AR1_FRAILTY: Fit a Functional Regression with AutoregressIve fraiLTY...

In FunSurv: Modeling Time-to-Event Data with Functional Predictors

Fit a Functional Regression with AutoregressIve fraiLTY (FRAILTY) model for Recurrent Event Data

Description

Jointly model longitudinal measurements and recurrent events, accommodating both scalar and functional predictors while capturing time-dependent correlations among events. The FRAILTY method employs a two-step estimation procedure. First, functional principal component analysis through conditional expectation (PACE) is applied to extract key temporal features from sparse, irregular longitudinal data. Second, the obtained scores are incorporated into a dynamic recurrent frailty model with an autoregressive structure to account for within-subject correlations across recurrent events. This function works only for univariate functional data.

Usage

AR1_FRAILTY(
  formula,
  sdat,
  fdat,
  para0 = c(0.5, 0.5),
  nbasis = 10,
  pve = 0.9,
  npc = NULL,
  makePD = FALSE,
  cov.weight.type = c("none", "counts"),
  iter.max = 50,
  eps = 1e-06
)

Arguments

formula	A formula, with the response on the left of a ~ operator being a Recur object as returned by function Recur in reda, and scalar covariates on the right.
sdat	A data frame containing subject IDs, time-to-event outcomes (starting time, end point, censoring time and event status), and scalar covariates
fdat	A data frame containing subject IDs, longitudinal measurements, and the corresponding time points for each measurement.
para0	A vector of initial values for \theta^2 and auto-regressive coefficient \rho. Both default to 0.5.
nbasis	An integer, representing the number of B-spline basis functions used for estimation of the mean function and bivariate smoothing of the covariance surface. Defaults to 10 (cf. fpca.sc in refund).
pve	A numeric value between 0 and 1, the proportion of variance explained: used to choose the number of principal components. Defaults to 0.9 (cf. fpca.sc in refund).
npc	An integer, giving a prespecified value for the number of principal components. Defaults to NULL. If given, this overrides pve (cf. fpca.sc in refund).
makePD	Logical: should positive definiteness be enforced for the covariance surface estimate? Defaults to FALSE (cf. fpca.sc in refund).
cov.weight.type	The type of weighting used for the smooth covariance estimate. Defaults to "none", i.e. no weighting. Alternatively, "counts" (corresponds to fpca.sc in refund) weights the pointwise estimates of the covariance function by the number of observation points.
iter.max	Maximum number of iterations for both inner iteration and outer iteration. Defaults to 50.
eps	Tolerance criteria for a possible infinite coefficient value. Defaults to 1e-6.

Details

Model specification:

Let T_{ij} denote the time of the jth event for subject i, and let C_i represent the censoring time. The observed event time, accounting for right censoring, is \widetilde{T}_{ij}=\min(T_{ij}, C_i), and \delta_{ij}=I(T_{ij}\leq C_i) serves as an indicator of whether the jth event for subject i is observed. The hazard function is specified as

h(t; \boldsymbol{Z}_i, {X}_i(\cdot))=h_{0}(t-t_{i,j-1}) \exp \left(\eta_{ij}\right),

where h_0(\cdot) is the baseline hazard function, and \eta_{ij} = \bm{\alpha}^{\top}\boldsymbol{Z}_i +\int_{t_{i, j-1}}^{t}{X}_{i}(s)\beta(s)ds + v_{ij}. Here, t_{i, j-1} is the previous event time with t_{i0} = 0. \bm{\alpha} is the fixed effect parameter associated with the time-invariant covariates \boldsymbol{Z}_i, and \beta(t) is a time-varying coefficient that captures the effect of functional predictor X_{i}(t) on the hazard rate of recurrent events.

Value

A funsurv object containing the following components:

beta	Estimation of coefficients of scalar covariates and FPC scores. Including estimated values, standard errors, and p-values
beta_vcov	Estimated variance-covariance of the estimates of beta
eAR	Estimation of variance components (\theta^2 and \rho)
eAR_vcov	Estimated variance of estimates of \theta^2 and \rho
frailties	Estimated frailty terms (random effects)
basesurv	Estimated baseline survival probability
time	Time points associated with baseline survival probability
FPC	Functional principal components

See Also

Recur

PACE

Examples

data(simDat)

fit <- AR1_FRAILTY(Recur(t_start %to% t_stop, id, status) ~ z1,
                   sdat = sdat, fdat = fdat, iter.max = 30)

summary(fit)

FunSurv documentation built on April 3, 2025, 9:13 p.m.