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 The INBREED Procedure

## Computational Details

This section describes the rules that the INBREED procedure uses to compute the covariance and inbreeding coefficients. Each computational rule is explained by an example referring to the fictitious population introduced in the "Getting Started" section.

### Coancestry (or Kinship Coefficient)

To calculate the inbreeding coefficient and the covariance coefficients, use the degree of relationship by descent between the two parents, which is called coancestry or kinship coefficient (Falconer and Mackay 1996, p.85), or coefficient of parentage (Kempthorne 1957, p.73). Denote the coancestry between individuals X and Y by f XY. For information on how to calculate the coancestries among a population, see the section "Calculation of Coancestry."

### Covariance Coefficient (or Coefficient of Relationship)

The covariance coefficient between individuals X and Y is defined by
Cov( X,Y) = 2f XY
where f XY is the coancestry between X and Y. The covariance coefficient is sometimes called the coefficient of relationship or the theoretical correlation (Falconer and Mackay 1996, p.153; Crow and Kimura 1970, p.134). If a covariance coefficient cannot be calculated from the individuals in the population, it is assigned to an initial value. The initial value is set to 0 if the INIT= option is not specified or to cov if INIT=cov. Therefore, the corresponding initial coancestry is set to 0 if the INIT= option is not specified or to (1/2)cov if INIT=cov.

### Inbreeding Coefficients

The inbreeding coefficient of an individual is the probability that the pair of alleles carried by the gametes that produced it are identical by descent (Falconer and Mackay 1996, Chapter 5; Kempthorne 1957, Chapter 5). For individual X, denote its inbreeding coefficient by F X. The inbreeding coefficient of an individual is equal to the coancestry between its parents. For example, if X has parents A and B, then the inbreeding coefficient of X is

F X = f AB

### Calculation of Coancestry

Given individuals X and Y, assume that X has parents A and B and that Y has parents C and D. For nonoverlapping generations, the basic rule to calculate the coancestry between X and Y is given by the following formula (Falconer and Mackay 1996, p.86):

f XY = (1/4) ( f AC + f AD + f BC + f BD )

And the inbreeding coefficient for an offspring of X and Y, called Z, is the coancestry between X and Y:

F Z = f XY

Figure 32.4: Inbreeding Relationship for Nonoverlapping Population

For example, in Figure 32.4, JIM' and MARK' from Generation 2 are progenies of MARK' and KELLY' and of MIKE' and KELLY' from Generation 1, respectively. The coancestry between JIM' and MARK' is

From the covariance matrix for Generation=1 in Figure 32.2 and the relationship that coancestry is half of the covariance coefficient,

f JIM, MARK = (1/4) ( [0.4688/2] + [0.5/2] + [0.25/2] + [1.125/2] ) = 0.29298

For overlapping generations, if X is older than Y, then the basic rule can be simplified to

F Z = f XY = (1/2) ( f XC + f XD )

That is, the coancestry between X and Y is the average of coancestries between older X with younger Y's parents. For example, in Figure 32.5, the coancestry between KELLY' and DAVID' is

f KELLY,DAVID = (1/2) ( f KELLY,MARK + f KELLY, KELLY )

Figure 32.5: Inbreeding Relationship for Overlapping Population

This is so because KELLY' is defined before DAVID'; therefore, KELLY' is not younger than DAVID', and the parents of DAVID' are MARK' and KELLY'. The covariance coefficient values Cov(KELLY,MARK) and Cov(KELLY,KELLY) from the matrix in Figure 32.1 yield that the coancestry between KELLY' and DAVID' is

f KELLY, DAVID = (1/2) ( [0.5/2] + [1.125/2] ) = 0.40625

The numerical values for some initial coancestries must be known in order to use these rule. Either the parents of the first generation have to be unrelated, with f=0 if the INIT= option is not specified in the PROC statement, or their coancestries must have an initial value of (1/2)cov, where cov is set by the INIT= option. Then the subsequent coancestries among their progenies and the inbreeding coefficients of their progenies in the rest of the generations are calculated using these initial values.

Special rules need to be considered in the calculations of coancestries for the following cases.

#### Self-Mating

The coancestry for an individual X with itself, f XX, is the inbreeding coefficient of a progeny that is produced by self-mating. The relationship between the inbreeding coefficient and the coancestry for self-mating is

f XX = (1/2) ( 1 + F X )

The inbreeding coefficient F X can be replaced by the coancestry between X's parents A and B, f AB, if A and B are in the population:

f XX = (1/2) ( 1 + f AB )

If X's parents are not in the population, then F X is replaced by the initial value (1/2)cov if cov is set by the INIT= option, or F X is replaced by 0 if the INIT= option is not specified. For example, the coancestry of JIM' with himself is

f JIM,JIM = (1/2) ( 1 + f MARK, KELLY )

where MARK' and KELLY' are the parents of JIM'. Since the covariance coefficient Cov(MARK,KELLY) is 0.5 in Figure 32.1 and also in the covariance matrix for GENDER=1 in Figure 32.2, the coancestry of JIM' with himself is

f JIM,JIM = (1/2) ( 1 + [0.5/2] ) = 0.625

When INIT=0.25, then the coancestry of JANE' with herself is

f JANE,JANE = (1/2) ( 1 + [0.25/2] ) = 0.5625
because JANE' is not an offspring in the population.

#### Offspring and Parent Mating

Assuming that X's parents are A and B, the coancestry between X and A is

f XA = (1/2) ( f AB + f AA )

The inbreeding coefficient for an offspring of X and A, denoted by Z, is

F Z = f XA = (1/2) ( f AB + f AA )

For example, MARK' is an offspring of GEORGE' and LISA', so the coancestry between MARK' and LISA' is

f MARK, LISA = (1/2) ( f LISA,GEORGE + f LISA, LISA )

From the covariance coefficient matrix in Figure 32.1, f LISA,GEORGE = 0.25/2 = 0.125, f LISA,LISA = 1.125/2 = 0.5625, so that

f MARK, LISA = (1/2) ( 0.125+0.5625 ) = 0.34375

Thus, the inbreeding coefficient for an offspring of MARK' and LISA' is 0.34375.

#### Full Sibs Mating

This is a special case for the basic rule given at the beginning of the section "Calculation of Coancestry". If X and Y are full sibs with same parents A and B, then the coancestry between X and Y is

f XY = (1/4) ( 2f AB + f AA + f BB )

and the inbreeding coefficient for an offspring of A and B, denoted by Z, is

F Z = f XY = (1/4) ( 2f AB + f AA + f BB )

For example, DAVID' and JIM' are full sibs with parents MARK' and KELLY', so the coancestry between DAVID' and JIM' is

f DAVID, JIM = (1/4) ( 2f MARK,KELLY + f MARK, MARK + f KELLY, KELLY )

Since the coancestry is half of the covariance coefficient, from the covariance matrix in Figure 32.1,

f DAVID,JIM = (1/4) ( 2 ×[0.5/2] + [1.125/2] + [1.125/2] ) = 0.40625

#### Unknown or Missing Parents

When individuals or their parents are unknown in the population, their coancestries are assigned by the value (1/2)cov if cov is set by the INIT= option or by the value 0 if the INIT= option is not specified. That is, if either A or B is unknown, then

f AB = (1/2) cov
For example, JANE' is not in the population, and since JANE' is assumed to be defined just before the observation at which JANE' appears as a parent (that is, between observations 4 and 5), then JANE' is not older than SCOTT'. The coancestry between JANE' and SCOTT' is then obtained by using the simplified basic rule:

f SCOTT,JANE = (1/2) ( f SCOTT,· + f SCOTT,· )

Here, dots (·) indicate JANE's unknown parents. Therefore, f SCOTT,· is replaced by (1/2)cov, where cov is set by the INIT= option. If INIT=0.25, then

f SCOTT,JANE = (1/2) ( [0.25/2] + [0.25/2] ) = 0.125

For a more detailed discussion on the calculation of coancestries, inbreeding coefficients, and covariance coefficients, refer to Falconer and Mackay (1996), Kempthorne (1957), and Crow and Kimura (1970).

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