Random effects or frailty in competing risk models consist of two underlying distributions:
the conditional distribution of the response variables (failure times), given the random effect
depending on the explanatory variables each with a failure type specific random effect; and
the distribution of the random effect in the population (frailty distribution). In this situation,
the distribution of interest is the unconditional distribution of the response variables which
may or may not have a tractable form. Bandeen-Roche and Liang (2002) described the
association of time to multivariate failure in the presence of a competing risk. As mentioned
above frailty models in the presence of competing risks involve the conditional distribution
failure time given the frailty and the distribution of the frailty. Fahrmeir and Tutz (1994) and
Oskrochi and Davies (1997) implemented the Cholesky decomposition for multivariate frailty
models. The Cholesky decomposition decomposes the frailty variance-covariance matrix into
triangular matrices to convert the integration over a multivariate distribution to multiple
Univariate data ID Yi δi xi1 · · · xip 1 Y1 δ1 x11 · · · x1p 2 Y2 δ2 x21 · · · x2p . . . . . . . . . . . . n Yn δn xn1 · · · xnp Multivariate data
ID Yij cluster δij xij1 · · · xijp
1 Y11 1 δ11 x111 x11p 2 Y21 1 δ21 x211 x21p . . . . . . . . . . . . n1 Yn11 1 δn11 xn111 · · · xn11p . . . . . . . . . . . . 1 Y1J J δ1J x1J 1 x1J p 2 Y2J J δ2J x2J 1 x2J p . . . . . . . . . . . . nJ YnJJ J δnJJ xnJJ 1 · · · xnJJ p
Competing risk data
ID Yi δi1 · · · δiK xi1 · · · xip 1 Y1 δ11 · · · δ1K x11 · · · x1p 2 Y2 δ21 · · · δ2K x21 · · · x2p . . . . . . . . . . . . n Yn δn1 · · · δnK xn1 · · · xnp
Fine and Gray (1999) proposed a novel semi-parametric proportional hazards model for
the sub-distribution. Using the partial likelihood principle and weighting techniques, they
derived estimation and inference procedures for the finite-dimensional regression parameters
under a variety of censoring scenarios. They gave a uniformly consistent estimator for
the predicted cumulative incidence for an individual with certain covariates, confidence
intervals and bands can be obtained analytically or with an easy-to-implement simulation
technique. Fine et al. (2001) considered semi-competing risk models, in which a terminal
event censors a non-terminal event but not vice versa. The joint distribution of the events is
formulated via Gamma frailty model with marginal distributions unspecified. They showed
that the dependence between morbidity and mortality can be estimated separately from their
marginals under a Gamma frailty copula in the region of observable data. Jiang et al. (2004)
considered semi-competing risk models where mortality and morbidity may be correlated
and mortality may censor morbidity. They proposed a semi-parametric estimator for the
survival function based on a joint model for the two time-to-event variables, which utilises
the Gamma frailty specification in the region of the observable data. They extended the
methods of Fine et al. (2001) for the left truncated semi-competing risk problem, developed
an estimator for the Gamma frailty parameter under left truncation and derived a closed
form estimator for the marginal distribution of the non-terminal event. Naskar et al. (2005)
proposed a dependent competing risks model where dependency is induced through the
mixture of various failure types and a frailty component. The frailty term is modelled non-
parametrically using a Dirichlet Process (DP). They considered a semi-parametric mixture
model for analysing clustered competing risks data. Conditional on cluster-specific quantities,
the joint distribution of the failure time and event indicator can be expressed as a mixture
of the distribution of time to failure due to a certain type (or specific cause), and the failure
(competing risks) are logistic functions of some covariates. The cluster-specific quantities are
subject to some unknown distribution that causes frailty. Lu and Tsiatis (2005) compared
two approaches for the competing risks model with missing cause of failure. Under the
assumption that the cause of death is missing at random, they compared the Goetghebeur
and Ryan (1995) partial likelihood approach with the one by Dewanji (1992). They showed
that the Dewanji partial likelihood estimator for the regression coefficients is consistent and
asymptotically Normal. While the Goetghebeur and Ryan estimator is more robust estimator
against mis-specification of proportional baseline hazards. Finkelstein and Esaulovac (2006,
2008) discussed the asymptotic behaviour of competing risks models with correlated frailty
in univariate and bivariate cases. They consider a set of absolutely continuous distributions
of a lifetime random variable.
In the next section, a review the correlated Gamma frailty and its application to competing
risks framework is given. In section 4.7, a new proposed correlated Inverse Gaussian frailty
model and its application to competing risks are presented. A general multivariate Inverse
Gaussian frailty model without any restriction on the correlation structure between the
frailties is given in section 4.8. In addition, a flexible multivariate frailty model that can
be applied whatever the original distribution of the frailty in section 4.10 is proposed. In this
model, a non-parametric multivariate frailty presented.