The overarching goal of this dissertation is to integrate heterogeneous data for the estimation of disease coheritability and subtyping. Chapter 2 focuses on the significance and estimation of heritability and coheritability, which quantify the proportion of phenotypic variation attributable to genetic factors and the genetic correlations between different traits, respectively. To achieve this, we develop robust statistical methods based on estimating equations that account for familial correlations and the computational challenges posed by large pedigrees and extensive datasets. Our methods are evaluated through simulations, demonstrating satisfactory consistency and robust inference properties. Compared to simpler methods performing separate trait analysis, our approaches show a greater power through joint analysis of multiple traits. An application to the analysis of heritability and coheritability in electronic health record (EHR) data reveals substantial genetic correlations between mental disorders and metabolic/endocrine measurements, suggesting shared genetic influences that warrant further investigation. These findings have implications for understanding these conditions' etiology, diagnosis, and treatment. Chapters 3 and 4 focus on the importance of patient subtyping for personalized mental health care, particularly relevant to the substantial variability observed in mental disorders. Chapter 3 develops methods for subtyping patients with mental disorders using various data modalities and variational inference. We propose latent mixture models inspired by the Item Response Theory to handle both binary and continuous data. We also introduce Black Box Variational Inference (BBVI) algorithms to overcome the challenges of numeric integration in nonlinear models. Our numerical experiments validate the proposed methods, demonstrating that variance-controlling techniques improve convergence speed and reduce iteration variance. However, the proposed algorithm encounters limitations with latent mixture models containing binary modalities due to approximations used in non-conjugate posterior distributions resulting from the non-exponential family likelihood function. Chapter 4 investigates multi-modal integration techniques for subtyping patients using data from the Adolescent Brain Cognitive Development (ABCD) study. We introduce a Bayesian hierarchical joint model with latent variables and utilize Pólya-Gamma augmentation for posterior approximation, which enables efficient Gibbs sampling and accurate estimation of model parameters. Extensive simulations confirm the consistency of estimators and the prediction accuracy of our method. Applying these methods to patient clustering in the ABCD study provides information for identifying potential clinical subtypes within mental health, which can inform the development of targeted psychological and educational interventions, ultimately improving mental health outcomes. Keywords: latent mixture model, integrative analysis, coheritability, multi-modality, disease subtyping, variational inference, Pólya-Gamma

(Truncated abstract) The aims of this thesis are: (i) To estimate the prevalence of psychiatric illness among persons with intellectual disability and, conversely, the prevalence of intellectual disability among persons with a psychiatric illness; (ii) To describe the disability and service utilisation profile of persons with conjoint disorder; (iii) To examine, in particular, intellectual disability co-occurring with schizophrenia; and (iv) To explore the role of hereditary and environmental (specifically obstetric) risk factors in the aetiology of (i) intellectual disability and (ii) intellectual disability co-occurring with psychiatric illness. This thesis has a special interest in the relationship between intellectual disability and schizophrenia. Where data and sample sizes permit, it explores that relationship at some depth and has included sections on the putative nature of the link between intellectual disability and schizophrenia in the introductory and discussion chapters. To realise its objectives, the thesis comprises a core study focusing on aims (i) – (iii) and a supplementary study whose focus is aim (iv). It also draws on work from an ancillary study completed prior to the period of candidacy...This thesis found that, overall, 31.7% of persons with an intellectual disability had a psychiatric illness; 1.8% of persons with a psychiatric illness had an intellectual disability. The rate of schizophrenia, but not bipolar disorder or unipolar major depression, was greatly increased among cases of conjoint disorder: depending on birth cohort, 3.7-5.2% of individuals with intellectual disability had co-occurring schizophrenia. Down syndrome was much less prevalent among conjoint disorder cases despite being the most predominant cause of intellectual disability while pervasive developmental disorder was over-represented. Persons with conjoint disorder had a more severe clinical profile including higher mortality rates than those with a single disability. The supplementary study confirmed the findings in the core body of work with respect to the extent of conjoint disorder, its severity, and its relationship with pervasive development disorder and Down syndrome. Moreover, the supplementary study and the ancillary influenza study indicated a role for neurodevelopmental insults including obstetric complications in the adverse neuropsychiatric outcomes, with timing of the insult a potentially critical element in defining the specific outcome. The supplementary study also added new information on familiality in intellectual disability. It found that, in addition to parental intellectual disability status and exposure to labour and delivery complications at birth, parental psychiatric status was an independent predictor of intellectual disability in offspring as well as a predictor of conjoint disorder. In conclusion, the facility to collect and integrate records held by separate State administrative health jurisdictions, and to analyse them within the one database has had a marked impact on the capacity for this thesis to estimate the prevalence of conjoint disorder among intellectually disabled and psychiatric populations, and to understand more about its clinical manifestations and aetiological underpinnings.