In Chapter 2 we analysed the process of genetic drift, or random change of allele frequencies in populations of small size due to the sampling of gametes, and in Chapter 4 we considered the inbreeding generated in these populations by the inevitable mating between relatives. We studied these phenomena under the simplified conditions of the ideal population of Wright–Fisher, which are described in Section 2.5. Under this simple model, we derived the expressions of the expected variance of allele frequencies by genetic drift (equations (2.8) and (2.9)) and the expected inbreeding coefficient and its rate of increase per generation (equations (4.12) and (4.13)), all of them being a function of the population census size, N. However, real populations may fail to meet one or more of the ideal conditions, so that the mentioned expressions would no longer hold.

Some heritable characteristics are qualitative, with an expression clearly identifiable in discrete classes. Such is the case of attributes like some differences in colour, shape or structure, by which individuals of a population or species can be classified. The analysis of this type of simple character was what allowed Mendel to describe the bases of inheritance and many other geneticists, later, to understand the relation between this and the chromosomal behaviour during reproduction, as well as the interactions between genes. However, most of the traits that we find in nature present a continuous variation. Even some of the seemingly discrete attributes, such as colour, may show gradual variation if analysed in detail. These types of characters with gradual variation are called quantitative traits and, sometimes, metric or continuous traits.

Mutation is the source of genetic variation of populations on which natural or artificial selection act to produce genetic changes leading to adaptive evolution or economic improvement of plants and animals. In the case of single loci affecting qualitative traits or genes of major effect on quantitative traits the estimation of the frequency or rate at which mutations appear per generation is relatively simple for dominant mutations, since it is based directly on the count. For example, if from 1 million births of phenotypically normal parents for the achondroplasia allele (a dominant mutation producing dwarfism) 10 individuals appear with the disease, the mutation rate per locus and generation will be u = 10/(2 × 106) = 0.5 × 10−5, where the factor 2 of the denominator stems from the fact that each individual carries two alleles.

As already indicated in Chapter 1, the phenotypic value (P) of an individual for a quantitative trait, deviated from the population mean, is decomposed into the genotypic value (G), determined by the genetic endowment of the individual, and the environmental deviation (E), that is, P = G + E.

Inbreeding is a consequence of the mating between relatives. This is an inevitable phenomenon in populations of small census size even when crossing between their individuals is ‘random’. But inbreeding can also exist in large populations, when relatives mate with each other naturally or forcedly. In this chapter we will analyse the conceptual and mathematical treatment of inbreeding, whose bases are largely due to Sewall Wright. We will address the concepts of coefficient of inbreeding and coancestry, the ways in which these are calculated from genealogical information and genetic markers data, as well as their modulation by the different population forces of change in the allelic frequencies that act in the populations.

The fit between observed (O) and expected (E) frequencies is statistically contrasted by a χ2 test, since the summation Σ(O − E)2/E is distributed as a χ2 distribution with a number of degrees of freedom equal to the number of genotypic classes that are compared less the number of parameters that are needed to obtain the expected values. In this case there are six classes and to obtain the expected values two of the three allele frequencies are needed (the third is determined once the first two are known) and the total number of individuals in the sample. Therefore, the number of degrees of freedom is d.f. = 6 − 2 − 1 = 3.