Shared gamma frailty model with marginal (population-averaged) output

Conditional intensity lambda_i(t | w_k) = w_k lambda_0(t) exp(beta' x_i), w_k ~ Gamma(1/theta, 1/theta) shared within cluster k (Vaupel, Manton & Stallard 1979). theta maximises the full marginal likelihood; the E-step posterior mean is (D_k + 1/theta) / (Lambda_k + 1/theta) (Klein 1992). The marginal survivor S_m(t|x) = (1 + theta Lambda_0(t) e^(beta x))^(-1/theta) is reported at the baseline (x = 0).

Usage

Shfrm(time, event, X, cluster, theta = NULL)

Arguments

time: Observed times.
event: Event indicator (1 = event, 0 = censored).
X: Covariate matrix.
cluster: Cluster labels.
theta: See Usage.

Value

List with estimate, se, theta, kendall_tau, frailty, baseline_times, baseline_cumhaz, marginal_survivor, loglik, method.

References

Vaupel, Manton and Stallard (1979), Demography 16(3), 439-454; Klein (1992), Biometrics 48, 795-806; Balan and Putter (2019), frailtyEM, Journal of Statistical Software 90(7).

Examples

# four clusters of two: within-cluster replication is what makes
# the frailty variance theta identifiable at all
Shfrm(time = c(1, 2, 3, 4, 5, 6, 7, 8), event = c(1, 1, 0, 1, 1, 0, 1, 1),
  X = c(0, 1, 0, 1, 0, 1, 0, 1), cluster = c(1, 1, 2, 2, 3, 3, 4, 4))
#> $estimate
#> [1] -0.5469267
#> 
#> $se
#> [1] 0.9162376
#> 
#> $theta
#> [1] 0.0001000002
#> 
#> $kendall_tau
#> [1] 4.999759e-05
#> 
#> $frailty
#> $frailty$`1`
#> [1] 1.000164
#> 
#> $frailty$`2`
#> [1] 1.00003
#> 
#> $frailty$`3`
#> [1] 0.999953
#> 
#> $frailty$`4`
#> [1] 0.9998531
#> 
#> 
#> $baseline_times
#> [1] 1 2 4 5 7 8
#> 
#> $baseline_cumhaz
#> [1] 0.1583556 0.3465116 0.6141855 0.9309274 1.5644428 3.2926311
#> 
#> $marginal_survivor
#> [1] 0.8535473 0.7071549 0.5410916 0.3942051 0.2092302 0.0371761
#> 
#> $loglik
#> [1] -13.53177
#> 
#> $method
#> [1] "Vaupel et al (1979) shared gamma frailty, full-marginal-likelihood profile (Balan-Putter eq. 3), Klein (1992) EM"
#>