Cox model with a shared gamma frailty — morie_cox

One multiplicative random effect per cluster, gamma distributed with mean 1 and variance theta, fitted by alternating a Cox fit at the current frailties with the closed-form posterior update, all inside a profile over theta.

Usage

morie_cox_frailty(
  time,
  event,
  X,
  cluster,
  theta = NULL,
  max_iter = 30L,
  tol = 1e-06,
  ties = "efron"
)

Arguments

time: follow-up times, non-negative.
event: 1 for an event, 0 for right-censored.
X: covariate matrix, one row per subject.
cluster: cluster label per subject.
theta: frailty variance. Profiled when NULL.
max_iter, tol: inner-loop controls.
ties: "efron" or "breslow".

Value

list with beta, se, hazard_ratio, theta, kendall_tau, frailty, clusters.

Details

Theta is profiled over the MARGINAL likelihood rather than estimated from the fitted frailties, and that is not a stylistic choice. The posterior means are shrunk toward 1, so their sample variance is badly biased downward – it collapses to zero and takes the frailty with it, leaving a model that reports independence no matter how clustered the data are.

Ignoring clustering leaves standard errors too SMALL, which is the direction that produces false findings. Kendall's tau for this model is theta / (theta + 2).

References

Therneau, T. M. and Grambsch, P. M. (2000). Modeling Survival Data, Ch. 9. Springer.

Examples

# \donttest{
set.seed(4)
X <- matrix(rnorm(120), ncol = 1)
cl <- rep(1:20, each = 6)
tt <- rexp(120, exp(X * 0.5) * rep(rgamma(20, 2, 2), each = 6))
round(morie_cox_frailty(tt, rep(1, 120), X, cl)$kendall_tau, 3)
#> [1] 0.032
# }