Log-normal frailty Cox model (penalized partial likelihood + REML)

Hazard lambda_ij(t | b_i) = lambda_0(t) exp(beta' x_ij + b_i) with b_i ~ N(0, sigma^2) per cluster. For fixed sigma^2 the estimates maximise the penalized Breslow partial likelihood l(beta, b) - b'b / (2 sigma^2); sigma^2 is updated by McGilchrist's REML formula (b'b + trace of the frailty block of the inverse penalized information) / q, and the steps alternate.

Usage

Lnfrm(time, event, X, cluster, max_outer = 50L, tol = 1e-07)

Arguments

time

Observed times.

event

Event indicator (1 = event, 0 = censored).

X

Covariate matrix.

cluster

Cluster labels.

max_outer, tol

Outer-loop controls.

Value

List with estimate, se, frailty, sigma2, loglik_penalized, n_outer, n_newton, method.

References

McGilchrist, C. A. (1993), Biometrics 49(1), 221-225; McGilchrist and Aisbett (1991), Biometrics 47(2), 461-466; Therneau, Grambsch and Pankratz (2003), Journal of Computational and Graphical Statistics 12(1), 156-175.

Examples

Lnfrm(time = c(2.5, 1.0, 3.5, 4.0, 2.0, 5.5, 3.0, 6.5), event = c(0, 1, 0, 1, 1, 0, 1, 0),
  X = c(1, 2, 3, 4, 5, 6, 7, 8), cluster = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] -0.2153522
#> 
#> $se
#> [1] 0.3001958
#> 
#> $frailty
#> $frailty$`1`
#> [1] -0.3128393
#> 
#> $frailty$`2`
#> [1] 0.5545354
#> 
#> $frailty$`3`
#> [1] -0.3906566
#> 
#> $frailty$`4`
#> [1] -0.07502709
#> 
#> $frailty$`5`
#> [1] 0.5017709
#> 
#> $frailty$`6`
#> [1] -0.4079018
#> 
#> $frailty$`7`
#> [1] 0.4265444
#> 
#> $frailty$`8`
#> [1] -0.296426
#> 
#> 
#> $sigma2
#> [1] 0.800307
#> 
#> $loglik_penalized
#> [1] -5.483809
#> 
#> $n_outer
#> [1] 50
#> 
#> $n_newton
#> [1] 5
#> 
#> $method
#> [1] "McGilchrist (1993) log-normal frailty, penalized PL + REML variance"
#>