Fast Variational Bayesian Inference for Correlated Survival Data: An Application to Invasive Mechanical Ventilation Duration Analysis

ABSTRACT

Correlated survival data are prevalent in various clinical settings and have been extensively discussed in the literature. A common example is clustered survival data, where survival times are associated due to shared characteristics within clusters. In our study, we analyze invasive mechanical ventilation data collected from multiple intensive care units (ICUs) across Ontario, Canada. Patients within the same ICU exhibit similarities in clinical profiles and mechanical ventilation settings, leading to a correlation in their ventilation durations. To address this association, we introduce a shared frailty log‐logistic accelerated failure time model that accounts for intra‐cluster correlation through a cluster‐specific random intercept. We present a novel, fast variational Bayes (VB) algorithm for parameter inference and evaluate its performance using simulation studies varying the number of clusters and their sizes. We further compare the performance of our proposed VB algorithm with the h‐likelihood method and a Markov Chain Monte Carlo (MCMC) algorithm. The proposed algorithm delivers satisfactory results and demonstrates computational efficiency over the MCMC algorithm. We apply our method to ICU ventilation data from Ontario to investigate the ICU‐site random effect on ventilation duration.

1. Introduction

Correlated survival data are commonplace in various research contexts and have been extensively studied in the literature [1, 2, 3]. One such context of correlated survival data is observed in clustered survival data, which are derived from multiple entities such as families or hospitals. Within each cluster, survival data exhibit correlation due to shared environmental factors, primarily cluster‐level characteristics such as hospital type (e.g., public or private hospitals in the US, teaching or non‐teaching hospitals in Canada). Random effects are commonly introduced to capture these shared characteristics and to model unmeasured heterogeneity between clusters that cannot be directly accounted for by cluster‐level covariates. Conditional on the random effects (frailty), the survival times within a cluster can be assumed independent, which leads to a shared frailty model to account for cluster‐level uncertainty [4, 5, 6].

Our research is motivated by data from intensive care units (ICUs) across multiple clinical centers in Ontario, Canada. The ICU data were provided by the Critical Care Information System (CCIS) Ontario database. In Canadian ICUs, invasive mechanical ventilation is prevalent, with approximately one‐third of patients requiring ventilation during their ICU stay [7]. The duration of ventilation has significant implications for clinical outcomes and is associated with an increased risk of complications [7]. Analysis on the ventilation duration for ICU patients utilizing the associated risk factors, such as patient categories (e.g., medical or surgical), admission diagnoses, and patient severity, can provide insights into interventions and treatments that may reduce time‐to‐extubation from initiation. Such insights can indirectly support capacity planning and effective clinical resource management by optimizing the allocation of ventilator‐equipped beds. The severity of patients in ICU is often assessed using the Multiple Organ Dysfunction Score (MODS) [8] that evaluates organ function. Kobara et al. [9] conducted a study on ventilation duration for ICU patients from Ontario using a survival analysis framework. In their study, patient ventilation duration is considered as a time‐to‐event (survival) outcome and parametric accelerated failure time (AFT) models are applied to predict the ventilation time. If a patient is transferred to another facility without subsequent follow‐up information, the ventilation time is considered right censored. Kobara et al. found that the log‐logistic AFT model well describes the association between risk factors and patients' ventilation duration in Ontario ICUs. However, Kobara et al. did not consider a possible correlation of ventilation duration times among patients within the same ICU. Previous studies have indicated that patient outcomes within the same ICU site may be correlated, and ignoring this hierarchical structure can result in flawed prediction models [10, 11].

As aforementioned, shared frailty can be used to accommodate cluster‐level association among patients within the cluster. Gorfine and Zucker [6] recently provided a comprehensive review of shared frailty methods for complex survival data. To incorporate risk factors in survival data, the Cox proportional hazards (PH) model with a multiplicative shared frailty on the hazard rates has been widely developed [6]. As an alternative to the Cox PH model, AFT models offer an intuitive interpretation of covariate effects on survival time. Lambert et al. [12] developed shared frailty AFT models with different parametric distributions assumed for the survival time and conducted maximum likelihood estimation by integrating out the unobserved frailties. They empirically found that the choice of distribution for the shared frailty is not critical, recommending the normal distribution. Do Ha et al. [13, 14] considered a shared frailty log‐normal AFT model and used the hierarchical‐likelihood (h‐likelihood) approach to estimate the model parameters. The h‐likelihood method obtains fixed effects estimates by maximizing the h‐likelihood, while utilizing the restricted maximum likelihood estimate for the estimation of the variance of the shared frailty [14]. An R package, frailtyHL [15] has been developed to implement the h‐likelihood estimation in the shared frailty log‐normal AFT model. Building on this, Park and Do Ha [16] introduced a penalized variable selection technique for the shared frailty log‐normal AFT model. Additionally, Zhou et al. [17] developed a Markov chain Monte Carlo (MCMC)‐based algorithm, called survregBayes, to estimate the shared frailty AFT model, and an R package, spBayesSurv [18], is available for its implementation. To our knowledge, there is no work on variational Bayesian inference for shared frailty AFT models.

As an alternative to MCMC methods, which are the gold standard for obtaining posterior distributions under a Bayesian framework, variational inference (VI) has gained popularity due to its favourable results and lower computational cost than MCMC. Recently, several types of VI algorithms have been developed, including mean‐field VI [19], stochastic VI [20], and black‐box VI [21]. A special case of mean‐field VI, called mean‐field variational Bayes (VB), arises when the Kullback–Leibler (KL) divergence is utilized to quantify the dissimilarity between exact and approximated posterior distributions. In addition, the approximated posterior distribution, referred to as the variational posterior, is assumed to belong to a mean‐field variational family. Under the mean‐field VB framework, the solutions to minimizing the KL divergence can be obtained by utilizing the coordinate ascent algorithm [19, 22]. Mean‐field VB has been widely applied to regression models such as the generalized additive model [23], nonparametric regression with measurement error [24], count response semiparametric regression [25], high‐dimensional linear regression [26], multilevel regression modeling [27], B‐spline regression mixture model for functional data clustering [28], basis selection for functional data representation [29], among others. Applications of other types of VI can be found in a comprehensive review of VI from a statistical perspective by Blei et al. [30].

In this study, we propose a shared frailty log‐logistic AFT model to account for the correlation among patient ventilation durations within the ICU sites. Ventilation duration is defined as the time‐to‐extubation [31] for patients after initiation on invasive mechanical ventilation. Similar to Kobara et al. [9], we treat time‐to‐extubation from initiation as a survival time outcome, considering it right‐censored if the patient is transferred to another facility while still on a ventilator with no follow‐up data available. We develop a novel and fast mean‐field VB algorithm to infer the model parameters. By applying the piece‐wise approximation techniques proposed by Xian et al. [32] to avoid intractable calculations, we obtain closed‐form posterior distributions. We conduct extensive simulation studies with various numbers of clusters and cluster sizes to evaluate the performance of the proposed method, and compare the performance of our VB algorithm with the h‐likelihood method [14] and the MCMC‐based algorithm survregBayes [18]. We also evaluate the performance of the VB algorithm using simulated data under scenarios involving survival distribution misspecification and imbalanced cluster sizes. Finally, we apply our methodology to investigate ventilation duration for ICU patients using the same dataset as Kobara et al. [9]. This study was approved by the Research Ethics Review Committee at King's University College at Western University. Our method is implemented as part of the R package survregVB available at CRAN (https://cran.r‐project.org/web/packages/survregVB/index.html).

The remainder of the paper is organized as follows. Section 2 presents the log‐logistic AFT model with a shared frailty under the Bayesian framework. We introduce our proposed VB algorithm in Section 3. In Section 4, simulation studies are conducted to evaluate the performance of our VB algorithm. Section 5 illustrates the application of the proposed method to the ICU ventilation duration data. A discussion is provided in Section 6.

2. Bayesian Log‐Logistic AFT Model With a Shared Frailty

Let $T_{ij}$ and $C_{ij}$ be the survival and censoring times, respectively, of the $j\text{th}$ subject from the $i\text{th}$ group (i.e., cluster) in a sample, $i=1,\dots ,K$ and $j=1,\dots ,n_i$. Let $t_{ij}=\min (T_{ij},C_{ij})$ and ${\delta }_{ij}=\mathbb{1}(T_{ij}\le C_{ij})$ be the subject's observed time and the indicator for right censoring, respectively. We consider a log‐logistic AFT model with shared frailty (a random intercept) specified as follows:

$$\log (T_{ij})={\gamma }_i+{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }+b{ϵ}_{ij}$$ (1)

where ${\mathbf{\text{X}}}_{ij}$ is a column vector with length $p,p\ge 2,$ containing $p-1$ fixed effects (covariates) and a constant one to incorporate the constant intercept (i.e., ${\mathbf{\text{X}}}_{ij}={(1,x_{ij1},\dots ,x_{ij(p-1)})}^T$), $\mathit{\beta }$ is the corresponding vector of coefficients for the fixed effects, where ${\gamma }_i$ is a random intercept for the $i\text{th}$ cluster and ${ϵ}_{ij}$ is a random variable following a standard logistic distribution, and $b$ is a scale parameter. We further assume that ${\gamma }_i\stackrel{\text{iid}}{\sim }N(0,{\sigma }_{\gamma }^2)$ represents the discrepancy between clusters with iid denoting identically and independently distributed. The survival time $T_{ij}$ and censoring time $C_{ij}$ are assumed independent given the covariates ${\mathbf{\text{X}}}_{ij}$. Our model follows a structure similar to the one presented by Robison (1991) [33] and Nolan et al. [34] for a Gaussian linear mixed‐effects model.

In our proposed Model (1), we incorporate the unknown and unobserved shared risk through cluster‐specific random intercepts and estimate the model parameters, $\mathit{\beta }$, $b$, and ${\sigma }_{\gamma }^2$ using a Bayesian framework by further assuming the following independent marginal prior distributions:

• $\mathit{\beta }\sim N_p({\mathit{\mu }}_0,{\sigma }_0^2{\mathbf{\text{I}}}_{p\times p})\text{with precision}{v}_0=1/{\sigma }_0^2$ and ${\mathbf{\text{I}}}_{p\times p}$ being a $p\times p$ identity matrix

• $b\sim \text{Inverse‐Gamma}({\alpha }_0,{\omega }_0)$
• ${\gamma }_i|{\sigma }_{\gamma }^2\stackrel{\text{iid}}{\sim }N(0,{\sigma }_{\gamma }^2),i=1,\dots ,K$
• ${\sigma }_{\gamma }^2\sim \text{Inverse‐Gamma}({\lambda }_0,{\eta }_0)$

Where ${\mu }_0,v_0,{\alpha }_0$, ${\omega }_0$, ${\lambda }_0$ and ${\eta }_0$ are known hyperparameters (parameters of the prior distributions).

3. Variational Bayes Algorithm

In what follows, we outline our methodology for deriving a mean‐field VB algorithm for Model (1). We summarize the resulting VB algorithm in Algorithm 1.

ALGORITHM 1. Variational Bayesian inference of correlated survival data using a shared frailty log‐logistic AFT model.

Given the observed data $\mathbf{\text{D}}:=\{(t_{ij},{\delta }_{ij},{\mathbf{X}}_{ij}),i=1,\dots ,K,j=1,\dots ,n_i\}$, we denote the complete data log‐likelihood by $\log p(\mathbf{\text{D}},\mathit{\beta },\mathit{\gamma },b,{\sigma }_{\gamma }^2)$, where $\mathit{\gamma }=({\gamma }_1,\dots ,{\gamma }_K)$. Our objective is to derive a VB algorithm to approximate the exact posterior joint distribution of $\mathit{\beta }$, $b$, $\mathit{\gamma }$ and ${\sigma }_{\gamma }^2$ given the data $\mathbf{\text{D}}$ by maximizing the evidence lower bound (ELBO) defined as

$$\text{ELBO}(q)={𝔼}_q[\log p(\mathbf{\text{D}},\mathit{\beta },\mathit{\gamma },b,{\sigma }_{\gamma }^2)]-{𝔼}_q[\log q(\mathit{\beta },b,\mathit{\gamma },{\sigma }_{\gamma }^2)]$$ (2)

where $q(\mathit{\beta },b,\mathit{\gamma },{\sigma }_{\gamma }^2)$ is the approximated posterior joint distribution, which is also called the variational density, and the expectation is taken with respect to the variational density [30]. We consider the mean‐field variational family which assumes that $q(\mathit{\beta },b,\mathit{\gamma },{\sigma }_{\gamma }^2)=q(\mathit{\beta })q(b)q({\sigma }_{\gamma }^2){\prod }_{i=1}^{K}q({\gamma }_i)$, and apply the coordinate ascent variational inference (CAVI) algorithm [19] to obtain each variational component (e.g., $q(\mathit{\beta })$ in $q(\mathit{\beta },b,\mathit{\gamma },{\sigma }_{\gamma }^2)$). Under our Model (1), the complete data log‐likelihood $\log p(\mathbf{\text{D}},\mathit{\beta },\mathit{\gamma },b,{\sigma }_{\gamma }^2)$ can be obtained by

$$\begin{array}{ll}\hfill \log p(\mathbf{\text{D}},\mathit{\beta },\mathit{\gamma },b,{\sigma }_{\gamma }^2)& =\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)+\log p(\mathit{\beta })+\log p(b)\\ \hfill & +\sum _{i=1}^K\log p({\mathit{\gamma }}_i|{\sigma }_{\gamma }^2)+\log p({\sigma }_{\gamma }^2)\end{array}$$ (3)

where

$$\begin{array}{ll}\hfill & \log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)\\ \hfill & =-\delta \log b+\sum _{i=1}^K\sum _{j=1}^{n_i}\left[{\delta }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right.\\ \hfill & \left.-(1+{\delta }_{ij})\log \left\{1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)\right\}\right]\end{array}$$ (4)

with $\delta ={\sum }_{i=1}^K{\sum }_{j=1}^{n_i}{\delta }_{ij}$ being the number of observed uncensored times and $y_{ij}=\log (t_{ij}),i=1,\dots ,K,j=1,\dots ,n_i,$ being the log observed survival time.

By maximizing the ELBO, the following solutions are provided by the CAVI algorithm:

$$\log q^{\ast }(\mathit{\beta })\stackrel{+}{\approx }{𝔼}_{-\mathit{\beta }}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)+\log p(\mathit{\beta })]$$

$$\log q^{\ast }({\gamma }_i)\stackrel{+}{\approx }{𝔼}_{-{\gamma }_i}\left[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)+\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]$$

$$\log q^{\ast }(b)\stackrel{+}{\approx }{𝔼}_{-b}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)+\log p(b)]$$

$$\log q^{\ast }({\sigma }_{\gamma }^2)\stackrel{+}{\approx }{𝔼}_{-{\sigma }_{\gamma }^2}\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)+\log p({\sigma }_{\gamma }^2)\right]$$

where we use $\stackrel{+}{\approx }$ to denote equality up to a constant additive factor for convenience, and $-\mathit{\beta }$ indicates the expectation is taken with respect to the variational density of other latent variables but $\mathit{\beta }$, same for other solutions. To achieve conjugacy and tractable expectation calculation of $\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)$ as specified in (4), we apply the proposed method by Xian et al. [32], piecewise approximations of the function, $f(x)=\log (1+\exp (x)),x\in (-\infty ,\infty )$, embedded in deriving the update equations for each parameter. As in Nolan et al. [34] and Lee and Wand [35], we are also interested in the posterior distribution of the random effects, $q^{\ast }({\gamma }_i),i=1,\dots ,K$, in our proposed VB framework.

3.1. Update Equation for Each Variational Density

The update equations to obtain the optimal variational densities of $\mathit{\beta }$, ${\gamma }_i$, $b$ and ${\sigma }_{\gamma }^2$ denoted by $q^{\ast }(\mathit{\beta })$, $q^{\ast }({\gamma }_i)$, $q^{\ast }(b)$ and $q^{\ast }({\sigma }_{\gamma }^2)$, respectively, which are the corresponding approximated posterior distributions, are presented as follows within the paper while their derivations are given in Appendix A. The calculations of expectations in the update equations are given in Section 3.2. In the update equations, ${\phi }_{ij}$, ${\zeta }_{ij}$, and ${\rho }_{ij}$ represent the piece‐wise approximation coefficients proposed by Xian et al. [32] for the $j\text{th}$ subject in the $i\text{th}$ cluster. Detailed information regarding these coefficients is also provided in Appendix A.

(1) Update equation for

$q^{\ast }(\mathit{\beta })$

$q^{\ast }(\mathit{\beta })$ is an $N_p({\mathit{\mu }}^{\ast },{\sum }^{\ast })$, where

$${\sum }^{\ast }={\left[v_0{\mathbf{\text{I}}}_{p\times p}+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{i=1}^K\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}{\mathbf{\text{X}}}_{ij}{\mathbf{\text{X}}}_{ij}^T\right]}^{-1}$$

and

$$\begin{array}{ll}\hfill {\mathit{\mu }}^{\ast }& =\left[\left\{v_0{\mathit{\mu }}_0^T+\sum _{i=1}^K\sum _{j=1}^{n_i}\left(\left({𝔼}_{q(b)}\left(\frac{1}{b}\right)\left(-{\delta }_{ij}+\left(1+{\delta }_{ij}\right){\rho }_{ij}\right)\right.\right.\right.\right.\\ \hfill & {\left.\left.\left.\left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{𝔼}_{q({\gamma }_i)}({\gamma }_i))\right){\mathbf{\text{X}}}_{ij}^T\right)\right\}{\sum }^{\ast }\right]}^T\end{array}$$

(2) Update equation for $q^{\ast }({\gamma }_i)$

$q^{\ast }({\gamma }_i)$ is an $N({\tau }_i^{\ast },{\sigma }_i^{2\ast })$, where

$$\begin{array}{ll}\hfill {\tau }_i^{\ast }& ={\sigma }_i^{2\ast }\sum _{j=1}^{n_i}\left[{𝔼}_{q(b)}\left(\frac{1}{b}\right)(-{\delta }_{ij}+(1+{\delta }_{ij}){\rho }_{ij})\right.\\ & \left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}\mathit{\beta })\right]\end{array}$$

and

$${\sigma }_i^{2\ast }={\left[{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}\right]}^{-1}$$

(3) Update equation for $q^{\ast }(b)$

$q^{\ast }(b)$ is an $\text{Inverse‐Gamma}({\alpha }^{\ast },{\omega }^{\ast })$, where ${\alpha }^{\ast }={\alpha }_0+\delta$ and

$$\begin{array}{ll}\hfill {\omega }^{\ast }& ={\omega }_0-\sum _{i=1}^K\sum _{j=1}^{n_i}({\delta }_{ij}-(1+{\delta }_{ij}){\phi }_{ij})(y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}(\mathit{\beta })-{𝔼}_{q({\gamma }_i)}{\gamma }_i)\end{array}$$

(4) Update equation for $q^{\ast }({\sigma }_{\gamma }^2)$

$q^{\ast }({\sigma }_{\gamma }^2)$ is an $\text{Inverse‐Gamma}({\lambda }^{\ast },{\eta }^{\ast })$, where ${\lambda }^{\ast }={\lambda }_0+K/2$ and

$${\eta }^{\ast }={\eta }_0+\frac{1}{2}\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}[{\gamma }_i^2]$$

3.2. ELBO Calculation

The ELBO under Model (1) is defined in (2) with the complete‐data log‐likelihood calculated by (3) and note that $q(\mathit{\beta },b,\mathit{\gamma },{\sigma }_{\gamma }^2)=q(\mathit{\beta })q(b)q({\sigma }_{\gamma }^2){\prod }_{i=1}^{K}q({\gamma }_i)$. Let

$${\mathit{\text{diff}}}_{\mathit{\beta }}={𝔼}_q[\log p(\mathit{\beta })]-{𝔼}_q[\log q(\mathit{\beta })]$$

$${\mathit{\text{diff}}}_{\mathit{\gamma }}={𝔼}_q\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]-{𝔼}_q\left[\sum _{i=1}^K\log q({\gamma }_i)\right]$$

$${\mathit{\text{diff}}}_b={𝔼}_q[\log p(b)]-{𝔼}_q[\log q(b)]$$

$${\mathit{\text{diff}}}_{{\sigma }_{\gamma }^2}={𝔼}_q[\log p({\sigma }_{\gamma }^2)]-{𝔼}_q[\log q({\sigma }_{\gamma }^2)]$$

Then we can calculate the ELBO as follows with proof details provided in Appendix B.

$$\text{ELBO}(q)={𝔼}_q[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]+{\mathit{\text{diff}}}_{\mathit{\beta }}+{\mathit{\text{diff}}}_{\mathit{\gamma }}+{\mathit{\text{diff}}}_b+{\mathit{\text{diff}}}_{{\sigma }_{\gamma }^2}$$ (5)

where

$$\begin{array}{ll}\hfill & {𝔼}_q[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]\\ \hfill & \stackrel{+}{\approx }-\delta {𝔼}_{q(b)}(\log b)+{𝔼}_{q(b)}\left(\frac{1}{b}\right)\sum _{i=1}^K\sum _{j=1}^{n_i}({\delta }_{ij}-(1+{\delta }_{ij}){\phi }_{ij})\\ \hfill & \times (y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}(\mathit{\beta })-{𝔼}_{q({\gamma }_i)}{\gamma }_i)\end{array}$$

$${\mathit{\text{diff}}}_{\mathit{\beta }}\stackrel{+}{\approx }-\frac{1}{2}{v}_0[\text{trace}({\sum }^{\ast })+{({\mathit{\mu }}^{\ast }-{\mathit{\mu }}_0)}^T({\mathit{\mu }}^{\ast }-{\mathit{\mu }}_0)]+\frac{1}{2}\log (|{\sum }^{\ast }|)$$

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_{\mathit{\gamma }}& \stackrel{+}{\approx }-\frac{K}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)-\frac{1}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}({\gamma }_i^2)-\frac{1}{2}\sum _{i=1}^K\left(\log {\sigma }_i^{2\ast }\right)\end{array}$$

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_b& \stackrel{+}{\approx }({\alpha }^{\ast }-{\alpha }_0){𝔼}_{q(b)}(\log b)+({\omega }^{\ast }-{\omega }_0){𝔼}_{q(b)}\left(\frac{1}{b}\right)-{\alpha }^{\ast }\log {\omega }^{\ast },\text{and}\end{array}$$

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_{{\sigma }_{\gamma }^2}& \stackrel{+}{\approx }({\lambda }^{\ast }-{\lambda }_0){𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)+({\eta }^{\ast }-{\eta }_0){𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)-{\lambda }^{\ast }\log {\eta }^{\ast }\end{array}$$

We now present the calculation of the expectations in the update equations and the ELBO calculation. All the expectations are taken with respect to the approximated variational distributions. Since $q^{\ast }(b)$ is an Inverse‐$\text{Gamma}({\alpha }^{\ast },{\omega }^{\ast })$, we have

$${𝔼}_{q(b)}\left(\frac{1}{b}\right)=\frac{{\alpha }^{\ast }}{{\omega }^{\ast }}$$

$$\begin{array}{ll}\hfill {𝔼}_{q(b)}\left(\frac{1}{b^2}\right)& ={𝔼}_{q(b)}\left[{\left(\frac{1}{b}\right)}^2\right]={\mathrm{Var}}_{q(b)}\left[\left(\frac{1}{b}\right)\right]+{\left[{𝔼}_{q(b)}\left(\frac{1}{b}\right)\right]}^2\\ \hfill \hspace{0.17em}& =\frac{{\alpha }^{\ast }}{{\omega }^{\ast 2}}+\frac{{\alpha }^{\ast 2}}{{\omega }^{\ast 2}}=\frac{{\alpha }^{\ast }+{\alpha }^{\ast 2}}{{\omega }^{\ast 2}},\text{and}\end{array}$$

$${𝔼}_{q(b)}(\log b)=\log ({\omega }^{\ast })-\psi ({\alpha }^{\ast })$$

where $\psi$ is the digamma function defined as $\psi (x)=\frac{d}{dx}\log \mathrm{\Gamma }(x)$.

Similarly,

$${𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)=\frac{{\lambda }^{\ast }}{{\eta }^{\ast }}\text{and}{𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)=\log ({\eta }^{\ast })-\psi ({\lambda }^{\ast })$$

4. Simulation Studies

4.1. Simulation Study 1: Different Number of Clusters and Cluster Sizes

4.1.1. Design of Simulation Study 1

In simulation study 1, we evaluate the performance of our proposed VB algorithms across different scenarios by varying the number of clusters and the number of observations within each cluster.

We generate the log of survival time for the $j\text{th}$ subject in the $i\text{th}$ cluster, $\log (T_{ij})$, $i=1,\dots ,K$ and $j=1,\dots ,n_i$ as follows:

$$\log T_{ij}=0.5+{\beta }_1{x}_{ij1}+{\beta }_2{x}_{ij2}+{\gamma }_i+b{ϵ}_{ij}$$ (6)

where $x_{ij1}$, $x_{ij2}$, and ${ϵ}_{ij}$ are mutually independently generated with $x_{ij1}\sim N(1,0.2^2)$, $x_{ij2}\sim \text{Bernoulli}(0.5)$ and ${ϵ}_{ij}\sim \text{logistic}(0,1)$. The values of ${\beta }_1$, ${\beta }_2$ and $b$ are chosen as 0.2, 0.8 and 0.8, respectively. The random intercept for the $i\text{th}$ cluster, ${\gamma }_i$, is generated from $N(0,{\sigma }_{\gamma }^2)$ with ${\sigma }_{\gamma }^2=1$. The censoring time for the $j\text{th}$ subject in the $i\text{th}$ cluster, $C_{ij}$, is generated from a uniform distribution, $\text{uniform}(0,d)$, where $d$ is a positive value controlling the percentage of censoring. Then $t_{ij}=\min (T_{ij},C_{ij})$ and ${\delta }_{ij}=\mathbb{1}(T_{ij}\le C_{ij})$. Take $d=48$ to achieve a 15% censoring rate in our simulations. Investigation of the effect of censoring rates can be found in Xian et al. [32], where it is shown that as the censoring rate increases, an increase in the mean squared error (MSE) of estimating parameters in an AFT model using a VB algorithm is observed.

We explore various scenarios by varying the number of clusters, $K$, and the number of observations within each cluster, $n_i=n$ for all $i=1,\dots ,K$. Across the experiments, we consider $K$ values from the set $\{15,30,50,\text{and}80\}$ and $n$ values from $\{5,15,30,\text{and}50\}$, resulting in a total of 16 unique scenarios. Our objective is to assess the estimation performance of the VB algorithm concerning the variations in $K$ and $n$. We consider a prior setting with ${\mathit{\mu }}_0={(0,0,0)}^T$, $v_0=0.1$, ${\alpha }_0={\lambda }_0=3$, and ${\omega }_0={\eta }_0=2$, which indicates no strong prior information on the parameters. The ELBO convergence threshold is set as 0.01, which is the default recommendation [36], and the maximum number of iterations is 100.

We conduct $N=500$ runs (replicates) in each considered scenario and apply our proposed VB algorithm to derive the approximated posterior distribution of each parameter for each run. The mean of each approximated posterior distribution is used as the parameter estimate for the corresponding parameter. The empirical bias and sample standard deviation (SD) as well as the empirical MSE for each estimate are obtained, where

$$\text{MSE}=\frac{{\sum }_{i=1}^N{({\theta }_0-{\widehat{\theta }}_i)}^2}{N}$$

and ${\widehat{\theta }}_i$ is the estimate of parameter $\theta$ in the $i\text{th}$ replicate, and ${\theta }_0$ is the true value. In addition, for each parameter of interest, we report the empirical 95% coverage rate (CR) calculated as

$$\text{CR}=\frac{{\sum }_{i=1}^N{I}_i}{N}$$

where $I$ is the indicator variable that equals 1 if the true parameter value ${\theta }_0$ falls within the corresponding 95% credible interval, and 0 otherwise.

4.1.2. Results of Simulation Study 1

The empirical bias, sample SD, empirical MSE, and empirical CR of estimating ${\beta }_1$, ${\beta }_2$, $b$, and ${\sigma }_{\gamma }^2$ from 500 replicates are summarized in Table 1. Considering ${\beta }_1$, ${\beta }_2$, and $b$, we observe that when the number of clusters $K$ is fixed, increasing the cluster size $n$ from 5 to 50 does not always significantly affect the empirical bias. However, there is a noticeable decrease in the sample SD, leading to a pronounced reduction in the empirical MSE. A similar trend is observed when the cluster size is fixed while increasing $K$. Figure 1 visually illustrates this asymptotic property through boxplots. Our primary focus lies in understanding the impact of $K$ and $n$ on estimating the variance of the random intercept, denoted as ${\sigma }_{\gamma }^2$. In most of the scenarios, when the number of clusters $K$ is fixed, we observe that the bias of estimating ${\sigma }_{\gamma }^2$ decreases as the cluster size increases, while the sample SD remains relatively stable. However, when $K$ is increased, both the empirical bias and the sample SD decrease noticeably, except for the case when $n=30$ as $K$ grows from 15 to 30. Particularly noteworthy is the scenario with $K=80$ and $n=50$, where we achieve a less biased and less variable estimation of the variance of the random intercept. In addition, the empirical CRs corresponding to a 95% credible interval for each parameter across all scenarios are close to the nominal level of 95%, with a mean of 94.1% and a standard deviation of 0.014. Furthermore, based on a nominal level of 95% with 500 replicates, the coverage with two standard deviations is $95\%\pm 2\text{SD}$ = $[93.1\%,97.0\%]$ since $\text{SD}=\sqrt{0.95\times 0.05/500}\approx 0.0097$. We observe that the majority of our empirical CRs are within the range of $[93.1\%,97.0\%]$.

TABLE 1.

Simulation study 1: Numerical estimation results (point estimate using the mean of the corresponding posterior distribution) including the empirical Bias, sample SD, empirical MSE, and coverage rate (CR) for parameters in each scenario with different numbers of clusters $K$ and cluster sizes $n$ from the proposed VB algorithm.

		${\beta }_1$				${\beta }_2$				$b$				${\sigma }_{\gamma }^2$
$K$	$n$	Bias	SD	MSE	CR	Bias	SD	MSE	CR	Bias	SD	MSE	CR	Bias	SD	MSE	CR
15	5	0.066	0.764	0.587	94.2	−0.026	0.352	0.124	90.0	0.005	0.095	0.009	94.5	−0.101	0.312	0.107	91.4
	15	0.018	0.489	0.239	93.2	0.007	0.192	0.037	94.0	0.006	0.052	0.003	94.8	−0.081	0.287	0.089	93.1
	30	0.006	0.354	0.125	93.2	−0.017	0.139	0.019	92.4	−0.006	0.036	0.001	96.0	0.003	0.290	0.087	95.5
	50	0.009	0.254	0.065	95.6	−0.001	0.104	0.011	94.8	0.003	0.029	0.001	94.7	−0.040	0.299	0.100	94.2
30	5	0.004	0.589	0.346	92.8	−0.015	0.257	0.066	92.8	0.015	0.071	0.005	94.4	−0.092	0.274	0.084	92.6
	15	0.002	0.360	0.129	93.0	−0.018	0.140	0.020	93.0	0.008	0.038	0.001	96.1	−0.057	0.242	0.062	91.1
	30	0.012	0.239	0.057	94.6	−0.004	0.088	0.008	95.7	0.005	0.026	0.001	95.6	−0.036	0.248	0.063	92.4
	50	−0.007	0.192	0.037	93.0	−0.005	0.074	0.005	95.0	0.003	0.020	$<0.001$	95.8	−0.026	0.228	0.052	95.2
50	5	−0.011	0.470	0.220	94.3	−0.033	0.199	0.041	91.4	0.014	0.051	0.003	95.1	−0.062	0.235	0.059	93.2
	15	0.003	0.253	0.064	95.2	−0.001	0.104	0.011	94.8	0.018	0.029	0.001	94.2	−0.033	0.203	0.042	92.4
	30	−0.006	0.192	0.037	92.3	−0.006	0.074	0.006	94.4	0.003	0.020	$<0.001$	95.6	−0.027	0.191	0.037	93.3
	50	−0.002	0.143	0.020	94.4	−0.004	0.057	0.003	95.6	0.002	0.016	$<0.001$	95.7	0.016	0.201	0.040	93.4
80	5	0.013	0.384	0.147	93.2	−0.005	0.161	0.026	91.3	0.014	0.041	0.002	95.2	−0.048	0.197	0.041	94.4
	15	−0.008	0.223	0.050	93.4	−0.005	0.090	0.008	92.4	0.008	0.023	0.001	95.5	−0.023	0.160	0.026	92.2
	30	0.008	0.141	0.020	95.8	−0.005	0.058	0.003	95.1	−0.003	0.015	$<0.001$	95.6	0.003	0.161	0.026	92.8
	50	−0.005	0.108	0.012	95.4	−0.002	0.043	0.002	95.5	0.003	0.013	$<0.001$	95.5	0.014	0.157	0.025	94.4

FIGURE 1.

Boxplots of estimates of (a) ${\beta }_1$, (b) ${\beta }_2$, (c) $b$, and (d) ${\sigma }_{\gamma }^2$. Estimates correspond to the posterior means from 500 replicates under various scenarios with different numbers of clusters $K$ and cluster sizes $n$ based on our proposed VB algorithm. The horizontal dashed line on each plot represents the true value of the corresponding parameter used when generating the data.

To assess the significance of the cluster‐specific random intercepts in the AFT model, we visualize the estimated random effects and their 95% credible intervals for one simulated dataset from the scenario with $K=80$ clusters and $n=30$ observations per cluster (Figure 2, right panel). For comparison, we also generate data under the same setting but without random intercepts ${\gamma }_i,i=1,2,\dots ,80$, and fit the shared frailty log‐logistic AFT model using our VB method (Figure 2, left panel). In both panels, the estimated random intercepts are ordered by magnitude for clarity. In the no‐frailty case (Figure 2, left panel), all credible intervals for the estimated random effects overlap and contain zero, suggesting no significant between‐cluster variation. In contrast, when the data include random intercepts, the estimated effects show greater variation across clusters than in the case without frailty. In addition, there is less overlap across credible intervals, along with intervals that do not contain zero, indicating the presence of significant frailty effects.

FIGURE 2.

Estimated cluster‐specific random effects and their 95% credible intervals using the proposed VB algorithm under the shared frailty log‐logistic AFT model with $K=80,n=30$. The left panel (a) shows results from a dataset generated without random intercepts; the right panel (b) shows results from a dataset generated with random intercepts. Random effects are ordered by magnitude to facilitate visualization.

Since the random intercepts are assumed a priori to have a mean of zero, their influence on the point estimates of the regression coefficients (fixed effects) and the overall goodness‐of‐fit is minimal. However, ignoring random effects when they are present distorts statistical inference on the fixed effects. In our simulated dataset with random intercepts under $K=80$, $n=30$, and ${\beta }_1=0.2$, ${\beta }_2=0.8$, the VB‐based shared frailty AFT model yields ${\widehat{\beta }}_1=0.290$ with a posterior standard deviation (SD) of 0.120, and a 95% credible interval (CI) of [0.054, 0.526], and ${\widehat{\beta }}_2=0.803$ with SD = 0.057, 95% CI = [0.690, 0.915]. In contrast, the VB‐based AFT model without shared frailty gives ${\widehat{\beta }}_1=0.289$ (SD = 0.181, 95% CI = [−0.065, 0.643]) and ${\widehat{\beta }}_2=0.787$ (SD = 0.070, 95% CI = [0.649, 0.925]). We observe that ignoring shared frailty inflates uncertainty, widens the credible intervals, and may make some of the fixed effects statistically insignificant.

The left part of Figure 3 presents the computational run time of 500 replicates in minutes against the number of clusters ($K$) with different colors for different cluster sizes $n$. Fixing $K$, as $n$ increases, the used time increases. The scatter plot on the right of Figure 3 shows the run time against the sample size ($K\times n$). As anticipated, the computational cost rises with increasing sample size.

FIGURE 3.

(a): The run time in minutes used for 500 replicates under various scenarios with different numbers of clusters $K$ and cluster sizes $n$ based on our proposed VB algorithm. (b): The run time in minutes used for 500 replicates in different sample sizes based on VB.

Furthermore, among those considered 16 scenarios, we select three scenarios ($K=30$ with $n=5$, $K=50$ with $n=15$, and $K=80$ with $n=30$) and conduct a comparative analysis of estimation results obtained from the proposed VB algorithm with those from two alternative methods which we introduced in Section 1: the h‐likelihood method proposed by Do Ha et al. [14] and the MCMC‐based survregbayes developed by Zhou et al. [17]. Summaries from the MCMC‐based survregbayes algorithm are derived from one Markov chain, subsampled every 5 iterations to achieve a final chain size of 2000 after a burn‐in period of 5000 iterations [17]. We further refer to our VB method and the h‐likelihood method as survregVBfrailty and survregHL, respectively. Numerical estimation results are presented in Table 2, where we focus on the comparison of estimation for ${\beta }_1$, ${\beta }_2$ and ${\sigma }_{\gamma }^2$ since the MCMC‐based survregbayes does not directly return the estimate for $b$. In the scenario with $K=30$ and $n=5$, the proposed survregVBfrailty exhibits a larger empirical bias compared to both the survregHL and survregbayes methods. However, as $K$ and $n$ increase, the discrepancy in empirical bias among the three methods becomes negligible. In terms of the sample SD, on average, across the three parameters ${\beta }_1$, ${\beta }_2$, and ${\sigma }_{\gamma }^2$, survregVBfrailty yields a 16.5% smaller sample SD when $K=30$ and $n=5$, a 7.6% smaller sample SD when $K=50$ and $n=15$, and a 3.7% smaller sample SD when $K=80$ and $n=30$, compared to survregHL. Compared with the survregBayes algorithm, we observe a decrease in sample SD of 7.8% in the $K=30$, $n=5$ scenario, mainly due to ${\beta }_1$, and only 2.0% in the $K=50$, $n=15$ scenario. However, in the scenario with $K=80$, $n=30$, no average difference in sample SD is observed between VB and MCMC. Therefore, the survregVBfrailty algorithm generally yields a smaller SD, resulting in a lower MSE compared to the other two methods in most cases. On average, across the three parameters ${\beta }_1$, ${\beta }_2$, and ${\sigma }_{\gamma }^2$, survregVBfrailty achieves a 27.2% lower MSE when $K=30$, $n=5$, a 12.0% lower MSE when $K=50$, $n=15$, and a 13.4% lower MSE when $K=80$, $n=30$, compared to survregHL. Compared with survregBayes, we observe a reduction in MSE of 12.4% in the $K=30$, $n=5$ scenario, mainly because of ${\beta }_1$, and only 3.0% in the $K=50$, $n=15$ scenario. However, in the scenario with $K=80$, $n=30$, no difference in MSE is observed between VB and MCMC. In addition, we compare the posterior densities for ${\beta }_1$, ${\beta }_2$, and ${\sigma }_{\gamma }^2$ obtained from both survregVBfrailty and the MCMC‐based survregBayes using a simulated dataset with $K=50$ and $n=15$. These densities present a strong consistency between the VB and MCMC algorithms as illustrated in Figure 4, which is also reflected in the results presented in Table 2.

TABLE 2.

A comparison of numerical estimation results, including the empirical Bias, sample SD and MSE, from our VB method survregVBfrailty, the h‐likelihood method survregHL and MCMC‐based survregbayes method in each scenario ($K$: number of clusters, $n$: number of observations in each cluster).

		survregVBfrailty			survregHL			survregbayes
Scenario	Parameter	Bias	SD	MSE	Bias	SD	MSE	Bias	SD	MSE
$K=30$, $n=5$	${\beta }_1$	0.004	0.589	0.346	0.001	0.664	0.440	0.001	0.664	0.440
	${\beta }_2$	−0.015	0.257	0.066	−0.002	0.270	0.073	−0.002	0.266	0.071
	${\sigma }_{\gamma }^2$	−0.092	0.274	0.084	0.001	0.412	0.170	−0.042	0.300	0.092
$K=50$, $n=15$	${\beta }_1$	0.003	0.253	0.064	0.001	0.268	0.072	−0.005	0.259	0.067
	${\beta }_2$	−0.001	0.104	0.011	0.007	0.107	0.011	−0.004	0.104	0.011
	${\sigma }_{\gamma }^2$	−0.033	0.203	0.042	0.011	0.237	0.056	−0.001	0.211	0.044
$K=80$, $n=30$	${\beta }_1$	0.008	0.141	0.020	0.009	0.143	0.021	−0.009	0.142	0.020
	${\beta }_2$	−0.005	0.058	0.003	0.001	0.060	0.004	0.001	0.058	0.003
	${\sigma }_{\gamma }^2$	0.003	0.161	0.026	0.007	0.172	0.029	0.001	0.160	0.026

FIGURE 4.

A comparison of posterior densities of (a) ${\beta }_1$, (b) ${\beta }_2$ and (c) ${\sigma }_{\gamma }^2$ obtained from our VB method survregVBfrailty and MCMC‐based survregbayes from a simulated data set in the scenario with $K=50$ and $n=15$.

To assess the computational efficiency of the proposed VB algorithm, we compare the run time in minutes for 500 replicates in these three scenarios across survregVBfrailty, survregHL, and survregbayes methods and present the results in Table 3. In Table 3, we observe that in the scenario with a small sample size ($K=30$ and $n=5$), there is minimal difference in the run times between the VB and h‐likelihood methods. However, as both $K$ and $n$ increase, the h‐likelihood method requires progressively longer computation times to obtain estimates. Particularly in the scenario with $K=80$ and $n=30$, the computational time is over 10 times longer than that required by the VB algorithm. Comparatively, the VB algorithm demonstrates significantly higher computational efficiency when contrasted with the MCMC‐based survregbayes algorithm. Notably, in the scenario with $K=80$ and $n=30$, the survregVBfrailty algorithm runs approximately 150 times faster than the survregbayes algorithm. All algorithms were implemented within R version 4.3.2, and simulations were conducted on a computer operating the Mac OS X platform, equipped with a 4.05 GHz CPU and 8 GB RAM.

TABLE 3.

Times in minutes for 500 replicates from our VB algorithm survregVBfrailty, the h‐likelihood method survregHL and the MCMC‐based survregbayes algorithm, respectively, under each scenario.

Scenario	survregVBfrailty	survregHL	survregbayes
$K=30$, $n=5$	0.61	0.67	47.65
$K=50$, $n=15$	1.70	9.59	232.31
$K=80$, $n=30$	8.47	124.53	1302.80

4.2. Simulation Study 2: Misspecification of the Survival Distribution

We conduct a simulation study to evaluate the performance of the proposed VB algorithm for parameter estimation under the log‐logistic AFT model when data are generated from a misspecified survival distribution. We use the same simulation setup as in Section 4.1, with 50 clusters and 30 observations per cluster (i.e., $K=50,n=30$). However, unlike in Section 4.1, we independently generate ${ϵ}_{ij},i=1,\dots ,K;j=1,\dots ,n$, in (6), from either a standard extreme value distribution or a standard normal distribution. This results in survival times following a Weibull or log‐normal distribution, respectively, both of which are commonly used in survival analysis. We then apply the proposed VB algorithm under the log‐logistic AFT model to the simulated data for parameter estimation.

Table 4 presents a comparison of the numerical estimation results for the regression coefficients ${\beta }_1,{\beta }_2$ and the variance of the random intercept, ${\sigma }_{\gamma }^2$, based on the posterior means. The comparison includes empirical bias, sample standard deviation (SD), mean squared error (MSE), and credible interval coverage rate under both the correctly specified survival distribution (log‐logistic) and the misspecified models (Weibull or log‐normal AFT). The results in Table 4 demonstrate the robustness of the proposed VB algorithm to survival model misspecification. Across all three model specifications—log‐logistic (correct), Weibull (misspecified), and log‐normal (misspecified)—the estimated regression coefficients (${\beta }_1,{\beta }_2$) exhibit minimal bias. The standard deviations (SD) and mean squared errors (MSE) remain relatively stable, indicating that the VB algorithm provides reliable parameter estimates even under model misspecification. The variance of the random intercept (${\sigma }_{\gamma }^2$) shows slightly higher bias under the Weibull and log‐normal models compared to the correctly specified log‐logistic model. However, the credible interval coverage rates (CR) remain close to the nominal level across all models, further supporting the robustness of the VB approach.

TABLE 4.

Simulation study 2: Comparison of numerical estimation results, including empirical bias, sample SD, MSE, and coverage rate (CR), under the shared frailty log‐logistic (correct), Weibull (misspecified), and log‐normal (misspecified) AFT models.

	Log‐logistic				Weibull				Log‐normal
Parameter	Bias	SD	MSE	CR	Bias	SD	MSE	CR	Bias	SD	MSE	CR
${\beta }_1$	−0.006	0.192	0.037	92.3	−0.003	0.133	0.018	92.8	0.001	0.113	0.013	92.4
${\beta }_2$	−0.006	0.074	0.006	94.4	−0.007	0.052	0.003	93.0	−0.002	0.045	0.002	94.4
${\sigma }_{\gamma }^2$	−0.027	0.191	0.037	93.3	0.054	0.211	0.047	93.6	0.050	0.201	0.043	92.8

4.3. Simulation Study 3: Imbalanced Cluster Sizes

In this section, we evaluate the performance of the proposed VB algorithm under scenarios with imbalanced cluster sizes. We conduct a simulation study with $K=80$ clusters across two scenarios. In Scenario 1, cluster sizes are randomly and uniformly sampled from the set $\{5,15,30,50\}$, introducing a high degree of heterogeneity. In Scenario 2, 75 clusters have a size of 50, while the remaining 5 clusters have a much smaller size of 10, resulting in a highly unbalanced structure. All other simulation settings are identical to those described in Section 4.1.

Table 5 presents the empirical bias, SD, MSE, and CR for estimating ${\beta }_1$, ${\beta }_2$, $b$, and ${\sigma }_{\gamma }^2$ based on 500 replicates for both scenarios. In Scenarios 1 and 2, the proposed VB algorithm performs well in estimating the regression coefficients ${\beta }_1$, ${\beta }_2$, and the scale parameter $b$, showing similar bias, SD, MSE, and CR to those observed in the balanced setting with $K=80$ in Section 4.1 (see Table 1).

TABLE 5.

Simulation study 3: Numerical estimation results including the empirical Bias, SD, MSE and coverage rate (CR) for parameters in two considered scenarios with imbalanced cluster sizes.

	Scenario 1				Scenario 2
Parameter	Bias	SD	MSE	CR	Bias	SD	MSE	CR
${\beta }_1$	0.005	0.162	0.026	94.4	−0.008	0.120	0.015	93.8
${\beta }_2$	−0.002	0.067	0.004	93.6	−0.003	0.046	0.002	95.4
$b$	0.004	0.016	$<0.001$	95.4	0.004	0.012	$<0.001$	95.2
${\sigma }_{\gamma }^2$	0.002	0.169	0.029	91.0	0.022	0.159	0.026	94.8

Although imbalanced cluster sizes can pose challenges in estimating the variance of the random intercept (i.e., ${\sigma }_{\gamma }^2$), the VB algorithm remains robust. Specifically, the MSEs for ${\sigma }_{\gamma }^2$ are 0.029 in Scenario 1 and 0.026 in Scenario 2, which are comparable to those reported in Section 4.1 under balanced settings. More concretely, the estimation results in Scenario 1 are similar to those in the case with $K=80$, $n=30$, while Scenario 2 yields results comparable to the case with $K=80$, $n=15$ in Section 4.1. In conclusion, the proposed VB algorithm maintains satisfactory estimation performance even under substantial imbalance in cluster sizes.

5. Application to Ventilation Duration Analysis

In this section, we apply the shared frailty log‐logistic AFT model with the proposed VB algorithm to the ICU data as we described in the Introduction section and conduct a retrospective study on ventilation duration time. We aim to investigate the ICU site‐specific random effect on patients' ventilation duration. We extend the work by Kobara et al. [9] by incorporating the uncertainty from the ICU sites via a group‐specific random intercept under a Bayesian analysis framework. It is important to note that, unlike Kobara et al., we exclude ICU patients who did not survive (approximately 18% of the 49,467 patients) to eliminate competing risks from the analysis and ensure a clearer assessment of ventilation duration using our current proposed methodology.

The CCIS ICU data, excluding deaths, were collected between July 2015 and December 2016 and contained 40,426 patients receiving invasive mechanical ventilation upon arrival to the ICU. About 3% of these patients were discharged/transferred to a Complex Continuing Care Facility, other hospitals, the Level 3 Unit, and outside the ICU while still on a ventilator, and therefore, their ventilation time was considered as right‐censored data [9]. The data were from 66 ICU sites (centers), and each site has a unique site code, for example, 3970. In the CCIS dataset, the median number of ventilated patients per ICU site over the 1.5‐year study period is 307.5, with an interquartile interval of $[165.8,588.0]$. The median ventilation duration (defined as time‐to‐extubation from initiation) per ICU patient is 2 days, with an interquartile interval of $[1,5]$ days. We consider the significant covariates investigated by Kobara et al. [9] as fixed effects which are admission source (e.g., from operation rooms), admission diagnosis, patient type (medical or surgical), scheduled admission (yes or no), scheduled surgery (yes or no), referring physician specialty, other interventions (yes or no), central venous line (CVL, yes or no), arterial line (AL, yes or no), intra‐cranial pressure monitor (IPM, yes or no), extracorporeal membrane oxygen (EMO, yes or no), intra‐aortic balloon pump (IABP, yes or no), age group (18‐39, 40‐80 or above 80 years of age), pre‐LOS (no more than 1 day, between 2 and 7 days, or no less than 7 days), and the MODS score (none with MODS $\le 1$, minimal with $1-4$ scores, mild with $4-8$ scores, moderate with $8-12$ scores or, severe with scores $>12$).

To apply our proposed VB algorithm in real data analysis, we consider the same weak prior setting in our simulation study described in Section 4: ${\mathit{\mu }}_0=\mathbf{\text{0}}$, $v_0=0.1$, ${\alpha }_0={\lambda }_0=3$, and ${\omega }_0={\eta }_0=2$. Table C1 in Appendix C displays the estimated regression coefficients from the fitted shared frailty log‐logistic AFT model. For comparative analysis, we also employed the h‐likelihood method survregHL and the MCMC‐based survregbayes to model the data, incorporating shared frailty. In the absence of shared frailty, we used the likelihood‐based survreg from the survival package in R, and the VB method survregVB proposed by Xian et al. [32]. Furthermore, for the Bayesian methods (survregVBfrailty, survregVB, and survregbayes), we report the 95% credible intervals, whereas for the likelihood‐based methods, we provide the 95% confidence intervals. In Table C1, we first observe that there is no significant difference in the estimated regression coefficients between survregVBfrailty and survregbayes. However, some of the estimated coefficients from survregHL, such as scheduled surgery, CVL, AL, IPM, and EMO, are different from those based on survregVBfrailty and survregbayes, which is further discussed in Section 6. In comparison to the models with frailty, the survregVB and survreg methods, which do not account for frailty, exhibit some differences in estimating certain coefficients. However, they maintain a strong overall consistency in estimating the regression coefficients with the results from the frailty models. This consistency is expected, as the inclusion of frailty does not affect the fixed effects, implying that the regression lines from models with or without frailty should be parallel. To illustrate the difference in interval estimation between models with and without frailty, we calculated the mean overlap percentage of the two 95% credible intervals from VB with or without frailty. Using the 95% credible intervals from VB without frailty (survregVB) as references, we found a 77.3% overlap on average. This means that if we fit a model using VB without considering the ICU site as a shared frailty, about 77% of the credible interval will fall within the corresponding credible interval obtained from the model that includes the shared frailty.

In what follows, we summarize the fixed effects on ventilation duration based on estimates from our proposed VB algorithm, survregVBfrailty. For the baseline (reference) patient group—defined as patients aged 18 to 39, admitted from downstream units, with a cardiovascular diagnosis, without scheduled ICU admissions or surgeries, under medical referral physician service, without any specific treatment interventions (e.g., CVL, AL, IPM, EMO, and IABP), with $\le 1$ day pre‐ICU LOS, and with a MODS score $\le 1$—the estimated mean ICU ventilation duration is 1.40 days, with a 95% credible interval of [1.32, 1.48]. Regarding the admission source, patients arriving from the emergency department (ED), operating room (OR), or ward have a time‐to‐extubation from initiation in days that are $(1-\exp (-0.117))100\%=11.04\%$ shorter (credible interval (CI): [7.32%, 14.61%]), 28.04% shorter (CI: [24.80%, 31.20%]), and 13.32% shorter (CI: [9.52%, 16.89%]), respectively, compared to patients admitted from a downstream unit (baseline). Conversely, patients admitted from home or another hospital have ventilation durations that are 10.19% (CI: [−1.78%, 23.61%]) and 11.29% (CI: [6.40%, 16.30%]) longer, respectively. Since the CI for patients admitted from home includes zero, we conclude that admission from home is not a statistically significant factor. Additionally, patients from other sources, such as from outside the province, have ventilation durations that are 19.81% longer (CI: [6.82%, 34.31%]).

Compared to cardiovascular patients, those with gastrointestinal, neurological, and trauma diagnoses experience significantly longer ventilation times. Specifically, ventilation duration increases by 30.60% (CI: [25.73%, 35.53%]) for gastrointestinal patients, 32.05% (CI: [27.51%, 36.89%]) for neurological patients, and 74.37% (CI: [65.86%, 83.31%]) for trauma patients. Patient categories (surgical or medical) do not show significant differences in ventilation time. However, scheduled ICU admissions or surgeries are important factors. Patients with a scheduled ICU admission have a 23.66% shorter ventilation time (CI: [20.23%, 26.88%]), and those with a scheduled surgery have a 12.37% shorter ventilation time (CI: [8.33%, 16.31%]) compared to patients without scheduled admissions or surgeries. The referral physician service is also a significant risk factor. Compared with medical referrals, surgical referrals result in a 6.29% shorter ventilation time (CI: [2.66%, 9.88%]), while respirology referrals result in a 17.94% longer ventilation time (CI: [11.96%, 24.23%]).

Specific treatment interventions upon ICU arrival significantly impact ventilation duration. Patients receiving a CVL, AL, IPM, EMO, or IABP have increased ventilation times by 20.32% (CI: [17.70%, 23.00%]), 21.90% (CI: [19.12%, 24.86%]), 71.60% (CI: [60.48%, 83.49%]), 143.03% (CI: [120.12%, 168.59%]), and 46.81% (CI: [37.16%, 57.15%]), respectively. Age is also a significant factor. Compared to patients aged 18 to 39, those aged 40 to 80 and those over 80 have longer ventilation periods, increasing by 13.54% (CI: [10.30%, 16.88%]) and 7.68% (CI: [4.08%, 11.40%]), respectively.

Both pre‐ICU LOS and patients' severity scores, as measured by MODS, are significant risk factors for longer ventilation duration. Longer pre‐ICU LOS correlates with longer ventilation times. Patients with a pre‐ICU LOS of 2‐7 days or $\ge 7$ days experience increases in ventilation duration by 4.19% (CI: [1.82%, 6.61%]) and 12.98% (CI: [10.19%, 15.72%]), respectively, compared to those with $\le 1$ day pre‐ICU LOS. Compared to patients with a MODS score $\le 1$, those with minimal (1–4), mild (4–8), moderate (8–12), or severe ($>12$) MODS scores experience increases in ventilation duration by 9.97% (CI: [7.04%, 13.09%]), 23.12% (CI: [19.96%, 26.49%]), 29.30% (CI: [25.11%, 33.64%]), and 12.08% (CI: [4.71%, 20.08%]), respectively.

We now turn to the analysis of variance in terms of the variations from individual patients and from ICU sites. Using the mean of the posterior distribution for ${\sigma }_{\gamma }^2$, the estimated variance (${\sigma }_{\gamma }^2$) for the shared ICU site effect from survregVBfrailty is 0.097, with a square root of 0.311, which indicates that the average spread of the ventilation time among ICU sites is $\exp (0.311)$, or 1.37. The estimated scale parameter $b$ is 0.442, resulting in an estimated variance of the logarithms of individual patient ventilation times of $0.442^2\times {\pi }^2/3$, or 0.643, where ${\pi }^2/3$ is the variance of a standard logistic distribution. To measure the strength of the correlation between patients within the same ICU site, the intra‐class correlation coefficient between the logarithms of ventilation time can be estimated by $0.097/(0.097+0.643)\approx 0.131$.

On the left of Figure 5, we observe a strong consistency in estimating the variance of the ICU‐site specific random effect between survregVBfrailty and the MCMC‐based survregbayes. We further visualize the estimated ICU‐site random effects with the 95% credible intervals based on their posterior distributions obtained from survregVBfrailty, as shown on the right of Figure 5, where the random effects are ranked from smallest to largest. Wider credible intervals correspond to ICUs with larger numbers of patients. As discussed in Lambert et al. [12] and demonstrated in our simulation study, non‐overlapping credible intervals suggest the presence of significant cluster‐specific random effects. We observe that some of the intervals do not overlap, indicating that these ICU sites perform differently in terms of patient ventilation duration.

FIGURE 5.

(a): Posterior distribution of the variance of the random intercept from VB and MCMC. (b): Estimated ICU site‐specific random effects with their 95% credible interval from the proposed VB algorithm. The random effects have been ranked in an increasing order.

To assess the goodness‐of‐fit of the proposed log‐logistic AFT model in our ICU ventilation duration analysis, we computed individual Cox–Snell residuals [37] and evaluated their empirical survival function using the Kaplan–Meier (K‐M) estimator. If the model is appropriate, the K‐M survival curve of the residuals should closely follow the Exponential(1) survival function. As shown in Figure 6, the K‐M survival curve (black) closely aligns with the theoretical exponential curve (red), indicating a good fit of the log‐logistic AFT model to the data.

FIGURE 6.

Kaplan–Meier (K‐M) survival curve (black) of the Cox–Snell residuals compared to the theoretical Exponential(1) survival curve (red), based on the ICU ventilation duration analysis.

For reference on computational efficiency, the run times in minutes for each method are as follows: survregVBfrailty took 1.15 min, h‐likelihood took 86.40 min, MCMC‐based survregbayes took 208.26 min, survreg took 0.01 min, and survregVB took 0.08 min.

6. Discussion

In this paper, we have proposed a fast variational Bayesian algorithm, called survregVBfrailty, for statistical inference using a shared frailty log‐logistic AFT model, which can be applied to analyze clustered survival data. We demonstrated that our proposed survregVBfrailty algorithm achieves satisfactory estimation performance through simulation studies under various scenarios with different numbers of clusters and cluster sizes. Additionally, we further investigate the robustness of our approach by evaluating its performance under model misspecification, conducting simulation studies where survival times are generated from Weibull and log‐normal AFT models with a shared frailty component. To better reflect real‐world clinical settings, where hospitals often have varying patient capacities, we also introduce two additional simulation scenarios to assess the impact of imbalanced cluster sizes on model performance. Our results confirm the reliability and efficiency of the proposed algorithm, with strong consistency observed between VB and MCMC methods in both simulations and real‐world ICU ventilation data analysis. Moreover, the survregVBfrailty algorithm is significantly more computationally efficient, running over 150 times faster than the MCMC‐based survregbayes algorithm.

The h‐likelihood method proposed by Do Ha et al. [13] for analyzing clustered survival data in a log‐normal AFT model has been shown to be robust against violations of the normality assumption (e.g., extreme value distribution) for the logarithm of survival time. In our simulation study, we investigated the performance of the h‐likelihood method when the survival data, given the covariates in a specific cluster, follow a log‐logistic distribution. We compared its estimation results to those obtained using VB and MCMC algorithms. We found that the h‐likelihood method, survregHL, results in higher mean squared errors (MSEs) compared to VB and MCMC algorithms. Specifically, survregHL produced a 49.1% higher MSE than VB and a 41.2% higher MSE than MCMC for estimating the variance of the random intercept in our simulation study. Additionally, in our application to ICU ventilation duration analysis, we observed significant differences in some regression coefficients between h‐likelihood and VB or MCMC methods. Therefore, our proposed shared frailty log‐logistic AFT model using variational Bayes can be viewed as a better approach to the shared frailty log‐normal AFT model using h‐likelihood. While our proposed shared frailty log‐logistic AFT model using VB offers a computationally efficient alternative with satisfactory estimation performance, it does not currently account for competing risks such as death. It is important to note that the absence of a competing risks framework has varying implications depending on the ICU context. In high‐mortality ICUs, failing to account for death as a competing event may impact model applicability, whereas in low‐mortality settings, or when patient transfers are reasonably treated as censoring events, the current model remains appropriate. With the VB algorithm now established, extending it to incorporate competing risks is a compelling direction for future work.

Our application of the proposed method to the CCIS ICU data for ventilation duration analysis reveals a moderate correlation of 0.134 among patients within the same ICU site. By incorporating the ICU site‐specific random effect as an unknown shared frailty, we further validate the significant risk factors including the patient severity score MODS identified in the study by Kobara et al. [9] based on the 95% credible intervals obtained from the posterior distributions of each regression coefficient. We observe a similar trend regarding the effect of MODS on ventilation duration as reported by Kobara et al. [9]. Specifically, when MODS reaches a severe level, the increase in ventilation duration becomes less pronounced. This may be due to patient mortality, which warrants further investigation in future studies. We demonstrate that different ICU sites have varying effects on patient ventilation duration, as observed through differences in the estimated ICU site‐specific random effects. Our research provides valuable insights and practical implications for clinical practice and resource management. Specifically, ICUs with smaller estimated random intercepts tend to have shorter overall ventilation durations compared to other ICUs. Understanding the patient characteristics and ventilation practices within these ICUs may help improve clinical performance. For ICUs with larger estimated random intercepts, equipping more ventilators could enhance the overall efficiency of the ICU ventilation procedures.

While Cox‐type frailty models remain a widely used tool in survival analysis (even in the presence of competing risks) due to their flexibility and semi‐parametric nature, they are inherently hazard‐based and rely on the PH assumption. However, this assumption may not hold in practice. For instance, Kobara et al. [9] found violations of the PH assumption in our Ontario ICU ventilation duration data, indicating that the assumption is not reasonable. This issue is further compounded when modeling both cause‐specific and subdistribution hazards, as the PH structure generally cannot hold for both simultaneously in the presence of competing risks [38, 39]. In contrast, AFT models provide a complementary framework by directly modeling survival time on the log scale, offering an intuitive interpretation of covariate effects in terms of time acceleration or deceleration. Therefore, our proposed VB approach under the shared frailty AFT framework offers a practical alternative, especially in settings where the PH assumption is in doubt, or when researchers seek direct inference on survival time. It complements, rather than replaces, Cox‐type models and broadens the toolkit available for applied researchers handling complex clustered survival data.

We present several open problems and directions for future work related to our proposed methodology. As discussed, we assume that the survival time from a specific cluster follows a log‐logistic distribution. Therefore, extending the current VB algorithm to accommodate other survival distributions, such as the Weibull distribution, could be an interesting avenue of research. Integrating competing risks into the shared frailty AFT modeling framework using variational inference also presents an interesting research direction. A possible approach is to extend the cause‐specific or crude‐risk AFT model [38] to a shared frailty version, which allows for modeling competing events. In our current framework, we account for cluster‐level uncertainty using a random intercept. This approach could be extended to a more general model that includes cluster‐level covariates (e.g., the ICU type, general or specialized), resulting in a mixed‐effects model. Specifically, one could consider a log‐logistic AFT model with random effects specified as follows:

$$\log (T_{ij})={\mathbf{\text{X}}}_{ij}^T\mathit{\beta }+{\mathbf{\text{Z}}}_{ij}^T{\mathit{\gamma }}_i+b{ϵ}_{ij},i=1,\dots ,K;j=1,\dots ,n_i$$

where ${\mathbf{\text{Z}}}_{ij}$ is a column vector with length $l,l\ge 1$, containing $l-1$ random effects (cluster‐level covariates, not including the random intercept), and ${\mathit{\gamma }}_i$ is the corresponding vector of coefficients for the random effects from $i\text{th}$ cluster. We further assume that ${\mathit{\gamma }}_i\stackrel{\text{iid}}{\sim }{N}_l(\mathbf{\text{0}},\mathrm{\Omega })$ where $\mathrm{\Omega }$ is an $l\times l$ covariance matrix. A prior distribution can be specified for $\mathrm{\Omega }$ as $\mathrm{\Omega }\sim \text{Inverse‐Wishart}({\mathrm{\Psi }}_0,{\eta }_0)$ with ${\mathrm{\Psi }}_0$ being a known $l\times l$ positive definite matrix and ${\eta }_0>l-1$ is a known hyperparameter. Such a general model can be used to better assess differences in performance across ICU sites regarding the duration of invasive mechanical ventilation. Another potential area for development is the integration of variable selection techniques within the VB algorithm, see Park and Do Ha [16] as a reference.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. Update Equations

(1) Update equation for $q^{\ast }(\mathit{\beta })$

$$\log q^{\ast }(\mathit{\beta })\stackrel{+}{\approx }{𝔼}_{-\mathit{\beta }}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]+{𝔼}_{-\mathit{\beta }}[\log p(\mathit{\beta })]$$

where

$$\begin{array}{ll}\hfill & {𝔼}_{-\mathit{\beta }}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]\\ \hfill & ={𝔼}_{-\mathit{\beta }}\left[-\delta \log b+\sum _{i=1}^K\sum _{j=1}^{n_i}\left\{{\delta }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right.\right.\\ \hfill & \left.\left.-(1+{\delta }_{ij})\log \left\{1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)\right\}\right\}\right]\end{array}$$ (A1)

with $\delta ={\sum }_{i=1}^K{\sum }_{j=1}^{n_i}{\delta }_{ij}$ being the number of observed survival times, and

$$\begin{array}{ll}\hfill {𝔼}_{-\mathit{\beta }}[\log p(\mathit{\beta })]& \stackrel{+}{\approx }\frac{p}{2}\log v_0-\frac{1}{2}{v}_0{(\mathit{\beta }-{\mathit{\mu }}_0)}^T(\mathit{\beta }-{\mathit{\mu }}_0)\\ \hfill & \stackrel{+}{\approx }-\frac{1}{2}{v}_0[{\mathit{\beta }}^T\mathit{\beta }-2{\mathit{\mu }}_0^T\mathit{\beta }]=v_0{\mathit{\mu }}_0^T\mathit{\beta }-\frac{1}{2}{v}_0{\mathit{\beta }}^T\mathit{\beta }\end{array}$$

In Equation (A1), we use the quadratic piece‐wise approximation proposed by Xian et al. to approximate $1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)$:

$$\begin{array}{ll}\hfill 1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)& \approx {\rho }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}+{\zeta }_{ij}{\left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)}^2\end{array}$$ (A2)

where ${\rho }_{ij}:=0^{{\nu }_{ij1}}\times 0.1696^{{\nu }_{ij2}}\times 0.5^{{\nu }_{ij3}}\times 0.8303^{{\nu }_{ij4}}\times 1^{1-{\sum }_{k=1}^4{\nu }_{ijk}}$ and ${\zeta }_{ij}:=0^{{\nu }_{ij1}}\times 0.0189^{{\nu }_{ij2}}\times 0.1138^{{\nu }_{ij3}}\times 0.0190^{{\nu }_{ij4}}\times 0^{1-{\sum }_{k=1}^4{\nu }_{ijk}}$ with ${\nu }_{ij1}=\mathbb{1}(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le -5)$, ${\nu }_{ij2}=\mathbb{1}(-5<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le -1.7)$, ${\nu }_{ij3}=\mathbb{1}(-1.7<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le 1.7)$, and ${\nu }_{ij4}=\mathbb{1}(1.7<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le 5)$. Then, we obtain,

$$\begin{array}{ll}\hfill \log q^{\ast }(\mathit{\beta })& \stackrel{+}{\approx }\left[v_0{\mathit{\mu }}_0^T+\left(\sum _{i=1}^K\sum _{j=1}^{n_i}\left({𝔼}_{q(b)}\left(\frac{1}{b}\right)(-{\delta }_{ij}+(1+{\delta }_{ij}){\rho }_{ij})\right.\right.\right.\\ \hfill & \left.\left.\left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{𝔼}_{q({\gamma }_i)}({\gamma }_i))\right){\mathbf{\text{X}}}_{ij}^T\right)\right]\mathit{\beta }\\ \hfill & -\frac{1}{2}{\mathit{\beta }}^T\left(v_0\mathbf{\text{I}}+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{i=1}^K\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}{\mathbf{\text{X}}}_{ij}{\mathbf{\text{X}}}_{ij}^T\right)\mathit{\beta }\end{array}$$

Let

$${\sum }^{\ast }:={\left[v_0\mathbf{\text{I}}+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{i=1}^K\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}{\mathbf{\text{X}}}_{ij}{\mathbf{\text{X}}}_{ij}^T\right]}^{-1}$$

and

$$\begin{array}{ll}\hfill {\mathit{\mu }}^{\ast }& :=\left[\left\{v_0{\mathit{\mu }}_0^T+\sum _{i=1}^K\sum _{j=1}^{n_i}\left({𝔼}_{q(b)}\left(\frac{1}{b}\right)\left(-{\delta }_{ij}+(1+{\delta }_{ij}){\rho }_{ij}\right)\right.\right.\right.\\ \hfill & {\left.\left.\left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{𝔼}_{q({\gamma }_i)}({\gamma }_i))\right){\mathbf{\text{X}}}_{ij}^T\right\}{\sum }^{\ast }\right]}^T\end{array}$$

Then, $q^{\ast }(\mathit{\beta })$ is $N_p({\mathit{\mu }}^{\ast },{\sum }^{\ast })$.

(2) Update equation for $q^{\ast }({\gamma }_i)$

$$\log q^{\ast }({\gamma }_i)\stackrel{+}{\approx }{𝔼}_{-{\gamma }_i}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]+{𝔼}_{-{\gamma }_i}\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]$$

where

$$\begin{array}{ll}& \hfill {𝔼}_{-{\gamma }_i}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]\\ & \stackrel{+}{\approx }{𝔼}_{-{\gamma }_i}\left[\sum _{j=1}^{n_i}\left\{{\delta }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}-(1+{\delta }_{ij})\log \left(1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)\right)\right\}\right]\end{array}$$ (A3)

and

$${𝔼}_{-{\gamma }_i}\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]\stackrel{+}{\approx }-\frac{1}{2}{\gamma }_i^2{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)$$

We again apply the quadratic approximation in (A2) to (A3), and obtain

$$\begin{array}{ll}\hfill \log q^{\ast }({\gamma }_i)& \stackrel{+}{\approx }{\gamma }_i\left(\sum _{j=1}^{n_i}\left[{𝔼}_{q(b)}\left(\frac{1}{b}\right)(-{\delta }_{ij}+(1+{\delta }_{ij}){\rho }_{ij})\right.\right.\\ & \left.\left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}\mathit{\beta })\right]\right)\\ & -\frac{1}{2}{\gamma }_i^2\left({𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}\right)\end{array}$$

Let

$${\sigma }_i^{2\ast }={\left[{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)\sum _{j=1}^{n_i}(1+{\delta }_{ij}){\zeta }_{ij}\right]}^{-1}$$

and

$$\begin{array}{ll}\hfill {\tau }_i^{\ast }& ={\sigma }_i^{2\ast }\sum _{j=1}^{n_i}\left[{𝔼}_{q(b)}\left(\frac{1}{b}\right)(-{\delta }_{ij}+(1+{\delta }_{ij}){\rho }_{ij})\right.\\ & \left.+2{𝔼}_{q(b)}\left(\frac{1}{b^2}\right)(1+{\delta }_{ij}){\zeta }_{ij}(y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}\mathit{\beta })\right]\end{array}$$

Then, $q^{\ast }({\gamma }_i)$ is $N({\tau }_i^{\ast },{\sigma }_i^{2\ast })$.

(3) Update equation for $q^{\ast }(b)$

$$\log q^{\ast }(b)\stackrel{+}{\approx }{𝔼}_{-b}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]+{𝔼}_{-b}[\log p(b)]$$

where

$$\begin{array}{ll}\hfill & {𝔼}_{-b}[\log p(\mathbf{\text{D}}|\mathit{\beta },\mathit{\gamma },b)]\\ \hfill & ={𝔼}_{-b}\left[-\delta \log b+\sum _{i=1}^K\sum _{j=1}^{n_i}\left\{{\delta }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right.\right.\\ \hfill & \left.\left.-(1+{\delta }_{ij})\log \left\{1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)\right\}\right\}\right]\end{array}$$ (A4)

and

$$\begin{array}{ll}\hfill \hspace{0.17em}& {𝔼}_{-b}[\log p(b)]\stackrel{+}{\approx }-({\alpha }_0+1)\log b-\frac{{\omega }_0}{b}\end{array}$$

since we assume the prior b∼ Inverse‐Gamma (α_0,ω_0). In Equation (A4), we apply the linear piece‐wise approximation proposed by Xian et al. to approximate $1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)$ to achieve conjugacy:

$$1+\exp \left(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\right)\approx {\phi }_{ij}\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}$$

where ${\phi }_{ij}=0^{{\eta }_{ij1}}\times 0.0426^{{\eta }_{ij2}}\times 0.3052^{{\eta }_{ij3}}\times 0.6950^{{\eta }_{ij4}}\times 0.9574^{{\eta }_{ij5}}\times 1^{1-{\sum }_{k=1}^5{\eta }_{ijk}}$ with ${\eta }_{ij1}=\mathbb{1}(\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le -5)$, ${\eta }_{ij2}=\mathbb{1}(-5<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le -1.701)$, ${\eta }_{ij3}=\mathbb{1}(-1.701<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le 0)$, ${\eta }_{ij4}=\mathbb{1}(0<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le 1.702)$, and ${\eta }_{ij5}=\mathbb{1}(1.702<\frac{y_{ij}-{\mathbf{\text{X}}}_{ij}^T\mathit{\beta }-{\gamma }_i}{b}\le 5)$.

Then, we obtain

$$\begin{array}{ll}\hfill \log q^{\ast }(b)& \stackrel{+}{\approx }-({\alpha }_0+\delta +1)\log b-\frac{1}{b}\left[{\omega }_0-\sum _{i=1}^K\sum _{j=1}^{n_i}({\delta }_{ij}-(1+{\delta }_{ij}){\phi }_{ij})\right.\\ \hfill & \left.\times (y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}(\mathit{\beta })-{𝔼}_{q({\gamma }_i)}{\gamma }_i)\right]\end{array}$$

Let ${\alpha }^{\ast }={\alpha }_0+\delta$ and

$$\begin{array}{ll}\hfill {\omega }^{\ast }& ={\omega }_0-\sum _{i=1}^K\sum _{j=1}^{n_i}({\delta }_{ij}-(1+{\delta }_{ij}){\phi }_{ij})(y_{ij}-{\mathbf{\text{X}}}_{ij}^T{𝔼}_{q(\mathit{\beta })}(\mathit{\beta })-{𝔼}_{q({\gamma }_i)}{\gamma }_i)\end{array}$$

we have $q^{\ast }(b)$ is $\text{Inverse‐Gamma}({\alpha }^{\ast },{\omega }^{\ast })$.

(4) Update equation for $q^{\ast }({\sigma }_{\gamma }^2)$

$$\log q^{\ast }({\sigma }_{\gamma }^2)\stackrel{+}{\approx }{𝔼}_{-{\sigma }_{\gamma }^2}\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]+{𝔼}_{-{\sigma }_{\gamma }^2}[\log p({\sigma }_{\gamma }^2)]$$

where

$$\begin{array}{ll}\hfill {𝔼}_{-{\sigma }_{\gamma }^2}\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]& ={𝔼}_{-{\sigma }_{\gamma }^2}\left[\sum _{i=1}^K\log \left(\frac{1}{\sqrt{2\pi {\sigma }_{\gamma }^2}}\exp (-\frac{1}{2{\sigma }_{\gamma }^2}{\gamma }_i^2)\right)\right]\\ \hfill & \stackrel{+}{\approx }-\frac{K}{2}\log {\sigma }_{\gamma }^2-\frac{1}{2{\sigma }_{\gamma }^2}\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}{\gamma }_i^2\end{array}$$

and ${𝔼}_{-{\sigma }_{\gamma }^2}[\log p({\sigma }_{\gamma }^2)]=-({\lambda }_0+1)\log {\sigma }_{\gamma }^2-{\eta }_0/{\sigma }_{\gamma }^2$.

Therefore,

$$\log q^{\ast }({\sigma }_{\gamma }^2)\stackrel{+}{\approx }-\left({\lambda }_0+\frac{K}{2}+1\right)\log {\sigma }_{\gamma }^2-\left[{\eta }_0+\frac{1}{2}\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}{\gamma }_i^2\right]/{\sigma }_{\gamma }^2$$

Let ${\lambda }^{\ast }={\lambda }_0+K/2$ and

$${\eta }^{\ast }={\eta }_0+\frac{1}{2}\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}{\gamma }_i^2$$

$q^{\ast }({\sigma }_{\gamma }^2)$ is an $\text{Inverse‐Gamma}({\lambda }^{\ast },{\eta }^{\ast })$.

Appendix B. ELBO Calculation

We now present details of calculating ${\mathit{\text{diff}}}_{\mathit{\beta }}+{\mathit{\text{diff}}}_{\mathit{\gamma }}+{\mathit{\text{diff}}}_b+{\mathit{\text{diff}}}_{{\sigma }_{\gamma }^2}$ in the ELBO defined in (5).

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_{\mathit{\beta }}& ={𝔼}_q[\log p(\mathit{\beta })]-{𝔼}_q[\log q(\mathit{\beta })]\\ \hfill & \stackrel{+}{\approx }{𝔼}_q\left[-\frac{1}{2}{v}_0{(\mathit{\beta }-{\mathit{\mu }}_0)}^T(\mathit{\beta }-{\mathit{\mu }}_0)\right]-{𝔼}_q\left[-\frac{1}{2}\log (|{\sum }^{\ast }|)\right.\\ \hfill & \left.-\frac{1}{2}{(\mathit{\beta }-{\mathit{\mu }}^{\ast })}^T{({\sum }^{\ast })}^{-1}(\mathit{\beta }-{\mathit{\mu }}^{\ast })\right]\\ \hfill & \stackrel{+}{\approx }-\frac{1}{2}{v}_0[\text{trace}({\sum }^{\ast })+{({\mathit{\mu }}^{\ast }-{\mathit{\mu }}_0)}^T({\mathit{\mu }}^{\ast }-{\mathit{\mu }}_0)]+\frac{1}{2}\log (|{\sum }^{\ast }|)\end{array}$$

Note that

$${𝔼}_q\left[\frac{1}{2}{(\mathit{\beta }-{\mathit{\mu }}^{\ast })}^T{({\sum }^{\ast })}^{-1}(\mathit{\beta }-{\mathit{\mu }}^{\ast })\right]=\frac{p}{2}$$

which is always a constant at each iteration and therefore we ignore it.

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_{\mathit{\gamma }}& ={𝔼}_q\left[\sum _{i=1}^K\log p({\gamma }_i|{\sigma }_{\gamma }^2)\right]-{𝔼}_q\left[\sum _{i=1}^K\log q({\gamma }_i)\right]\\ \hfill & \stackrel{+}{\approx }-\frac{K}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)-\frac{1}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}({\gamma }_i^2)\\ \hfill & -\frac{1}{2}\sum _{i=1}^K\left[\log {\sigma }_i^{2\ast }-\frac{1}{{\sigma }_i^{2\ast }}({𝔼}_{q({\gamma }_i)}({\gamma }_i^2)-{\mu }_i^{\ast 2})\right]\\ \hfill & \stackrel{+}{\approx }-\frac{K}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)-\frac{1}{2}{𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)\sum _{i=1}^K{𝔼}_{q({\gamma }_i)}({\gamma }_i^2)-\frac{1}{2}\sum _{i=1}^K\left(\log {\sigma }_i^{2\ast }\right)\end{array}$$

Since $({𝔼}_{q({\gamma }_i)}({\gamma }_i^2)-{\mu }_i^{\ast 2})/{\sigma }_i^{2\ast }=1$ for $i=1,\dots K$, and therefore,

$$\left(\sum _{i=1}^K({𝔼}_{q({\gamma }_i)}({\gamma }_i^2)-{\mu }_i^{\ast 2})\right)/(2{\sigma }_i^{2\ast })=K/2$$

which is a constant, and we can ignore it.

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_b& ={𝔼}_q[\log p(b)]-{𝔼}_q[\log q(b)]\\ \hfill & \stackrel{+}{\approx }{𝔼}_q\left[-({\alpha }_0+1)\log b-\frac{{\omega }_0}{b}\right]\\ \hfill & -{𝔼}_q\left[{\alpha }^{\ast }\log {\omega }^{\ast }-\log (\mathrm{\Gamma }({\alpha }^{\ast }))-({\alpha }^{\ast }+1)\log b-\frac{{\omega }^{\ast }}{b}\right]\\ \hfill & =({\alpha }^{\ast }-{\alpha }_0){𝔼}_{q(b)}(\log b)+({\omega }^{\ast }-{\omega }_0){𝔼}_{q(b)}\left(\frac{1}{b}\right)-{\alpha }^{\ast }\log {\omega }^{\ast }\end{array}$$

Since ${\alpha }^{\ast }$ does not change at each iteration, we remove $\log (\mathrm{\Gamma }({\alpha }^{\ast }))$ in the calculation of the ELBO.

$$\begin{array}{ll}\hfill {\mathit{\text{diff}}}_{{\sigma }_{\gamma }^2}& ={𝔼}_q[\log p({\sigma }_{\gamma }^2)]-{𝔼}_q[\log q({\sigma }_{\gamma }^2)]\\ \hfill & \stackrel{+}{\approx }{𝔼}_q\left[-({\lambda }_0+1)\log {\sigma }_{\gamma }^2-\frac{{\eta }_0}{{\sigma }_{\gamma }^2}\right]\\ \hfill & -{𝔼}_q\left[{\lambda }^{\ast }\log {\eta }^{\ast }-\log (\mathrm{\Gamma }({\lambda }^{\ast }))-({\lambda }^{\ast }+1)\log {\sigma }_{\gamma }^2-\frac{{\eta }^{\ast }}{{\sigma }_{\gamma }^2}\right]\\ \hfill & =({\lambda }^{\ast }-{\lambda }_0){𝔼}_{q({\sigma }_{\gamma }^2)}(\log {\sigma }_{\gamma }^2)+({\eta }^{\ast }-{\eta }_0){𝔼}_{q({\sigma }_{\gamma }^2)}\left(\frac{1}{{\sigma }_{\gamma }^2}\right)-{\lambda }^{\ast }\log {\eta }^{\ast }\end{array}$$

Since ${\lambda }^{\ast }$ does not change at each iteration, we remove $\log (\mathrm{\Gamma }({\lambda }^{\ast }))$ in the calculation of the ELBO.

Appendix C. Real Data Analysis

TABLE C1.

Results for ICU ventilation duration analysis: Posterior means (Mean) with 95% credible intervals (95% Cred. Int.) from the VB algorithms and MCMC‐based survregbayes, respectively. Point estimates (Est.) with 95% confidence interval (95% Conf. Int.) from the h‐likelihood survregHL and survreg methods.

	With frailty						Without frailty
	survregVBfrailty		survregHL		survregbayes		survreg		survregVB
	Mean	95% Cred. Int.	Est.	95% Conf. Int.	Mean	95% Cred. Int.	Est.	95% Conf. Int.	Mean	95% Cred. Int.
Admission Source
Downstream (ref)	—	—	—	—	—	—	—	—	—	—
ED	−0.117	[−0.158, −0.076]	−0.112	[−0.160, −0.064]	−0.114	[−0.160, −0.068]	−0.126	[−0.168, −0.085]	−0.125	[−0.166, −0.083]
Home	0.097	[−0.018, 0.212]	0.115	[−0.019, 0.249]	0.108	[−0.023, 0.240]	0.082	[−0.037, 0.201]	0.080	[−0.038, 0.197]
Hospital	0.107	[0.062, 0.151]	0.093	[0.042, 0.144]	0.103	[0.072, 0.134]	0.084	[0.040, 0.129]	0.087	[0.042, 0.132]
OR	−0.329	[−0.374, −0.285]	−0.305	[−0.358, −0.252]	−0.325	[−0.357, −0.294]	−0.342	[−0.387, −0.296]	−0.341	[−0.387, −0.296]
Other	0.181	[0.066, 0.295]	0.170	[0.043, 0.297]	0.171	[0.059, 0.283]	0.205	[0.088, 0.322]	0.210	[0.094, 0.325]
Ward	−0.143	[−0.185, −0.100]	−0.133	[−0.181, −0.084]	−0.143	[−0.161, −0.124]	−0.145	[−0.187, −0.102]	−0.146	[−0.188, −0.103]
Diagnosis
Cardiovascular (ref)	—	—	—	—	—	—
Gastrointestinal	0.267	[0.229, 0.304]	0.252	[0.206, 0.298]	0.258	[0.206, 0.308]	0.271	[0.232, 0.309]	0.271	[0.232, 0.309]
Neurological	0.278	[0.243, 0.314]	0.276	[0.234, 0.318]	0.273	[0.234, 0.313]	0.293	[0.258, 0.329]	0.292	[0.256, 0.328]
Other	0.381	[0.355, 0.407]	0.357	[0.326, 0.388]	0.387	[0.358, 0.416]	0.384	[0.358, 0.410]	0.383	[0.357, 0.410]
Trauma	0.556	[0.506, 0.606]	0.515	[0.456, 0.574]	0.535	[0.482, 0.588]	0.565	[0.514, 0.616]	0.568	[0.517, 0.619]
Patient type
Medical (ref)	—	—	—	—	—	—
Surgical	−0.006	[−0.049, 0.036]	−0.009	[−0.058, 0.041]	−0.007	[−0.058, 0.044]	−0.012	[−0.055, 0.031]	−0.015	[−0.058, 0.029]
Scheduled Admission
No (ref)	—	—	—	—	—	—
Yes	−0.270	[−0.313, −0.226]	−0.286	[−0.340, −0.231]	−0.278	[−0.333, −0.223]	−0.299	[−0.343, −0.254]	−0.287	[−0.331, −0.242]
Scheduled Surgery
No (ref)	—	—	—	—	—	—
Yes	−0.132	[−0.178, −0.087]	−0.121	[−0.176, −0.065]	−0.128	[−0.169, −0.087]	−0.131	[−0.177, −0.084]	−0.134	[−0.180, −0.087]
Referral
Medical (ref)	—	—	—	—	—	—
Other	−0.022	[−0.055, 0.011]	−0.022	[−0.062, 0.018]	−0.023	[−0.060, 0.014]	−0.041	[−0.075, −0.008]	−0.042	[−0.076, −0.009]
Respirology	0.165	[0.113, 0.217]	0.155	[0.091, 0.219]	0.160	[0.101, 0.220]	0.177	[0.124, 0.230]	0.178	[0.126, 0.231]
Surgical	−0.065	[−0.104, −0.027]	−0.065	[−0.111, −0.019]	−0.068	[−0.119, −0.017]	−0.048	[−0.087, −0.009]	−0.049	[−0.088, −0.010]
Other Interventions
No (ref)	—	—	—	—	—	—
Yes	0.085	[0.068, 0.102]	0.090	[0.068, 0.111]	0.087	[0.066, 0.108]	0.060	[0.043, 0.078]	0.060	[0.042, 0.077]
CVL
No (ref)	—	—	—	—	—	—
Yes	0.185	[0.163, 0.207]	0.173	[0.147, 0.199]	0.191	[0.161, 0.221]	0.198	[0.176, 0.220]	0.200	[0.178, 0.222]
AL
No (ref)	—	—	—	—	—	—
Yes	0.198	[0.175, 0.222]	0.175	[0.147, 0.203]	0.188	[0.162, 0.214]	0.201	[0.178, 0.225]	0.204	[0.180, 0.228]
IPM
No (ref)	—	—	—	—	—	—
Yes	0.540	[0.473, 0.607]	0.487	[0.413, 0.562]	0.570	[0.513, 0.628]	0.551	[0.483, 0.618]	0.550	[0.482, 0.618]
EMO
No (ref)	—	—	—	—	—	—
Yes	0.888	[0.789, 0.988]	0.810	[0.705, 0.915]	0.896	[0.765, 1.028]	0.963	[0.862, 1.064]	0.964	[0.862, 1.066]
IABP
No (ref)	—	—	—	—	—	—
Yes	0.384	[0.316, 0.452]	0.338	[0.260, 0.416]	0.350	[0.264, 0.436]	0.362	[0.293, 0.432]	0.356	[0.286, 0.426]
Age
18‐39 (ref)	—	—	—	—	—	—
40‐80	0.127	[0.098, 0.156]	0.111	[0.077, 0.145]	0.133	[0.106, 0.159]	0.109	[0.080, 0.138]	0.110	[0.080, 0.139]
$>80$	0.074	[0.040, 0.108]	0.064	[0.023, 0.105]	0.069	[0.030, 0.108]	0.055	[0.020, 0.090]	0.056	[0.021, 0.091]
Pre‐LOS
$\le 1$ day (ref)	—	—	—	—	—	—
2‐7 days	0.041	[0.018, 0.064]	0.044	[0.014, 0.073]	0.041	[0.010, 0.072]	0.045	[0.021, 0.069]	0.044	[0.021, 0.068]
$\ge 7$ days	0.122	[0.097, 0.146]	0.141	[0.110, 0.172]	0.144	[0.119, 0.169]	0.132	[0.107, 0.158]	0.131	[0.106, 0.157]
MODS
None (ref)	—	—	—	—	—	—
Minimal	0.095	[0.068, 0.123]	0.086	[0.050, 0.122]	0.093	[0.062, 0.124]	0.077	[0.049, 0.105]	0.076	[0.048, 0.104]
Mild	0.208	[0.182, 0.235]	0.187	[0.152, 0.223]	0.198	[0.165, 0.231]	0.179	[0.152, 0.206]	0.176	[0.149, 0.203]
Moderate	0.257	[0.224, 0.290]	0.240	[0.198, 0.282]	0.245	[0.202, 0.288]	0.236	[0.202, 0.269]	0.234	[0.201, 0.268]
Severe	0.114	[0.046, 0.183]	0.100	[0.023, 0.177]	0.114	[0.038, 0.191]	0.111	[0.043, 0.179]	0.107	[0.039, 0.175]

Xian C., P. E. de Souza C., He W., Rodrigues F. F., and Tian R., “Fast Variational Bayesian Inference for Correlated Survival Data: An Application to Invasive Mechanical Ventilation Duration Analysis,” Statistics in Medicine 44, no. 15‐17 (2025): e70198, 10.1002/sim.70198.

Data Availability Statement

The ICU data analyzed in Section 5 were from Critical Care Information System (CCIS) Ontario database. Under the authors' data use agreement, they are unable to share the files directly.