The classical twin design relies on a number of strong number of assumptions in order to yield unbiased estimates of heritability. This includes the equal environments assumption — that monozygotic and dizygotic twins experience similar degrees of environmental similarity — an assumption that is likely to be violated in practice for many traits of interest. An alternative method of estimating heritability that does not suffer from many of these limitations is to model trait similarity between sibling pairs as a function of their empirical genome-wide identity by descent sharing, estimated from genetic markers. In this review, I recount the story behind Nick Martin’s and my development of this method, our first attempts at applying it in a human population and more recent studies using the original and related methods to estimate trait heritability.

Two related individuals are identical by descent at a genetic locus if they share the same gene copy at that locus due to inheritance from a recent common ancestor. We consider idealized continuous identity by descent (IBD) data in which IBD status is known continuously along chromosomes. IBD data contains information about the relationship between the two individuals, and about the underlying crossover processes. We present a Monte Carlo method for calculating probabilities for IBD data. The method is not restricted to Haldane's Poisson process model of crossing-over but may be used with other models including the chi-square, Kosambi renewal and Sturt models. Results of a simulation study demonstrate that IBD data can be used to distinguish between alternative models for the crossover process.

A biological population with local random mating, migration, and mutation is studied which exhibits clustering at several different levels. The migration is determined by the clustering rather than actual geographic or physical distance. Darwinian selection is assumed to be absent, and population densities are such that nearby individuals have a probability of being related. An expression is found for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance. Asymptotics for a small mutation rate u are discussed for both a finite number of clustering levels (and of total population size), and for an infinite number of levels. A natural example is discussed in which the probability of heterozygosity varies as u to a power times a periodic function of log(l/u).

A branching random field is considered as a model of either of two situations in genetics in which migration or dispersion plays a role. Specifically we consider the expected number of individuals NA in a (geographical) set A at time t, the covariance of NA and NB for two sets A, B, and the probability I(x, y, u) that two individuals found at locations x, y at time t are of the same genetic type if the population is subject to a selectively neutral mutation rate u. The last also leads to limit laws for the average degree of relationship of individuals in various types of branching random fields. We also find the equations that the mean and bivariate densities satisfy, and explicit formulas when the underlying migration process is Brownian motion.