382 
Journal of Agricultural Research 
Vol. XXXI, No. 4 
Multiplying by (%) n+nl+1 (1 fF a ) gives (3^) w+ * +1 (1 + F a ) as the 
contribution to be assigned a tie in this case. A tie in the complete 
portion of the pedigree on both sides of course makes the unmodified 
contribution of (H) n+W1+1 (1 + F a ). In dealing with partially com¬ 
plete pedigrees, care is of course necessary to see which pairs of 
random lines are eliminated from consideration by ties involving the 
complete part of the pedigree. 
We are now in a position to consider more fully the method of 
calculating the average inbreeding of a group of animals. The ties 
between the sires and dams should be tabulated by the common 
ancestor involved. Certain of these animals may be found to be 
responsible for a large number of ties, others for only a few each. 
The pedigrees of those common ancestors which recur frequently 
should then be tabulated completely for a considerable number of 
generations in order to obtain a good estimate of their individual 
inbreeding by the method just discussed. For those which recur 
infrequently it is usually sufficient to assume an average degree of 
inbreeding equal to that of the breed as a whole at about the same 
time. The values adopted for the inbreeding of the various common 
ancestors ( F a ) can then be used in the term (1 + F a ) in calculating the 
total coefficient. For example, the closest connection between the 
sire and dam of the bull Millionaire (79438), in Table I, is the bull 
Favourite (252). Favourite happens to be responsible for many 
more ties in Shorthorn pedigrees than is any other animal. As he 
came early in the history of the breed, his complete coefficient of 
inbreeding can be calculated without difficulty, and it turns out to be 
19.2 per cent. Millionaire (79438) thus must be assigned a weight of 
0.50 (1+0.192) =0.596 in finding the average degree of inbreeding 
of any tabulation of Shorthorns in which he is included. 
The probable error must be rated up to the final value. If p is the 
proportion of ties observed, g ( = 1—p) the proportion of pairs of 
lines which do not contain ties, and N is the number of pairs of lines 
compared, the probable error of F x is 0.6745-^ x a ® rs ^ i a PP rox “ 
imation. For two-column pedigrees, N is of course simply the 
number of animals chosen from the breed. If four-column pedigrees 
are used, there are four possible ties in each pedigree, so that N is four 
times the number of animals. With eight-column pedigrees, N is 
16 times the number of animals, etc. The probable error is reduced 
somewhat if partially complete pedigrees are used instead of ones in 
which all lines back of sire and dam respectively are purely random. 
There is no element of chance in the ties in the complete portion of such 
pedigrees, so that the contribution of those ties has no probable 
error. In applying the formula to partially complete pedigrees, p 
should include only ties involving random lines, and its probable 
error should be multiplied by the ratio of the portion of F XJ due to 
such ties to p. 
These formulae give the probable error of the approximate formula 
for the particular animals chosen. Strictly, the probable error can 
be applied to a larger group of which these animals are a random sam¬ 
ple only in case two-column pedigrees are used. As applied to single 
individuals, the formula gives the probable error of the approximate 
coefficient merely as a measure of the complete coefficient. It does 
