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The INBREED Procedure |
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.
And the inbreeding coefficient for an offspring of X and Y, called Z, is the coancestry between X and Y:
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,
For overlapping generations, if X is older than Y, then the basic rule can be simplified to
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
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
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.
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
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:
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
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
When INIT=0.25, then the coancestry of `JANE' with herself is
Assuming that X's parents are A and B, the coancestry between X and A is
The inbreeding coefficient for an offspring of X and A, denoted by Z, is
For example, `MARK' is an offspring of `GEORGE' and `LISA', so the coancestry between `MARK' and `LISA' is
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
Thus, the inbreeding coefficient for an offspring of `MARK' and `LISA' is 0.34375.
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
and the inbreeding coefficient for an offspring of A and B, denoted by Z, is
For example, `DAVID' and `JIM' are full sibs with parents `MARK' and `KELLY', so the coancestry between `DAVID' and `JIM' is
Since the coancestry is half of the covariance coefficient, from the covariance matrix in Figure 32.1,
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
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
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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