Semiparametric hierarchical proportional hazards models with applications to animal health data

This thesis discusses and applies hierarchical models for survival data in the field of veterinary medicine. The focus is on hierarchical proportional hazards models when the baseline hazard is left completely unspecified. Parameter estimation for these models is explored and the performance of their estimation methods is investigated in terms of statistical properties such as unbiasedness, robustness, and probability coverage. The thesis is formed by manuscripts of four studies. The first study compares, via simulation, the performance of different estimation methods for estimating a random slope Cox model with and without covariance between the random effects. The simulation is built to mimic real animal health data. The aim of the study is to establish some practical guidelines for the choice of appropriate statistical estimation methods for modeling random slopes in 2-level hierarchical data. Results show that estimating the full covariance matrix for random effects is always preferable in the analysis and Poisson maximum likelihood estimation is an adequate approach for this task. The second study explores the feasibility of a full hierarchical survival analysis for a large dataset with three levels of hierarchy and time-dependent predictors and coefficients. To this end, a log-normal nested frailty Cox model is applied to Canadian Bovine Mastitis Research Network (CBMRN) data to identify risk factors associated with the hazard of clinical mastitis (CM) during cow lactations. This nested frailty model is estimated by the Poisson maximum likelihood approach with Gaussian quadrature. The performance, in terms of bias and efficiency of estimates, of the Poisson maximum likelihood approach (estimated using either Gaussian quadrature or Laplace approximation) is compared with the performance of the penalized partial likelihood approach. The Poisson maximum likelihood with Gaussian quadrature produces fairly robust and adequate estimates while the penalized partial likelihood and the Poisson maximum likelihood with Laplacian approximation are found to have substantial drawbacks. Further, the research indicates that some of the herd managerial factors combined with cow characteristics influence the hazard of CM during the lactation period; some of these effects are different earlier as compared to later in the lactation. The third study involves analyzing a dataset on calf loss and mortality in beef cattle in Western Canada. This dataset has a cross-classified and multiple membership structure which is a special type of data structure that has only been accounted for in the analyses of linear and generalized linear models but not in survival analysis. The study objectives are twofold: the first is to explore and demonstrate the use of Poisson generalized linear mixed models (GLMMs) in the Bayesian framework for estimating a Cox model with cross-classified and multiple membership frailties. The second, is to simultaneously examine the individual, herd management, and environmental factors associated with beef calf mortality in Western Canada and to estimate the age period where calves are most at risk. Finally, a simulation study with settings similar to the real data is carried out to evaluate the estimation approach. The simulation results gave evidence that the approach used provides valid estimates. In the fourth study, the robustness of Poisson maximum likelihood estimation was assessed, through simulation, for a Cox model with normal random effects under misspecification of the random-effects distribution. The impact of misspecifying the distribution of random effects is assessed based on two different non-normal distributions for random effects and three different model designs. Some of the factors that might affect the estimation are also investigated. The study shows that the Poisson maximum likelihood approach yields robust estimates under misspecification of the random-effects distribution for within-group fixed effects and in a wide range of situations for betweengroup fixed effects. For variance components, the approach produces robust estimation under model misspecification as long as the magnitude of heterogeneity is small, though misspecification may become a matter of concern when the magnitude of heterogeneity and group sizes become large.