text:
Billesley and Hall Green
Wedded bliss 61 years ago today: But where are they now?
Appeal to find couple pictured in mystery wedding album found dumped in a skip
Catherine Lillington
Rose and Patrick Roach on their wedding day
It's the lost wedding album that shows a beaming bride and dapper groom celebrating their marriage, 61 years ago today.
Rose Jane Lillian Scott and Patrick Walter Roach tied the knot at Hall Green Parish Church in Birmingham on a sunny Saturday afternoon on June 28, 1952.
But their treasured collection of memories was later found dumped in a skip – and now an appeal has gone out to help reunite them with their nostalgic snaps.
As befitting her name, the bride carried a bouquet of roses and wore an elegant lace gown on her big day.
Rose and Patrick beamed for the camera surrounded by family and friends outside the church and are seen laughing together as they cut into a stunning three-tier wedding cake.
On another shot, Rose clutches lucky horseshoes and the pair are covered in confetti as they are shown getting into their car.
Saleswoman Tina Bradley said her late dad Tom found the album in a skip at his former workplace Saville Gordon, a steel metal company in Tyseley.
Tina, 56, from Olton, said: "When I was clearing out my mum and dad's house I found it in the shed. I'm not sure how it ended up there.
The wedding album of Rose Scott and Patrick Roach
"I think my dad found it on a skip and thought it was a shame because it was somebody's wedding album. He must have brought it home to reunite it with its owners like I'm trying to do now."
Tina said her dad had fought in Japan in the Second World War. He died aged 83.
She said: "He was a war veteran and would always be on the right side of everything."
Inside the Cupid-embossed wedding album are the names of the best man, Arnold Beddow and bridesmaids, Audrey Wright, Audrey Martin and Patricia Whitehouse.
Also married in 1952 were former American President Ronald Reagan and Nancy Reagan. It was the year George VI died and was succeeded by his daughter, Elizabeth.
The year also saw the release of movies like Singin' in the Rain and The Quiet Man.
* Do you know the couple or were you one of the wedding guests? Call our newsdesk on 0121 234 5430.
Tyseley


With the Reiki level I empowerment you have a powerful tool to help yourself and others. It provides you with the necessary energy and strength to face your own imbalances, to remove resistances stopping you from manifesting the beautiful being you already are. The focus is on your healing journey with a very strong emphasis on self Reiki treatment.


AnonD-417533, 28 Jul 2015not so much, and plus, on this price in europe we need to pay about 22-25% of TAXDude... that pricing is inclusive of tax! So thats exactly how much u need to pay!
Racist is a very strong term, and just as others have said in the comments, they will ship to countries with larger market, not a larger population. Meaning that it is possible they only sell a few thousand in Pakistan while they sell up to tens of thousands in Denmark.
Anonymous, 28 Jul 2015Zenfone 2 is better than this with 4gb ram, 64 gb rom and sd card slot at 23000 inr.WHAT?!!
Asus is an option if you need cheap expandable storage. But the OIS camera? 4K video?
Wait for a bit - by the time most people get invites the price should be closer to INR 21K.
Eske Rahn, 28 Jul 2015"12 major markets" Major? ...One of these is Denmark with a population of only 6M...so?
if they sell f.e. 50 000 pices in Denmark, and only 10 000 in Australia, which is bigger market for them!?
and they do know, where did they sell first phone, so they know what can they axpect.
and this is more like wher tey will have resellers, and now we all know how much will it cost if we live in some coutries of those on list, or near them.
Price is much higher.should b arnd 17k and 19k.
same old design.No metal,glass or anything.
i wud prefer 5inch display.
Anonymous, 28 Jul 2015Zenfone 2 is better than this with 4gb ram, 64 gb rom and sd card slot at 23000 inr.Consider the design factor and the camera factor, this is much much worth than asus zenfone 4. for a normal user, I think 64Gb is more than enough, I see people using 16Gb with no card slot.
And they won't ship it to Pakistan with a population of 180 million, still Denmark with a population of 6 million will get. Eff you OnePlus for being so racist.
Zenfone 2 is better than this with 4gb ram, 64 gb rom and sd card slot at 23000 inr.
Pricing done right for India. But availability only through the cumbersome invite system? Not so intuitive and won't sell that fast.
Pricing ok but a flagship phone with no card slot or waterproofing?
And still only invite only?
Where is china rates? Chines compony not launching phone in china or what?


text:
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redpajama_set_name: RedPajamaGithub


What Does the AUDIT Section of the CPA Exam Involve?
Each tests the examinee on a different aspect of the accounting profession, from management accounting to auditing and business valuations. In this article we'll take a closer look at the Auditing and Attestation (AUD) section. The AUD section includes knowledge of ethics and independence, auditing procedures and standards, and the skills needed to apply that knowledge. Like other sections of the CPA Exam, the AUD section consists of three "testlets" of multiple choice questions. Each testlet consists of 30 multiple choice questions. The testlets get harder or easier based on how well you answered the previous testlet. There is also one testlet containing 7 task-based simulation questions. The AUD section, however, is often considered the hardest section of the CPA Exam. This is mainly because you're not picking the correct answer in the multiple choice section, but rather the best answer. All of the answers you're presented with may be correct. But only one is the best answer. For many, this makes the choice between answers a difficult one. You have four hours to complete the AUD section of the exam, so take your time to find the best answer, but also be mindful of the clock. The AUD section isn't testing you on memorized knowledge, like other sections, but requires you to apply that knowledge to solve problems. Therefore you need to not only study the concepts but learn how to apply them. Take as many sample exams as you can and try applying the concepts to different scenarios. Finally, when you're tackling the task-based simulations, keep the research tab for the end. Most research tab sections are only worth three or four points and can be quite complicated to answer. Instead of spending a lot of time on this tab, paste some relevant answers and move on to the next section. If you have time, come back and try to get as many points as you can. As with any section of the CPA Exam, preparing through proper study, getting a good night's rest and eating breakfast can help you be more comfortable and alert during the exam. The CPA Exam is a hurdle to your future profession and should be prepared for wholly.
Readers of this article also read: Best Test Taking Strategies for the CPA Exam & What Order Should I Take the Exams.


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Gated & Guarded community with perimeter fencing. Post your messages here about the security of Alam Sanctuary.
All matters involving MPSp can be posted here.


text:
\section{Introduction}
It is common to observe longitudinal markers censored by a terminal event in health research with interest in the analysis of the longitudinal marker trajectory and the risk of the terminal event as well as their relationship. It includes the analysis of CD4 lymphocytes counts and AIDS survival \citep{wulfsohn1997joint}, prostate-specific antigen dynamics and the risk of cancer recurrence \citep{proust2009development}, cancer tumor dynamics and the risk of death \citep{Rustand20}, dynamics of aortic gradient and aortic regurgitations and their relationship with the competing risks of death or reoperation in the field of cardiac surgery \citep{Andrinopoulou14} or cognitive markers's relationship with the the time to onset of Alzheimer's disease \citep{Kang22}, to name but a few.\\

Longitudinal markers are usually endogenous (i.e., their values are affected by a change in the risk of occurrence of the event) and measured with error, which prevents their inclusion as standard time-varying covariates in a survival model \citep{prentice1982covariate}. In this context, joint models were introduced to simultaneously analyze longitudinal and event time processes assuming an association structure between these two outcomes, usually through correlated or shared random effects \citep{faucett1996simultaneously, wulfsohn1997joint, henderson2000joint}. Joint models address measurement error, missing data and informative right-censoring in the longitudinal process. They are useful to understand and quantify the association between the longitudinal marker and the risk of the terminal event, and to predict the risk of terminal event based on the history of the longitudinal marker. From an inferential point of view, they also reduce bias in parameter estimation and increase efficiency by utilizing all the available information from both the longitudinal and survival data simultaneously (e.g., to compare two treatment lines in a clinical trial), see \citet{rizopoulos2012joint}.\\

The joint modeling framework has been extended to account for multivariate longitudinal outcomes \citep{lin2002maximum, hickey2016joint} and competing risks of terminal events \citep{elashoff2008joint, hickey2018comparison} as usually encountered in health research. This implies however a substantial increase in the number of random effects to capture the correlation between all the processes in play and leads to a heavy computational burden, particularly due to the multidimensional integral over the random effects in the likelihood. Despite the frequent interest in multiple longitudinal endogenous markers measured over time in health-related studies, their simultaneous analysis in a comprehensive joint model has been limited so far by the available inference techniques and associated statistical software as specifically stated in \citet{hickey2016joint} and \citet{li2021flexible}. \citet{Mehdizadeh21} justifies the use of a two-stage inference approach for the joint analysis of a longitudinal marker with competing risks of events because ``the implementation of joint modeling involves complex calculations''. Also recently, \citet{MURRAY2022107438} discussed about the necessity to develop the multivariate joint modeling framework towards non-Gaussian longitudinal outcomes and competing risks of events.\\

The rapid evolution of computing resources and the development of statistical software made the estimation of a variety of joint models possible, initially using likelihood approaches \citep{rizopoulos2010jm, hickey2018joinerml}, with increasing reliance on Bayesian estimation strategies (e.g., Monte Carlo methods) \citep{JSSv072i07, rstan20}. Sampling-based methods are convenient for scaling up to more and more complex data. They have been proposed to fit multivariate joint models for longitudinal and survival data but always limited to relatively small samples and model specifications (i.e., small number of outcomes and simple regression models for each outcome), see for example \cite{ibrahim2004bayesian}. Moreover, these also have slow convergence properties. The Integrated Nested Laplace Approximation (INLA) algorithm has been introduced as an efficient and accurate alternative to Markov Chain Monte Carlo (MCMC) methods for Bayesian inference of joint models \citep{Rue09, JSSv100i02}. This algorithm requires to formulate the joint models as latent Gaussian models and takes advantage of the model structure to provide fast approximations of the exact inference \citep{van2019joint}. The potential of INLA to provide fast and accurate estimation of joint models has been previously demonstrated in the specific context of a two-part joint model in comparison with maximum likelihood estimation with a Newton-like algorithm \citep{rustand2021bayesian}. This inference technique is thus a very strong candidate for the deployment of joint models in increasingly complex settings encountered in health research where a large number of longitudinal markers and competing clinical events may be under study simultaneously.\\

The main contribution of this paper is the introduction of joint models for multivariate longitudinal markers with different distributions and competing risks of events, along with a fast and reliable inference technique applicable in practice. The good properties of this inference technique are demonstrated in simulation studies and the flexibility of the approach is illustrated in a randomized placebo controlled trial for the treatment of primary biliary cholangitis where the associations of 5 longitudinal markers of different nature with competing risks of death and liver transplantation are simultaneously assessed.\\

The paper is organized as follows. In Section \ref{metsec} we describe the multivariate joint model structure and the estimation strategy. In Section \ref{simsec} we present simulation studies to demonstrate the properties of this estimation strategy compared to the available alternatives. In Section \ref{appsec} we describe the application in primary biliary cholangitis before concluding with a discussion in Section \ref{dissec}.

\section{Methods}
\label{metsec}
\subsection{The joint model for multivariate longitudinal data and a multiple-cause terminal event}

Let $Y_{ijk}$ denote the $k^{th}$ $(k=1, ..., K)$ observed longitudinal measurement for subject $i$ $(i=1, ..., n)$ measured at time points $t_{ijk}$ with $j$ the occasion $(j=1, ..., n_{ik})$. The joint model for multivariate longitudinal data and a multiple-cause terminal event considers mixed effects models to fit the biomarkers's evolutions over time with link functions $g(\cdot)$ to handle various types of distributions, and a cause-specific proportional hazard model for the terminal event.\\

The model for the multivariate longitudinal part is defined as follows:
\[
\begin{array}{lc}
  \mathbb{E}[Y_{ijk}]=g^{-1}\left(\eta_{ik}(t_{ijk})\right)=g^{-1}\left(\bm X_{ik}(t_{ijk})^\top \bm \beta_k + \bm Z_{ik}(t_{ijk})^\top \bm b_{ik}\right) & \forall k=1,...,K
   \end{array}
\]
\noindent where $\eta_{ik}(t)$ is the linear predictor corresponding to individual $i$ and marker $k$ at time $t$, $\bm X_{ik}(t)$ and $\bm Z_{ik}(t)$ are vectors of covariates (possibly different for each marker $k$) associated with the fixed effects parameters $\bm \beta_k$ and the multivariate normal vector of random effects $\bm b_{i}$ which accounts for the intra-and-inter-marker correlation between the repeated measurements of an individual:
\begin{equation*}
\bm b_i = \begin{bmatrix}
\bm b_{i1}\\
\vdots\\
\bm b_{iK}
\end{bmatrix}
\sim MVN \left( \begin{bmatrix}
\bm{0} \\
\vdots \\
\bm{0}
\end{bmatrix} ,
\begin{bmatrix}
\bm{\Sigma}_{\bm b_{i1}} & \cdots &\bm{\Sigma}_{\bm b_{i1} \bm b_{iK}} \\
\vdots & \ddots & \vdots\\
\bm{\Sigma}_{\bm b_{i1} \bm b_{iK}} & \cdots & \bm{\Sigma}_{\bm b_{iK}}
\end{bmatrix}\right).
\label{eq2}
\end{equation*}
There are several distributions available to fit different types of longitudinal markers using mixed effects, including linear and lognormal models, generalized linear models (i.e., exponential family), proportional odds models for an ordinal outcome, zero-inflated models (Poisson, Binomial, negative Binomial and BetaBinomial) and two-part models for a semicontinuous outcome.\\

Additionally, let $T_i$ denote the positive continuous response variable that represents the elapsed time between the beginning of the follow-up and an event of interest for subject $i$. We can define the couple ($T_i^*, \delta_i$) where the actual observed time is $T_i^*=\textrm{min}(T_i, C_i)$ with $C_i$ the censoring time of individual $i$, and the indicator variable $\delta_i$ specifies the nature of the observed time, either cause $m$ ($m=1,...,M$) or censoring with $\delta_i = m \times I_{T_i<C_i}$. The model for the cause-specific survival part is defined as follows:
\[
\begin{array}{lc}
  \lambda_{im}(t)=\lambda_{0m}(t)\ \textrm{exp}\left(\bm W_{im}^\top(t) \bm \gamma_m +  h_m(\eta_{i1}(t), ..., \eta_{iK}(t)) \bm \varphi_m \right) & \forall m=1,...,M, \forall t \in \mathbb{R^+}.
   \end{array}
\]
The vector of possibly time dependent covariates $\bm W_{im}(t)$ is associated with the risk of each cause $m$ of terminal event through regression parameters $\bm \gamma_m$ (see \citet{niekerk2021competing} for more details on competing risks joint models with \textbf{R-INLA}). The multivariate function $h_m(\eta_{i1}(t), ..., \eta_{iK}(t))$ defines the nature of the association with the risk of each cause $m$ of terminal event through the vector of parameters $\bm \varphi_m$. It can be specified as any linear combination of the linear predictors's components from the longitudinal submodels (therefore including shared random effects, current value and current slope parameterizations).\\

The baseline risk of event, $\lambda_{0m}(t)$, for each cause $m$, can be specified with parametric functions (e.g., exponential or Weibull) but it is usually preferred to avoid parametric assumptions on the shape of the baseline risk when it is unknown. We can define random walks of order one or two corresponding to smooth spline functions based on first and second order differences, respectively. The number of bins are not influential (as opposed to knots of other splines) since an increase in the number of bins only results in an estimate closer to the stochastic model. The second order random walk model provides a smoother spline compared to first order since the smoothing is then done on the second order. See \citet{martino2011approximate} and \citet{vanSankhya} for more details on the use of these random walk models as Bayesian smoothing splines. We use these second order random walk smooth splines for the baseline hazard in the simulations and application sections (Sections \ref{simsec} and \ref{appsec}).

\subsection{Bayesian estimation with R-INLA}
\label{BayInf}
Let $\bm \lambda$ denote the parameters associated to the baseline risk functions $\lambda_{0m}(t)$, then the full vector of unknown parameters in the model is $\bm \theta_i=(\bm \beta_{1}, \cdots \bm \beta_{K}, \bm b_{i1} \cdots, \bm b_{iK}, \bm \lambda, \bm{\gamma}, \bm{\varphi})$. Let $\boldsymbol{D_i} \equiv \{T_i^*, \delta_i, Y_{ijk}: i=1, \cdots, n; j=1,\cdots, n_i, k=1, \cdots, K \}$ denote the observation variables. The Bayesian inference provides an estimation of the posterior distribution $p(\boldsymbol{\theta_i}|\boldsymbol{D_i})$ as defined by Bayes's theorem:
$$p(\bm \theta_i | \boldsymbol{D_i}) = \frac{p(\boldsymbol{D_i}| \bm \theta_i) p(\bm \theta_i)}{p(\boldsymbol{D_i})} \propto p(\boldsymbol{D_i} | \bm \theta_i) p(\bm \theta_i),$$
where $p(\boldsymbol{D_i} | \bm \theta_i)$ is the likelihood and $p(\bm \theta_i)$ is the joint prior. The marginal likelihood $p(\boldsymbol{D_i})=\int_{\bm \theta_i} p(\boldsymbol{D_i} | \bm \theta_i) p(\bm \theta_i) \mathrm{d}\bm \theta_i$ acts as a normalizing constant. The posterior distribution is often not analytically tractable and sampling-based methods like MCMC are used. Approximate methods like INLA provide exact approximations to the posterior at lower cost than sampling-based methods \citep{rue2017bayesian}.

\subsubsection{Likelihood function}

Assuming $\bm \theta_i =(\bm\phi, \bm b_i)$ where $\bm\phi=(\bm \beta_{1}, \cdots \bm \beta_{K}, \bm \lambda, \bm{\gamma}, \bm{\varphi})$ the vector of the fixed effects of the model and $\bm b_i = (\bm b_{i1} \cdots, \bm b_{iK})$ the random effects, we can derive the full likelihood of this model:
\begin{align*}
        p(\bm D_i|\bm\phi) = \int_{\bm b_{i}}  &\left[p(T_i^*, \delta_i | \bm \theta_i) \prod_{k=1}^{K} \prod_{j=1}^{n_i} p(Y_{ijk} | \bm \theta_i) p(\bm{b}_{i}) \right] \mathrm{d}\bm b_{i},
\end{align*}
where $p(T_i^*, \delta_i | \bm \theta_i) = \prod_{m=1}^M \lambda_{im}(T_i^*|\bm \theta_i)^{\mathbb{1}_{\delta_i=m}} \prod_{m=1}^M \exp\left( - \int_{0}^{T_i^*} \lambda_{im}(t|\bm \theta_i) \mathrm{d}t \right)$, $p(Y_{ijk} | \bm\theta_i)$ is the density of the $k^{th}$ longitudinal submodel and $p(\bm{b}_{i})$ is the density of the random effects.

\subsubsection{Integrated Nested Laplace Approximation}
In order to describe INLA's methodology, we need to formulate the joint model with a hierarchical structure with three layers where the first layer is the likelihood function of the observed data assumed conditionally independent. The second layer is formed by the latent Gaussian field $\bm u$ defined by a multivariate Gaussian distribution conditioned on the hyperparameters and the third layer corresponds to the prior distribution assigned to the hyperparameters $\bm \omega$, such that the vector of unknown parameters $\bm \theta$ from Section \ref{BayInf} is decomposed into two separate vectors, $\bm u$ and $\bm \omega$. For example the vector of fixed effects for longitudinal marker $k$, $\bm \beta_{k}$ have a prior defined as $\bm \beta_k \sim \mathcal{N}(0, \tau_{\beta_k} \bm I)$. The Gaussian parameters $\bm \beta_k$ belong to vector $\bm u$ and the corresponding hyperparameters $\tau_{\beta_k}$ belong to vector $\bm \omega$.\\

This model formulation is referred to as a Latent Gaussian Model (LGM). This general model formulation has computational advantages, particularly due to the sparsity of the precision matrix of the latent Gaussian field in the second layer of the LGM. We assume an inverse-Wishart prior distribution for the covariance matrix of the random effects and Gaussian priors for the fixed effects. Finally, a penalizing complexity prior \citep{simpson2017penalising} is assumed for the precision parameter of the stochastic spline model for the log baseline hazard function.\\

The first step of INLA's methodology is an approximation of the marginal posterior distribution of the hyperparameters using the Laplace approximation:
$$p(\bm\omega|\pmb{D})\approx \frac{p(\bm\omega)p(\pmb{u}|\bm\omega)p(\pmb{D}|\pmb{u},\bm\omega)}{\tilde{p_G}(\pmb{u}|\bm\omega,\pmb{D})}|_{\bm u=\bm u^*(\bm\omega)},$$
where $\bm u^*(\bm\omega)$ denotes the mode of the full conditional for a given value of the vector of hyperparameters $\bm \omega$ and ${\tilde{p_G}(\pmb{u}|\bm\omega,\pmb{D})}$ is the Gaussian approximation obtained by matching the mode and curvature at the mode of the full joint density ${p(\pmb{u}|\bm\omega,\pmb{D})}$. The second step is an approximation of the conditional posterior distributions of the latent field:
		$$
		p(u_i|\pmb{\omega},\pmb{D}) \propto \frac{p(\pmb{u},\bm\omega|\pmb{D})}{p(\pmb{u}_{-i}|u_i,\bm\omega,\pmb{D})},
		$$
In the last step, a numerical integration is used to approximate the marginal posterior distributions of the latent field from the first two steps:
		$$
		p(u_i|\pmb{D})\approx \sum_{h=1}^{H}\tilde{p}(u_i|\bm\omega_h^*,\pmb{D})\tilde{p}(\bm\omega^*_h|\pmb{D})\Delta_h,
		$$
where the integration points $\{\bm\omega^*_1, ..., \bm\omega^*_H\}$ are selected from a rotation using polar coordinates and based on the density at these points and $\Delta_h$ are the corresponding weights. The approximation of the posterior marginal for each element of the latent field and each hyperparameter, using numerical integration, forms the ``integrated'' part of INLA algorithm while the first two steps above correspond to the ``nested Laplace'' approximation steps of INLA. In summary, INLA is a combination of nested numerical approximations described in \citet{NiekerkAvenue}. Originally, INLA uses a series of two Laplace approximations, nested. The recent development suggests to use a low-rank variational Bayes correction to match the full Laplace approximation with no significant additional computational cost to the first Gaussian approximation. Additionally, the ``smart gradient" approach is used to more efficiently find the mode, see details in \citet{Esmail22}. Moreover, using numerical methods to deal with sparse precision matrix computations instead of dense covariance is crucial. Finally, \textbf{R-INLA} initially parameterized multivariate random effects with their precision and correlation but since the application of INLA to multivariate joint models involves random effects with high dimension, we developed a more stable way to deal with this in recent INLA versions by using the Cholesky parameterization which improves the computational stability and reduces the computational burden.\\

An alternative strategy referred to as empirical Bayes only uses the mode at step 1 (i.e., not the curvature at the mode that informs about uncertainty), which speeds up computations but cannot be considered as fully Bayesian. Pictured as a trade-off between frequentist and Bayesian estimation strategies, we show in our simulation studies (Section 3) that the empirical Bayes estimation stategy has frequentist properties as good as the full Bayesian strategy.\\

In the following we use \textbf{R-INLA} version INLA\_22.10.07 with the \textbf{PARDISO} library that provides a high performance computing environment with parallel computing support using OpenMP (\citet{JSSv100i02}). Additionally, we use the R package \textbf{INLAjoint} v22.10.17 (\citet{Rustand_INLAjoint}) that provides a user-friendly interface to fit joint models with \textbf{R-INLA}. See github.com/DenisRustand/MultivJoint for examples of codes that fit multivariate joint models with \textbf{INLAjoint}.

\section{Simulations}
\label{simsec}
\subsection{Settings}

We underwent an extensive simulation study with two purposes: (i) validate the inference approach based on INLA algorithm, (ii) compare its performances with those obtained by two alternative existing inference techniques:
\begin{itemize}
\item Maximum likelihood estimation with Monte Carlo expectation maximisation algorithm as implemented in the R package \textbf{joineRML} (version 0.4.5). This package can fit joint models of time-to-event data and multivariate longitudinal data in a frequentist framework using a recently introduced fast approximation that combines the expectation maximisation algorithm to quasi-Monte Carlo \citep{hickey2018joinerml, philipson2020faster}. However, this method is restricted to Gaussian markers and the association structure between the longitudinal and survival part is limited to the linear combination of the random effects (equivalent to current level when longitudinal and survival parts do not include the same covariates). We used quasi-Monte Carlo with Sobol sequence and same options as described in \citet{philipson2020faster} for the simulations.
\item Bayesian inference via Markov Chain Monte Carlo sampling as implemented in the R package \textbf{rstanarm} (version 2.21.1) which relies on the Hamiltonian Monte Carlo algorithm implemented in \textbf{Stan} \citep{rstan20}. It can fit joint models of time-to-event data and multivariate longitudinal data with various types of distributions (i.e., generalized linear mixed effects models) but is limited to a maximum of 3 longitudinal outcomes (we limited our simulation studies to a maximum of 3 longitudinal markers for this reason). Alternatively, the R package \textbf{JMbayes2} (version 0.3-0) was recently introduced, it can fit joint models of time-to-event data and multivariate longitudinal data using MCMC implemented in C++ \citep{JMbayes222} with various options for the distribution of the longitudinal outcomes. It handles competing risks and multi-state models for survival outcomes.
\end{itemize}

\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{SimSchem.pdf}
\caption{Data generation settings for the simulation studies}
\label{Fig1}
\end{figure}

We are interested in the frequentist properties of the different estimation strategies and we therefore report the absolute bias, the standard deviation of the absolute bias and the frequentist coverage probability of the 95\% credible intervals for Bayesian inference and 95\% confidence intervals for the frequentist estimation implemented in \textbf{joineRML}. We use the same prior distributions for the fixed effects parameters of the Bayesian estimations, namely the default priors defined in \textbf{rstanarm} (Gaussian distributions with mean 0 and scale 2.5). In contrast, \textbf{JMbayes2} defines prior distributions using frequentist univariate regression models fitted separately for each outcome. The mean of the Gaussian prior is defined as the maximum likelihood estimate (MLE) and the precision is defined as the inverse of 10 times the variance of the estimate from the univariate model (only the prior for the association parameter matches the other Bayesian estimation strategies since it is not given by univariate models). Regarding the prior distributions for the multivariate Gaussian random effects variance and covariance parameters, \textbf{R-INLA} uses the Inverse Wishart prior distribution, \textbf{JMbayes2} uses gamma priors with mean defined as the MLE from univariate models while \textbf{rstanarm}'s default prior is the Lewandowski-Kurowicka-Joe (LKJ) distribution (\citet{LEWANDOWSKI20091989}), described as faster than alternative priors in the package documentation. Starting values for \textbf{R-INLA} and \textbf{joineRML} are kept as default (i.e., non-informative) while \textbf{JMbayes2} and \textbf{rstanarm} use MLE values from univariate models as initial values for both fixed and random effects. We note that these initial values provide an important advantage for \textbf{JMbayes2} and \textbf{rstanarm} but we decided to keep them because this is part of the procedure when using these packages while \textbf{R-INLA} does not rely on submodels MLE values, although informative priors and initial values can also be considered. The baseline hazard is modeled with a different approach for each package: \textbf{R-INLA} defines Bayesian smoothing spline for the log baseline hazard with 15 bins, \textbf{rstanarm} uses cubic B-splines for the log baseline hazard with 2 internal knots, \textbf{JMbayes2} uses quadratic B-splines with 9 internal knots and the frequentist implementation from \textbf{joineRML} keeps the baseline hazard unspecified.\\

The design of simulation is fully described in Figure \ref{Fig1}. We considered 10 scenarios depending on the number of longitudinal markers (from 1 to 3) and the nature of the markers (Gaussian, Poisson or Binomial). We considered linear trajectories for each marker, and the linear predictor of each marker at the time of risk assessment for the structure of association between the longitudinal and event processes (as the common available association structure accross softwares). The time of event was generated by the methodology of Sylvestre and Abrahamowicz implemented in the R package PermAlgo \citep{sylvestre2008comparison}. The parameter values of the generation models were based on the real data analysis (Section \ref{appsec}). For each simulation scenario, we generated 300 datasets that included 500 individuals each where approximately 40\% of the sample observed the terminal event before the end of the follow-up, and each longitudinal marker had approximately 3000 total repeated measurements (min $\simeq 1$, max $\simeq 17$, median $\simeq 5$ per individual), except for simulation scenarios including at least one binary outcome. When considering binary outcomes, we simplified the longitudinal models by removing the random slope and increased the number of repeated measurements per individuals since binary outcomes are less informative than continuous and count outcomes, leading to approximately 8500 repeated measurements with min $\simeq 1$, max $\simeq 50$ and median $\simeq 13$ per individual. Indeed, results were not satisfying (i.e., important bias and poor coverage for the variance-covariance of the random effects) regardless of the estimation strategy when using random intercept and slope as well as a limited number of repeated measurements per individuals.\\

We considered two estimation strategies for \textbf{R-INLA}, namely the empirical Bayes strategy and full Bayesian strategy (see Section \ref{metsec}). We also tested two different estimation strategies for \textbf{rstanarm}; the first one includes 1 chain and 1000 MCMC iterations (including a warmup period of 500 iterations that is discarded) while the other one has default specifications (4 chains and 2000 iterations, including a warmup period of 1000 iterations that is discarded). This illustrates the impact of the choice of the number of chains and iterations on the parameter estimates and on the computation time. We kept the default specifications with \textbf{JMbayes2} since it provides a good trade-off between speed and accuracy (3 chains and 3500 iterations including a warmup period of 500 iterations that is discarded). Note that the default is increased to 6500 iterations for binary outcomes but we kept 3500 iterations for those scenarios as it was increasing the computation time for a negligible improvement in the results. All the computation times in the results are given with parallel computations over 15 CPUs (Intel Xeon Gold 6248 2.50GHz).\\

A replication script is available at github.com/DenisRustand/MultivJoint; it includes the generation and estimation codes for scenario 10 with 3 longitudinal markers of different nature.

\subsection{Results}

Since \textbf{joineRML} is restricted to Gaussian markers, we first report the results of the scenarios with Gaussian continuous outcomes only, and then report those with counts and binary outcomes.

\begin{table}[ht]
\centering
\caption{Simulations with $K$=3 continuous longitudinal markers}
\footnotesize
{\tabcolsep=1pt
\hskip-0.2cm\begin{tabular}{@{}lccccccccccccccccccccccccccccc@{}}
\hline
Approach: &  \multicolumn{3}{c}{R-INLA 1} & \multicolumn{3}{c}{R-INLA 2} & \multicolumn{3}{c}{joineRML} & \multicolumn{3}{c}{JMbayes2} & \multicolumn{3}{c}{rstanarm 1} & \multicolumn{3}{c}{rstanarm 2}\\
 & \multicolumn{3}{c}{(Empirical Bayes)} & \multicolumn{3}{c}{(Full Bayesian)} & & & & & & & \multicolumn{3}{c}{(1 chain / 1000 iter.)} & \multicolumn{3}{c}{(4 chains / 2000 iter.)}\\
True value & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP  \\
\hline
$\beta_{10}$=0.2 & 0 & (0.055) & 94\% & 0 & (0.055) & 94\% & -0.002 & (0.055) & 96\% & 0 & (0.055) & 94\% & -0.002 & (0.054) & 93\% & 0.001 & (0.055) & 93\% \\
 $\beta_{11}$=-0.1 & 0 & (0.023) & 96\% & 0 & (0.023) & 96\% & -0.007 & (0.023) & 97\% & 0.001 & (0.023) & 96\% & 0 & (0.024) & 96\% & 0 & (0.024) & 96\% \\
 $\beta_{12}$=0.1 & -0.004 & (0.046) & 92\% & -0.004 & (0.046) & 92\% & -0.006 & (0.045) & 92\% & -0.004 & (0.046) & 92\% & -0.002 & (0.046) & 91\% & -0.003 & (0.046) & 92\% \\
 $\beta_{13}$=-0.2 & 0.004 & (0.042) & 94\% & 0.004 & (0.042) & 95\% & 0.007 & (0.042) & 96\% & 0.005 & (0.042) & 95\% & 0.008 & (0.042) & 95\% & 0.004 & (0.043) & 94\% \\
 $\sigma_{\varepsilon 1}$=0.4 & -0.001 & (0.006) & 95\% & -0.001 & (0.006) & 95\% & 0 & (0.006) & 90\% & 0 & (0.006) & 95\% & 0.001 & (0.005) & 95\% & 0.001 & (0.006) & 94\% \\
 $\beta_{20}$=0.2 & -0.002 & (0.062) & 95\% & -0.002 & (0.062) & 95\% & -0.002 & (0.062) & 95\% & -0.002 & (0.062) & 95\% & 0.002 & (0.059) & 96\% & -0.003 & (0.061) & 96\% \\
 $\beta_{21}$=-0.1 & 0.001 & (0.031) & 94\% & 0.001 & (0.031) & 94\% & 0.016 & (0.031) & 91\% & 0.001 & (0.031) & 94\% & 0.002 & (0.03) & 93\% & 0 & (0.031) & 94\% \\
 $\beta_{22}$=0.1 & 0.002 & (0.052) & 95\% & 0.002 & (0.052) & 94\% & 0.004 & (0.052) & 94\% & 0.003 & (0.052) & 94\% & -0.001 & (0.05) & 96\% & 0.004 & (0.051) & 95\% \\
 $\beta_{23}$=-0.2 & -0.001 & (0.051) & 94\% & -0.001 & (0.051) & 94\% & -0.005 & (0.051) & 95\% & -0.001 & (0.051) & 94\% & -0.002 & (0.053) & 93\% & -0.001 & (0.052) & 93\% \\
 $\sigma_{\varepsilon 2}$=0.4 & 0 & (0.006) & 94\% & 0 & (0.006) & 95\% & -0.001 & (0.006) & 91\% & 0 & (0.006) & 95\% & 0 & (0.006) & 95\% & 0 & (0.006) & 94\% \\
 $\beta_{30}$=0.2 & -0.003 & (0.06) & 96\% & -0.003 & (0.059) & 96\% & -0.003 & (0.059) & 97\% & -0.002 & (0.059) & 96\% & -0.002 & (0.061) & 94\% & -0.003 & (0.06) & 96\% \\
 $\beta_{31}$=-0.1 & 0.003 & (0.024) & 97\% & 0.003 & (0.024) & 97\% & -0.007 & (0.023) & 96\% & 0.002 & (0.024) & 98\% & 0.003 & (0.024) & 97\% & 0.003 & (0.023) & 98\% \\
 $\beta_{32}$=0.1 & 0.006 & (0.047) & 95\% & 0.006 & (0.047) & 95\% & 0.004 & (0.047) & 95\% & 0.006 & (0.047) & 94\% & 0.007 & (0.049) & 94\% & 0.007 & (0.048) & 95\% \\
 $\beta_{33}$=-0.2 & 0.001 & (0.052) & 95\% & 0.001 & (0.052) & 94\% & 0.004 & (0.051) & 97\% & 0 & (0.052) & 95\% & -0.002 & (0.052) & 95\% & 0 & (0.051) & 95\% \\
 $\sigma_{\varepsilon 3}$=0.4 & -0.001 & (0.006) & 95\% & -0.001 & (0.006) & 95\% & 0 & (0.006) & 89\% & 0 & (0.006) & 95\% & 0 & (0.006) & 94\% & 0.001 & (0.006) & 95\% \\
 $\sigma_{b10}^2$=0.16 & 0.005 & (0.017) & 96\% & 0.004 & (0.015) & 96\% & -0.003 & (0.015) & 94\% & 0.002 & (0.015) & 99\% & 0.001 & (0.017) & 91\% & 0.001 & (0.016) & 92\% \\
 $\sigma_{b11}^2$=0.16 & 0.005 & (0.016) & 95\% & 0.005 & (0.016) & 95\% & -0.003 & (0.016) & 94\% & -0.001 & (0.016) & 94\% & 0.002 & (0.016) & 94\% & 0.001 & (0.017) & 93\% \\
 $\sigma_{b20}^2$=0.25 & 0.005 & (0.02) & 97\% & 0.005 & (0.019) & 97\% & 0 & (0.019) & 98\% & 0.005 & (0.02) & 96\% & 0.005 & (0.02) & 96\% & 0.005 & (0.02) & 96\% \\
 $\sigma_{b21}^2$=0.25 & 0.002 & (0.023) & 96\% & 0.003 & (0.028) & 97\% & -0.004 & (0.023) & 96\% & 0.004 & (0.023) & 93\% & 0.007 & (0.025) & 92\% & 0.006 & (0.024) & 93\% \\
 $\sigma_{b30}^2$=0.25 & 0.003 & (0.021) & 96\% & 0.003 & (0.021) & 96\% & -0.003 & (0.022) & 94\% & 0.002 & (0.022) & 95\% & 0.005 & (0.023) & 93\% & 0.004 & (0.022) & 94\% \\
 $\sigma_{b31}^2$=0.16 & 0.005 & (0.017) & 95\% & 0.004 & (0.016) & 94\% & -0.004 & (0.016) & 93\% & 0.001 & (0.016) & 94\% & 0.003 & (0.016) & 94\% & 0.003 & (0.016) & 95\% \\
 $\textrm{cov}_{b10,b11}$=0.08 & -0.006 & (0.012) & 92\% & -0.006 & (0.012) & 91\% & -0.001 & (0.011) & 96\% & -0.003 & (0.011) & 95\% & -0.002 & (0.011) & 94\% & -0.001 & (0.011) & 94\% \\
 $\textrm{cov}_{b10,b20}$=0.02 & 0.001 & (0.013) & 97\% & 0.001 & (0.012) & 98\% & 0 & (0.012) & 95\% & 0.001 & (0.012) & 95\% & 0 & (0.012) & 94\% & 0 & (0.012) & 93\% \\
 $\textrm{cov}_{b10,b21}$=0.04 & -0.003 & (0.014) & 96\% & -0.003 & (0.014) & 96\% & -0.002 & (0.014) & 96\% & -0.002 & (0.013) & 95\% & -0.002 & (0.014) & 93\% & -0.002 & (0.014) & 94\% \\
 $\textrm{cov}_{b10,b30}$=0 & -0.001 & (0.013) & 96\% & -0.001 & (0.013) & 96\% & 0 & (0.012) & 96\% & 0 & (0.012) & 95\% & 0 & (0.012) & 95\% & 0 & (0.012) & 96\% \\
 $\textrm{cov}_{b10,b31}$=-0.04 & 0.002 & (0.012) & 95\% & 0.001 & (0.012) & 95\% & 0.001 & (0.012) & 94\% & 0.002 & (0.011) & 93\% & 0.001 & (0.012) & 93\% & 0.001 & (0.012) & 94\% \\
 $\textrm{cov}_{b11,b20}$=0.04 & -0.003 & (0.014) & 94\% & -0.004 & (0.014) & 94\% & -0.002 & (0.014) & 94\% & -0.002 & (0.013) & 94\% & -0.001 & (0.014) & 92\% & -0.001 & (0.014) & 93\% \\
 $\textrm{cov}_{b11,b21}$=0 & 0.001 & (0.013) & 97\% & 0.001 & (0.014) & 96\% & -0.002 & (0.013) & 97\% & -0.001 & (0.013) & 95\% & -0.001 & (0.013) & 96\% & -0.001 & (0.013) & 96\% \\
 $\textrm{cov}_{b11,b30}$=-0.08 & 0.001 & (0.015) & 95\% & 0.001 & (0.016) & 94\% & -0.001 & (0.015) & 95\% & 0.003 & (0.014) & 92\% & 0 & (0.015) & 93\% & 0 & (0.015) & 92\% \\
 $\textrm{cov}_{b11,b31}$=-0.08 & 0.002 & (0.012) & 95\% & 0.002 & (0.012) & 95\% & 0 & (0.012) & 96\% & 0.004 & (0.012) & 92\% & 0.001 & (0.012) & 95\% & 0.002 & (0.012) & 94\% \\
 $\textrm{cov}_{b20,b21}$=0.1 & -0.002 & (0.016) & 97\% & -0.001 & (0.017) & 97\% & 0 & (0.016) & 98\% & -0.002 & (0.015) & 96\% & -0.001 & (0.014) & 98\% & 0 & (0.015) & 97\% \\
 $\textrm{cov}_{b20,b30}$=0.05 & 0.002 & (0.015) & 97\% & 0.001 & (0.015) & 97\% & 0 & (0.015) & 97\% & 0.001 & (0.015) & 95\% & 0.002 & (0.015) & 96\% & 0.001 & (0.015) & 97\% \\
 $\textrm{cov}_{b20,b31}$=0.02 & -0.001 & (0.013) & 97\% & -0.002 & (0.013) & 97\% & 0.001 & (0.013) & 98\% & 0 & (0.012) & 98\% & 0 & (0.013) & 97\% & 0.001 & (0.013) & 98\% \\
 $\textrm{cov}_{b21,b30}$=0.05 & 0 & (0.017) & 95\% & 0 & (0.017) & 95\% & 0.001 & (0.017) & 96\% & -0.001 & (0.017) & 95\% & 0.002 & (0.015) & 96\% & 0.001 & (0.015) & 97\% \\
 $\textrm{cov}_{b21,b31}$=-0.04 & 0.002 & (0.016) & 95\% & 0.001 & (0.014) & 95\% & 0.001 & (0.014) & 96\% & 0.001 & (0.014) & 95\% & -0.002 & (0.014) & 93\% & 0 & (0.014) & 95\% \\
 $\textrm{cov}_{b30,b31}$=0.1 & -0.003 & (0.014) & 96\% & -0.004 & (0.014) & 95\% & 0.003 & (0.014) & 95\% & -0.003 & (0.014) & 95\% & -0.001 & (0.015) & 94\% & -0.001 & (0.014) & 95\% \\
 $\varphi_1$=0.5 & 0.008 & (0.095) & 96\% & 0.006 & (0.095) & 96\% & 0.001 & (0.098) & 97\% & 0.017 & (0.097) & 98\% & 0.019 & (0.092) & 96\% & 0.015 & (0.098) & 95\% \\
 $\varphi_2$=-0.5 & -0.001 & (0.076) & 94\% & -0.001 & (0.075) & 95\% & 0.001 & (0.076) & 98\% & -0.011 & (0.077) & 96\% & -0.009 & (0.075) & 93\% & -0.011 & (0.074) & 94\% \\
 $\varphi_3$=0.5 & -0.003 & (0.087) & 95\% & -0.004 & (0.087) & 95\% & -0.009 & (0.087) & 97\% & 0.006 & (0.089) & 96\% & 0.011 & (0.09) & 95\% & 0.01 & (0.088) & 96\% \\
 Conv. rate & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{0.65} & \multicolumn{3}{c}{0.83} \\
 Comp. time (sec.) & \multicolumn{3}{c}{53.38 (7.2)} & \multicolumn{3}{c}{88.75 (8.58)} & \multicolumn{3}{c}{51.89 (14.61)} & \multicolumn{3}{c}{166.41 (62.4)} & \multicolumn{3}{c}{1015.24 (405.83)} & \multicolumn{3}{c}{1247.33 (688.58)}\\
 \arrayrulecolor{black}\hline
\end{tabular}}
\label{sim3Y}
\end{table}

\subsubsection{Gaussian longitudinal markers only}

The summary of the estimations provided by the two \textbf{R-INLA} strategies, the frequentist estimation with \textbf{joineRML} and the Bayesian estimations with \textbf{JMbayes2} and \textbf{rstanarm} are reported in Table 1 for 3 longitudinal Gaussian markers, and in Tables S1 and S2 of the supporting information for 1 and 2 markers, respectively. All the methods provided correct inference with negligible bias and correct coverage rate of the 95\% credible or confidence intervals. However, methods differed in terms of computation time and convergence rate. While \textbf{R-INLA}, \textbf{joineRML} and \textbf{JMbayes2} systematically converged, convergence issues appeared for \textbf{rstanarm} with the increasing number of markers (up to 65\% and 83\% in the scenario with 3 markers when running 1 and 4 chains, respectively).\\

Computing times are summarized in Figure \ref{Fig2}. With 1 marker, \textbf{R-INLA} methodology presented the lowest computation time (5.52 and 5.65 seconds on average per dataset for the empirical bayes strategy and full Bayesian strategy, respectively) followed by \textbf{joineRML} (21.62s. on average) and \textbf{JMbayes2} (47.09s. on average). When the number of markers increased, \textbf{joineRML} had the best scaling with similar computation times as the empirical Bayes version of \textbf{R-INLA} for 3 markers while the full Bayesian strategy of \textbf{R-INLA} was slightly longer. The computation time with \textbf{JMbayes2} for 2 and 3 markers is increased compared to \textbf{R-INLA} and \textbf{joineRML}. Finally, whatever the number of markers, \textbf{rstanarm} was much longer. With 3 markers, the 1 chain/1000 iterations strategy was approximately 19 times longer than the empirical Bayes INLA approach, and the 4 chain/2000 iterations strategy was approximately 23 times longer (ratios were even larger in smaller dimension).

\begin{table}[ht]
\caption{Simulations with $K$=3 longitudinal markers with a mixture of distributions (continuous, count and binary)}
\footnotesize
\centering
{\tabcolsep=2.25pt
\begin{tabular}{@{}lccccccccccccccccccccccccccccc@{}}
\hline
Approach: &  \multicolumn{3}{c}{R-INLA 1} & \multicolumn{3}{c}{R-INLA 2} & \multicolumn{3}{c}{JMbayes2} & \multicolumn{3}{c}{rstanarm 1} & \multicolumn{3}{c}{rstanarm 2}\\
 & \multicolumn{3}{c}{(Empirical Bayes)} & \multicolumn{3}{c}{(Full Bayesian)} & & & & \multicolumn{3}{c}{(1 chain / 1000 iter.)} & \multicolumn{3}{c}{(4 chains / 2000 iter.)}\\
True value & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP & Bias & (SD) & CP  \\
\hline
$\beta_{10}$=0.2 & 0.001 & (0.049) & 95\% & 0.001 & (0.049) & 96\% & 0.002 & (0.049) & 95\% & 0.006 & (0.051) & 93\% & -0.001 & (0.048) & 96\% \\
 $\beta_{11}$=-0.1 & 0.003 & (0.019) & 96\% & 0.003 & (0.019) & 95\% & 0.002 & (0.019) & 95\% & 0.003 & (0.021) & 93\% & -0.001 & (0.019) & 94\% \\
 $\beta_{12}$=0.1 & 0 & (0.04) & 95\% & 0 & (0.04) & 94\% & -0.001 & (0.04) & 94\% & -0.004 & (0.041) & 93\% & 0.004 & (0.039) & 97\% \\
 $\beta_{13}$=-0.2 & -0.001 & (0.034) & 97\% & 0 & (0.034) & 98\% & 0 & (0.034) & 98\% & -0.003 & (0.035) & 96\% & -0.007 & (0.033) & 98\% \\
 $\sigma_{\varepsilon 1}$=0.4 & 0 & (0.003) & 94\% & 0 & (0.003) & 94\% & 0 & (0.003) & 94\% & 0.002 & (0.01) & 92\% & 0.001 & (0.004) & 95\% \\
 $\beta_{20}$=3 & -0.002 & (0.053) & 96\% & -0.002 & (0.053) & 95\% & 0.002 & (0.053) & 96\% & 0.001 & (0.049) & 98\% & -0.008 & (0.047) & 98\% \\
 $\beta_{21}$=-0.1 & -0.002 & (0.02) & 96\% & -0.002 & (0.02) & 96\% & -0.001 & (0.02) & 96\% & -0.002 & (0.019) & 97\% & -0.003 & (0.019) & 98\% \\
 $\beta_{22}$=0.1 & 0.002 & (0.043) & 96\% & 0.002 & (0.043) & 95\% & 0 & (0.043) & 96\% & 0.001 & (0.041) & 95\% & 0.005 & (0.041) & 95\% \\
 $\beta_{23}$=-0.2 & 0.004 & (0.045) & 94\% & 0.004 & (0.045) & 94\% & 0.003 & (0.045) & 95\% & 0.002 & (0.045) & 94\% & 0.009 & (0.04) & 96\% \\
 $\beta_{30}$=1 & -0.014 & (0.092) & 95\% & -0.014 & (0.091) & 95\% & -0.005 & (0.093) & 94\% & 0.001 & (0.096) & 94\% & 0.026 & (0.102) & 97\% \\
 $\beta_{31}$=-1 & 0.008 & (0.031) & 93\% & 0.009 & (0.031) & 93\% & -0.001 & (0.031) & 95\% & -0.001 & (0.032) & 94\% & -0.007 & (0.031) & 96\% \\
 $\beta_{32}$=1 & -0.006 & (0.074) & 96\% & -0.007 & (0.073) & 95\% & 0.004 & (0.075) & 97\% & -0.002 & (0.076) & 94\% & -0.008 & (0.071) & 98\% \\
 $\beta_{33}$=-1 & 0.01 & (0.07) & 95\% & 0.011 & (0.071) & 95\% & -0.001 & (0.071) & 95\% & -0.003 & (0.068) & 96\% & -0.018 & (0.077) & 85\% \\
 $\sigma_{b10}^2$=0.16 & 0.003 & (0.016) & 95\% & 0.002 & (0.014) & 94\% & -0.001 & (0.014) & 99\% & -0.007 & (0.029) & 94\% & -0.008 & (0.021) & 96\% \\
 $\sigma_{b11}^2$=0.09 & 0.003 & (0.01) & 94\% & 0.004 & (0.009) & 95\% & 0 & (0.009) & 93\% & 0.008 & (0.038) & 88\% & 0.011 & (0.025) & 94\% \\
 $\sigma_{b20}^2$=0.25 & 0.002 & (0.019) & 95\% & 0.001 & (0.017) & 96\% & 0.002 & (0.017) & 95\% & 0.004 & (0.017) & 96\% & 0.002 & (0.016) & 95\% \\
 $\sigma_{b21}^2$=0.16 & 0.002 & (0.014) & 95\% & 0.002 & (0.013) & 95\% & 0.003 & (0.014) & 94\% & 0.002 & (0.012) & 97\% & 0.007 & (0.016) & 82\% \\
 $\sigma_{b30}^2$=0.25 & -0.01 & (0.036) & 96\% & -0.009 & (0.037) & 96\% & 0 & (0.041) & 93\% & 0.004 & (0.041) & 96\% & 0.002 & (0.037) & 97\% \\
 $\textrm{cov}_{b10,b11}$=0.03 & -0.002 & (0.009) & 96\% & -0.004 & (0.009) & 94\% & -0.001 & (0.007) & 97\% & -0.001 & (0.01) & 96\% & -0.003 & (0.009) & 96\% \\
 $\textrm{cov}_{b10,b20}$=0.02 & 0 & (0.011) & 95\% & 0 & (0.011) & 95\% & 0 & (0.01) & 96\% & -0.001 & (0.01) & 94\% & -0.003 & (0.011) & 97\% \\
 $\textrm{cov}_{b10,b21}$=0.04 & -0.001 & (0.01) & 96\% & -0.001 & (0.01) & 96\% & 0 & (0.009) & 96\% & -0.002 & (0.011) & 94\% & 0 & (0.009) & 97\% \\
 $\textrm{cov}_{b10,b30}$=0 & 0 & (0.019) & 96\% & -0.003 & (0.017) & 97\% & 0.001 & (0.016) & 95\% & -0.002 & (0.016) & 94\% & -0.001 & (0.017) & 95\% \\
 $\textrm{cov}_{b11,b20}$=0.03 & -0.001 & (0.01) & 96\% & -0.001 & (0.01) & 96\% & -0.001 & (0.009) & 96\% & 0 & (0.011) & 94\% & -0.002 & (0.01) & 97\% \\
 $\textrm{cov}_{b11,b21}$=0 & 0 & (0.008) & 95\% & 0 & (0.008) & 96\% & 0 & (0.008) & 95\% & 0.002 & (0.01) & 92\% & 0 & (0.008) & 96\% \\
 $\textrm{cov}_{b11,b30}$=-0.06 & 0.005 & (0.014) & 92\% & 0.006 & (0.013) & 94\% & 0.002 & (0.014) & 93\% & 0 & (0.015) & 95\% & -0.001 & (0.014) & 97\% \\
 $\textrm{cov}_{b20,b21}$=0.08 & 0 & (0.012) & 95\% & 0 & (0.012) & 95\% & -0.001 & (0.011) & 94\% & 0 & (0.011) & 96\% & 0.002 & (0.011) & 96\% \\
 $\textrm{cov}_{b20,b30}$=0.05 & -0.001 & (0.019) & 95\% & -0.002 & (0.019) & 95\% & -0.003 & (0.018) & 94\% & -0.001 & (0.019) & 96\% & 0.001 & (0.018) & 96\% \\
 $\textrm{cov}_{b21,b30}$=0.04 & -0.001 & (0.017) & 95\% & -0.001 & (0.017) & 96\% & -0.003 & (0.016) & 94\% & 0.009 & (0.019) & 92\% & 0.011 & (0.018) & 95\% \\
 $\varphi_1$=0.5 & -0.003 & (0.101) & 95\% & -0.003 & (0.102) & 94\% & 0 & (0.103) & 98\% & 0.009 & (0.113) & 94\% & 0.021 & (0.114) & 96\% \\
 $\varphi_2$=-0.2 & 0.006 & (0.074) & 93\% & 0.005 & (0.073) & 94\% & 0.005 & (0.074) & 97\% & 0.002 & (0.077) & 93\% & 0.006 & (0.072) & 94\% \\
 $\varphi_3$=0.3 & -0.001 & (0.094) & 94\% & 0.001 & (0.092) & 94\% & -0.004 & (0.092) & 94\% & -0.006 & (0.093) & 94\% & -0.003 & (0.087) & 95\% \\
 Conv. rate & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{1} & \multicolumn{3}{c}{0.51} & \multicolumn{3}{c}{0.64} & \\
 Comp. time (sec.) & \multicolumn{3}{c}{43.55 (5.58)} & \multicolumn{3}{c}{69.8 (44.59)} & \multicolumn{3}{c}{295.73 (20.22)} & \multicolumn{3}{c}{3766.41 (1429.06)} & \multicolumn{3}{c}{10861.22 (3735.02)}\\
 \arrayrulecolor{black}\hline
\end{tabular}}
\label{simGLM3Y}
\end{table}

\begin{figure}[!ht]
\centering
\includegraphics[scale=0.85]{CompTime_7x10.pdf}
\caption{Computation times and convergence rates for the 9 simulations scenarios with K=1,2 and 3 longitudinal markers with continuous, count and binary distributions.}
\label{Fig2}
\end{figure}

\subsubsection{Non-Gaussian markers}

Results of scenarios with $k=1, 2$ and $3$ count longitudinal markers are presented in Tables S3, S4 and S5 of the supporting information, respectively. Results of scenarios with $k=1, 2$ and $3$ binary longitudinal markers are presented in Tables S6, S7 and S8 of the supporting information, respectively. Finally results of scenario 10 with one continuous Gaussian marker, one count Poisson marker and one binary binomial marker are summarized in Table \ref{simGLM3Y}.\\

On these 7 scenarios, the two strategies of INLA systematically provided correct inference with 100\% convergence, no bias and coverage probabilities close to the nominal value. The MCMC strategy of \textbf{JMbayes2} also provided unbiased estimates with standard deviations that match with \textbf{R-INLA} and coverage probabilities close to the nominal values for scenarios with counts longitudinal markers and the last scenario considering a mixture of markers. When considering only binary longitudinal markers, \textbf{JMbayes2} was not able to fit the scenario with only one marker because it requires to have at least two random effects in the model (while this scenario only included a random intercept). Additionally, with two and three binary longitudinal markers, \textbf{JMbayes2} was associated to a higher bias compared to \textbf{R-INLA} and \textbf{rstanarm} for the association parameters. Note that we reduced \textbf{JMbayes2}'s default number of iterations for scenarios with binary outcomes to match the number of iterations of other scenarios. When using the default number of iterations (i.e., 6500) this bias was slightly reduced but remained higher compared to \textbf{R-INLA} and \textbf{rstanarm} (e.g., for $\varphi_1=0.3$, bias$=0.296$ with 3500 iterations (Table S8) and bias$=0.227$ with 6500 iterations for a computation time that increased from 147s. per model to 262s. per model (not shown)). The two MCMC strategies of \textbf{rstanarm} provided no bias and coverage probabilities close to the nominal value in the scenarios with only binary outcomes, and in the case of a mixture of markers, two cases where more repeated information had been considered. In contrast, with count data, the performances were poor with a small rate of convergence (between 41\% and 62\% for the 4 chains / 2000 iterations, between 25\% and 61\% for 1 chain / 1000 iterations), a higher bias and standard deviations together with low coverage probabilities, particularly with 1 chain and 1000 iterations.\\

Regarding computation time (Figure \ref{Fig2} for all counts and all binary outcomes, and Figure S1 for the mixed outcomes), as already observed with Gaussian markers, the two strategies of INLA were much faster than the MCMC strategies. For instance when considering a mixture of markers, the computation time with \textbf{R-INLA} was less than one minute on average with empirical Bayes strategy (43.55s.) and slightly over one minute (69.8s.) with full Bayesian strategy while \textbf{JMbayes2} fitted one model in 5 minutes on average (295.73s.), \textbf{rstanarm} fitted a model on average in one hour (3766s.) with 1 chain and 1000 iteration and three hours (10861s.) with 4 chains and 2000 iterations.

The computation times were also between 40 and 300 times smaller with \textbf{R-INLA} compared to \textbf{rstanarm} and between 3 and 13 times smaller with \textbf{R-INLA} compared to \textbf{JMbayes2} in the count case, and between 20 and 75 times smaller with \textbf{R-INLA} compared to \textbf{rstanarm} and between 5 and 8 times smaller with \textbf{R-INLA} compared to \textbf{JMbayes2} in the binary case. Of note, the error messages following non-convergence with \textbf{rstanarm} indicated that running the chains for more iterations may help to meet the convergence criteria. However, the computation time was already so high that the most complex scenario (i.e., 3 markers with different distributions) required around 37 days straight to fit the 300 datasets with 4 chains and 2000 iterations despite parallel computing.

\section{Application to PBC}
\label{appsec}
\subsection{Description}
We illustrate the flexibility of our inference approach using \textbf{R-INLA} with an application to the study of primary biliary cholangitis (PBC), a rare chronic liver disease that often causes death. We leveraged the longitudinal data from 312 PBC patients followed at the Mayo Clinic between 1974 and 1988 who received either a placebo or D-penicillamine \citep{murtaugh1994primary}. These data are publicly available in several software including the R package \textbf{JM} \citep{rizopoulos2010jm}. During the follow-up, 140 patients died and 29 patients received a liver transplantation which we consider here as a competing event of death. In addition, repeated measures of various longitudinal markers potentially associated with the disease progression were collected. Among them, we considered:
\begin{itemize}
\item 4 continuous markers: serum bilirubin (in mg/dl, log scale), serum aspartate aminotransferase (in U/ml, log scale), serum albumin (in gm/dl) and prothrombin time in seconds,
\item 2 count markers: platelet (per cubic ml / 1000) and alkaline phosphatase (in U/liter)
\item 3 binary markers: spiders (presence of blood vessel malformations in the skin), ascites (presence of abnormal accumulation of fluid in the abdomen) and hepatomegaly (presence of enlarged liver).
\end{itemize}

\noindent The number of individual repeated measurements for these markers was 6.2 on average (minimum 1, maximum 16, median 5).

\subsection{Objective and strategy of analysis}

Our objective was to evaluate the association between the 9 longitudinal markers of progression and the competing risks of death and liver transplantation. We thus built a joint model for analyzing simultaneously the candidate markers and the 2 causes of event. The final joint model was built step-by-step. We first considered each marker separately and fitted joint models for one longitudinal marker and 2 competing causes of event in order to define the appropriate shape of the marker trajectory, choosing between linear and natural cubic splines with 2 knots at quantiles 0.33 (1 year of follow-up) and 0.66 (4 years of follow-up). The models systematically included random intercept and slopes on each function of time (i.e., two or four correlated random effects for linear or splines, respectively) as well as treatment (placebo or D-penicillamine) and its interactions with time functions. Then, we defined the structure of association with the two causes of event, choosing between current level (i.e. shared linear predictor) and current slope (or both). At this stage, we dropped the marker alkaline because it was not found associated to the two competing risks of events and we also dropped prothrombin, ascites and hepatomegaly because they were giving unstable results (i.e., vague posterior distribution that mostly reflects the non-informative prior). We then built the final model including simultaneously the longitudinal markers found associated with the events in the previous step. Based on this strategy, the final model included 5 longitudinal markers, 3 continuous, 1 count and 1 binary. It was defined as follows for any time $t$:

\[
\left\{
\setlength\arraycolsep{0pt}
\begin{array}{ c @{{}{}} l  @{{}{}} r }
\log(Y_{i1}(t)) &= \eta_{i1}(t) + \varepsilon_{i1}(t) & \hspace{-6cm} \textbf{\textit{(bilirubin - lognormal)}}\\
&=(\beta_{10} + b_{i10})+\beta_{11}X_{i}+(\beta_{12}+b_{i11})\textrm{NS}_1(t)+(\beta_{13}+b_{i12})\textrm{NS}_2(t)+\\
& \ \ \ \hspace{0.05cm} (\beta_{14}+b_{i13})\textrm{NS}_3(t)+\beta_{15}X_{i}\textrm{NS}_1(t)+\beta_{16}X_{i}\textrm{NS}_2(t)+\beta_{17}X_{i}\textrm{NS}_3(t) + \varepsilon_{i1}(t) \\ \ \\

\log(Y_{i2}(t)) &= \eta_{i2}(t) + \varepsilon_{i2}(t) & \hspace{-6cm} \textbf{\textit{(aspartate aminotransferase - lognormal)}}\\
&= (\beta_{20} + b_{i20})+\beta_{21}X_{i}+(\beta_{22}+b_{i21})\textrm{NS}_1(t)+(\beta_{23}+b_{i22})\textrm{NS}_2(t)+\\
& \ \ \ \hspace{0.05cm} (\beta_{24}+b_{i23})\textrm{NS}_3(t)+\beta_{25}X_{i}\textrm{NS}_1(t)+\beta_{26}X_{i}\textrm{NS}_2(t)+\beta_{27}X_{i}\textrm{NS}_3(t) + \varepsilon_{i2}(t) \\ \ \\

Y_{i3}(t)&=\eta_{i3}(t) + \varepsilon_{i3}(t) & \hspace{-6cm} \textbf{\textit{(albumin - normal)}}\\
&= (\beta_{30} + b_{i30})+\beta_{31}X_{i}+(\beta_{32}+b_{i31})t+\beta_{33}X_{i}t+ \varepsilon_{i3}(t)\\ \ \\

\log(E[Y_{i4}(t)])&=\eta_{i4}(t) & \hspace{-6cm} \textbf{\textit{(platelet - Poisson)}}\\
&= (\beta_{40} + b_{i40})+\beta_{41}X_{i}+(\beta_{42}+b_{i41})\textrm{NS}_1(t)+(\beta_{43}+b_{i42})\textrm{NS}_2(t)+\\
& \ \ \ \hspace{0.05cm} (\beta_{44}+b_{i43})\textrm{NS}_3(t)+\beta_{45}X_{i}\textrm{NS}_1(t)+\beta_{46}X_{i}\textrm{NS}_2(t)+\beta_{47}X_{i}\textrm{NS}_3(t) \\ \ \\

\textrm{logit}(E[Y_{i5}(t)])&=\eta_{i5}(t) & \hspace{-6cm} \textbf{\textit{(spiders - binomial)}}\\
&= (\beta_{50} +b_{i50})+\beta_{51}X_{i}+(\beta_{52}+b_{i51})t+\beta_{53}X_{i}t \\ \ \\

\lambda_{i1}(t)&=\lambda_{01}(t)\ \textrm{exp}\left(\gamma_1 X_i + \eta_{i1}(t)\varphi_1 + \eta_{i1}'(t)\varphi_3 + \eta_{i2}(t)\varphi_4 \hspace{0.05cm} + \right. & \hspace{-6cm} \textbf{\textit{(death risk)}}\\
& \hspace{3.25cm} \left.\eta_{i3}(t)\varphi_5 + \eta_{i4}(t)\varphi_7 + \eta_{i5}(t)\varphi_9 \right) \\ \ \\

\lambda_{i2}(t)&=\lambda_{02}(t)\ \textrm{exp}\left(\gamma_2 X_i + \eta_{i1}(t)\varphi_2 + \eta_{i3}(t)\varphi_6 + \eta_{i4}(t)\varphi_8 \right) & \hspace{-6cm} \textbf{\textit{(transplantation risk)}}
\end{array}
\right.
\]

The independent Gaussian measurement errors for the first three markers is captured by $\varepsilon_{ik}(t)$, variable $X_i$ corresponds to the treatment received (placebo vs. D-penicillamine) and $\textrm{NS}_1(t), \textrm{NS}_2(t), \textrm{NS}_3(t)$ are the natural cubic splines with internal knots at 1 and 4 years. We assume independent random effects between longitudinal markers to avoid too many covariance parameters, resulting in 16 random effects and 20 covariance parameters (the full variance-covariance matrix of the random effects if we assumed correlation between markers would have contained 120 covariance parameters). Compared to independent univariate joint models, this model has the asset of allowing the estimation of markers associations that are adjusted for the other markers, which is an essential feature, particularly in case of confounding as longitudinal markers can capture some individual heterogeneity.

\subsection{Results}
The parameter estimates as well as their standard deviations and credible intervals are given in Table S9 of the supporting information. We illustrate the results with a plot of the average linear predictor trajectory conditional on treatment for each marker and the baseline risk of death and transplantation curves in Figure \ref{Fig3}. The computation time for this model (with the same resources as in the simulation studies) was 276 seconds.\\

Our results show that the linear predictor describing the level of serum bilirubin increases but with a reducing rate of change over time while platelets counts slowly decreases but with an increasing rate of change over time. The evolution of aspartate aminotransferase levels depends on the drug received; for patients receiving the placebo drug, it increases in the beginning of the follow-up and then stabilizes while for patients receiving D-penicillamine, it slightly reduces in the early follow-up and then also stabilizes. Finally, we observe an overall decrease in albumin concentration and an increase of the spiders's levels.\\

Regarding the association with clinical endpoints, our results exhibited a strong association between the log serum bilirubin and the risk of death through both its current level (log risk of death increased by 1.22 (95\% CI 1.05, 1.37) for each unit increase of the current value), and its current slope (0.89 (95\% CI 0.30, 1.53) increase of the log risk of death for each unit increase in the current slope of the log serum bilirubin) adjusted for treatment and all the other markers. Moreover, a unit increase in this biomarker's current value was associated to a 1.15 (95\% CI 0.96, 1.36) increase in the log risk of transplantation. The level of aspartate aminotransferase was slightly associated with decreased risks of death (association with log risk of death of -0.35, 95\% CI -0.66, -0.01). There was also a strong negative asssociation between the current level of albumin and both the risk of death (-1.82 (95\% CI -2.17, -1.46) decrease in the log risk of death for a one unit increase) and the risk of transplantation ( -1.14 (95\% CI -1.61, -0.63) decrease in the log risk of transplantation for a one unit increase). After adjustment for treatment and the other markers, a higher current level of platelet counts was associated with a decreased risk of death (association with log risk of death of -0.63, 95\% CI -0.79, -0.47). Finally, the marker spiders were not found associated to the risks of events, indicating that the association observed in the univariate model was fully explained by other markers included in the full model.

\begin{figure}[!ht]
\centering
\includegraphics[scale=0.8]{Plot_res_8x8.pdf}
\caption{Average linear predictor trajectories for each longitudinal marker and baseline risk curves. The 95\% bands for the uncertainty of the marker's trajectory is obtained by sampling from the joint posterior distributions (1000 samples). The association parameters between each longitudinal marker and the survival submodels are given in each plot by $\varphi^{*.*}$, where the first letter is ``L'' for current level of the linear predictor and ``S'' for current slope while the second letter corresponds to ``D'' for the effect on the risk of death and ``T'' for the risk of transplantation.}
\label{Fig3}
\end{figure}

Our results regarding the treatment are consistent with the literature, for example a meta-analysis evaluating the effect of placebo vs. D-penicillamine in randomized-clinical trials of primary biliary cholangitis patients by \citet{gong2006systematic} found no effect of this drug on the risk of death and the risk of transplantation. Also this meta-analysis found an association between D-penicillamine and a decrease of serum alanine aminotransferase, which is consistent with our finding where D-penicillamine is associated to a decrease of serum aspartate aminotransferase in early follow-up (serum alanine aminotransferase and aspartate aminotransferase are both enzymes that indicate liver damage). The R code for this application with \textbf{INLAjoint} package is available at github.com/DenisRustand/MultivJoint (``PBC\_INLA.R'').

\section{Discussion}
\label{dissec}

In this paper, we proposed a fast and accurate inference approach which makes it possible to fit flexible joint models for multivariate longitudinal and survival data. Based on the Integrated Nested Laplace Approximation Bayesian algorithm implemented in the R package \textbf{R-INLA}, it alleviates the computational burden of iterative estimation strategies implemented in classical software (i.e., maximum likelihood estimation or Bayesian inference with MCMC sampling), and thus allows the estimation of multivariate joint models with much less restrictions as illustrated in our simulation studies and application to primary biliary cholangitis. Several models were developed for the analysis of this dataset but they were often limited to one or few markers, did not account for the competing risks of events or use a different approach than joint modeling to reduce the complexity (e.g., \citet{philipson2020faster, devaux2021individual, andrinopoulou2021reflections, Hughes21, MURRAY2022107438}).\\

We demonstrated that the estimation of this class of joint models with \textbf{R-INLA} is faster and more reliable than any alternative estimation strategy (very good inference properties, computation time and convergence rate). Compared to alternative joint modeling software and inference techniques, the estimation of multivariate joint models with \textbf{R-INLA} has several other assets. There is no limitation for the number of longitudinal markers that can be included in the joint model and various distributions are available to describe the marker's distributions. For instance, it handles in the longitudinal part: zero-inflated models, two-part model or proportional odds model that are often of interest in health research (but currently not available in joint modeling software). It is also possible to consider competing risks of events as illustrated in the application (currently not possible with \textbf{joineRML} and \textbf{rstanarm}). Additionally, beyond competing risks \textbf{R-INLA} can handle various survival models in the context of joint modeling, including frailty models, multi-state models and mixture cure models. Finally, several parameterizations for the link between longitudinal and survival outcomes are available with \textbf{R-INLA}: shared random effects, current value and current slope of the biomarker or any linear combination of the biomarkers's linear predictors; and future extensions may allow for a non-linear effect of the biomarker on the risk of event. Since the \textbf{R-INLA} package is developed to fit a wide variety of statistical models beyond joint models, we used the user-friendly interface implemented in the R package \textbf{INLAjoint} that facilitates the use of INLA for joint modeling, as illustrated in the R codes ``MultivJoint.R'' and ``PBC\_INLA.R'' available at github.com/DenisRustand/MultivJoint that replicates the model fit for a simulated dataset and for the application Section of this paper, respectively.\\

There are however some limitations with the joint models that INLA can fit at the moment. This inference technique is focused on models that can be expressed as latent Gaussian models. In addition, although the number of random effects is not limited, the number of correlated random effects is currently limited to 10 although this limit is practical and could be extended if necessary. As illustrated in the application, it does not preclude the inclusion of multiple independent groups of correlated random effects in a model. Also there is often interest in deriving individual prediction from the fitted model. This is not specifically implemented yet but the Bayesian properties of the model and the function available to sample from the joint posterior distribution makes it easy to compute any prediction based on the model's output.\\

Our simulation studies mimic the design proposed in the function \textit{simData()} of the R package \textbf{joineRML} (see \citet{hickey2018joinerml}) but we extended it to non-Gaussian outcomes. Therefore the design is not specifically chosen to fit with \textbf{R-INLA} but rather based on previous simulations for multivariate joint models and limited by \textbf{joineRML} and \textbf{rstanarm}'s range of models that can be fitted. Our comparison of the available methodologies to fit multivariate joint models suggests that the iterative algorithms reach some limitations when fitting complex models, compared to the Bayesian approach implemented in \textbf{R-INLA}. For the MCMC approach, our simulations show that \textbf{JMbayes2} has much better properties compared to \textbf{rstanarm}, both in terms of frequentist properties and computation time. It is the best alternative to \textbf{R-INLA} available at the moment since it is able to fit models with similar features for a moderate increase in computation time. Despite not being able to fit the same model as proposed in our application Section, \textbf{JMBayes2} is able to fit a similar model that includes a full variance-vovariance matrix for the all the random effects of the model (while our model for the application assume independence between random effects accross longitudinal markers).

The comparison between the software implementations in the simulation studies was not totally fair because the prior distributions accross Bayesian strategies were not perfectly matching and because \textbf{JMbayes2} and \textbf{rstanarm} used MLE from univariate submodels to define initial values and \textbf{JMbayes2} also used those MLE to define some of the prior distributions, note that it is possible to use the same non-informative priors with \textbf{JMbayes2} as with \textbf{rstanarm} and \textbf{R-INLA}. However, the purpose of our simulations was to illustrate the properties of the estimations strategies as they are meant to be used and therefore we decided to keep those differences to prevent a misuse of these softwares in the context of our comparison with \textbf{R-INLA}.\\

Until now, joint modeling software have only used iterative opimization (see for instance \citet{papageorgiou2019overview, furgal2019review} and \citet{alsefri2020bayesian}). It induces strong limitations in the applicability of joint modeling. With a deterministic inference strategy, INLA offers a very promising approach for the development and application of joint models for multivariate longitudinal data and survival times as highlighted in this work.

redpajama_set_name: RedPajamaArXiv


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text:
Q: Jackrabbit clustering with a H2 database in mixed mode I have a Magnolia CMS system and I try to configure a clustered repository for a workspace with user generated content.
We will have an author and a public instance, each with a dedicated MYSQL database for the magnolia default repository.
The user generated content should be stored in a h2 database which will be located in a shared directory.
We can't start the h2 database directly in server mode, therefore, the connections to the h2 database will be configured as mixed mode with the flag AUTO_SERVER=true (see http://www.h2database.com/html/features.html#auto_mixed_mode).
The initializing of the repositories is flawless. Once data is stored in the clustered repository the next reboot stops/hangs during the loading phase of the clustered repository (see log).
Did i miss a configuration? Is it even possible to use a h2 in mixed mode for jackrabbit clustering?
repository.xml:
<JCR>
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        <Map name="userGeneratedContent" repositoryName="magnoliacluster" workspaceName="userGeneratedContent" />
        <Map name="website" repositoryName="magnolia" workspaceName="website" />
        <Map name="config" repositoryName="magnolia" workspaceName="config" />
        <Map name="users" repositoryName="magnolia" workspaceName="users" />
        <Map name="userroles" repositoryName="magnolia" workspaceName="userroles" />
        <Map name="usergroups" repositoryName="magnolia" workspaceName="usergroups" />
        <Map name="mgnlSystem" repositoryName="magnolia" workspaceName="mgnlSystem" /> <!-- System internal data -->
        <Map name="mgnlVersion" repositoryName="magnolia" workspaceName="mgnlVersion" /> <!-- magnolia version workspace -->
    </RepositoryMapping>

    <!-- magnolia default repository -->
    <Repository name="magnolia" provider="info.magnolia.jackrabbit.ProviderImpl" loadOnStartup="true">
        <param name="configFile" value="${magnolia.repositories.jackrabbit.config}" />
        <param name="repositoryHome" value="${magnolia.repositories.home}/magnolia" />
        <!-- the default node types are loaded automatically
            <param name="customNodeTypes" value="WEB-INF/config/repo-conf/nodetypes/magnolia_nodetypes.xml" />
        -->
        <param name="contextFactoryClass" value="org.apache.jackrabbit.core.jndi.provider.DummyInitialContextFactory" />
        <param name="providerURL" value="localhost" />
        <param name="bindName" value="${magnolia.webapp}" />
        <workspace name="website" />
        <workspace name="config" />
        <workspace name="users" />
        <workspace name="userroles" />
        <workspace name="usergroups" />
        <workspace name="mgnlSystem" />
        <workspace name="mgnlVersion" />
    </Repository>

    <!-- magnolia cluster repository -->
    <Repository name="magnoliacluster" provider="info.magnolia.jackrabbit.ProviderImpl" loadOnStartup="true">
        <param name="configFile" value="${magnolia.repositories.jackrabbit.cluster.config}" />
        <param name="repositoryHome" value="${magnolia.repositories.home}/magnoliacluster" />
        <!-- the default node types are loaded automatically
            <param name="customNodeTypes" value="WEB-INF/config/repo-conf/nodetypes/magnolia_nodetypes.xml" />
        -->
        <param name="contextFactoryClass" value="org.apache.jackrabbit.core.jndi.provider.DummyInitialContextFactory" />
        <param name="providerURL" value="localhost" />
        <param name="bindName" value="cluster-${magnolia.webapp}" />
        <workspace name="userGeneratedContent" />
    </Repository>
</JCR>

clustered-jackrabbit-bundle-h2-search.xml:
<Repository>
    <FileSystem class="org.apache.jackrabbit.core.fs.db.DbFileSystem">
        <param name="url" value="jdbc:h2:/PATH/TO/db;AUTO_SERVER=TRUE"/>
        <param name="schemaObjectPrefix" value="fs_"/>
        <param name="user" value="sa"/>
        <param name="password" value="sa"/>
    </FileSystem>
    <Security appName="magnolia">
        <SecurityManager class="org.apache.jackrabbit.core.DefaultSecurityManager" />
        <AccessManager
            class="org.apache.jackrabbit.core.security.DefaultAccessManager">
        </AccessManager>
        <LoginModule
            class="info.magnolia.jaas.sp.jcr.JackrabbitAuthenticationModule">
        </LoginModule>
    </Security>
    <DataStore class="org.apache.jackrabbit.core.data.FileDataStore">
        <param name="url" value="jdbc:h2:/PATH/TO/db;AUTO_SERVER=TRUE"/>
        <param name="schemaObjectPrefix" value="datastore_"/>
        <param name="user" value="sa"/>
        <param name="password" value="sa"/>
    </DataStore>
    <Workspaces rootPath="${rep.home}/workspaces"
        defaultWorkspace="default" />
    <Workspace name="default">
        <FileSystem class="org.apache.jackrabbit.core.fs.local.LocalFileSystem">
            <param name="path" value="${wsp.home}/default" />
        </FileSystem>
        <PersistenceManager class="org.apache.jackrabbit.core.persistence.pool.H2PersistenceManager">
            <param name="url" value="jdbc:h2:/PATH/TO/db;AUTO_SERVER=TRUE"/>
            <param name="schemaObjectPrefix" value="ws_${wsp.name}_"/>
            <param name="user" value="sa"/>
            <param name="password" value="sa"/>
        </PersistenceManager>
        <SearchIndex class="org.apache.jackrabbit.core.query.lucene.SearchIndex">
            <param name="path" value="${wsp.home}/index" />
            <param name="useCompoundFile" value="true" />
            <param name="minMergeDocs" value="100" />
            <param name="volatileIdleTime" value="3" />
            <param name="maxMergeDocs" value="100000" />
            <param name="mergeFactor" value="10" />
            <param name="maxFieldLength" value="10000" />
            <param name="bufferSize" value="10" />
            <param name="cacheSize" value="1000" />
            <param name="forceConsistencyCheck" value="false" />
            <param name="autoRepair" value="true" />
            <param name="queryClass" value="org.apache.jackrabbit.core.query.QueryImpl" />
            <param name="resultFetchSize" value="100" />
            <param name="extractorPoolSize" value="3" />
            <param name="extractorTimeout" value="100" />
            <param name="extractorBackLogSize" value="100" />
            <!-- needed to highlight the searched term -->
            <param name="supportHighlighting" value="true"/>
        </SearchIndex>
        <WorkspaceSecurity>
            <AccessControlProvider
                class="info.magnolia.cms.core.MagnoliaAccessProvider" />
        </WorkspaceSecurity>
    </Workspace>
    <Versioning rootPath="${rep.home}/version">
        <FileSystem class="org.apache.jackrabbit.core.fs.local.LocalFileSystem">
            <param name="path" value="${rep.home}/workspaces/version" />
        </FileSystem>
        <PersistenceManager
                class="org.apache.jackrabbit.core.persistence.bundle.H2PersistenceManager">
            <param name="url" value="jdbc:h2:${rep.home}/version/db" />
            <param name="schemaObjectPrefix" value="version_" />
        </PersistenceManager>
    </Versioning>
    <Cluster>
        <Journal class="org.apache.jackrabbit.core.journal.DatabaseJournal">
            <param name="driver" value="org.h2.Driver" />
            <param name="url" value="jdbc:h2:/PATH/TO/db;AUTO_SERVER=TRUE"/>
            <param name="schemaObjectPrefix" value="journal_"/>
            <param name="databaseType" value="h2"/>
            <param name="user" value="sa"/>
            <param name="password" value="sa"/>
        </Journal>
    </Cluster>
</Repository>

log:
---------------------------------------------
MAGNOLIA LICENSE
---------------------------------------------
Version number : 5.3.5
Build          : 4. November 2014 (rev. of UNKNOWN)
Edition        : Enterprise Edition
Provider       : Magnolia International Ltd. (info@magnolia-cms.com)
2016-07-27 11:17:59,919 INFO  info.magnolia.cms.beans.config.ConfigLoader       : Initializing content repositories
2016-07-27 11:17:59,922 INFO  info.magnolia.repository.DefaultRepositoryManager : Loading JCR
2016-07-27 11:17:59,931 INFO  info.magnolia.repository.DefaultRepositoryManager : Loading JCR magnolia
2016-07-27 11:17:59,951 INFO  info.magnolia.jackrabbit.ProviderImpl             : Loading repository at /PATH/author/repositories/magnolia (config file: /PATH/WEB-INF/config/repo-conf/jackrabbit-bundle-h2-search.xml) - cluster id: "<unset>"
Jul 27, 2016 11:18:00 AM org.apache.catalina.startup.HostConfig deployDirectory
INFO: Deploying web application directory /PATH/webapps/manager
Jul 27, 2016 11:18:00 AM org.apache.catalina.startup.TldConfig execute
INFO: At least one JAR was scanned for TLDs yet contained no TLDs. Enable debug logging for this logger for a complete list of JARs that were scanned but no TLDs were found in them. Skipping unneeded JARs during scanning can improve startup time and JSP compilation time.
Jul 27, 2016 11:18:00 AM org.apache.catalina.startup.HostConfig deployDirectory
INFO: Deployment of web application directory /PATH/webapps/manager has finished in 417 ms
2016-07-27 11:18:02,174 INFO  info.magnolia.repository.DefaultRepositoryManager : Loading JCR magnoliacluster
2016-07-27 11:18:02,174 INFO  info.magnolia.jackrabbit.ProviderImpl             : Loading repository at /PATH/author/repositories/magnoliacluster (config file: /PATH/WEB-INF/config/repo-conf/clustered-jackrabbit-bundle-h2-search.xml) - cluster id: "<unset>"

Ps.: I'm not sure why it says 'cluster id: ""'. Although I have it set as org.apache.jackrabbit.core.cluster.node_id in my properties file.

A: From the following wiki page:

The cluster id identifies the instance and is used to write changes to the journal as well as to load changes from the journal. Make sure this is a unique value and is not shared with the other nodes in the cluster.
Cluster id can be defined either in the properties file (most convenient way) or in the persistence manager in the cluster configuration (both ways are used in the attached files)

Since you haven't provided any cluster id in your jackrabbit configuration file above, could you please check that your magnolia.properties file contains one? If not, just set an unique value for each Magnolia instances accessing the JCR cluster.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Sympy lambdify how to write f(x1,x2, x3,....xn) I am trying to use lambdify. The code is:
expr = x+y
f = lambdify((x,y), expr, "numpy")

Now evaluate the function, for example: f(1,2)
if the x, y input is stored in a list, like [1,2] how can I pass the variable to the function f? assuming that f can be a function of thousand variables so manually input is not ideal.
Any suggestions?
Thanks


Marla Coppolino is on a mission to enlighten you to the plight of the lowly land snail through creative, intimate and unconventional means.
Life at a Snail's Pace is my graduation film from Goldsmiths, University of London. I knew I wanted to make a documentary film related to nature and I decided that it would be about snails. I am French and ate snails my whole life and it's only recently, after visiting a snail farm, that I realised that I actually didn't know anything about them. Finding Marla and listening to her passion for land snails convinced me to make Life at a Snail's Pace.
You should watch Life at a Snail's Pace because Marla is extraordinary, her advocacy for the lowly molluscs reaches branches that you couldn't even have imagined. The film not only teaches about land snails but also entertains by immersing you into the diverse inspirations they bring up to Marla.
This is a personal story, all centred around Marla and snails. I guess even though we don't all push our passions to this level, we all love some things that others can't understand. I think this is a personal theme that a lot of people will relate to. Then, the idea that humans are disconnected from nature and that their habits have harmful effects on the environment is now universal and Marla's message connects with that.
As it is a documentary, there was never a script as such, but defined sequences and themes that I wanted out of the film. These evolved throughout filming as I learned more about Marla and snails. For instance, I had no idea about the miniatures and that was a fantastic surprise. Those scenes were very helpful to connect other sequences together but also to mix in with the scientific facts to keep the audience interest.
Most people have never seen snails under this light before and actually don't really know much about them. Everyone seems to appreciate the cinematography and the macro shots.
For a graduation film this is a huge compliment, because it means that the film manages to capture the beauty of snails and explores Marla's perspective. I hope it helps to see through her eyes and have the potential to convince the audience to not be judgemental.
This is a great opportunity to communicate with the industry and the audience why Life at a Snail's Pace needs to be seen.
Anyone who would like to help the snail cause!
My hope is that the film could change some people's point of view of snails or want to learn more about them. Even if some people will never find them beautiful, at least understand that as every living being they are important for the whole ecosystem. In a similar line, even though some people may not believe in Marla's quest, I hope the film makes people less judgemental and admire her dedication.
Did you know that 90% of snails in North America are smaller than a lentil?
I would like to give a huge thank you to Marla for sharing her passion with me, thank you David Denyer for your amazing composition made with sole instrument: a snail shell, thank you to all the performing snails, thank you to Goldsmiths, University of London and thank you to everyone who contributed to this film.
Currently I am developing a short film about my grandmother's origins through the little known history of the sandalwood trade in the Pacific.
Alexandra Gaulupeau has a knack for finding amazing and unusual stories in the unlikeliest of places and looks for the links between us all, be it human or even mollusc!


From 11-16 August 2018, the Afghanistan Office of the President, supported by UN Women and UNDP, organized a visit of senior government officials from Afghanistan to Indonesia to learn about their experiences implementing gender-responsive budgeting. Indonesia's successful GRB story as helped the government to effectively distribute public funds to address the needs of women and men. The visit contributed to the development of a community of practice which opens dialogues between gender equality stakeholders from Afghanistan and Indonesia to share lessons-learned and best practices. Click to learn more about this South-South cooperation initiative.


The universe is a vast expanse of wonders. Ibra is a god of these marvels—distant galaxies, misty nebulae, all the spinning worlds that orbit countless suns, and the empty void that stretches between them. Science can explain all of these and more, but the magnificence of the cosmos knows no bounds, and there are always new miracles and mysteries waiting to be discovered, explored, and explained. Ibra itself is one of these cosmic enigmas. It has no gender or defined physical form, and it's unclear whether Ibra is a deity who sprang into being when the universe was born, some alien paragon who ascended to godhood, or a personification of the cosmos itself.
Ibra's followers believe that one can find truth, serenity, and meaning in the beauty of the cosmos, but it is up to the individual to decipher the universe's secrets to find these rewards. It is not Ibra's place to reveal the answers to creation's great questions, and the Inscrutable rewards those who push the boundaries of knowledge and science to explore the universe and bring its wonders to light. In Ibra's worldview, there is no law or chaos, no good or evil: only the questions raised by the shining light of a newborn star, the beautiful dance of heavenly bodies, and the perfect solitude of the comet as it hurtles through the dark, regardless of whether the answers to those questions are ever found.
Astronomers, astrophysicists, cosmologists, mathematicians, and interstellar explorers worship Ibra as their patron, and many mystics and solarians serve as the Inscrutable's priests. Ibra's temples are most frequently found in space rather than planetside, and they always include observatories where worshipers may study the cosmos and record their observations and hypotheses, some even directly exposed to vacuum. No one in the Pact Worlds knows what species first venerated Ibra as a god, as the oldest discovered monuments to the deity have all been massive temple complexes on otherwise uninhabited worlds, including Bretheda's tiny moon Chamari.
Ibra's holy symbol is as inscrutable as the god itself. Found on the oldest temples, the golden shape is often assumed to be an eye, a compass, or perhaps a planet, with an arrow leading the faithful ever onward into the unknown. More obvious is the six- star constellation inside the symbol, yet questions abound there as well. What is the constellation—a configuration unrecognizable from any known world—and what does it mark? What is the significance of the symbol's tiny, lesser stars? Some Ibran priests devote their lives to exploring new worlds in search of matching the viewpoint, and a popular rumor claims the constellation is only visible from Golarion, with the symbol the key to finding it.