Package 'FunSurv'

Construct an AR(1) correlation matrix

Description

Construct an AR(1) correlation matrix

Usage

ar1_cor(n, rho)

Arguments

n	number of events for each subject
rho	autoregressive correlation

Value

A n by n matrix

Examples

## Generate AR(1) structure
ar1_cor(n = 5, rho = 0.3)

## first derivative of the AR(1) structure with respect to rho
dar1_cor.drho(n = 5, rho = 0.3)

---

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 $j$th 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 $j$th 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)

---

A function to obtain the baseline survival function

Description

A function to obtain the baseline survival function

Usage

basesurv(object)

Arguments

object

A funsurv object

Value

A data frame including time and baseline survival

Examples

data(simDat)

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

plot(fit, what = "basesurv")

---

First derivative of AR(1) correlation matrix with respect to the auto-regressive coefficient

Description

First derivative of AR(1) correlation matrix with respect to the auto-regressive coefficient

Usage

dar1_cor.drho(n, rho)

Arguments

n	number of events for each subject
rho	autoregressive correlation

Value

A n by n inverse matrix

---

Plot method for 'funsurv' objects

Description

Plot method for 'funsurv' objects

Usage

## S3 method for class 'funsurv'
plot(x, what = c("beta", "fpc", "basesurv"), ...)

Arguments

x	A funsurv object
what	A character string specifying what to be plotted. Use what = "beta" to plot the estimated $\beta(t)$. Use what = "fpc" to plot the functional principal components associated with the the longitudinal measurements. Use what = "basesurv" to plot the baseline survival probabilities.
...	additional graphical parameters to be passed to methods.

Value

A ggplot object ...

Examples

data(simDat)
fit <- AR1_FRAILTY(Recur(t_start %to% t_stop, id, status) ~ z1,
                   sdat = sdat, fdat = fdat, iter.max = 30)
plot(fit, what = "beta")
plot(fit, what = "fpc")
plot(fit, what = "basesurv")

---

Simulated datasets for demonstration

Description

The dataset was generated based on the proposed model $h(t; \boldsymbol{Z}_i, {X}_i(\cdot))=h_{0}(t-t_{i,j-1}) \exp \left(\eta_{ij}\right)$, where $h_0(\cdot)$ is the baseline hazard function generated from a Weibull distribution. $\eta_{ij} = \bm{\alpha}^{\top}\boldsymbol{Z}_i +\int_{t_{i, j-1}}^{t}{X}_{i}(s)\beta(s)ds + v_{ij}$. $\bm{\alpha}$ 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. The simulated dataset is organized into two data frames: a survival data frame (sdat) and a functional data frame (fdat). The variables in each data frame are listed below:

Usage

data(simDat)

Format

A list with two data frame:

sdat

Survival data; a data frame with xxx rows and xxx variables:

id

Subjects identification

event

A sequence of the number of events per subject

t_start

Event starting time

t_end

Event end time

censoring_time

Event censoring time

status

Event status: status=1 if event is observed and status=0 if event is censored

z1

A univariate scalar covariates. Can be extended to multiple scalar covariates

fdat

Functional data; a data frame with xxx rows and xxx variables:

id

Subjects identification

time

Time points for each longitudinal measurement

x

Longitudinal measurements at distinct time points

Source

Simulated data

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