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.