The goal of FlexVarJM is to estimate joint model with subject-specific time-dependent variability.

The global function is ‘lsjm’. It handles to estimate joint model with a marker which has a subject-specific time-dependent variability and competing events with the possibility to take into account the left truncation. For more information you can read the corresponding article : https://arxiv.org/abs/2306.16785

You can install the development version of FlexVarJM from GitHub with:

This is an example in a simulated dataset where is a binary variable.

\[y_i(t_{ij}) = \color{blue}\tilde{y}_i(t_{ij}) \color{black} + \epsilon_{ij} = \beta_0 + b_{0i} + (\beta_1 + b_{1i})t_{ij} + \beta_2 * binary_i + \epsilon_{ij} \]

For the first risk, k = 1, we estimate the following risk function :

\[ \lambda_{i1}(t) = \lambda_{01}(t)\exp(\gamma_{11}*binary_i + \color{blue}\alpha_{11}\tilde{y}_i(t) + \color{red}\alpha_{\sigma 1} \sigma_i(t) \color{black}) \] And for the second risk, k = 2 : \[ \lambda_{i2}(t) = \lambda_{02}(t)\exp(\color{blue}\alpha_{21}\tilde{y_i}(t) + \color{blue}\alpha_{22}\tilde{y}'_i(t) + \color{red}\alpha_{\sigma 2} \sigma_i(t) \color{black}) \]

where :

\(\epsilon_{i}(t_{ij}) \sim \mathcal{N}(0, \color{red}\sigma_i^2\color{black})\) with \(\color{red}\log(\sigma_i(t_{ij})) = \mu_0 + \tau_{0i} + (\mu_1 + \tau_{1i})\times t_{ij} + \mu_2 * binary_i\)

with \(b_i=\left(b_{0i},b_{1i}\right)^{\top}\) and \(\tau_i=\left(\tau_{0i},\tau_{1i}\right)^{\top}\) assuming that the two sets of random effects \(b_i\) and \(\tau_i\) are not independent: \[(b_i, \tau_i)^\top \sim N(0, \Sigma)\]

\(\lambda_{0k}(t) = \kappa_k^2 t^{\kappa_k^2-1}e^{\zeta_{0k}}\) : Weibull function

\(\tilde{y}'_i(t)\) is the current slope of the marker \(y\)

```
example <- lsjm(formFixed = y~visit+binary,
formRandom = ~ visit,
formGroup = ~ID,
formSurv = Surv(time, event ==1 ) ~ binary,
timeVar = "visit",
data.long = Data_toy,
variability_hetero = TRUE,
formFixedVar =~visit+binary,
formRandomVar =~visit,
correlated_re = TRUE,
sharedtype = c("current value", "variability"),
hazard_baseline = "Weibull",
competing_risk = TRUE,
formSurv_CR = Surv(time, event ==2 ) ~ 1,
hazard_baseline_CR = "Weibull",
sharedtype_CR = c("slope", "variability"),
formSlopeFixed =~1,
formSlopeRandom = ~1,
indices_beta_slope = c(2),
S1 = 500,
S2 = 1000,
nproc = 5,
Comp.Rcpp = TRUE
)
summary(example)
```

You can access to the table of estimations and standard deviation with :

The computing time is given by :

The output of the marqLevAlg algorithm is in :

Finally, some elements of control are in :

You can check the goodness-of-fit of the longitudinal submodel and of the survival submodel by computing the predicted random effects :

You can compute the probability for (new) individual(s) to have event 1 or 2 between time s and time s+t years given that he did not experience any event before time s, its trajectory of marker until time s ans the set of estimated parameters. To have a ‘IC%’ confidence interval, the predictions are computed ‘nb.draws’ time and the percentiles of the predictions are computed. For example, for individuals 1 and 3 to experiment the event 1 at time 1.5, 2, and 3, given their measurements until time 1 :