lgspline (version 0.2.0)

weibull_glm_weight_function: Weibull GLM Weight Function for Constructing Information Matrix

Description

Computes diagonal weight matrix \(\textbf{W}\) for the information matrix \(\textbf{G} = (\textbf{X}^{T}\textbf{W}\textbf{X} + \textbf{L})^{-1}\) in Weibull accelerated failure time (AFT) models.

Usage

weibull_glm_weight_function(
  mu,
  y,
  order_indices,
  family,
  dispersion,
  observation_weights,
  status
)

Value

Vector of weights for constructing the diagonal weight matrix \(\textbf{W}\)

in the information matrix \(\textbf{G} = (\textbf{X}^{T}\textbf{W}\textbf{X} + \textbf{L})^{-1}\).

Arguments

mu

Predicted survival times

y

Observed response/survival times

order_indices

Order of observations when partitioned to match "status" to "response"

family

Weibull AFT family

dispersion

Estimated dispersion parameter (\(s^2\))

observation_weights

Weights of observations submitted to function

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

This function generates weights used in constructing the information matrix after unconstrained estimates have been found. Specifically, it is used in the construction of the \(\textbf{U}\) and \(\textbf{G}\) matrices following initial unconstrained parameter estimation.

These weights are analogous to the variance terms in generalized linear models (GLMs). Like logistic regression uses \(\mu(1-\mu)\), Poisson regression uses \(e^{\mu}\), and Linear regression uses constant weights, Weibull AFT models use \(\exp((\log y - \log \mu)/s)\) where \(s\) is the scale (= \(\sqrt{\text{dispersion}}\)) parameter.

Examples

## Demonstration of glm weight function in constrained model estimation
set.seed(1234)
n <- 1000
t1 <- rnorm(n)
t2 <- rbinom(n, 1, 0.5)

## Generate survival times
lambda <- exp(0.5 * t1 + 0.3 * t2)
yraw <- rexp(n, rate = 1/lambda)

## Introduce right-censoring
status <- rbinom(n, 1, 0.75)
y <- ifelse(status, yraw, runif(1, 0, yraw))

## Fit model demonstrating use of custom glm weight function
model_fit <- lgspline(y ~ spl(t1) + t2,
                      data.frame(y = y, t1 = t1, t2 = t2),
                      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,
                      shur_correction_function = weibull_shur_correction,
                      status = status,
                      opt = FALSE,
                      K = 1)

print(summary(model_fit))