Selection Index

The concept of a selection index was first described by a plant breeder, HFS Smith, in 1936. 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 individuals 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. 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.” Selection index as we know it today was formalized by Hazel in 1943. 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”:

The are the genetic values (or EPD) for each trait and the 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 are unknown, but let us assume we record an animal’s phenotype for several traits and combine them into a selection index:

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

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

The $G_{i}$ are the genetic values (or EPD) for each trait and the $u_{i}$ 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 are unknown, 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.