The shared frailty comepting risks model is fitted to family data with specified baseline hazard distributions and frailty distribution
Event 1:
h1(t|X,Z) = h01(t - t0) Z1 exp(βs1 * xs + βg1 * xg),
Event 2:
h2(t|X,Z) = h02(t - t0) Z2 exp(βs2 * xs + βg2 * xg),
where h01(t) and h02(t) are the baseline hazard functions for event 1 and event 2, respectively, which can be specified by base.dist. t0 is a minimum age of disease onset, Z1 and Z2 are frailties shared within families for each event and follow either a gamma, log-normal, correlateg gamma, or correlated log-normal distributions, which can be specified by frailty.dist. xx and xg indicate male (1) or female (0) and carrier (1) or non-carrier (0) of a main gene of interest, respectively. Additional covariates can be added to formula1 for event 1 and formula2 for event 2 in the model.
Choice of frailty distributions for competing risk models
frailty.dist = "gamma" shares the frailties within families generated from a gamma distribution independently for each competing event, where
Zj follows Gamma(kj, 1/kj).
frailty.dist = "lognormal" shares the frailties within families generated from a log-normal distribution independently for each competing event, where
Zj follows log-normal distribution with mean 0 and variance 1/kj.
frailty.dist = "cgamma" shares the frailties within families generated from a correlated gamma distribution to allow the frailties between two events to be correlated, where the correlated gamma frailties (Z1, Z2) are generated with three independent gamma frailties (Y0, Y1, Y2) as follows:
Z1 = k0/(k0 + k1) Y0 + Y1 ;
Z2 = k0/(k0 + k2) Y0 + Y2 ,
where Y0 from Gamma(k0, 1/k0),
Y1 from Gamma(k1, 1/(k0 + k1)),
Y2 from Gamma(k2, 1/(k0 + k2)).
frailty.dist = "clognormal" shares the frailties within families generated from a correlated log-normal distribution where
log(Zj) follows a normal distribution with mean 0, variance 1/kj and correlation between two events k0.
depend should specify the values of related frailty parameters: c(k1, k2) with frailty.dist = "gamma" or frailty.dist = "lognormal"; c(k1, k2, k0) for frailty.dist = "cgamma" or frailty.dist = "clognormal".
More details about the competing risks model for family data arising from population-based study designs (design="pop", "pop+" and their inference procedure based on the ascertainment-corrected likelihood approach can be found in Choi et al., 2021.
Note that the baseline parameters include lambda and rho, which represent the scale and shape parameters, respectively, and eta, additional parameter to specify for "logBurr" distribution. For the "lognormal" baseline distribution, lambda and rho represent the location and scale parameters for the normally distributed logarithm, where lambda can take any real values and rho > 0. For the other baselinse distributions, lambda > 0, rho > 0, and eta > 0. When a piecewise constant distribution is specified for the baseline hazards, base.dist="piecewise", baseparm should specify the initial interval-constant values, one more than the cut points specified bycuts.
Transformed baseline parameters are used for estimation; log transformation is applied to both scale and shape parameters (\(\lambda, \rho\)) for "Weibull", "loglogistic", "Gompertz" and "gamma" baselines, to (\(\lambda, \rho, \eta\)) for "logBurr" and to the piecewise constant parameters for a piecewise baseline hazard. For "lognormal" baseline distribution, the log transformation is applied only to \(\rho\), not to \(\lambda\), which represents the location parameter for the normally distributed logarithm.
Calculations of penetrance estimates and their standard errors and 95% confidence intervals at given ages can be obtained by penetrance function via Monte-Carlo simulations of the estimated penetrance model.