An Evaluation of Inbreeding

An Investigation into Wright's Equation

and Hardiman's Method

By James R. Hardiman  Contact: highflyer@supanet.com

Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding, Coefficients of inbreeding,

Wright's Equation and Hardiman's Method are both based on the principle that the inbreeding of an individual is one half the relationship of its sire and dam, however the calculations involve different data and so the inbreeding coefficients produced by them are not interchangeable and should not be compared with each other. Wright's Equation is haphazardly calculated to any number of generations, whereas Hardiman's Method is always calculated to five generations. Wright's Equation considers duplicated ancestors only if they are common to both sire and dam, whereas Hardiman's Method considers all duplicated ancestors. Wrights Equation considers inbred ancestors only if they are duplicated ancestors, whereas Hardiman's Method considers all inbred ancestors. It should be noted here that an inbreeding coefficient is of little value without a standard with which it can be compared, so this proposed Register of Inbreeding Coefficients will be used to calculate a breed average for comparison.

STANDARD INBREEDING COEFFICIENT - includes only the subject's inbreeding in the first five generations

CUMULATIVE INBREEDING COEFFICIENT - includes all inbreeding in the subject's pedigree back as far as records allow

WRIGHT'S EQUATION FOR CALCULATING THE COEFFICIENT OF INBREEDING

As devised by Sewell Wright in 1922

Fx is the inbreeding coefficient of the horse in question, Fa is the inbreeding coefficient of the common ancestor, n1 is the number of generations from the sire to the common ancestor, and n2 is the number of generations from the dam to the common ancestor.

Cedric is the result of the mating of half-sibs, both his parents being by Walton. If Walton carries a gene with two different alleles, the one which is passed to Phantom has a 50% probability of being passed to Cedric. There is also a 50% probability that the Walton mare will receive the same allele from Walton and a 50% probability of it being passed to Cedric. The probability that Cedric will be homozygous for this allele is 0.5 x 0.5 x 0.5 = 0.125 or 12.5%. If Walton were the only common ancestor, the inbreeding coefficient for Cedric would be 12.5%.

Simply put, Cedric's inbreeding coefficient is found by calculating ½ to the power n, where n is the number of individuals from Cedric to the common ancestor Walton and back to Cedric on the other side of the pedigree, less Cedric. This makes n equal to Phantom - Walton - Walton mare, or 3. The calculation therefore is (½)³ or (½ x ½ x ½), which equals one eighth or 12.5%.

In the equation quoted above n1 = 1 (Phantom) and n2 = 1 (Walton mare). Walton, the common ancestor, is represented by 1. If Walton is also inbred, then Cedric's inbreeding coefficient would be 12.5 x (1 plus the inbreeding coefficient of Walton, the common ancestor). If the common ancestor is not inbred then Fa = 0, therefore this part of the equation can be ignored because (1+Fa) = 1.
If Cedric had more than one common ancestor then calculations would be made for each and added together to give his inbreeding coefficient.

On the face of it Wright's Equation works, however closer examination reveals inherent defects.

In the above pedigree Omar is inbred to Godolphin Arabian (1 x 3), so the value for n in the equation equals Godolphin Arabian - Lath - Lath mare or 3. This gives an inbreeding coefficient of 12.5%, the same as that for Cedric. The problem is Omar is more closely inbred than Cedric. Cedric inherited 25% of his genes from Walton through Phantom and 25% of his genes from Walton through Walton mare, making a total of 50%, which when divided by four gives an inbreeding coefficient of 12.5%. Omar, on the other hand, inherited 50% of his genes from Godolphin Arabian as his sire and 12.5% of his genes from Godolphin Arabian through Lath mare, making a total of 62.5%, which when divided by four gives an inbreeding coefficient of 15.625%.

The value of n in Wright's Equation does not take into consideration the positions of the common ancestor in the pedigree.

If Cedric's grandsire Walton was inbred to A (2 x 2), Cedric's inbreeding coefficient would be .125 x 1.125 = .140625 or 14.0625%.The increase of 1.5625% represents the total influence of A which appears four times in the fourth generation of the pedigree of Cedric. Cedric inherited 25% of his genes from A, which when divided by four gives an inbreeding coefficient of 6.25%. Cedric's inbreeding coefficient would therefore be 12.5% + 6.25% = 18.75%.

If Omar's sire Godolphin Arabian was inbred to B (2 x 2), Omar's inbreeding coefficient would also be .125 x 1.125 = .140625 or 14.0625%. The increase of 1.5625% represents the total influence of B which appears twice in the third generation and twice in the fifth generation of the pedigree of Omar. Omar inherited 25% of his genes from B in the third generation and 6.25% of his genes from B in the fifth generation, making a total of 31.25%, which when divided by four gives an inbreeding coefficient of 7.8125%. Omar's inbreeding coefficient would therefore be 15.625% + 7.8125% = 23.4375%.

Multiplying an inbreeding coefficient by one plus the inbreeding coefficient of a common ancestor severely underestimates the total amount of inbreeding in a pedigree.

W E Jones in Genetics of the Horse makes the following statements concerning Wright's Equation:

Page 189 - "To determine if there is any inbreeding, one observes the material on the sire's side of the pedigree to see if any animal is common with what is on the dam's side of the pedigree".

Page 189 - "The amount of inbreeding is usually expressed in a percentage, which can be considered to be an estimate of the percentage of the total genes that have been put in the homozygous state. This estimate is always less than the actual because our records are never extensive enough to show all of the relationships that ever existed".

Page 190 - "The equation for calculating the coefficent of inbreeding as proposed by Wright may look somewhat complicated to one without much background in mathematics. The coefficient of inbreeding is a fraction which when multiplied by 100 gives the percentage of inbreeding. The formula is based on the principle that the inbreeding of an individual is one half the relationship of its sire and dam".

In fact, an inbreeding coefficient calculated using Wright's Equation is neither an estimation of the number of genes put into the homozygous state nor an estimation of the percentage of inbreeding. It is merely the probability that identical alleles will be inherited from ancestors common to both sire and dam.

Wright's Equation considers duplicated ancestors only if they are common to both sire and dam, but if the inbreeding of an individual is one half the relationship of its sire and dam, then duplicated ancestors wholly contained within the pedigrees of either the sire or the dam should also be considered because ultimately they will trace to ancestors common to both sire and dam. The following pedigree is an example:

HIGH BIRD has no ancestors in the first five generations which are common to both sire and dam.

The sire HIGH TIME is inbred to DOMINO (3 x 3) x 2
The paternal grandsire ULTIMUS is inbred to DOMINO 2 x 2

The dam Billie Dove is inbred to CYLLENE 3 x 3, ST SIMON 3 x 4 and BONA VISTA 4 x (4 x 4)
The maternal grandam Polistena is inbred to BONA VISTA 3 x 3

The following inbreeding coefficients are produced by Wright's Equation using only the information contained in this five generation pedigree:

HIGH BIRD = 0% HIGH TIME = 12.5% ULTIMUS = 12.5% Billie Dove = 6.25% Polistena = 3.125%

The inbreeding coefficient of 0% for HIGH BIRD produced by Wright's Equation is impossible. If his pedigree is extended to nine generations the following ancestors are common to both sire and dam:

Banter (9 x 9 x 9 x 9 x 9) x (9 x 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9)
BAY MIDDLETON (9 x 9 x 8 x 7) x (8 x 9 x 9)
BIRDCATCHER (8 x 9 x 9 x 9 x 8 x 7) x (8 x 9 x 9 x 9)
CAMEL (9 x 9 x 9 x 9 x 9 x 9) x (9 x 9 x 9 x 9 x 8 x 9 x 9)
CATTON 9 x 9
Cerberus mare (9 x 9) x 9
Decoy (9 x 8) x 8
DONCASTER 6 x (7 x 7 x 7)
ECONOMIST 9 x (8 x 9)
GLENCOE (9 x 9 x 9 x 9 x 9 x 9 x 9 x 8 x 9 x 9 x 8) x (8 x 9)
Guiccioli (9 x 9 x 9 x 8 x 9 x 8) x 9
ISONOMY 5 x (6 x 6)
KING TOM 8 x (6 x 8 x 8 x 7 x 8)
LANERCOST (9 x 9 x 8) x (9 x 8)
LOTTERY 8 x 9
MELBOURNE (7 x 9) x (9 x 8 x 8 x 9)
MULEY (9 x 9 x 8) x (9 x 9)
ORLANDO (8 x 8 x 9 x 7 x 7) x (9 x 9 x 8)
PALMYRA 7 x (9 x 9)
PANTALOON (9 x 9 x 9 x 9 x 8 x 8 x 7) x (9 x 9 x 9 x 8 x 9 x 9 x 9 x 9)
Pasquinade (9 x 8) x 9
Phryne 7 x 9
Pocahontas (9 x 9 x 9 x 8 x 8 x 8 x 7) x (7 x 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9 x 8 x 9 x 9)
Rebecca 7 x 9
RIFLEMAN 9 x 8
SCOTTISH CHIEF 7 x 6
SIR HERCULES (9 x 9 x 9 x 8 x 9 x 8) x 9
STOCKWELL (8 x 8 x 7 x 7 x 7 x 6) x (8 x 8 x 9 x 8 x 8 x 8 x 8)
SULTAN (9 x 9 x 9 x 8 x 8) x (9 x 9)
TAURUS2 8 x 9
THE BARON (9 x 9 x 8 x 8 x 8 x 7) x (9 x 9 x 9 x 9 x 9 x 9 x 9)
THE FLYING DUTCHMAN 9 x (7 x 9 x 8)
THORMANBY (8 x 5) x (8 x 8 x 8 x 8 x 8)
TOUCHSTONE (9 x 8 x 9 x 9 x 9 x 9 x 8 x 8 x 8) x (8 x 8 x 9 x 9 x 9 x 9 x 8 x 9 x 8 x 9 x 8)
VELOCIPEDE 9 x 9
WHALEBONE (9 x 9) x 9
WHISKER (9 x 8) x 9

The method I am proposing produces the following inbreeding coefficients using only the information contained in this five generation pedigree:

HIGH BIRD = 18.75% HIGH TIME = 18.75% ULTIMUS = 12.5% Billie Dove = 18.75% Polistena = 6.25%

HARDIMAN'S METHOD FOR CALCULATING THE COEFFICIENT OF INBREEDING

As devised by James R Hardiman in 2000

Cedric's inbreeding coefficient of 12.5% is calculated for an ancestor which appears twice in the second generation of his pedigree, therefore each appearance in the second generation is worth 6.25%. The total for each generation is 25%, which is the probability of an allele being passed from Walton to Phantom to Cedric. Since the total for each generation is 25, the value for each position in the pedigree can be summarised as follows:

1st Generation = 12.5
2nd Generation= 6.25
3rd Generation = 3.125
4th Generation = 1.5625
5th Generation = .78125

There are 32 ancestors in the fifth generation, each contributing .78125 of the whole. Since at least two positions in the pedigree are required for inbreeding and the maximum of 25 cannot be reached, that leaves 30 possible basic inbreeding coefficients within five generations. Each basic coefficient, which is a multiple of .78125, represents the number of ancestors in the 5th generation that are contained within the pedigree of the common ancestor. It should be noted that there are many more combinations of inbreedings, but there are only 30 possible basic inbreeding coefficients.

The inbreeding coefficients have to be modified if the common ancestor is also inbred. The movement of an ancestor back one generation in a pedigree halves its influence, therefore the inbreeding coefficient of an inbred ancestor has to be halved to compensate. The value of an inbreeding coefficient for each generation is calculated as follows:

Generation 0 - Inbreeding Coefficient
Generation 1 - Divide by 2
Generation 2 - Divide by 4
Generation 3 - Divide by 8
Generation 4 - Divide by 16
Generation 5 - Divide by 32

In the above pedigree D is inbred to A (1 x 2). The inbreeding coefficient for D is therefore 12.5 + 6.25 = 18.75.

A, however, is inbred to B (1 x 2) and appears in two different generations. The inbreeding coefficient for A is also 18.75.

The movement of an ancestor back one generation in a pedigree halves its influence, so the modified coefficient for A in generation 1 will be 9.375 and in generation 2 it will be 4.6875. The influence of A in the pedigree is therefore 9.375 + 4.6875 = 14.0625. The inbreeding coefficient for D will be 18.75 + 14.0625 = 32.8125.

In the above pedigree B appears once in generation 2, twice in generation 3 and once in generation 4. The values for these positions in the pedigree are 6.25, 3.125, 3.125 and 1.5625. The addition of these figures gives 14.0625, which is the influence of A in the pedigree.

AN EXAMPLE OF THE CALCULATION OF A COEFFICIENT OF INBREEDING USING HARDIMAN'S METHOD

THE COMPLETE PEDIGREE OF JIGG-OF-JIGGS

The inbreeding coefficient for Jigg-of-Jiggs is calculated as follows:

Details of each specific inbreeding:

JIGG-OF-JIGGS inbred to Heneage's Jigg mare 2 x 1 = 18.75
JIGG-OF-JIGGS inbred to Darcy's Pet Mare 5 x 5 = 1.5625
JIGG-OF-JIGGS inbred to GREY HAUTBOY 5 x 5 = 1.5625
Total SIC for JIGG-OF-JIGGS = 21.875

HUNT'S JIGG inbred to Darcy's Pet Mare 4 x 5 = 2.34375
HUNT'S JIGG inbred to GREY HAUTBOY 4 x 5 = 2.34375
HUNT'S JIGG inbred to SPANKER 5 x 5 = 1.5625
Total SIC for HUNT'S JIGG = 6.25

Heneage's Jigg mare inbred to Charming Jenny 4 x 3 = 4.6875
Heneage's Jigg mare inbred to BYERLEY'S TURK=LISTER'S TURK 4 x 3 = 4.6875
Heneage's Jigg mare inbred to Old Morocco Mare 5 x (5 x 4) = 3.125
Total SIC for Heneage's Jigg mare = 12.5

BOLTON GOLIAH inbred to HAUTBOY 4 x 4 = 3.125
BOLTON GOLIAH inbred to Darcy's Grey Royal=Lonsdale Arabian Mare (5 x 4) x 5 = 3.125
BOLTON GOLIAH inbred to DARCY'S WHITE TURK (5 x 4) x 5 = 3.125
Total SIC for BOLTON GOLIAH = 9.375

FOX inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (5 x 4 x 5) x (5 x 4) = 5.46875
FOX inbred to DARCY'S YELLOW TURK (5 x 4) x 4 = 3.90625
FOX inbred to Old Morocco Mare (5 x 4) x 4 = 3.90625
FOX inbred to Old Bald Peg 5 x (5 x 4) = 3.125
Total SIC for FOX = 16.40625

SON OF JIGG inbred to Old Morocco Mare (4 x 3) x (5 x 4) = 7.03125
SON OF JIGG inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (5 x 4) x (5 x 4 x 5) = 5.46875
SON OF JIGG inbred to DARCY'S YELLOW TURK 4 x (5 x 4) = 3.90625
Total SIC for SON OF JIGG = 16.40625

CLUMSEY inbred to Darcy's Grey Royal=Lonsdale Arabian Mare 3 x 2 = 9.375
CLUMSEY inbred to DARCY'S WHITE TURK 3 x 2 = 9.375
CLUMSEY inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (4 x 5) x (3 x 4) = 7.03125
Total SIC for CLUMSEY = 25.78125

Grey Wilkes inbred to Darcy's Grey Royal=Lonsdale Arabian Mare 3 x 2 = 9.375
Grey Wilkes inbred to DARCY'S WHITE TURK 3 x 2 = 9.375
Grey Wilkes inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (4 x 5) x (3 x 4) = 7.03125
Total SIC for Grey Wilkes = 25.78125

Charming Jenny inbred to Old Morocco Mare 2 x 1 = 18.75
Total SIC for Charming Jenny = 18.75

HAUTBOY inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK 2 x 3 = 9.375
Total SIC for HAUTBOY = 9.375

Darcy's Pet Mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK 2 x 3 = 9.375
Total SIC for Darcy's Pet Mare = 9.375

Spanker mare inbred to FAIRFAX'S MOROCCO BARB 3 x 2 = 9.375
Spanker mare inbred to Old Bald Peg 3 x 2 = 9.375
Total SIC for Spanker mare = 18.75

Hautboy mare inbred to DARCY'S YELLOW TURK 3 x 3 = 6.25
Hautboy mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (3 x 4) x 4 = 6.25
Total SIC for Hautboy mare = 12.5

METHOD ONE
Working forwards through the pedigree

METHOD TWO
Working backwards through the pedigree

METHOD THREE
Register of Inbreeding Coefficients

The figure of 50.732421875, arrived at by both Method One and Method Two above, is the cumulative inbreeding coefficent for Jigg-of-Jiggs and includes all the inbreeding in his pedigree back as far as records allow. The standard inbreeding coefficient for Jigg-of-Jiggs of 21.875 is less than half of this total, so it is clear that the inbreeding carried forward from previous generations is still valid but the methods of calculating is cumbersome and time consuming.

It is proposed that a Register of Inbreeding Coefficients of all stallions and mares be compiled, so that any Thoroughbred's cumulative inbreeding coefficient may be found by first calculating the standard inbreeding coefficient and adding to it half the value of each of the parents cumulative inbreeding coefficients as taken from the register.

Using this method and beginning with the earliest inbred ancestor, the cumulative inbreeding coefficient for Jigg-of-Jiggs would be calculated as follows:

Charming Jenny inbred to Old Morocco Mare 2 x 1 = 18.75
Total SIC and CIC for Charming Jenny = 18.75

Spanker mare inbred to FAIRFAX'S MOROCCO BARB 3 x 2 = 9.375
Spanker mare inbred to Old Bald Peg 3 x 2 = 9.375
Total SIC and CIC for Spanker mare = 18.75

HAUTBOY inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK 2 x 3 = 9.375
Total SIC and CIC for HAUTBOY = 9.375

Darcy's Pet Mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK 2 x 3 = 9.375
Total SIC and CIC for Darcy's Pet Mare = 9.375

CLUMSEY inbred to Darcy's Grey Royal=Lonsdale Arabian Mare 3 x 2 = 9.375
CLUMSEY inbred to DARCY'S WHITE TURK 3 x 2 = 9.375
CLUMSEY inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (4 x 5) x (3 x 4) = 7.03125
Total SIC for CLUMSEY = 25.78125

Half CIC for GREY HAUTBOY = 2.34375
Half CIC for Darcy's Pet Mare = 4.6875
Total CIC for CLUMSEY = 32.8125

Grey Wilkes inbred to Darcy's Grey Royal=Lonsdale Arabian Mare 3 x 2 = 9.375
Grey Wilkes inbred to DARCY'S WHITE TURK 3 x 2 = 9.375
Grey Wilkes inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (4 x 5) x (3 x 4) = 7.03125
Total SIC for Grey Wilkes = 25.78125

Half CIC for GREY HAUTBOY = 2.34375
Half CIC for Darcy's Pet Mare = 4.6875
Total CIC for Grey Wilkes = 32.8125

Hautboy mare inbred to DARCY'S YELLOW TURK 3 x 3 = 6.25
Hautboy mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (3 x 4) x 4 = 6.25
Total SIC for Hautboy mare = 12.5

Half CIC for HAUTBOY = 4.6875
Total CIC for Hautboy mare = 17.1875

SON OF JIGG inbred to Old Morocco Mare (4 x 3) x (5 x 4) = 7.03125
SON OF JIGG inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (5 x 4) x (5 x 4 x 5) = 5.46875
SON OF JIGG inbred to DARCY'S YELLOW TURK 4 x (5 x 4) = 3.90625
Total SIC for SON OF JIGG = 16.40625

Half CIC for PELHAM'S JIGG = 4.6875
Half CIC for Grey Wilkes = 16.40625
Total CIC for SON OF JIGG = 37.5

FOX inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (5 x 4 x 5) x (5 x 4) = 5.46875
FOX inbred to DARCY'S YELLOW TURK (5 x 4) x 4 = 3.90625
FOX inbred to Old Morocco Mare (5 x 4) x 4 = 3.90625
FOX inbred to Old Bald Peg 5 x (5 x 4) = 3.125
Total SIC for FOX = 16.40625

Half CIC for CLUMSEY = 16.40625
Half CIC for Bay Peg = 4.6875
Total CIC for FOX = 37.5

BOLTON GOLIAH inbred to Darcy's Grey Royal=Lonsdale Arabian Mare (5 x 4) x 5 = 3.125
BOLTON GOLIAH inbred to DARCY'S WHITE TURK (5 x 4) x 5 = 3.125
BOLTON GOLIAH inbred to HAUTBOY 4 x 4 = 3.125
Total SIC for BOLTON GOLIAH = 9.375

Half CIC for FOX = 18.75
Half CIC for Champion mare = 2.1484375
Total CIC for BOLTON GOLIAH = 30.2734375

Heneage's Jigg mare inbred to BYERLEY'S TURK=LISTER'S TURK 4 x 3 = 4.6875
Heneage's Jigg mare inbred to Charming Jenny 4 x 3 = 4.6875
Heneage's Jigg mare inbred to Old Morocco Mare 5 x (5 x 4) = 3.125
Total SIC for Heneage's Jigg mare = 12.5

Half CIC for HENEAGE'S JIGG = 9.375
Half CIC for Brother to Snake mare = 2.34375
Total CIC for Heneage's Jigg mare = 24.21875

HUNT'S JIGG inbred to Darcy's Pet Mare 4 x 5 = 2.34375
HUNT'S JIGG inbred to GREY HAUTBOY 4 x 5 = 2.34375
HUNT'S JIGG inbred to SPANKER 5 x 5 = 1.5625
Total SIC for HUNT'S JIGG = 6.25

Half CIC for BOLTON GOLIAH = 15.13671875
Half CIC for Heneage's Jigg mare = 12.109375
Total CIC for HUNT'S JIGG = 33.49609375

JIGG-OF-JIGGS inbred to Heneage's Jigg mare 2 x 1 = 18.75
JIGG-OF-JIGGS inbred to Darcy's Pet Mare 5 x 5 = 1.5625
JIGG-OF-JIGGS inbred to GREY HAUTBOY 5 x 5 = 1.5625
Total SIC for JIGG-OF-JIGGS = 21.875

Half CIC for HUNT'S JIGG = 16.748046875
Half CIC for Heneage's Jigg mare = 12.109375
Total CIC for JIGG-OF-JIGGS = 50.732421875

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