Calculating individual inbreeding values often requires information that a farmer or hatchery manager does not have-pedigrees. Those who culture fish usually do not know a fish's mother or father. Even if this information is known, it is unlikely that grandparents or cousins can be identified. This information is routinely gathered in livestock husbandry, and family pedigrees are recorded on forms or are entered into computer data bases.
It is far easier to record this information for cattle, pigs, sheep, and goats, because these animals are large, live on land, and can be branded or tagged with little effort at birth. This enables those who work with the marked livestock to identify each individual, to record matings and births, and thus to create pedigrees. Few aquaculturists record such information with fish, because fish are difficult to mark, and marking techniques often kill a large percentage of the fish. Additionally, while this information has been considered important for centuries with livestock, it is still considered to be of little value by most aquaculturists.
Even though it is impossible for most farmers or research scientists to determine individual inbreeding values, it is important to know how these values are calculated and to learn the techniques that are used. An understanding of the methods and protocols needed to determine individual inbreeding values explains how inbreeding is created. It helps explain how an individual can become homozygous by descent, which is what occurs when relatives mate and produce offspring. Learning how to trace family trees and how to calculate individual inbreeding values demonstrates how an ancestral allele can be inherited, both maternally and paternally, from a common ancestor. An understanding of these techniques also shows how inbreeding can be prevented, demonstrates how inbreeding can be reduced to zero, and helps lay the groundwork for the next chapter, the determination of average inbreeding values in hatchery populations.
Two methods can be used to determine individual inbreeding values: path analysis and covariance analysis. Both methods produce the same answer, but some feel more comfortable with one than the other. Both methods use simple arithmetic. Path analysis is quicker and requires fewer steps, but it is easier to make a mistake using this technique. Covariance analysis can take far more time if the pedigree is complicated, but once complete, the information generated by covariance analysis can be used to quickly predict inbreeding values that will be produced from any possible mating between individuals in the pedigree.
Those who are not interested in learning how to calculate individual inbreeding values can skip this chapter and go to Chapter 4, which describes the techniques that are used to determine average inbreeding values in hatchery populations.
The first step in determining an individual fish's inbreeding is to create its pedigree. A pedigree is a family tree. It lists an individual's brothers and sisters, parents, aunts and uncles, grandparents, great-grandparents, nephews and nieces, and cousins. The information is usually streamlined; often, only direct ancestors are included; and the family tree is often traced back four or fewer generations, only going back to an individual's great-grandparents or occasionally including the great-great-grandparents.
More generations can be listed, but with each additional ancestral generation, the genetic contribution of each fish in the oldest generation to the present generation decreases to the point where it becomes genetically meaningless. This is because an animal's genetic contribution to its descendants is halved each generation: Each parent contributes 50% of a individual's genes; each grandparent contributes 25%; each great-grandparent contributes 12.5%; and each great-great-grandparent contributes 6.25%. The contribution of each great-great-great-great-great-grandparent (seven ancestral generations) is only 0.78%. These values are determined by the number of direct ancestors an individual has in each ancestral generation: two parents, four grandparents, eight great-grandparents, and 16 great-great-grandparents. All fish have 128 great-great-great-great-great-grandparents, which means the contribution from each is so small (0.78%) that it is of no value in most breeding programmes.
Figure 8 contains two pedigrees. Traditionally, males are represented by squares and females are represented by circles. Various symbols can be added to the squares and circles to convey phenotypic information, such as body colour, genetic diseases, etc., but this information is not needed to determine inbreeding. Other information that can be added to a pedigree are: names, birth and death dates, and weights at specific periods. When letters are used to designate individuals, the letter F is usually omitted, because F is the symbol for coefficient of inbreeding; its omission prevents confusion.
Figure 8. Examples of two pedigrees. The top pedigree uses squares to represent males and circles to represent females. Fish name is inside each square or circle. The other pedigree simply uses letter symbols to represent fish. Information about the fish, such as birth date, weight, etc., would be kept in ledgers or on data sheets. When letters are used to designate individuals, the letter F is usually omitted, because F is the symbol for coefficient of inbreeding; this prevents confusion.
In path analysis, a pedigree is converted to a path diagram (Figure 9). A path diagram differs from a pedigree in that arrows between parents and offspring replace the tier-step decent brackets that are used in a pedigree. Each arrow represents a gamete, which means that each arrowhead represents the path or way an individual received 50% of its genes, and the arrow shaft touches the parent that contributed those genes. Thus, an arrow shows how a fish received one homologue of each chromosome pair, which means it also shows how a fish received one allele of each allelic pair (each gene).
Each fish in a path diagram can be touched by zero, one, or two arrowheads. If a fish's parents are unknown, no arrowhead will touch it; if one parent is known, a single arrowhead will touch the fish; if both parents are known, the fish will be touched by two arrowheads. A fish cannot be touched by more than two arrowheads, because no fish can have more than two biological parents. If this occurs, you have made an error. (Eventually, various biotechnological manipulations will enable research scientists to create fish that have three or four parents; however, for everyday practical breeding work, this situation can be ignored.) Many arrows can leave an individual, if that fish produced a number of offspring that are listed in the family tree that is being analyzed.
The arrows, or paths, flow from the oldest to the youngest generation, and they show how the genes were transmitted from generation to generation. They do not always flow in a straight line and sometimes cross or go sideways, depending on the complexity of the family tree and the matings that have occurred. Because of this, it is crucial to make the path diagram large enough to house the arrows and to accurately place the arrows. If arrows are incorrectly drawn, inbreeding values generated by the path diagram will be incorrect.
An individual's inbreeding value is calculated by determining all possible paths that the individual has with one or more common ancestors. A common ancestor is an individual that occurs on both sides of the family tree or pedigree; i.e., an individual that contributes genes through both the mother and the father. If just one common ancestor exists, the individual's parents are related and the fish is inbred. If there are no common ancestors, the inbreeding of that fish is zero.
Obviously, if family trees are traced back far enough, all species of fish, like all humans, are related though a common ancestor. However, for practical breeding purposes, common ancestors that are five or six ancestral generations removed are of little importance in any breeding programme, so the genetic contribution of such distant relatives is usually ignored. In most western societies, the legal distance for consanguineous marriages is second cousins; i.e., it is illegal to marry a first cousin or someone more related than that, but is lawful to marry a second cousin or someone less related than that. Western societal norms figure that the inbreeding produced by second cousin marriages is insignificant and therefore acceptable, while that produced by first cousin marriages is not. The inbreeding that is produced by matings between various relatives is listed in Figure 23 (page 57).
The following formula is used to determine individual inbreeding values from a path diagram:
FX = ∑[(0.5)N(1 + FA)]
| where: | FX | = | the inbreeding of an individual; |
| ∑ | = | the symbol for “sum of ” or “add”; | |
| N | = | the number of individuals in a path that is determined by tracing a path from one parent back to the common ancestor and forward from the common ancestor to the other parent; if more than one common ancestor exists, the term (0.5)N is repeated for each common ancestor; if more than one path exists between the individual and a common ancestor, the term (0.5)N is repeated for each unique path; | |
| FA | = | the inbreeding of the common ancestor. |
If the inbreeding value of the common ancestor is zero or unknown, in which case you must assume that it is zero, the formula is simplified to:
FX = ∑(0.5)N
| PEDIGREE | PATH DIAGRAM |
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PATH FROM G TO COMMON ANCESTOR B
Figure 9. A simple pedigree and the path diagram that can be used to calculate inbreeding. The path that is used to calculate inbreeding of fish G is shown.
The formula may look like complicated math, but it is simple arithmetic that can be easily done with basic, inexpensive hand-held calculators; the term (0.5)N can be determined by using the “yx” button.
In Figure 9, fish G is the only fish that has a common ancestor. An inspection of the pedigree shows that fish B is on both sides of fish G's family tree; this means that fish G is inbred. No other fish depicted in the pedigree has a common ancestor.
To determine an individual's inbreeding, a path is traced from one parent back to the common ancestor and from the common ancestor up to the other parent. Consequently, fish G's (Figure 9) inbreeding is determined by tracing the path that can be drawn from fish G's parents to fish B; i.e., from fish D (one parent) to fish B (the common ancestor) and from fish B to fish E (the other parent). Therefore, the path that is traced goes:
D-B-E
There are three individuals in this path, so N = 3. Since fish B (the common ancestor) is not inbred, the simplified formula can be used to calculate the inbreeding value of fish G:
FG = (0.5)3
FG = 0.125
Thus, FG = 0.125 or 12.5%. This means that fish G is expected to have 12.5% more homozygous loci than the average fish in the population. This also means that 12.5% of the loci that were heterozygous became homozygous as as result of inbreeding.
Figure 10 shows a more complicated pedigree and its path diagram. Fish J and fish N are inbred, because both have common ancestors. Fish A is the common ancestor of fish J. The inbreeding of fish J is determined by tracing a path to fish A through fish J's parents (fish H and fish I). The path is:
H-A-I
There are three individuals in the path, so N = 3. Since fish A (the common ancestor) is not inbred, we can use the simplified formula, and the inbreeding value for fish J is:
FJ = (0.5)3
FJ = 0.125
Thus, FJ = 0.125 or 12.5%.
Fish N has more than one common ancestor. This means that more than one path will determine the inbreeding of fish N. Fish A and fish C are both common ancestors of fish N. A glance at the pedigree reveals that fish A and fish C are the only fish to appear on both sides of fish N's pedigree. Fish J is inbred, but since it does not appear on both sides of fish N's pedigree, its inbreeding value does not contribute to fish N's inbreeding value.
The path that can be traced from fish J to fish M (the parents of fish N) though common ancestor fish C is:
J-I-C-K-M
There are five individuals in this path, so N = 5.
Two different paths can be drawn from fish J to fish M (the parents of fish N) though common ancestor fish A:
| PEDIGREE | PATH DIAGRAM |
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PATH FROM J TO COMMON ANCESTOR A
PATH FROM N TO COMMON ANCESTOR C
PATHS FROM N TO COMMON ANCESTOR A
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Figure 10. A pedigree and the path diagram that is used to calculate inbreeding. The paths that are used to calculate inbreeding in fish J and fish N are shown.
There are four individuals in each path, so N = 4 for both. Since neither common ancestor is inbred, the simplified formula can be used. The only difference that occurs when there is more than one common ancestor or when more than one path exits for a particular common ancestor is that the inbreeding value is determined by adding the value derived from each path:
FN = (0.5)5 + (0.5)4 + (0.5)4
FN = 0.03125 + 0.0625 + 0.0625
FN = 0.15625
Thus, FN = 0.15625 or 15.62%.
There are two important rules about tracing paths:
Rule 1:
You cannot retrace a path; i.e., you cannot go though any individual twice in a given path.
That is why you cannot use the path J-I-C-I-A-M to determine inbreeding of fish N in Figure 10. Fish I occurs twice in that path.
Rule 2:
A path is traced backward from one parent to the common ancestor, and then forward from the common ancestor to the other parent.
The part of the path that goes from one parent to the common ancestor will travel only in a backward direction; i.e., the path it traces will always start at the arrowhead and go towards the shaft. The part of the path that returns from the common ancestor to the other parent will travel only in a forward direction; i.e., the path it traces will always start at an arrow's shaft and go to the arrowhead. A path that travels in both directions, either on the way to the common ancestor or on the way back, is an erroneous path; i.e, two arrowheads or the ends of two shafts cannot touch in a given path. This is why you cannot use the path J-H-A-I-C-K-M when determining the inbreeding of fish N in Figure 10. This path goes forward and backward while going from common ancestor A to parent M; two arrowheads touch when going A-I-C.
In covariance analysis, a pedigree is converted to a covariance table, which is illustrated in Figure 11. Each individual in the pedigree is listed at the top of each column and to the left of each row. The parents of each individual are listed at the far left of the table. If one or both parents are unknown, that information is represented by a dash in the parents' column.
The cells that make up a covariance table will contain either the covariance values between two individuals or the covariance value of an individual. The cells that form the diagonal that starts with the upper left cell and ends with the lower right cell and that are the intersections produced by each individual's row and column (e.g., cells AA, BB, CC, etc.) are the cells that contain individual covariance values. The cells that lie below this diagonal are the cells that contain covariance values between two individuals.
Because all individuals are listed both in the rows and in the columns, the cells above the diagonal and those below the diagonal will contain the same information. It is not necessary to record the information twice, so the cells above the diagonal are left blank.
COVARIANCE TABLE
FORMULAE USED TO COMPLETE COVARIANCE TABLE
COVARIANCE VALUES BETWEEN INDIVIDUALS (BELOW-DIAGONAL CELLS 
)
INDIVIDUAL COVARIANCE VALUES (DIAGONAL CELLS 
)
Figure 11. A covariance table that can be used to calculate individual inbreeding values. Each fish is listed in a row and in a column. The parents that produced each fish are listed to the left of each row. If a parent is not known, it is represented by a dash. The formula that determines covariance values between individuals will be used to fill in the below-diagonal cells (dotted cells). The formula that determines individual covariance values will be used to fill in the diagonal cells (lined cells). The cells above the diagonal contain the same information as those below the diagonal, so they are not used.
Two simple math formulae are needed to calculate the covariance values. The formula that determines covariance values between two individuals (the values below the diagonal) is:
where: CovBI = the covariance between two individuals;
Cov Ind 1 with Sire Ind 2 = the covariance of individual 1 with the sire (father) of individual 2;
Cov Ind 1 with Dam Ind 2 = the covariance of individual 1 with the dam (mother) of individual 2.
The formula that determines individual covariance values (the values along the diagonal) is:
| where: | CovInd | = the covariance of an individual; |
| Cov of the Parents | = CovBI of the parents. |
A few rules are needed to complete the table:
Rule 1 for determining CovBI values:
If individual 2's sire or dam is not known, the covariance of individual 1 with the unknown
parent = 0.0.
Rule 2 for determining CovBI values:
CovBI values range from 0.0 to 2.0.
Rule 1 for determining CovInd values:
If one or both parents are unknown, Covparents = 0.0. Consequently, CovInd = 1.0.
Rule 2 for determining CovInd values: CovInd values range from 1.0 to 2.0.
CovInd values (the diagonal values) can be used to determine individual inbreeding values. Inbreeding is determined by subtracting 1.0 from CovInd:
FInd = CovInd - 1.0
Thus, an individual that is 100% (1.0) inbred will have a CovInd = 2.0:
F = 2.0 - 1.0 = 1.0
while one with no inbreeding will have a CovInd = 1.0:
F = 1.0 - 1.0 = 0.0
Rule 1 for determining CovInd values states that if you do not know both parents Covparents = 0.0. The practical aspect of this rule means that if you do not know both parents, an individual automatically has inbreeding of zero. This assumption may be false, but it must be made. This rule simplifies calculations for many covariance tables because most hatcheries do not have good breeding records.
The covariance table for the pedigree and path diagram illustrated in Figure 9 is shown in Figure 12. The calculations that were used to complete that covariance table are given in Figure 13.
Figure 12. Covariance table for the pedigree given in Figure 9.
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Figure 13. Calculations used to complete the covariance table in Figure 12.
This method is somewhat tedious, but the arithmetic is easy, and covariance values from the top half of the table are used to determine those in the bottom half. This method is more time consuming than a path diagram, because most of the table must be completed in order to determine the inbreeding of a fish at the end of a pedigree. But this method uses simple arithmetic, and when path diagrams become complicated, it is easy to overlook a path or to make a mistake when tracing a path.
In the covariance table in Figure 12, the covariance values for fish A, fish B, and fish C (CovAA, CovBB, and CovCC) are all 1.0 because we do not know their parents. The covariance values for fish D (CovDD) and fish E (CovEE) are both 1.0 because their parents are not related.
The CovBI values in the covariance table shown in Figure 12 describe the relationships between two individuals (also shown in the pedigree in Figure 9): CovAD and CovBD = 0.5 because fish A and fish B are the parents of fish D, and each parent contributes half (0.5) of fish D's genes. CovDE = 0.25, because fish D and fish E are half-sibs in that they share a single parent-fish B. (If fish D and fish E had shared both parents [full-sibs], CovDE would have been 0.5, or twice that for half-sibs.) CovAG = 0.25, because fish A is fish G's grandparent, and a grandparent contributes 25% of an individual's genes. Fish G receives genes from grandparent B via both parents (fish G is a double grandchild of fish B), which is why CovBG = 0.5, while CovAG and CovCG (fish A and fish C are fish G's other grandparents) = 0.25. CovDG = 0.625 because fish G is more related to that parent (fish D) than is normally the case (0.5): fish B appears on both sides of fish G's pedigree (fish B is fish G's maternal and paternal grandmothers). Fish G is related to its mother (fish D) in the normal manner, but it is also related to her through its father (fish E). CovBC = 0.0 because fish B and fish C are not related.
To determine inbreeding values, 1.0 is subtracted from CovInd values. Fish A, B, C, D, and E are not inbred (F = 0.0), because all have CovInd = 1.0. Fish G is inbred. Its inbreeding is:
FG = CovGG - 1.0
FG = 1.125 - 1.0
FG = 0.125
Thus, FG = 0.125 or 12.5%.
The covariance table and calculations needed to complete the table for the pedigree listed in Figure 10 are shown in Figures 14 and 15.
One tremendous advantage that a covariance table has over a path diagram is that a covariance table can be used to predict inbreeding from any mating among the individuals listed in the pedigree. The inbreeding that would be produced by any mating is simply half the CovBI value for the two fish. This information can then be used to decide if certain matings should be avoided. For example, using the covariance table in Figure 14, the inbreeding that would be produced by mating fish H and fish N would be 21.875% (CovHN = 0.4375). The inbreeding that would be produced by mating fish D with fish J would be 0% (CovDJ = 0.0).
If a farmer creates a table of covariance values for all female and male brood fish, he can tell at a glance what the inbreeding of any possible mating would be. An example of such a table is shown in Table 1. (Note: a table of covariance values is not a covariance table. Individuals are not listed in both rows and columns. In a table of covariance values, males are listed in columns and females are listed in rows [or vice versa].) All values in a table of covariance values are CovBI values, which means the inbreeding that will be produced by any mating combination is half the CovBI value for those individuals. For example, the mating of female 1 with male 2 (Table 1) will produce offspring with F = 6.25% (Covfemale 1 with male 2 = 0.125). If the farmer wants to produce a generation of fish with F = 0%, all he has to do is mate fish where the covariance value = 0.0. Thus, he could mate female 1 with male 1, female 2 with male 3, etc.
Figure 14. Covariance table for the pedigree given in Figure 10.
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Figure 15 (part 1). Calculations used to complete the covariance table in Figure 14.
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Figure 15 (part 2). Calculations used to complete the covariance table in Figure 14.
Table 1. Example of a table of covariance values for brood fish at a fish station. Males are listed across the top, and females are listed along the left. This table is a compilation of CovBI values among all brood fish, so all cells are filled; the information above the diagonal is not the same as that below the diagonal, and the diagonal values are not CovInd values. This table can be used to predict the inbreeding that will be produced by any mating combination. The inbreeding that will be produced by any mating is simply half the CovBI value. The table lists only 10 males and 10 females.
| Females | Males | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 0.0 | 0.125 | 0.125 | 0.125 | 0.0 | 0.5 | 0.5 | 0.5 | 0.25 | 0.25 |
| 2 | 0.75 | 0.75 | 0.0 | 0.0 | 0.25 | 0.0 | 0.625 | 0.625 | 0.0 | 0.25 |
| 3 | 0.0 | 0.0 | 0.0 | 0.25 | 0.0 | 0.0625 | 0.0625 | 0.625 | 0.0 | 0.25 |
| 4 | 0.125 | 0.125 | 0.5 | 0.5 | 0.5 | 0.0625 | 0.0625 | 0.25 | 0.25 | 0.25 |
| 5 | 0.0 | 0.25 | 0.0 | 0.0 | 0.75 | 0.75 | 0.75 | 0.0 | 0.625 | 0.0 |
| 6 | 0.0 | 0.125 | 0.125 | 0.25 | 0.0 | 0.625 | 0.625 | 0.625 | 0.625 | 0.0 |
| 7 | 0.25 | 0.125 | 0.125 | 0.0 | 0.5 | 0.0 | 0.0 | 0.625 | 0.625 | 0.625 |
| 8 | 0.0 | 0.25 | 0.25 | 0.0 | 0.5 | 0.0 | 0.0 | 0.625 | 0.0625 | 0.0625 |
| 9 | 0.0 | 0.25 | 0.25 | 0.0 | 0.5 | 0.0 | 0.0 | 0.625 | 0.125 | 0.125 |
| 10 | 0.0 | 0.025 | 0.25 | 0.0 | 0.0 | 0.625 | 0.625 | 0.0 | 0.5 | 0.5 |
Individual inbreeding values can be determined only if a farmer has detailed breeding records which can be turned into pedigrees. There are many reasons why farmers and hatchery managers should maintain breeding records. The ability to determine individual inbreeding values is but one reason. If such information exists, it is easy to calculate individual inbreeding values and to use them to predict future inbreeding values from various mating combinations.
Individual inbreeding values can be determined by two techniques: path analysis and covariance analysis. Both techniques provide accurate results, so the method that is used is determined by personal choice. Path analysis requires less work, but following paths can be tricky, so it is easier to make a mistake using this method. Covariance analysis is more tedious, but the math is very simple, and the completed table can be used to quickly predict any inbreeding that would be produced by mating any fish that are included in the table.
Both techniques clearly demonstrate an important fact: An individual is inbred if and only if its parents are related through one or more common ancestors. Both parents can be highly inbred, but if they are not related, their offspring will have F = 0.0. Figure 16 clearly illustrates this concept. Fish J and fish U are inbred; FJ = 0.25 and FU = 0.375. However, the mating of fish J and fish U will produce offspring with no inbreeding, because fish J and fish U are not related through a common ancestor. FZ = 0.0 because no fish appears on both the maternal and paternal sides of fish Z's pedigree.
This fact has important consequences for brood stock management and for breeding programmes. Inbreeding can be reduced to zero in a single generation by mating unrelated fish. Even if inbreeding has reached levels that cause problems, inbreeding can be eliminated and prevented in subsequent generations simply by examining pedigrees and by mating unrelated fish.
This can also be be accomplished by acquiring fish from another hatchery and by making hybrids. If the fish from the two hatcheries are unrelated, the F1 hybrids will have inbreeding of zero. F1 hybrids produced by this technique always have inbreeding of zero, provided the parents are unrelated. In fact, many hybrid seeds that are produced commercially are produced by crossbreeding highly inbred, but unrelated, lines. This reduces inbreeding to zero in the plants that will be grown and also produces some hybrid vigour, as well as uniform crops.
Figure 16. Pedigree illustrating the fact that the mating of two inbred, but unrelated, fish will produce offspring with no inbreeding: FJ = 25% and FU = 37.5%, but because they are not related, FZ = 0%.
