Table 3 Association structures for joint model

Random effect

Univariate

W_i(t) = αb_i

[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

\( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k{b}_{ik} \)

[20, 30, 36, 41, 50, 82, 85,86,87]

Current Value parameterisation

Univariate

W_i(t) = αm_i(t)

[13, 24, 39, 40, 43, 46, 47, 52, 65, 67, 71,72,73, 76]

Multivariate

\( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k{m}_{ik}(t) \)

[14, 21, 26, 28, 32, 34, 45, 49, 57, 63, 83, 84, 87]

Correlated random effect

Univariate

W_i(t) = φ

with  ϐ_i = {b_i, φ_i}~H

[16, 18]

Multivariate

W_i(t) = φ

with \( {\mathrm{\ss}}_i=\left\{{b}_i,{\varphi}_i\right\}\sim {H}_{\alpha_k} \)

[57]

Random effect with fixed effect

Multivariate

\( {W}_i(t)={\sum}_{k=1}^K{\alpha}_k\left({\beta}_k+{b}_{ik}\right) \)

[29, 57]

Time-dependent slope

Univariate

\( {W}_i(t)={\alpha}^{(1)}{m}_i(t)+{\alpha}^{(2)}\ \frac{d\ }{dt}{m}_i(t) \)

[87]

Multivariate

\( {W}_i(t)={\sum}_{k=1}^K\left\{{\alpha_k}^{(1)}\ {m}_{ik}(t)+{\alpha_k}^{(2)}\ \frac{d\ }{dt}{m}_{ik}(t)\right\} \)

[45, 81]

Cumulative effect

Univariate

\( {W}_i(t)=\alpha {\int}_0^t{m}_i(s) ds \)

[45]