Prediction of annual genetic gain and improvement lag between populations

R.ésumé Prédiction du progrès génétique annuel et du décalage génétique entre sous-populations. Une approche du calcul de l’espérance du progrès génétique et du décalage entre sous-populations, basée sur l’algèbre matricielle, est proposée. Il s’agit d’une généralisation de la formule classique de Rendel et Robertson (1950), dont la caractéristique principale est de comparer les valeurs moyennes des générations successives. Un exemple simple est donné.


INTRODUCTION
The formula of Rendel and Robertson (1950) for estimating the annual genetic gain is well suited to closed homogeneous populations. It may be used directly when there is only one type of breeding animal per sex. In other cases, such as progeny test designs where known and tested males are both reproducing, the formula has to be adapted (Lindh6, 1968). Bichard (1971) showed how to process a hierarchical population and how to estimate the improvement lag between subpopulations. These methods are based on comparisons between the mean additive genetic values of successive generations. More recently, iterative methods (Hill, 1974;Elsen and Mocquot, 1974;Elsen 1980;Ducrocq and (auaas, 1988) have been developed in order to take account of the year by year change of genetic values. They are well fitted for the description of hierarchical populations.
In the present paper, we propose a new method for estimating the genetic progress and improvement lag between subpopulations, which can be applied to these heterogeneous populations. Like the method of Rendel and Robertson (1950), our procedure is based on a comparison of successive generations and is, thus, of a simpler formulation than iterative methods.

Description of the population
We consider only stable populations where the selection policy (selection pressure, organisation of matings) and structure are constant.
The population is subdivided into groups of breeding animals. Let X i be the mean genetic value of the ith group. The model gives the value of the vector X of X i given the mean values at the previous generation, Yi.
The groups are defined in the following way. Two individuals belong to the same group, i: -if they are of the same sex; -if the probabilities that their respective parents of the same sex belong to the same group j (Pij) are equal; -if they have equal probabilities of being parents of individuals belonging to the same group of the next generation.
Generations are defined here in a relative way: let us consider the population at a given time. All individuals belonging to group i at this moment constitute a generation of this group. By definition their parents which belong to group j are from the previous (parental) generation of group j relative to i. With this definition, this parental generation of group j does not comprise the same individuals if considering their offspring from group i or from group i'. Its mean genetic value will be noted 5 j ; for group i.
Computations of the mean genetic values X i and Y ji are made by considering the individuals, at birth, prior to any selection. The principle of the method is to write relationships between groups of one generation and groups of the previous one. For group i, we have: where M and 7 are male and female breeding animals respectively, and where 0!! is the deviation between the mean value of group j at birth, Y!i, and the value of those individuals from this group which will actually be parents of group i.
On the other hand, due to the genetic progress (OG per year), we have: where L ji is the generation interval between group j and group i.

Solution
Pooling relations [1] and !2!, we get Or, in matrix notation, where A is the matrix of a ij and H the vector of E a!A! &mdash; L ji AG), which we j shall write H = A -LAG, A and L being the vectors of L a ij a ji and E a ij l ji , j j respectively.

Case of a single population
In this case matrix A is stochastic. Indeed, the events &dquo;sire (or dam) of an individual of group i belongs to group j&dquo;, defined over the different groups j of the population, form a complete system of events, and The largest eigenvalue of A is 1. Let V (transpose V T ) be the eigenvector corresponding to 1. This vector V may easily be found by substitution since V T A = V T (note that to simplify this substitution one of the elements of Y may be fixed to 1). ..
Knowing V, the annual genetic gain is easily deduced, using:

Case of a composite population
The general equation is still of type (4!, but here we have: where h X, hh!A and h H are vectors and matrices specific to subpopulations h and h'. In particular, an annual genetic gain AG h , specific to population h, is found for each h H.
As a whole, the matrix A is still stochastic, but each of its hh A elements is not necessarily of this type.
Thus, we have: where U is the column eigenvector (made of 1's) and V T the row eigenvector corresponding to the eigenvalue 1. The T matrix is such that RT = TR = 0.
The lag between 2 groups k and k' is given by g T X, where g T is a row vector with all its elements zero, except the elements corresponding to the groups k(g k = 1) and k'(g!! _ -1). The lag between 2 subpopulations, which could be defined by the difference in mean values of productive animals (milking cows, slauthtered lambs...), will most often be given by the lag between 2 groups belonging respectively to these 2 subpopulations and defined on an equivalent basis. Nevertheless, one can imagine that in some instances the level of a subpopulation may be characterized by a weighted sum L 9k X k . k Thus, as RX is a vector all of whose elements are equal, g T RX = 0 and the lag E is:

Model
Let a population comprise a nucleus and a base. In the nucleus, as in the base, the selection is on maternal performance only. Good females (selection pressures O F1F , in the nucleus, f::1 F2F2 in the base) are dams of young females and natural service males. A fraction d 1 of the nucleus female replacement is made through artificial insemination. The AI sires are sons of elite dams (selection pressure &eth;-F1A1) and AI sires.
Among the sires used in the base, a fraction d 2 was born in the nucleus, given the diffusion of genetic gain. These males are chosen from among those born from artificial insemination, with a selection pressure of O F , AZ on maternal value.
The mean values of the breeding animals will be denoted: . for the nucleus: 9 for the base: Noting that tlF1F1 = tl F1B1 and tl F2F2 = tl F2B2' equation [4] is The eigenvector 1 V of the submatrix 11 A may be written, fixing its first element to 1, The annual genetic gain becomes .
Noting that the eigenvector V of the matrix A is (1V T , 0), we find that E is easily deduced.

Numerical application
We consider the simple situation where all the dam-progeny generation lengths (LF1Fl1LF1Al1LF1Bl1LF1A2,LF2F2,LF2B2) are 5 years and sire-progeny generation It is also assumed that the females are not recorded in the base (O FZF2 = O FZ s 2 = 0), and that, in the nucleus, the dam-daughter are the best 50%, the dam-AI sire are the best 10% and the dam-natural mating sire, the next 20%. Given a common accuracy h = 0.5 for the dam, the selection differentials, in standard deviation units, are given by: where i is the selection intensity function, assuming the trait normally distributed.
With these assumptions, we find The lag E = w!H is then The figure 1 shows the behaviour of the genetic gain and the improvement lag with varying fractions d l and d 2 .

CONCLUSION
The main difficulty of the method is the definition of groups. A particular population may be analysed in different ways. The smaller the number of groups, the more easily the eigenvector V and the inverse matrix M-1 will be found, but the more difficult will be the correct writing of matrix A and vectors 0 and L. There are