A mixed model approach for structured hazard regression

The classical Cox proportional hazards model is a benchmark
approach to analyze continuous survival times in the presence of
covariate information. In a number of applications, there is a need
to relax one or more of its inherent assumptions, such as linearity
of the predictor or the proportional hazards property. Also, one is
often interested in jointly estimating the baseline hazard together
with covariate effects or one may wish to add a spatial component
for spatially correlated survival data. We propose an extended Cox
model, where the (log-)baseline hazard is weakly parameterized
using penalized splines and the usual linear predictor is replaced
by a structured additive predictor incorporating nonlinear effects
of continuous covariates and further time scales, spatial effects,
frailty components, and more complex interactions. Inclusion of
time-varying coefficients leads to models that relax the
proportional hazards assumption. Nonlinear and time-varying effects
are modelled through penalized splines, and spatial components are
treated as correlated random effects following either a Markov
random field or a stationary Gaussian random field. All model
components, including smoothing parameters, are specified within a
unified framework and are estimated simultaneously based on mixed
model methodology. The estimation procedure for such general mixed
hazard regression models is derived using penalized likelihood for
regression coefficients and (approximate) marginal likelihood for
smoothing parameters. Performance of the proposed method is studied
through simulation and an application to leukemia survival data in
Northwest England.