# Table 3 Association structures for joint model

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%)



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%)



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%)



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 