weibull_schur_correction: Correction for the Variance-Covariance Matrix for Uncertainty...

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Correction for the Variance-Covariance Matrix for Uncertainty in Scale

Description

Computes the Schur complement \textbf{S} such that \textbf{G}^* = (\textbf{G}^{-1} + \textbf{S})^{-1} properly accounts for uncertainty in estimating the Weibull scale parameter when estimating the variance-covariance matrix. Otherwise, the variance-covariance matrix is optimistic and assumes the scale is known, when it was in fact estimated. Note that the parameterization adds the output of this function elementwise (not subtract) so for most cases, the output of this function will be negative or a negative definite/semi-definite matrix.

Usage

weibull_schur_correction(
  X,
  y,
  B,
  dispersion,
  order_list,
  K,
  family,
  observation_weights,
  status
)

weibull_shur_correction(
  X,
  y,
  B,
  dispersion,
  order_list,
  K,
  family,
  observation_weights,
  status
)

Arguments

X	Block-diagonal matrices of spline expansions
y	Block-vector of response
B	Block-vector of coefficient estimates
dispersion	Scalar, estimate of dispersion (sigma^2 = scale^2). The lgspline framework stores and passes dispersion (sigma^2); the Weibull scale (sigma) matching survreg$scale is sqrt(dispersion).
order_list	List of partition orders
K	Number of partitions minus 1 (K)
family	Distribution family
observation_weights	Optional observation weights (default = 1)
status	Censoring indicator (1 = event, 0 = censored) Indicates whether an event of interest occurred (1) or the observation was right-censored (0). In survival analysis, right-censoring occurs when the full survival time is unknown, typically because the study ended or the subject was lost to follow-up before the event of interest occurred.

Details

Adjusts the variance-covariance matrix unscaled for coefficients to account for uncertainty in estimating the Weibull scale parameter, that otherwise would be lost if simply using \textbf{G}=(\textbf{X}^{T}\textbf{W}\textbf{X} + \textbf{L})^{-1}. This is accomplished using a correction based on the Schur complement so we avoid having to construct the entire variance-covariance matrix, or modifying the procedure for lgspline substantially. For any model with nuisance parameters that must have uncertainty accounted for, this tool will be helpful.

This both provides a tool for actually fitting Weibull accelerated failure time (AFT) models, and boilerplate code for users who wish to incorporate Lagrangian multiplier smoothing splines into their own custom models.

Value

List of blockwise Schur-complement corrections \textbf{S}_k to be elementwise added to each block of the information matrix before inversion, with 0 returned for empty partitions.

Examples

## Minimal example of fitting a Weibull Accelerated Failure Time model
# Simulating survival data with right-censoring
set.seed(1234)
t1 <- rnorm(1000)
t2 <- rbinom(1000, 1, 0.5)
yraw <- rexp(exp(0.01*t1 + 0.01*t2))
# status: 1 = event occurred, 0 = right-censored
status <- rbinom(1000, 1, 0.25)
yobs <- ifelse(status, runif(length(yraw), 0, yraw), yraw)
df <- data.frame(
  y = yobs,
  t1 = t1,
  t2 = t2
)

## Fit model using lgspline with Weibull Schur correction
model_fit <- lgspline(y ~ spl(t1) + t2,
                      df,
                      unconstrained_fit_fxn = unconstrained_fit_weibull,
                      family = weibull_family(),
                      need_dispersion_for_estimation = TRUE,
                      dispersion_function = weibull_dispersion_function,
                      glm_weight_function = weibull_glm_weight_function,
                      schur_correction_function = weibull_schur_correction,
                      status = status,
                      opt = FALSE,
                      K = 1)

print(summary(model_fit))

## Fit model using lgspline without Weibull Schur correction
naive_fit <- lgspline(y ~ spl(t1) + t2,
                      df,
                      unconstrained_fit_fxn = unconstrained_fit_weibull,
                      family = weibull_family(),
                      need_dispersion_for_estimation = TRUE,
                      dispersion_function = weibull_dispersion_function,
                      glm_weight_function = weibull_glm_weight_function,
                      status = status,
                      opt = FALSE,
                      K = 1)

print(summary(naive_fit))

lgspline documentation built on May 8, 2026, 5:07 p.m.