Fit Joint Model for Longitudinal and Survival Data Using ODEs

Fits a joint model that simultaneously analyzes longitudinal biomarker trajectories and time-to-event outcomes using ordinary differential equations (ODEs). This approach captures complex temporal dynamics in biomarker evolution while quantifying their association with survival through shared parameters.

Usage

JointODE(
  longitudinal_formula,
  longitudinal_data,
  survival_formula,
  survival_data,
  id = "id",
  time = "time",
  spline_baseline = list(degree = 3, n_knots = 5, knot_placement = "quantile",
    boundary_knots = NULL),
  spline_index = list(degree = 3, n_knots = 4, knot_placement = "quantile",
    boundary_knots = NULL),
  control = list(),
  ...
)

Arguments

longitudinal_formula

Formula for the longitudinal submodel. Left side specifies the response; right side includes time and covariates (e.g., v ~ x1).

longitudinal_data

Data frame containing longitudinal measurements. Must include multiple observations per subject with columns for subject ID, measurement times, and response values.

survival_formula

Formula for the survival submodel. Must use Surv(time, status) on the left side; right side specifies baseline covariates.

survival_data

Data frame containing survival/event data. Must have exactly one row per subject with event time and status.

id

Character string naming the subject ID variable. Must exist in both datasets (default: "id").

time

Character string naming the time variable in longitudinal data (default: "time").

spline_baseline

List of B-spline configuration for baseline hazard:

degree: Degree of B-spline basis (default: 3)
n_knots: Number of interior knots (default: 5)
knot_placement: Method for knot placement: "quantile" (based on event times) or "equal" (default: "quantile")
boundary_knots: Boundary knots, if NULL uses event range (default: NULL)

spline_index

List of B-spline configuration for single index model:

degree: Degree of B-spline basis (default: 3)
n_knots: Number of interior knots (default: 4)
knot_placement: Method for knot placement: "quantile" (based on index values) or "equal" (default: "quantile")
boundary_knots: Boundary knots, if NULL uses index range (default: NULL)

control

List of optimization control parameters:

method: Optimization algorithm (default: "BFGS")
maxit: Maximum iterations (default: 1000)
tol: Convergence tolerance (default: 1e-6)
verbose: Print progress (default: FALSE)

...

Additional arguments passed to fitting functions.

Value

Object of class "JointODE" containing:

coefficients: Estimated model parameters including longitudinal, survival, and association parameters
logLik: Log-likelihood at convergence
AIC: Akaike Information Criterion
BIC: Bayesian Information Criterion
convergence: Optimization convergence details
fitted: Fitted values for both submodels
residuals: Model residuals
data: Original input data
call: Matched function call

Details

The joint ODE model links longitudinal and survival processes through shared parameters. The longitudinal trajectory is modeled using ODEs to capture non-linear dynamics, while the survival hazard incorporates features of the trajectory (level, slope, or cumulative burden).

Model estimation uses maximum likelihood with numerical integration over random effects via adaptive Gauss-Hermite quadrature.

Note

Input data are automatically validated and processed before fitting. For large datasets or complex ODE systems, consider adjusting control parameters to improve convergence.

Examples

if (FALSE) { # \dontrun{
sim <- simulate()
fit <- JointODE(
  longitudinal_formula = v ~ x1 + x2,
  longitudinal_data = sim$longitudinal_data,
  survival_formula = Surv(time, status) ~ w1 + w2,
  survival_data = sim$survival_data
)
summary(fit)
} # }