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\nveterinary medicine. The focus is on hierarchical proportional hazards models when the\nbaseline hazard is left completely unspecified. Parameter estimation for these models is\nexplored and the performance of their estimation methods is investigated in terms of\nstatistical properties such as unbiasedness, robustness, and probability coverage.\nThe thesis is formed by manuscripts of four studies. The first study compares, via\nsimulation, the performance of different estimation methods for estimating a random\nslope Cox model with and without covariance between the random effects. The\nsimulation is built to mimic real animal health data. The aim of the study is to establish\nsome practical guidelines for the choice of appropriate statistical estimation methods for\nmodeling random slopes in 2-level hierarchical data. Results show that estimating the full\ncovariance matrix for random effects is always preferable in the analysis and Poisson\nmaximum likelihood estimation is an adequate approach for this task.\nThe second study explores the feasibility of a full hierarchical survival analysis for a\nlarge dataset with three levels of hierarchy and time-dependent predictors and\ncoefficients. To this end, a log-normal nested frailty Cox model is applied to Canadian\nBovine Mastitis Research Network (CBMRN) data to identify risk factors associated with\nthe hazard of clinical mastitis (CM) during cow lactations. This nested frailty model is\nestimated by the Poisson maximum likelihood approach with Gaussian quadrature. The\nperformance, in terms of bias and efficiency of estimates, of the Poisson maximum\nlikelihood approach (estimated using either Gaussian quadrature or Laplace approximation) is compared with the performance of the penalized partial likelihood\napproach. The Poisson maximum likelihood with Gaussian quadrature produces fairly\nrobust and adequate estimates while the penalized partial likelihood and the Poisson\nmaximum likelihood with Laplacian approximation are found to have substantial\ndrawbacks. Further, the research indicates that some of the herd managerial factors\ncombined with cow characteristics influence the hazard of CM during the lactation\nperiod; some of these effects are different earlier as compared to later in the lactation.\nThe third study involves analyzing a dataset on calf loss and mortality in beef cattle in\nWestern Canada. This dataset has a cross-classified and multiple membership structure\nwhich is a special type of data structure that has only been accounted for in the analyses\nof linear and generalized linear models but not in survival analysis. The study objectives\nare twofold: the first is to explore and demonstrate the use of Poisson generalized linear\nmixed models (GLMMs) in the Bayesian framework for estimating a Cox model with\ncross-classified and multiple membership frailties. The second, is to simultaneously\nexamine the individual, herd management, and environmental factors associated with\nbeef calf mortality in Western Canada and to estimate the age period where calves are\nmost at risk. Finally, a simulation study with settings similar to the real data is carried out\nto evaluate the estimation approach. The simulation results gave evidence that the\napproach used provides valid estimates.\nIn the fourth study, the robustness of Poisson maximum likelihood estimation was\nassessed, through simulation, for a Cox model with normal random effects under\nmisspecification of the random-effects distribution. The impact of misspecifying the\n\ndistribution 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\naffect the estimation are also investigated. The study shows that the Poisson maximum\nlikelihood approach yields robust estimates under misspecification of the random-effects\ndistribution for within-group fixed effects and in a wide range of situations for betweengroup\nfixed effects. For variance components, the approach produces robust estimation\nunder model misspecification as long as the magnitude of heterogeneity is small, though\nmisspecification may become a matter of concern when the magnitude of heterogeneity\nand group sizes become large.

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