Fit a Transformed Hazards Model by SMKLE

Fit a transformed hazards model for survival data with sparsely and intermittently observed longitudinal covariates using the sieve maximum kernel-weighted log-likelihood estimator (SMKLE).

Usage

skmle(
  formula,
  data,
  id,
  obs_times,
  s = 0,
  h = NULL,
  nknots = 3,
  lq_nodes = 64,
  maxeval = 10000,
  xtol_rel = 1e-06,
  one_sided = TRUE,
  weight = NULL,
  optimizer = c("slsqp", "barrier")
)

Arguments

formula: A model formula. The left-hand side must be a survival::Surv() response and the right-hand side must contain at least one covariate.
data: Data frame containing the variables used in formula, id, and obs_times.
id: Subject identifier. Non-numeric identifiers are allowed and are internally converted to integer subject codes.
obs_times: Longitudinal observation times aligned row-wise with data. Times may be on any scale; the sieve basis and the cumulative-hazard quadrature are built on the observed follow-up, so there is no need to rescale to the unit interval first. h must be on the same scale.
s: Box-Cox transformation parameter, defaulting to 0. s = 0 is the proportional hazards model, s = 1 is the additive hazards model, and values in between interpolate. If you do not have a reason to choose otherwise, the default is the familiar Cox model.
h: Positive kernel bandwidth. If omitted, one is read off the observation times as a rule of thumb and reported in a message. Use skmle_cv() to choose it from the data.
nknots: Number of interior knots used in the sieve approximation of the baseline component. Knots are placed at (1:nknots)/(nknots + 1). The basis is a natural cubic spline (splines::ns); its order is fixed, which is why there is no norder argument.
lq_nodes: Number of Legendre-Gauss quadrature nodes used in numerical integration.
maxeval: Maximum number of optimizer evaluations.
xtol_rel: Relative convergence tolerance passed to the optimizer.
one_sided: Logical. TRUE (the default) uses a half kernel: only covariate observations strictly before the event or quadrature time inform that time, which is the risk-set restriction and the estimator as published. FALSE uses a full, two-sided kernel, smoothing the covariate path from both sides. The switch applies to the risk-set averages inside the C++ backend as well as to the row weights, so the two are always consistent.
weight: Weight function of the standardised lag \((t - r)/h\), or NULL (the default) for the Epanechnikov kernel implied by one_sided. Any R function will do; see kernel-weights.
optimizer: "slsqp" (the default) maximises the objective with NLopt's SLSQP, constraining 1 + s * eta >= 0 at the event-time rows and flooring the hazard elsewhere, which is the estimator as published. "barrier" uses an interior-point (log-barrier) Newton method that keeps 1 + s * eta > 0 at every point the objective evaluates, event rows and quadrature pairs alike, so no floor is needed. For 0 <= s <= 1 that problem is concave and the barrier fit is its global optimum. The two coincide at s = 0, where there are no constraints. maxeval and xtol_rel apply to "slsqp" only.

Value

An object of class skmle containing:

coefficients: regression coefficient estimates,
var: estimated variance-covariance matrix for the regression coefficients,
gamma: estimated spline coefficients for the baseline component,
loglik: maximized log-likelihood,
convergence: optimizer status code,
scores: per-subject contributions to the score for the coefficients (one row per subject), from which the sandwich's B is formed,
model metadata such as n, s, h, and call.

Details

skmle() is the main model-fitting function in the package. It combines:

kernel weighting to handle intermittently observed longitudinal covariates,
a sieve approximation for the unknown baseline component, and
a C++-backed numerical optimizer for the joint estimation problem.

The returned object follows the usual R model pattern: print the fitted coefficients with print(), obtain inferential output with summary(), and visualize the estimated baseline component with plot().

References

Sun, Dayu, Zhuowei Sun, Xingqiu Zhao, and Hongyuan Cao. "Kernel Meets Sieve: Transformed Hazards Models with Sparse Longitudinal Covariates." Journal of the American Statistical Association (2025): 1-12.

Examples

# \donttest{
library(survival)

set.seed(123)
dat <- sim_skmle_data(
  n = 80,
  mu = function(tt) 8 * (0.75 + (0.5 - tt)^2),
  mu_bar = 8,
  alpha = function(tt) 0.5 * 0.75 + 0.75 * (tt * (1 - sin(2 * pi * (tt - 0.25)))),
  beta = c(1, -0.5),
  s = 0,
  cen = 0.7
)

fit <- skmle(
  Surv(X, delta) ~ covariates,
  data = dat,
  id = id,
  obs_times = obs_times,
  s = 0,
  h = 0.5,
  nknots = 3
)

summary(fit)
#> Call:
#> skmle(formula = Surv(X, delta) ~ covariates, data = dat, id = id, 
#>     obs_times = obs_times, s = 0, h = 0.5, nknots = 3)
#> 
#>   n= 80
#> 
#>             Estimate Std. Error z value  Pr(>|z|)    
#> covariates1  1.02973    0.29390  3.5037 0.0004588 ***
#> covariates2 -0.41579    0.31618 -1.3151 0.1884888    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Log-likelihood: -0.1331 
# }