Modelling strategies to improve genetic evaluation for the New Zealand sheep industry

The question of how best to optimise the accuracy of genetic evaluation for livestock populations has been given new life by the advent of genomics. Therefore we will investigate methods of evaluating and/or improving the accuracy of genetic evaluation in ways applicable to genotyped populations, while trying to maximise computational efficiency. We will explore modelling strategies with utility outside animal breeding, including examples of these potential non-animal breeding applications. In Chapter 1, we will introduce linear mixed models, which is the basis for genetic evaluation in livestock populations, and highlight some properties of these models. We will go on in Chapters 2 and 3 to discuss the concept of genetic connectedness, which in statistical terms is related to reliability of random effect inference in a linear mixed model. In Chapter 2, we will consider comparability of predicted random effects when the model is correctly specified. This will involve investigating and describing the properties of existing diagnostics that we consider assume correct model specification. We will go on show that the types of random effect contrasts, commonly used to determine connectedness by animal breeders, are equal to a contrast of fixed effects. As approximate diagnostics based on simpler fixed effect contrasts have already been proposed in the literature, we will explore the bias introduced when using simpler approximate fixed effect contrasts rather than the fixed effect contrast equivalent to the random effect contrast of interest. Chapter 3 extends the connectedness concept to the case of model misspeciii iii fication. This is a broader topic, which forms the basis for the remainder of the thesis. From an animal breeding perspective, we will describe how model misspecification is typically associated with an incorrect covariance structure for the random effect. We will argue that common beliefs about the causes of incorrect covariance structure naturally suggest a genetic group model. For a genotyped population, we will demonstrate the similarity between the genetic group and population structure concepts. From this, we postulate a probabilistic model for genotype data, contained in the set of multivariate binomial latent variable models, that can be used to define genetic groups. A probabilistic model for population structure is only a tiny subset of latent variable models applicable to multivariate binomial data. In Chapter 4, we will move away from animal breeding in order to explore latent variable models of a principal component/factor analysis type applicable to multivariate binomial and binary data generally. Here we will provide insight in the expected distributions of the observed data, properties of the trial sizes, identification and representation of the link function, develop a computationally efficient framework for model fitting, and fit the proposed models to simulated and one real demographic dataset. Chapter 5 will use the insights gained in Chapter 4 to develop an appropriate latent variable model to analyse genotype data, taking into account biological properties. Using real datasets, and as in Chapter 4 looking outside animal breeding by considering human populations, we will demonstrate moving from an observed to a latent scale produces subtle differences in the clustering of individuals. In Chapters 6 and 7, we will conclude by returning to animal breeding applications. Chapter 6 will investigate genetic group models, where genetic group is defined from the output of pre-specified model. In Chapter 7, we conclude by investigating the relationship matrix (covariance structure for the random effect in genetic evaluation) implied by the latent variable model of Chapter 5 and provide further insight into what kinds of population this model would be appropriate. Acknowledgements To Beef and Lamb Genetics New Zealand, thank you for your generous funding. Your support has made the last three years a much less stressful time than it might otherwise have been. To my fellow postgraduate students in the statistics department, thank you for being friends as well as colleagues, for providing advice, for reminding ourselves that we can achieve our aims and agreeing to teach STAT 110 at Summer School. To the technical and administrative support staff, thank you for all the assistance and the interesting conversations. To Dr Austina Clark and Associate Professor John Harraway, thank you for letting me tutor your courses. The continual refreshing of knowledge on study design and multivariate statistics has strengthened this thesis immensely. To my supervisors. In hindsight, I must thank Professor Richard Barker for insisting I take a full undergraduate statistics program after completing my mathematics degree. It has proven to be one of the best decisions I have ever made, and this thesis is better for it. To Dr Matthew Schofield, thank you for being approachable and willing to discuss my questions at a moments notice. To Dr Ken Dodds and the wider group of scientists at Invermay, you have been welcoming, inclusive and open with your knowledge, from which I have learned much. Lastly, I must thank Dr Michael Lee for seeing potential in a summer student back in 2012. Without your championing and persuading, this PhD would never have happened. iv