# Selection Index

The concept of a selection index was first described by a plant breeder, HFS Smith, in 1936[1]. He understood that the characters with which a plant breeder is principally concerned with are quantitative in nature, and that the interplay of genetics and environment made it difficult to discern the genotypic values of individual plants. Furthermore, he also understood that several desirable traits determined a line’s value (e.g., grain size, ear size, yield, protein content, etc.). Most importantly he opined that “…the actual worth to be attributed to each character is usually unknown.” He applied a statistical technique called a Discriminant Function to the traits he observed to help best indicate the “genetic value” of a plant.

The notion of a single function composed of multiple traits in livestock was first described by Hazel and Lush in 1942[2]. In this paper the index approach was described as a “total score” method that “…will permit extra merit in one characteristic to offset slight defects in another.” As above, the authors noted that “The greatest practical obstacle to the total score method is the difficulty of knowing how much importance should be given each trait in making up the score.”

## Economically Optimal Index Construction

Selection index as we know it today was formalized by Hazel in 1943[3]. He first defined the concept of aggregate genotype (H), or breeding objective in terms of the “…net genetic improvement which can be brought about by selecting among a group of animals is the sum of the genetic gains made for the several traits which have economic importance. It is logical to weight the gain made for each trait by the relative economic value of that trait”:

$H=a_{1}G_{1}+a_{2}G_{2}+...+a_{m}G_{m}$

The $G_{i}$ are the genetic values (or EPD) for each trait and the $a_{i}$ represent the marginal economic value (mev) for each trait. Hazel noted that the mev for each trait “…depends on the amount by which profit may be expected to increase for each unit improvement in that trait.” In other words, the mev measures the increase in profit that can be generated by adding an extra unit of input but holding all other values constant. This is a very similar approach to calculating derivatives of equations (as we will see later).

In practice, the genetic values may be unknown (e.g., no EPD are available), but let us assume we record an animal’s phenotype for several traits and combine them into a selection index:

$I=b_{1}x_{1}+b_{2}x_{2}+...+b_{n}x_{n}$

The $b_{i}$ are known as selection index weights and the $x_{i}$ are phenotypes. How are the $b_{i}$ estimated? According to theory we want to estimate the $b_{i}$ so that when used as a selection criterion $I$ will maximize the response in the aggregate genotype or breeding objective ($H$). This can be achieved by estimating the $b_{i}$ using the following equation:

$b=P^{-1}Ga$

For m objective traits and n information sources (or indicator traits), $P$ is an n×n matrix of phenotypic (co)variances, $G$ is an n×m matrix of genetic (co)variances, $a$ is an mx1 vector of economic values and $b$ is an mx1 vector of selection index weights.

It is worth noting at this point that the traits recorded and which appear in the index do not need to be, and often are not, the same traits as those that appear in the aggregate genotype.

In the previous section, index development was based purely on phenotypic records. In reality, selection indexes developed by breed associations or other entities utilize Expected Progeny Differences (EPD) weighted by economic values. EPD for each trait from genetic evaluation are first estimated, where all possible phenotypic information is utilized. Under this framework, it is assumed that multi-trait genetic evaluations are utilized to account for covariance between traits. The EPD are then combined with the economic values to derive the index.

## Desired Gains Approach

Performance recording and genetic evaluation services should be flexible, enabling the definition of breeding objectives in different ways, according to traits’ suitability and to breeders’ preferences. An optimal index is the sum of expected progeny differences (EPDs) for each economically relevant trait weighted by economic values and summed over all traits.

There will be instances in which the most appropriate course of action does not necessarily involve explicit use of economic values. For example, a breeding organization may have a perceived flaw, that when corrected, would improve the marketability of its members' seed stock (e.g., docility, scrotal circumference, etc.). In these situations, a Desired Gains approach would be the most effective and technically sensible method to achieve genetic improvement. [4][5] Desired Gains indexes do not require economic weights, and they do not maximize the correlation between the breeding objective and the index as optimum indexes do. An index constructed using pre-determined desired genetic gain will achieve breeding goals within a minimum number of generations. That said, the use of desired gains should not be a problem for breeders who are fully aware of the relative merits and potential shortcomings of their breeding stock.

## Ad Hoc Index

With these thoughts in mind, one’s ability to simply apply a set of proportional weightings, where the weightings sum to one (e.g., 10% to calving ease, 25% to carcass weight, etc.) has no rational basis in genetic improvement. These ad hoc indexes lack efficiency. They fail to consider the differing amounts of variation and clearly defined value in the traits. Without defined breeding objectives there is a costly loss in potential for genetic improvement over time.

## Index Construction BIF Guideline

BIF recommends not using an ad hoc approach to the construction of selection indexes. Indexes should be created for seedstock and commercial producers that are economically optimal. If a breeding organization has a perceived marketing shortcoming, a well-defined desired gains approach should be used for breeding stock development.

## References

1. Smith, F.H. 1936. A discriminant function for plant selection. Ann. Eugen. 7:240–250. doi:10.1111/j.1469-1809.1936. tb02143.x
2. Hazel, L.N., and J.L. Lush. 1942. The efficiency of three methods of selection. J. Hered. 33:393–399. doi:10.1093/oxfordjournals. jhered.a105102
3. Hazel, L.N. 1943. The genetic basis for constructing selection indexes. Genetics 8:476–490.
4. Pesek, J. and R. J. Baker. 1969. Desired improvement to selection indexes. Canadian J. Plant Sci. 49:803-804.
5. Itoh, Y. and Y. Yamada. 1986. Re-examination of selection index for desired gains. Genet. Sel. Evol. 18(4), 499-504.