# Joint Latent Process Models

The R package JLPM implements in the jointLPM function the estimation of a joint shared random effects model.

The longitudinal data (\(y_1, y_2, \dots, y_K\)) can be continuous or ordinal and are modelled using a mixed model.

For the continuous Gaussian case:

\[ \forall k \in 1,\dots,K \hspace{1cm} y_k(t_{ijk}) = X_i(t_{ijk})\beta + Z_i(t_{ijk})u_i + \varepsilon_{ijk} \]

For the continuous non Gaussien case, a transformation \(H_k\) is estimated for each outcome:

\[ \forall k \in 1,\dots,K \hspace{1cm} H_k(y_k(t_{ijk}), \eta_k) = X_i(t_{ijk})\beta + Z_i(t_{ijk})u_i + \varepsilon_{ijk} \]

For the ordinal case, mixed models are combined to the Item Response Theory:

\[ \forall k \in 1,\dots,K \hspace{1cm} \mathbb{P}( y_k(t_{ijk}) = m) \Leftrightarrow \mathbb{P}(\eta_{m-1} < X_i(t_{ijk})\beta + Z_i(t_{ijk})u_i + \varepsilon_{ijk} < \eta_m) \]

where \(X_i(t_{ijk})\) and \(Z_i(t_{ijk})\) are vectors of covariates measured at time \(t_{ijk}\) for subject \(i\), \(\beta\) is the vector of fixed effects, \(u_i \sim \mathcal{N}(0,B)\), \(\varepsilon_{ijk} \sim \mathcal{N}(0,\sigma_k^2)\), \(\eta_k\) are the parameters of the link function \(H_k\) or the thresholds associated to the outcome \(y_k\).

Note that even with multiple longitudinal outcomes, a univariate mixed model is estimated.

The time-to-event data are modelled in a proportional hazard model where different associations between the longitudinal outcome and the event can be included.

With an association through the random effects of the longitudinal model:

\[ \alpha_i(t) = \alpha_0(t) \exp(\tilde{X}_i \gamma + \delta u_i) \]

With an association through the current level of the longitudinal model:

\[ \alpha_i(t) = \alpha_0(t) \exp(\tilde{X}_i \gamma + \delta (X_i(t)\beta + Z_i(t)u_i)) \]

With an association through the current slope of the longitudinal model:

\[ \alpha_i(t) = \alpha_0(t, \omega) \exp(\tilde{X}_i \gamma + \delta ( \frac{d}{dt}X_i(t)\beta + \frac{d}{dt}Z_i(t)u_i)) \]

with \(\alpha_0(t, \omega)\) the baseline risk function at time \(t\), parameterized with \(\omega\),\(\tilde{X}_i\) a vector of covariates, \(\gamma\) the fixed effects, \(\delta\) the association parameter.

The parameters \(\beta, B, \sigma_k, \eta_k, \omega, \gamma, \delta\) are estimated using a Marquardt-Levenberg algorithm.