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Table 3 Association structures for joint model

From: Bayesian joint modelling of longitudinal and time to event data: a methodological review

Parameterisation Latent association Number of articles (%)a Reference
Random effect Univariate Wi(t) = αbi 31(41%) [13, 15, 17, 19, 22, 23, 25, 27, 31, 33, 35, 37, 38, 44, 48, 51, 53, 54, 56, 59,60,61, 66, 69, 70, 75, 77,78,79,80, 82]
Multivariate \( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k{b}_{ik} \) 8(10.7%) [20, 30, 36, 41, 50, 82, 85,86,87]
Current Value parameterisation Univariate Wi(t) = αmi(t) 14(18.7%) [13, 24, 39, 40, 43, 46, 47, 52, 65, 67, 71,72,73, 76]
Multivariate \( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k{m}_{ik}(t) \) 13(17.3%) [14, 21, 26, 28, 32, 34, 45, 49, 57, 63, 83, 84, 87]
Correlated random effect Univariate Wi(t) = φ
with  ϐi = {bi, φi}~H
2(2.7%) [16, 18]
Multivariate Wi(t) = φ
with \( {\mathrm{\ss}}_i=\left\{{b}_i,{\varphi}_i\right\}\sim {H}_{\alpha_k} \)
1(1.3%) [57]
Random effect with fixed effect Multivariate \( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k\left({\beta}_k+{b}_{ik}\right) \) 2(2.7%) [29, 57]
Time-dependent slope Univariate \( {W}_i(t)={\alpha}^{(1)}{m}_i(t)+{\alpha}^{(2)}\ \frac{d\ }{dt}{m}_i(t) \) 1(1.3%) [87]
Multivariate \( {W}_i(t)={\sum}_{k=1}^K\left\{{\alpha_k}^{(1)}\ {m}_{ik}(t)+{\alpha_k}^{(2)}\ \frac{d\ }{dt}{m}_{ik}(t)\right\} \) 2(2.7%) [45, 81]
Cumulative effect Univariate \( {W}_i(t)=\alpha {\int}_0^t{m}_i(s) ds \) 1(1.3%) [45]
  1. Abbreviation: aThe number of article that used this association among all other articles with its percentage
  2. Notation: mik(t) denotes the true underlying value of the longitudinal outcome for individual i and outcome k; αk represents the association parameter for the k-th outcome; αk(1) and αk(2) denote the association parameters for the current value and the derivative from the mean trajectory function for the k-th longitudinal outcome respectively; bik denotes the random effect for individual i and outcome k; φ represents a random effect and H denotes joint distribution for the random effects; βk denotes the coefficient parameters