Advances in shared frailty models with application to twin data

This thesis focuses on advanced statistical methods for the analysis of correlated survival
times with a focus on investigation of twin data. The methods we will discuss were motivated
by the TwinsUK study to investigate the relationships between fracture incidence
with covariates. To model the correlation between twins we use shared frailty models.
However, there is a potential danger of bias in the estimation if the frailty distribution is
misspecified. Frailties are often assumed to follow a gamma distribution. To safeguard us
from the impact of the misspecification of this distribution, we consider flexible baseline
hazards, for instance, B-splines in addition to a parametric baseline hazard. We apply this
methodology to the TwinsUK cohort to predict the probability of experiencing a fracture
in the next five or ten years, given their bone mineral densities (BMD) and their health
status. The models with parametric and more flexible baseline hazards yield very close
results in estimating survival probabilities and thus a choice of parametric baseline hazard
is generally preferred. We find that bone mineral density is a statistically significant
predictor in the model whereas health status is not.

We then-via simulation studies-assess the consequences of frailty distribution misspecification
of estimation of parameters and survival probabilities. When the Weibull baseline
hazard is used, in most cases the scale parameter corrected for the wrong frailty. However,
for some extreme cases, it appears that the scale parameter cannot adjust and thus
parameters and survival probabilities are affected. However, using a flexible function for
the baseline hazard improves the estimation of parameters as well as survival probabilities in
the presence of frailty distribution misspecification.

Often age is preferred underlying time scale. However, participants have different ages at
entry hence using age result in delayed entry. An additional challenge is clustering in the
twins data. In this thesis we will develop methods to estimate models for the relationship
between a time-varying covariate and age to an event while adjusting for delayed entry.
Four approaches for modeling time varying covariates namely, last observation carried
forward, risk set regression calibration, ordinary regression calibration, and joint modelling
approaches will be adapted to the situation of clustered data with delayed entry.