J. C. BERRY AND J. L. LUSH 281
CASE OF COWS WITH UNEQUAL NUMBERS OF RECORDS
Table VI shows that the highest record made by a cow is about as likely to
occur in any one of the first five lactations as in any other. Assuming a per
fectly even distribution of the high record, it follows that the probability that
a cow with just one record has already made what would be the highest in five
if and when she completed the additional four is .20 ; in the case of a cow with
two records the probability is .40; for the three-record cow it is .60 ; for the
four-record cow, .80; and for the cow with five records, the probability is,
of course, 1.0. This shows clearly the unfairness of using the highest record
as a basis for comparing cows which have completed unequal numbers of
lactations.
The bias introduced by unequal numbers is evident also upon considering
the ratio range/σ. The standard deviation(σ) of records of the same cow,
as calculated from our data, is 81.6 pounds. Using Tippett's (6) values for
range /σ for varying values of n (the number of records) it is determined that
with a cow like the average of ours when n = 2 the high record is expected to
be 4.6.0 pounds of butter fat above the mean (473.0 pounds) of the whole
sample of 575 records. When n = 3 the high record is expected to be 69.0
pounds above the mean ; when n = 4, 84.0 pounds above ; and when n = 5, 95.0
pounds above. In other words, the expected value for the highest record of
two is 473+46=519 pounds ; of three, 542 ; of four, 557 ; and of five, 568
pounds. The actual values for the average of the highest of the first three
and of all five were calculated for the 115 cows of this study. They proved
to be 545 and 571 pounds respectively—values that agree well with those
expected theoretically. The bias of comparing the highest records only,
when cows have different numbers of records, is obvious.
Copeland thinks it impractical to compare lifetime records of daughters
and dams in proving bulls because the daughters and many of the dams will
not have completed their lifetime of production at the time when, by using
single records or by using only the highest records, it becomes possible to
' prove ' the bull. In using lifetime records for proving bulls, one merely
uses all the information available at the moment when a decision has to be
made. In using all of a cow's completed records as her average it is no more
necessary to wait until her lifetime is ended than it is necessary, when proving
a bull, to wait until records have been made by all of the daughters he will
ever sire. The greater the number of animals and the larger the number of
records completed by them, the less the error likely to be involved in the
' proof '. That the figures for the proof next year or the year afterward
may be a little different usually means only that the later figures, because of
the additional records included, are a bit more dependable. Breeders of
dairy cattle are continually faced with the necessity of making decisions con
cerning cows which have completed different numbers of lactations. The
most dependable procedure is to use all the lactation records available at the
time the decision needs to be made.
In comparing cows which have completed unequal numbers of records and
which have been selected for comparison because their averages are high (as
in choosing a son of a high producer), or because their averages are low (as
when deciding which of two cows to cull), it is necessary to take account of