\dots, y_{L1}(t_{ij})) \end{split}}\]

\[\scriptsize{\begin{split} \color{#2E8B57}(T_2 &,\color{#2E8B57}\delta_2) \\ & \color{#2E8B57}w_2 \\ \color{#D2691E}(y_{12}(t_{ij}), & \color{#D2691E}\dots, y_{L2}(t_{ij})) \end{split}}\]

\[\scriptsize{\begin{split} \color{#2E8B57}(T_n &,\color{#2E8B57}\delta_n) \\ & \color{#2E8B57}w_n \\ \color{#D2691E}(y_{1n}(t_{ij}), & \color{#D2691E}\dots, y_{Ln}(t_{ij})) \end{split}}\]

Multivariate longitudinal data

\[i\]

\[\require{color}\color{#2E8B57}\scriptsize{\begin{split} &(T_i ,\delta_i) = (6.43, 0) \\ w_i &= (\text{female}, \text{ D} \text{-peni} , 56) \end{split}}\]

\[\require{color}\color{#D2691E}\scriptsize{\begin{split} &(y_{1i}(t_{ij}), y_{2i}(t_{ij}) , y_{3i}(t_{ij})) = \\ (\log(\texttt{serBilir})_i&(t_{ij}), \texttt{albumin}_i(t_{ij}), \texttt{alkaline}_i(t_{ij})) \end{split}}\]

Methods

• Joint model:

\[ \scriptsize\require{color} \begin{cases} g_l\left(E(y_{li}(t)|\boldsymbol{b}_{li})\right)=\colorbox{#D2691E}{$\color{white}m_{li}(t)$}= \colorbox{#D2691E}{$\color{white}\boldsymbol x_{li}^\top(t)\boldsymbol\beta_l + \boldsymbol z_{li}^\top(t)\boldsymbol b_{li}$}, \quad l=1,\dots,L \\ \\ h_i(t)= h_0(t)\exp \left[ \, \colorbox{#2E8B57}{$\color{white}\boldsymbol w_i^\top \boldsymbol\gamma$} + \sum_{l=0}^L\alpha_l\, \colorbox{#D2691E}{$\color{white}m_{li}(t)$} \, \right] \\ \end{cases} \]

• We assume

\[ \scriptsize\require{color} \boldsymbol{b}_i = (\boldsymbol{b}_{1i}^\top, \boldsymbol{b}_{2i}^\top, \dots, \boldsymbol{b}_{Li}^\top)^\top \sim N\left( \boldsymbol{0} , \boldsymbol{D} \right) \]

• Conditional independence is assumed

\[ \large\require{color} (\, \textcolor{#D2691E}{\boldsymbol{y}} , \textcolor{#2E8B57}{T^*} , \boldsymbol{b}\, ) \]

\[ \large\require{color} (\, \textcolor{#D2691E}{\boldsymbol{y}} \mid \boldsymbol{b}\, ) (\,\textcolor{#2E8B57}{T^*} \mid \boldsymbol{b}\, ) \]

\[ \large\require{color} (\, \textcolor{#D2691E}{\boldsymbol{y_1}} \mid \boldsymbol{b_1}\, ) \cdots (\, \textcolor{#D2691E}{\boldsymbol{y_L}} \mid \boldsymbol{b_L}\, ) (\,\textcolor{#2E8B57}{T^*} \mid \boldsymbol{b}\, ) \]

• If \(L\) is large, fitting the model becomes computationally prohibitive

Dynamic predictions

• New subject \(j\) with \(\textcolor{#2E8B57}{w_j}\) and

\[ \require{color}\color{#D2691E}\boldsymbol{\mathcal{Y}_{j}^o(t)}=\left\{y_{lj}(t_{ik}) ; 0 \leq t_{ik} \leq t, k=1,\dots, n_{lj}, l=1,\dots,L\right\} \]

• Cumulative risk probabilities

\[ \normalsize\require{color} \pi_{j}(u \mid t)=\operatorname{P}\left(T_{j}^* \leq u \mid T_{j}^*>t, \boldsymbol{\color{#D2691E}{\mathcal{Y}}_{j}^o(t)}, \textcolor{#2E8B57}{w_j}; \boldsymbol{\theta}\right) \]

• Estimation based on posterior predictive distributions

• Monte Carlo scheme used in practice

Super learning

• Super learning (SL) is an ensemble method that combines prediction algorithms to obtain an optimal prediction
• Optimality is defined with respect a strictly proper scoring rule (a metric uniquely minimized by the true model)
• SL is built under cross-validation (CV) framework

Idea

• Decompose the multivariate joint model:

\[\normalsize\require{color} h_i(t)= h_0(t)\exp \left[ \, \boldsymbol w_i^\top \boldsymbol\gamma + \sum_{l=0}^L\alpha_l\, m_{li}(t) \, \right]\]

\[\normalsize\require{color} M_1:\, h_i(t)= h_0(t)\exp \left[ \, \boldsymbol w_i^\top \boldsymbol\gamma + \alpha_1\, m_{1i}(t) \, \right]\]

\[\normalsize\require{color} M_2:\,h_i(t)= h_0(t)\exp \left[ \, \boldsymbol w_i^\top \boldsymbol\gamma + \alpha_2\, m_{2i}(t) \, \right]\]

\[\huge\vdots\]

\[\normalsize\require{color} M_L:\, h_i(t)= h_0(t)\exp \left[ \, \boldsymbol w_i^\top \boldsymbol\gamma + \alpha_L\, m_{Li}(t) \, \right]\]

Super learning

Simulation study

• \(\require{color}\textcolor{#00BFFF}{L=4}\) longitudinal outcomes were considered
• Data-generating model: a multivariate joint model
• Super learning is applied to \(\require{color}\textcolor{#00BFFF}{\{M_1, M_2, M_3, M_4\}}\) with \(\require{color}\textcolor{#00BFFF}{V=3}\)
• Goal: to compare SL-based predictions with true model predictions

• Scenario I: Administrative censoring
• Scenario II: Random censoring
• Scenario III: Informative censoring
• Integrated Brier score (IBS) and expected cross-entropy (EPCE) as scoring rules

Results

• SL-based predictions closely approximate the true model
• SL accounts well for overfitting: similar results between training and testing data

Case study: PBC data

\[ \scriptsize\begin{cases} \log(\texttt{serBilir}(t_{ij})) & = \colorbox{#D2691E}{$\color{white}m_{1i}(t_{ij})$} + \varepsilon_{1i}(t_{ij}) \\ &= (\beta_0^1 + b_{0i}^1) + (\beta_{1}^1 + b_{1i}^1)t_{ij} + \beta_2^1\texttt{drug}_i + \varepsilon_{1i}(t_{ij}),\\ \texttt{albumin}(t_{ij}) & = \colorbox{#D2691E}{$\color{white}m_{2i}(t_{ij})$} + \varepsilon_{2i}(t_{ij}) \\ & = (\beta_0^2 + b_{0i}^2) + (\beta_1^2 + b_{1i}^2)t_{ij} + \beta_2^2\texttt{sex}_i + \varepsilon_{2i}(t_{ij}),\\ \texttt{alkaline}(t_{ij}) & = \colorbox{#D2691E}{$\color{white}m_{3i}(t_{ij})$} + \varepsilon_{3i}(t_{ij}) \\ & =(\beta_0^3 + b_{0i}^3) + (\beta_{1}^3 + b_{1i}^3)t_{ij} + \varepsilon_{3i}(t_{ij}),\\ \log\left( \frac{p(\texttt{ascites}(t_{ij})=1)}{1-p(\texttt{ascites}(t_{ij})=1)} \right) & = \colorbox{#D2691E}{$\color{white}m_{4i}(t_{ij})$} + \varepsilon_{4i}(t_{ij})\\ & = (\beta_0^4 + b_{0i}^4) + \beta_1^4t_{ij} + \varepsilon_{4i}(t_{ij}). \end{cases}\]

• We fitted a multivariate JM, and we applied SL to the library of univariate JMs \(\{M_1, M_2, M_3, M_4\}\)

\[\scriptsize\begin{split} h_{i}\left(t \mid \colorbox{#D2691E}{$\color{white}\boldsymbol{\mathcal{Y}}_{i}(t)$}, \colorbox{#2E8B57}{$\color{white}\boldsymbol{w}_{i}$}\right) =h_{0}(t) \exp \big( & \colorbox{#2E8B57}{$\color{white}\gamma\texttt{drug}_i$}+ \alpha_1\colorbox{#D2691E}{$\color{white}m_{1i}(t_{ij})$} + \alpha_2\colorbox{#D2691E}{$\color{white}m_{2i}(t_{ij})$} \\ & \alpha_3\colorbox{#D2691E}{$\color{white}m_{3i}(t_{ij})$} + \alpha_4\colorbox{#D2691E}{$\color{white}m_{4i}(t_{ij})$}\big) \end{split}\]

• We fitted a multivariate JM, and we applied SL to the library of univariate JMs \(\{M_1, M_2, M_3, M_4\}\)

\[\scriptsize\begin{cases} M1: & h_{i}\left(t \mid \colorbox{#D2691E}{$\color{white}\boldsymbol{\mathcal{Y}}_{i}(t)$}, \colorbox{#2E8B57}{$\color{white}\boldsymbol{w}_{i}$}\right) =h_{0}(t) \exp \big( \colorbox{#2E8B57}{$\color{white}\gamma\texttt{drug}_i$}+ \alpha_1\colorbox{#D2691E}{$\color{white}m_{1i}(t_{ij})$} \big) \\ M2: & h_{i}\left(t \mid \colorbox{#D2691E}{$\color{white}\boldsymbol{\mathcal{Y}}_{i}(t)$}, \colorbox{#2E8B57}{$\color{white}\boldsymbol{w}_{i}$}\right) =h_{0}(t) \exp \big( \colorbox{#2E8B57}{$\color{white}\gamma\texttt{drug}_i$}+ \alpha_2\colorbox{#D2691E}{$\color{white}m_{2i}(t_{ij})$} \big) \\ M3: & h_{i}\left(t \mid \colorbox{#D2691E}{$\color{white}\boldsymbol{\mathcal{Y}}_{i}(t)$}, \colorbox{#2E8B57}{$\color{white}\boldsymbol{w}_{i}$}\right) =h_{0}(t) \exp \big( \colorbox{#2E8B57}{$\color{white}\gamma\texttt{drug}_i$}+ \alpha_3\colorbox{#D2691E}{$\color{white}m_{3i}(t_{ij})$} \big) \\ M4: & h_{i}\left(t \mid \colorbox{#D2691E}{$\color{white}\boldsymbol{\mathcal{Y}}_{i}(t)$}, \colorbox{#2E8B57}{$\color{white}\boldsymbol{w}_{i}$}\right) =h_{0}(t) \exp \big( \colorbox{#2E8B57}{$\color{white}\gamma\texttt{drug}_i$}+ \alpha_4\colorbox{#D2691E}{$\color{white}m_{4i}(t_{ij})$} \big) \end{cases}\]

• Results:

  

• Results:

  

• Results:

  

Software

• Package available in CRAN
• Vignette on super learning available in JMBbayes2 Website

Discussion

• SL predictions successfully approximate multivariate JM
• SL has responded well to overfitting
• SL present advantages with respect established methods (such as Bayesian model averaging)
• SL allows us to derive dynamic predictions when \(L\) is large

• As a cross-validation method, SL remains computationally expensive
• SL is prediction-oriented and not designed for inferences
• A sensitivity analysis should be conducted

• Complete code for the simulations and application in the PBC data available in my GitHub
• Open access to my master’s thesis manuscript

Acknowledgements

• Grant PID2023-148033OB-C21: Statistics for Health Sciences: Advances in Survival Analysis and Clinical Trials. Financiado por MICIU/AEI /10.13039/501100011033 y por FEDER, UE.
• Grant 2021 SGR 01421: GRBIO: Grup de Recerca en Bioestadística i Bioinformàtica. Agència de Gestió d’Ajuts Universitaris i de Recerca.
• Projecte Iniciació a la Recerca (INIREC), convocatòria 5601.