## Presenting two statistical genetics problems: Are bones found in a carpark from a specified dead king & heritability analysis in the genome era

**Problem 1. How to evaluate the probability that bones found in a carpark are from a specified dead king: **Although the evidence for bones found under a Leicester UK carpark to have been those of King Richard III seemed extremely strong even before the full genetic data became available, there will always be sceptics seeking headlines on such a highly-publicised issue. Moreover, for reasons I will explain and contrary to popular conceptions, the genetic data were not decisive on their own. Therefore when publishing the full genetic data, we set out to quantify as much as we could of the disparate lines of evidence, both genetic and non-genetic, in order to come up with an overall summary of evidential weight. This involved a number of assumptions, judgment calls and presentational issues that it may be interesting to review in order to be well prepared when next faced with a dead-king-in-carpark problem.

**Problem 2. Heritability analysis in the genome era: **The heritability of an observed trait is the fraction of its variance explained by a matrix of relatedness doefficients, and is widely interpreted as how "genetic" a trait is. The usual statistical model assumes a random effect interpreted as a latent genetic contribution to the trait. Relatedness was traditionally measured from pedigrees, but in recent years we have become increasingly aware both of the limitations of traditional kinship coefficients and the possibility of doing better by measuring kinship directly from genomic data rather than pedigrees. However, there are many ways to measure genetic similarity from genome data and arguments for any one approach being canonical are not compelling. In many formulations, the random effects model is equivalent to a penalised regression model similar to ridge regression, in which every genetic marker is a predictor variable. I will explore some of the implications of the new approaches, which are both destructuve of some well-established ideas and constructive in suggesting new ways to investigate the genetic architecture of observed traits. This is joint work with Biostatistics Research Fellow Doug Speed (UCL), who is funded by the UK Medical Research Council.