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Table 1 Some association structures for joint models of time-to-event and multivariate longitudinal data

From: Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues

Parameterization Latent association Studiesi
A1: Current value (linear predictor) \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k{\mu}_{ik}(t)} \) [17, 19, 21, 29, 37, 39, 42, 48, 49, 52, 56, 58, 60, 62, 67, 70]
A2: Current value (expected value)a \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k{h}_k^{-1}\left({\mu}_{ik}\right(t\left)\right)} \) [62]
A3: Interactionb \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K\left\{{\alpha}_k{\mu}_{ik}(t)+{\sum}_{l=1}^p{x}_{il}^{(2)}{\mu}_{ik}(t){\gamma}_{kl}\right\}} \) [37]
A4: Lagged timec \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k{\mu}_{ik}\left(t-c\right)} \) [46]
A5: General vector functiond \( {W}_i(t)={a}^T\psi \left(t,,,{b}_i\right) \) [38]
A6: Time-dependent slopese \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K\left\{{\mu}_{ik}(t)+{\displaystyle {\sum}_{v=1}^V{\alpha}_{vk}\frac{d^v}{d{t}^v}{\mu}_{ik}(t)}\right\}} \) [29, 56]
A7: Cumulative effects \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k{\displaystyle {\int}_0^t}{\mu}_{ik}(s)ds} \) [56]
A8: Random effectsf \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k^T{b}_{ik}} \) [18, 29, 5355, 59, 62, 64, 65, 68]
A9: Generalised random effects + fixed effectsg \( {W}_i(t)={\displaystyle {\sum}_{k=1}^K{\alpha}_k^Tr\left({b}_{ik} + {\tilde {\beta}}_k^{(1)}\right)} \) [29, 50, 51, 59, 62]
A10: Correlated random effectsh \( {W}_i(t)={\theta}_i{\textstyle}\mathrm{with}{\textstyle }{\left({b}_i^T,,,{\theta}_i\right)}^T\sim {\mathrm{F}}_{\mathrm{a}} \) [29]
  1. Notation: μ ik (t) denotes the linear predictor term of the longitudinal submodel for subject i and outcome k; α k denotes the association parameter for the k-th outcome
  2. ah k () is the link function for the k-th outcome
  3. bx (2) il denotes the l-th baseline covariates for subject i (l = 1, …, p) with corresponding coefficient parameters γ kl for each outcome k. In practice, some γ kl coefficients will be set to zero
  4. cc is a lag time (with c = 0 returning the current value parameterization). In Albert and Shih [46], time was modelled discretely and a selection model adopted, such that W i (t j ) = ∑ Kk = 1 α k μ ik (tj − 1)
  5. dα is a vector of association parameters and ψ(tb i ) is a vector of time and random effects. It is assumed that ψ(tb i ) can be decomposed as ψ(tb i ) = ψ(t)b i . This general parameterization admits the current value parameterization as a special case, and leads to a number of extensions including interactions with time. In cases where ψ(tb i ) does not factorise, the authors propose using an approximation method
  6. eα vk denote additional association parameters for the ν-th derivative (with respect to time) for the k-th longitudinal outcome mean trajectory function
  7. fα k denotes a vector of association parameters of same dimension as the number of random effects for each outcome. In practice, some elements of α k might be forced to zero, e.g. if only random intercepts were used to link the model
  8. gas per the random effects parameterization α k denotes a vector of association parameters of same dimension as the number of random effects for each outcome. \( {\tilde {\beta}}_k^{(1)} \) denotes the subset of coefficient parameters from β (1) k that correspond to the random effect terms, and r() denotes a vector function. If r() is the identify function, then the standard random + fixed effects parameterization is returned
  9. hF α denotes a multivariate density function with parameters α to model correlation
  10. iRizopoulos [4] describes a general MVJM and notes that, in principle, the general association structures that are used in the R JM package [27] are applicable to the multivariate case. However, the model was only described without fitting or application, therefore we have not included these association structures here