Genetic mapping: placing genes and other genetic markers on chromosomes, ordering them in relation to one another and other landmarks (such as centromeres), and assigning genetic (also known as map) distances to adjacent pairs.
As we will see, statistical notions play a key role in constructing genetic maps. More sophisticated statistical analyses can (and these days do) get used, though they are not necessary. Apparently they were not embraced by the early Drosophila researchers, who simply used many progeny and many loci.
These ideas arose in the period 1913--1920. Before that, recombination fractions were obtained by counting recombinants in backcrosses. There was an interesting early paper applying the method of minimum chi-square (for the first time) to the estimation of a recombination fraction from a phase-known intercross involving completely dominant characters by Engledow and Yule, 1914, see Edwards (Ann Hum Genet 60 1996:237--249). However this paper appears to have had no influence on later developments.
Our subject really began with a paper by Sturtevant in 1913, whose title tells it all: ``The linear arrangement of six sex-linked factors in Drosophila, as shown by their mode of association". As you would expect, association was measured by recombination fractions. The factors in question all corresponded to visible phenotypes in Drosophila, eye color, body color and other similar features. Sturtevant suggested that the unit of length between two factors be such that on average, one crossover will occur between the loci in every 100 gametes formed. That is, percent crossover should be used as a distance measure. He saw that this measure was additive for small distances, but not so for larger ones, and attributed this to double crossovers. Before we pursue this issue a little more, consider the following:
Exercise 1:
Suppose that we have a set of points A, B, C, D, ...along a line
which has some unknown true distance along it. Suppose also that we have
a set of pairwise ``distances" 
, 
, ..., 
,...
which are monotone functions of true distances. Describe a procedure for
ordering the points along the line.
Now let us return to the double crossover issue. Suppose that we have loci A, B and C in that order along a chromosome. In any meiotic product there will be an even or an odd number of exchanges between A and B, and between B and C, with frequencies roughly as follows: Numbers of exchanges in intervals given below each interval.
Now we make the following observation: in the second and third cases, there is also a recombination between A and C, but not in the fourth. This suggests what Sturtevant observed: that
.
Shortly after his paper, a variety of relationships between recombination fractions appeared, including
(satisfactory for small rs, false for larger rs)
(the best initially)
(under independence, equivalent to
.)
This independence relation was also found to be incompatible with the data. Thus the best of these relationships was somewhere between the first and third.
Exercise 2: Prove that the third relation holds under the assumption of independence of exchanges in disjoint intervals. (How much more need we assume to conclude that the exchanges form a Poisson process?)