COEFFICIENT OF INBREEDING

AN INVESTIGATION INTO WRIGHT'S EQUATION AND HARDIMAN'S METHOD


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 the 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

See REGISTER OF INBREEDING COEFFICIENTS


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 PHANTOM WALTON
 
Walton mare WALTON
 

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 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 () or ( x x ), 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.

OMAR GODOLPHIN ARABIAN    
 
   
 
Lath mare LATH GODOLPHIN ARABIAN
 
   
 

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
1933
HIGH TIME
1916
ULTIMUS
1906
COMMANDO
1898
DOMINO HIMYAR
Mannie Gray
Emma C DAREBIN
Guenn
Running Stream
1898
DOMINO HIMYAR
Mannie Gray
Dancing Water ISONOMY
Pretty Dance
Noonday
1898
DOMINO
1891
HIMYAR ALARM
Hira
Mannie Gray ENQUIRER
Lizzie G
Sundown
1887
SPRINGFIELD ST ALBANS
Viridis
Sunshine THORMANBY
Sunbeam
Billie Dove
1922
ATHELING
1913
DESMOND
1896
ST SIMON GALOPIN
St Angela
L'Abesse de Jouarre TRAPPIST
Festive
Wood Daisy
1906
CYLLENE BONA VISTA
Arcadia
Mountain Daisy AYRSHIRE
Light of Other Days
Polistena
1912
POLYMELUS
1902
CYLLENE BONA VISTA
Arcadia
Maid Marian HAMPTON
Quiver
Imola
1901
ST HILAIRE ST SIMON
Distant Shore
Yola BONA VISTA
Doralice

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 PHANTOM (12.5%) WALTON (6.25%)
 
Walton mare (12.5%) WALTON (6.25%)
 

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.

Inbreeding notations
Basic coefficients
1 x 2 x 3 x 4 x 5 24.21875
1 x 2 x 3 x 4 23.4375
1 x 2 x 3 x 5 22.65625
1 x 2 x 3 21.875
1 x 2 x 4 x 5 21.09375
1 x 2 x 4 20.3125
1 x 2 x 5 19.53125
1 x 2 18.75
1 x 3 x 4 x 5 17.96875
1 x 3 x 4 17.1875
1 x 3 x 5 16.40625
1 x 3 15.625
1 x 4 x 5 14.84375
1 x 4 14.0625
1 x 5 13.28125
2 x 2 12.5
2 x 3 x 4 x 5 11.71875
2 x 3 x 4 10.9375
2 x 3 x 5 10.15625
2 x 3 9.375
2 x 4 x 5 8.59375
2 x 4 7.8125
2 x 5 7.03125
3 x 3 6.25
3 x 4 x 5 5.46875
3 x 4 4.6875
3 x 5 3.90625
4 x 4 3.125
4 x 5 2.34375
5 x 5 1.5625

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

0 1 2 3 4
D A B    
 
   
 
C B  
 
   
 
E A B  
 
C B
 
     
 
   
 

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

This pedigree contains new and conjectural information of a radical nature - See
CONFUSED PEDIGREES AND MISTAKEN IDENTITIES
DARCY'S YELLOW TURK, DARCY'S WHITE TURK AND HELMSLEY TURK
GARDINER MARE, HUTTON'S SPOT AND ALCOCK'S ARABIAN
LAYTON GREY BARB, ROCKWOOD, TAFFOLET BARB AND TREGONWELL'S BARB MARE
HIGHLANDER, MERLIN AND WOODCOCK
BLACK-A-TOP, OLD MONTAGUE MARE, PULLEINE'S CHESNUT ARABIAN AND WHITESHIRT MARE
SNAKE, CONEYSKINS, LISTER'S TURK AND BYERLEY'S TURK

 

 Old Bald Peg = FAIRFAX'S MOROCCO BARB
              |    [HELMSLEY TURK]
              |          |
       ----------------> | <-------------------------------------------------------------------------------------
       |                 |                                                                                      |
Leedes Bald Peg          |                                 Lawson's Barb Mare  =  DARCY'S YELLOW TURK  = Old Morocco Mare
[Young Bald Peg]         |                                 [Dodsworth's dam]   |       [DODSWORTH]     |        |
       |                 |                                 [Merlin's grandam]  |  [MERLIN'S GRANDSIRE] |        |
       |                 |                                                     |            |          |        |
   -----                 |                             -------------------------            |          |        ------
   |                     |                             |                                    |          |             |
   |             DARCY'S WHITE TURK       =   Pierson's Bay mare                Barb Mare   |          |             |
   |             [DARCY'S DIAMOND]        |                                         |       |          |             |
   |           [PLACE'S WHITE TURK]       |                                         |       |          |             |
   |           [BURNET'S WHITE BARB]      ------------------------------------      |       |          |             |
   |      [PLACE'S WHITE TRUGUNWELL TURK]                                    |      |       |          |             |
   |                     |                                                   |      |       |          |             |
   |                     |                 --------------------------------> | <--> | <---> | <---------             |
   |                     |                 |                                 |       \     /           |             |
   |                     |           Darcy's Grey Royal                     mare  =  BRIMMER        SPANKER          |
   |                     |         [Lonsdale Arabian Mare]                        |                    |             |
   |                     |                 |                                      |                    |             |
   |                     ------------    ---                                      |                    |             |
   |                                 \  /                                         |                    |             |
   |       -------------------------> | <---------------------------------------> | <-------------------             |
   |       |                          |                                           |                    |             |
   |       |        ----------------> | <------------------------------------------                    |             |
   |       |        |                 |                                                                |             |
   |       |        |                 ------------------------------------                             |             |
    \     /         |                 |                                  |                              \           /
Yellow Bald Peg   mare  =          HAUTBOY         Arlington Barb Mare   |     BYERLEY'S TURK     =     Charming Jenny
 [Chesnut Peg]          |         [ROCKWOOD]       [Layton Barb Mare]    |     [LISTER'S TURK]    |   [Wyvill's Roan Mare]
      |                 |  [PULLEINE'S WHITE TURK]          |            |   [RUTLAND BLACKLEGS]  |
      |                 |  [PULLEINE'S OLD ARABIAN]         |            |               1678     |
      |                 |             |                     |            |                        |
      |                 |             |      ----------------         ----                 -----------------------
      |                 |              \    /                         |                    |                     |
   Bay Peg             mare         GREY HAUTBOY       =       Darcy's Pet Mare       PELHAM'S JIGG       BROTHER TO SNAKE
  by Leedes             |        [LAYTON GREY BARB]    |       [Old Grey Royal]     [LISTER'S SNAKE]             |
   Arabian              |                              |    [Tregonwell Barb mare]         |                     |
      |                 |                              |  [Darcy's Oldest Royal Mare]      |                     |
      |                 |                              |         [Clubfoot]                |                     |
      |                 |                              |                                   |                     |
      ----------------> | <-----------------------     ------------------------            |                     -------
                        |                        |     |                      |            |                           |
  BLUE CAP              |                        |  CLUMSEY              Grey Wilkes       |      CURWEN'S BAY BARB   mare
      |                 |                        |     |                 [Old Wilkes]      |              |            |
      |                 |                        |     |                      |            |              |            |
      |                 |                        |     |                      |            |              |            |
      |                 |                         \   /                        \           /              |            |
     mare           CHAMPION                       FOX                          SON OF JIGG        =     mare          |
      |        by Harpham Arabian                   | 1714                [SMITH'S SON OF SNAKE]   |                   |
      |                 | 1707                      |                                              |                   |
      --------    -------                           |                                              |                   |
              \  /                                  |                                              |                   |
              mare                                  |                                       HENEAGE'S JIGG             |
               |                                    |                                              |                   |
               ----------------       ---------------                                              ---------    --------
                               \     /                                                                      \  /
                            BOLTON GOLIAH                                =                                  mare
                                    1730                                 |                                   |
                                                                         |                                   |
                                                                         |                                   |
                                                                    HUNT'S JIGG                              |
                                                                         | 1741                              |
                                                                         ------------             ------------
                                                                                     \           /
                                                                                     JIGG-OF-JIGGS
                                                                                             1745

 

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

0 1 2 3 4 5
JIGG-OF-JIGGS
28.125
17.236328125
12.109375
(57.470703125)
HUNT'S JIGG
03.125
08.056640625
06.0546875
(17.236328125)
BOLTON GOLIAH
02.34375
04.6875
01.025390625
(08.056640625)
FOX
02.05078125
02.05078125
00.5859375
(04.6875)
CLUMSEY
01.611328125
00.146484375
00.29296875
(02.05078125)
GREY HAUTBOY
(00.146484375)
Darcy's Pet Mare
(00.29296875)
Bay Peg
(00.5859375)
LEEDES ARABIAN

Yellow Bald Peg
(00.5859375)

Champion mare
(01.025390625)
CHAMPION
(01.025390625)
HARPHAM ARABIAN
Hautboy mare
00.732421875
00.146484375
00.146484375
(01.025390625)
Blue Cap mare BLUE CAP
 
Heneage's Jigg mare
(06.0546875)
HENEAGE'S JIGG SON OF JIGG PELHAM'S JIGG
Grey Wilkes
Curwen's Bay Barb mare
CURWEN'S BAY BARB
 
Brother to Snake mare BROTHER TO SNAKE LISTER'S TURK
Charming Jenny
   
 
Heneage's Jigg mare
06.25
04.6875
01.171875
(12.109375)
HENEAGE'S JIGG
(04.6875)
SON OF JIGG
02.05078125
00.5859375
02.05078125
(04.6875)
PELHAM'S JIGG
(00.5859375)
BYERLEY'S TURK
Charming Jenny
(00.5859375)
Grey Wilkes
01.611328125
00.146484375
00.29296875
(02.05078125)
GREY HAUTBOY
(00.146484375)
Darcy's Pet Mare
(00.29296875)
Curwen's Bay Barb mare
CURWEN'S BAY BARB  
 
   
 
Brother to Snake mare
(01.171875)
BROTHER TO SNAKE
(01.171875)
LISTER'S TURK  
 
Charming Jenny
(01.171875)
SPANKER
Old Morocco Mare
     
 
   
 
0 1 2 3 4 5

Details of each specific inbreeding:

JIGG-OF-JIGGS inbred to Heneage's Jigg mare 2 x 1 = 18.75
JIGG-OF-JIGGS inbred to BYERLEY'S TURK=LISTER'S TURK 5 x (5 x 4) = 3.125
JIGG-OF-JIGGS inbred to Charming Jenny 5 x (5 x 4) = 3.125
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 = 28.125

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 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

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

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

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

Hautboy mare inbred to DARCY'S DIAMOND=DARCY'S WHITE TURK 2 x 3 = 9.375
Hautboy mare inbred to DARCY'S YELLOW TURK=MERLIN'S GRANDSIRE 3 x (3 x 4) = 7.8125
Hautboy mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (3 x 4) x 4 = 6.25
Total SIC for Hautboy mare = 23.4375

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

Brimmer mare inbred to DARCY'S YELLOW TURK=MERLIN'S GRANDSIRE 2 x 3 = 9.375
Total SIC for Brimmer mare = 9.375

 

METHOD ONE
Working forwards through the pedigree

6 HAUTBOY = modified SIC of 0.146484375
6 Brimmer mare = modified SIC of 0.146484375
5 Darcy's Pet Mare = modified SIC of 0.29296875
5 Yellow Bald Peg = modified SIC of 0.5859375
5 Hautboy mare = modified SIC of 0.732421875 + modified SIC for HAUTBOY in generation 6 of 0.146484375 + modified SIC for Brimmer mare in generation 6 of 0.146484375 = 1.025390625
5 Charming Jenny = half the modified SIC in generation 4 of 1.171875 = 0.5859375
4 CLUMSEY = modified SIC of 01.611328125 + modified SIC for HAUTBOY in generation 6 of 0.146484375 + modified SIC for Darcy's Pet Mare in generation 5 of 0.29296875 = 02.05078125
4 Grey Wilkes = modified SIC of 01.611328125 + modified SIC for HAUTBOY in generation 6 of 0.146484375 + modified SIC for Darcy's Pet Mare in generation 5 of 0.29296875 = 02.05078125
4 Charming Jenny = modified SIC of 1.171875
3 FOX = modified SIC of 02.05078125 + modified CIC for CLUMSEY in generation 4 of 02.05078125 + modified SIC for Yellow Bald Peg in generation 5 of 0.5859375 = 04.6875
3 SON OF JIGG = modified SIC of 02.05078125 + modified SIC for Charming Jenny in generation 5 of 0.5859375 + modified CIC for Grey Wilkes in generation 4 of 02.05078125 = 04.6875
2 BOLTON GOLIAH = modified SIC of 02.34375 + modified CIC for FOX in generation 3 of 04.6875 + modified CIC for Hautboy mare in generation 5 of 1.025390625 = 08.056640625
2 Heneage's Jigg mare = half the modified CIC in generation 1 of 12.109375 = 06.0546875
1 HUNT'S JIGG = modified SIC of 03.125 + modified CIC for BOLTON GOLIAH in generation 2 of 08.056640625 + modified CIC for Heneage's Jigg mare in generation 2 of 06.0546875 = 17.236328125
1 Heneage's Jigg mare = modified SIC of 06.25 + modified CIC for SON OF JIGG in generation 3 of 04.6875 + modified SIC for Charming Jenny in generation 4 of 01.171875 = 12.109375
0 JIGG-OF-JIGGS = SIC of 28.125 + modified CIC for HUNT'S JIGG in generation 1 of 17.236328125 + modified CIC for Heneage's Jigg mare in generation 1 of 12.109375 = 57.470703125

 

METHOD TWO
Working backwards through the pedigree

  Inbred Horse SIC   Modified SIC
0 JIGG-OF-JIGGS 28.125   28.125
1 HUNT'S JIGG 6.25 2 03.125
1 Heneage's Jigg mare 12.5 2 06.25
2 BOLTON GOLIAH 9.375 4 02.34375
2 Heneage's Jigg mare 12.5 4 03.125
3 FOX 16.40625 8 02.05078125
3 SON OF JIGG 16.40625 8 02.05078125
4 CLUMSEY 25.78125 16 01.611328125
4 SON OF JIGG 16.40625 16 01.025390625
4 Grey Wilkes 25.78125 16 01.611328125
4 Charming Jenny 18.75 16 01.171875
5 Darcy's Pet Mare 9.375 32 00.29296875
5 Yellow Bald Peg 18.75 32 00.5859375
5 Hautboy mare 23.4375 32 00.732421875
5 Grey Wilkes 25.78125 32 00.8056640625
5 Charming Jenny 18.75 32 00.5859375
5 Charming Jenny 18.75 32 00.5859375
5 Darcy's Pet Mare 9.375 32 00.29296875
6 HAUTBOY 9.375 64 00.146484375
6 HAUTBOY 9.375 64 00.146484375
6 Brimmer mare 9.375 64 00.146484375
6 Charming Jenny 18.75 64 00.29296875
6 Darcy's Pet Mare 9.375 64 00.146484375
6 HAUTBOY 9.375 64 00.146484375
7 HAUTBOY 9.375 128 00.0732421875
Total CIC for JIGG-OF-JIGGS 57.470703125

 

METHOD THREE
Register of Inbreeding Coefficients

The figure of 57.470703125, 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 28.125 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 calculation are 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:

Brimmer mare
(9.375)
BRIMMER
(0)
DARCY'S YELLOW TURK *      
 
   
 
     
 
   
 
       
 
   
 
     
 
   
 
Darcy's Diamond mare
(0)
       
 
   
 
     
 
   
 
Pierson's Bay Mare MERLIN'S GRANDSIRE *    
 
   
 
     
 
   
 

Brimmer mare inbred to DARCY'S YELLOW TURK=MERLIN'S GRANDSIRE 2 x 3 = 9.375
Total SIC and CIC for Brimmer mare = 9.375


HAUTBOY
(09.375)
DARCY'S WHITE TURK
(0)
HELMSLEY TURK *      
 
   
 
     
 
   
 
       
 
   
 
     
 
   
 
Lonsdale Arabian Mare
(0)
       
 
   
 
     
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB *    
 
   
 
     
 
   
 

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


Hautboy mare
(32.8125)
HAUTBOY
(09.375)
DARCY'S WHITE TURK * HELMSLEY TURK **    
 
   
 
     
 
   
 
Lonsdale Arabian Mare DARCY'S YELLOW
TURK ***
   
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB **  
 
   
 
Brimmer mare
(09.375)
BRIMMER DARCY'S YELLOW
TURK ***
   
 
   
 
     
 
   
 
Darcy's Diamond mare DARCY'S DIAMOND * HELMSLEY TURK **  
 
   
 
Pierson's Bay Mare MERLIN'S GRANDSIRE***  
 
   
 

Hautboy mare inbred to DARCY'S DIAMOND=DARCY'S WHITE TURK 2 x 3 = 9.375
Hautboy mare inbred to DARCY'S YELLOW TURK=MERLIN'S GRANDSIRE 3 x (3 x 4) = 7.8125
Hautboy mare inbred to FAIRFAX'S MOROCCO BARB=HELMSLEY TURK (3 x 4) x 4 = 6.25
Total SIC for Hautboy mare = 23.4375
Half CIC for HAUTBOY = 4.6875
Half CIC for Brimmer mare = 4.6875
Total CIC for Hautboy mare = 32.8125


Yellow Bald Peg
(18.75)
SPANKER
(0)
       
 
   
 
     
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB *    
 
   
 
Old Bald Peg **    
 
   
 
Leedes Bald Peg
(0)
FAIRFAX'S MOROCCO BARB *      
 
   
 
     
 
   
 
Old Bald Peg **      
 
   
 
     
 
   
 

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


Darcy's Pet Mare
(09.375)
DARCY'S WHITE TURK
(0)
HELMSLEY TURK *      
 
   
 
     
 
   
 
       
 
   
 
     
 
   
 
Darcy's Grey Royal
(0)
       
 
   
 
     
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB *    
 
   
 
     
 
   
 

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


Charming Jenny
(18.75)
SPANKER
(0)
       
 
   
 
     
 
   
 
Old Morocco Mare *      
 
   
 
     
 
   
 
Old Morocco Mare *
(0)
       
 
   
 
     
 
   
 
       
 
   
 
     
 
   
 

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


Grey Wilkes
(32.8125)
GREY HAUTBOY
(04.6875)
HAUTBOY
(09.375)
DARCY'S WHITE TURK * HELMSLEY TURK ***  
 
   
 
Lonsdale Arabian Mare **    
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB ***
 
       
 
   
 
     
 
   
 
Darcy's Pet Mare
(09.375)
DARCY'S WHITE TURK * HELMSLEY TURK ***    
 
   
 
     
 
   
 
Darcy's Grey Royal **      
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB ***  
 
   
 

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


CLUMSEY
(32.8125)
GREY HAUTBOY
(04.6875)
HAUTBOY
(09.375)
DARCY'S WHITE TURK * HELMSLEY TURK ***  
 
   
 
Lonsdale Arabian Mare **    
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB ***
 
       
 
   
 
     
 
   
 
Darcy's Pet Mare
(09.375)
DARCY'S WHITE TURK * HELMSLEY TURK ***    
 
   
 
     
 
   
 
Darcy's Grey Royal **      
 
   
 
Old Morocco Mare FAIRFAX'S MOROCCO BARB ***  
 
   
 

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


SON OF JIGG
(37.5)
PELHAM'S JIGG
(9.375)
       
 
   
 
     
 
   
 
Charming Jenny
(18.75)
SPANKER DARCY'S YELLOW
TURK **
 
 
Old Morocco Mare * FAIRFAX'S MOROCCO BARB ***
 
Old Morocco Mare * FAIRFAX'S MOROCCO BARB ***  
 
   
 
Grey Wilkes
(32.8125)
GREY HAUTBOY HAUTBOY DARCY'S WHITE TURK HELMSLEY TURK ***
 
Lonsdale Arabian Mare DARCY'S YELLOW
TURK **
Old Morocco Mare *
     
 
   
 
Darcy's Pet Mare DARCY'S WHITE TURK HELMSLEY TURK ***  
 
   
 
Darcy's Grey Royal DARCY'S YELLOW
TURK **
 
 
Old Morocco Mare * FAIRFAX'S MOROCCO BARB ***
 

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
(37.5)
CLUMSEY
(32.8125)
GREY HAUTBOY HAUTBOY DARCY'S WHITE TURK HELMSLEY TURK *
 
Lonsdale Arabian Mare DARCY'S YELLOW
TURK **
Old Morocco Mare ***
     
 
   
 
Darcy's Pet Mare DARCY'S WHITE TURK HELMSLEY TURK *  
 
   
 
Darcy's Grey Royal DARCY'S YELLOW
TURK **
 
 
Old Morocco Mare *** FAIRFAX'S MOROCCO BARB *
Old Bald Peg ****
Bay Peg
(9.375)
       
 
   
 
     
 
   
 
Yellow Bald Peg
(18.75)
SPANKER DARCY'S YELLOW
TURK **
 
 
Old Morocco Mare *** FAIRFAX'S MOROCCO BARB *
Old Bald Peg ****
Leedes Bald Peg FAIRFAX'S MOROCCO BARB *  
 
Old Bald Peg ****  
 

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
(32.2265625)
FOX
(37.5)
CLUMSEY GREY HAUTBOY HAUTBOY * DARCY'S WHITE TURK **
Lonsdale Arabian Mare ***
   
 
Darcy's Pet Mare DARCY'S WHITE TURK **  
 
Darcy's Grey Royal ***  
 
       
 
   
 
     
 
   
 
Champion mare
(8.203125)
CHAMPION
(16.40625)
     
 
   
 
Hautboy mare
(32.8125)
HAUTBOY * DARCY'S WHITE TURK **
Lonsdale Arabian Mare ***
   
 
       
 
   
 
     
 
   
 

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 = 4.1015625
Total CIC for BOLTON GOLIAH = 32.2265625


Heneage's Jigg mare
(24.21875)
HENEAGE'S JIGG
(18.75)
SON OF JIGG
(37.5)
PELHAM'S JIGG BYERLEY'S TURK *  
 
Charming Jenny **  
Old Morocco Mare ***
     
 
   
 
       
 
   
 
     
 
   
 
Brother to Snake mare
(4.6875)
BROTHER TO SNAKE
(9.375)
LISTER'S TURK *    
 
   
 
Charming Jenny **
(18.75)
SPANKER  
Old Morocco Mare ***
Old Morocco Mare ***  
 
       
 
   
 
     
 
   
 

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
(34.47265625)
BOLTON GOLIAH
(32.2265625)
FOX CLUMSEY GREY HAUTBOY *  
 
Darcy's Pet Mare **  
 
Bay Peg    
 
Yellow Bald Peg SPANKER ***
 
       
 
   
 
     
 
   
 
Heneage's Jigg mare
(24.21875)
HENEAGE'S JIGG SON OF JIGG    
 
Grey Wilkes GREY HAUTBOY *
Darcy's Pet Mare **
     
 
   
 
Brother to Snake mare BROTHER TO SNAKE    
 
Charming Jenny SPANKER ***
 
     
 
   
 

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 = 16.11328125
Half CIC for Heneage's Jigg mare = 12.109375
Total CIC for HUNT'S JIGG = 34.47265625


JIGG-OF-JIGGS
(57.470703125)
HUNT'S JIGG
(34.47265625)
BOLTON GOLIAH FOX CLUMSEY GREY HAUTBOY **
Darcy's Pet Mare ***
   
 
     
 
   
 
Heneage's Jigg mare *      
 
   
 
Brother to Snake mare BROTHER TO SNAKE LISTER'S TURK ****
Charming Jenny *****
   
 
Heneage's Jigg mare *
(24.21875)
HENEAGE'S JIGG SON OF JIGG PELHAM'S JIGG BYERLEY'S TURK ****
Charming Jenny *****
Grey Wilkes GREY HAUTBOY **
Darcy's Pet Mare ***
     
 
   
 
Brother to Snake mare BROTHER TO SNAKE LISTER'S TURK ****  
 
Charming Jenny *****  
 
     
 
   
 

JIGG-OF-JIGGS inbred to Heneage's Jigg mare 2 x 1 = 18.75
JIGG-OF-JIGGS inbred to BYERLEY'S TURK=LISTER'S TURK 5 x (5 x 4) = 3.125
JIGG-OF-JIGGS inbred to Charming Jenny 5 x (5 x 4) = 3.125
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 = 28.125
Half CIC for HUNT'S JIGG = 17.236328125
Half CIC for Heneage's Jigg mare = 12.109375
Total CIC for JIGG-OF-JIGGS = 57.470703125


HARDIMAN'S COEFFICIENT OF INBREEDING CALCULATED TO TWO DECIMAL PLACES

It is proposed that Hardiman's Coefficient always be rounded to two decimal places. The following method is recommended:

1. Calculate the SIC in full and round to two decimal places.

2. Add together the parent's CIC's, divide by two, add to the SIC and then round to two decimal places.

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

Brimmer mare
Total SIC = 9.375 = 9.38
Total CIC = 0 + 0 = 0 2 = 0 + 9.38 = 9.38

HAUTBOY
Total SIC = 9.375 = 9.38
Total CIC = 0 + 0 = 0 2 = 0 + 9.38 = 9.38

Hautboy mare
Total SIC = 23.4375 = 23.44
Total CIC = 9.38 + 9.38 = 18.76 2 = 9.38 + 23.44 = 32.82

Yellow Bald Peg
Total SIC = 18.75
Total CIC = 0 + 0 = 0 2 = 0 + 18.75 = 18.75

Darcy's Pet Mare
Total SIC = 9.375 = 9.38
Total CIC = 0 + 0 = 0 2 = 0 + 9.38 = 9.38

Charming Jenny
Total SIC = 18.75
Total CIC = 0 + 0 = 0 2 = 0 + 18.75 = 18.75

Grey Wilkes
Total SIC = 25.78125 = 25.78
Total CIC = 4.69 + 9.38 = 14.07 2 = 7.035 + 25.78 = 32.815 = 32.82

CLUMSEY
Total SIC = 25.78125 = 25.78
Total CIC = 4.69 + 9.38 = 14.07 2 = 7.035 + 25.78 = 32.815 = 32.82

SON OF JIGG
Total SIC = 16.40625 = 16.41
Total CIC = 9.38 + 32.82 = 42.20 2 = 21.10 + 16.41 = 37.51

FOX
Total SIC = 16.40625 = 16.41
Total CIC = 32.82 + 9.38 = 42.20 2 = 21.10 + 16.41 = 37.51

BOLTON GOLIAH
Total SIC = 9.375 = 9.38
Total CIC = 37.51 + 8.21 = 45.72 2 = 22.86 + 9.38 = 32.24

Heneage's Jigg mare
Total SIC = 12.5
Total CIC = 18.76 + 4.69 = 23.45 2 = 11.725 + 12.5 = 24.225 = 24.23

HUNT'S JIGG
Total SIC = 6.25
Total CIC = 32.24 + 24.23 = 56.47 2 = 28.235 + 6.25 = 34.485 = 34.49

JIGG-OF-JIGGS
Total SIC = 28.125 = 28.13
Total CIC = 24.49 + 24.23 = 58.72 2 = 29.36 + 28.13 = 57.49

The difference between 57.49 and 57.470703125 of less than 0.02 is negligible.


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