Agronomy Journal 44: 215-216 (1952)
Predicting Yields of Missing Single Crosses of Corn1
Robert C. Eckhardt2

A STANDARD procedure in modern corn breeding is the testing of all possible single crosses among n inbred lines. Two-way tables, such as table 1, are prepared, and the total (or mean) values for the single crosses with one line in common are recorded at the side or bottom of the table. These values indicate the relative general combining ability among the n inbreds.

Data for one or more of the n (n-1)/2 single crosses frequently are missing or unreliable for one reason or another. One such loss eliminates comparison of the means of two inbreds, and much of the value of the two-way table is lost when data for two or three crosses are lacking. Sometimes the means for the inbreds with misses in certain combinations are computed from the available values, but this procedure can be grossly misleading under some conditions. Another difficulty is that related inbred lines are used in many single cross tests. The yields of such crosses are unduly low, and it is obviously unfair to use them in ascertaining general combining ability.

1Contribution from the Division of Cereal Crops and Diseases, B.P.I.S.A.E., A.R.A., U.S.D.A. and the Mississippi Agricultural Experiment Station cooperating. Journal article 253 of the Mississippi Agricultural Experiment Station, State College, Miss. Received for publication May 19, 1951.
3G. F. Sprague and Lloyd A. Tatum. General vs. specific combining ability in single crosses of corn. Jour. Amer. Soc. Agron. 34: 923-932. 1942.
The procedure that follows is a suggested method for estimating the yield of a missing single cross by minimizing the specific combining abilities of the two inbreds. Such an estimated yield can be averaged in with the remaining actual yields to get a usable estimate of the general combining ability.

Sprague and Tatum3 have given a formula to eliminate differences in general combining ability among single crosses in order to estimate specific combining ability. Their formula (p. 924, op. cit.) for estimating specific combining ability is

  (nā€‘2) (ab) -Ta-Tb + (2/n-1)T (1)

where n is the number of inbred lines in test, ab the yield of the cross a x b, Ta and Tb the row totals of a and b respectively in a two-way table, and 2T the total of Ta + Tb + ... Tn.

In a two-way table with data on ab missing, the row totals of a and b lack the yield data of ab, and these totals will be designated T'a and T'b in this paper. Therefore, Ta = T'a + ab, Tb = T'b + ab, and 2T = 2T' + 2ab. We can now substitute in the appropriate places, equate (1) to zero to eliminate specific combining ability (2), and solve for ab (3).

(nā€‘2) ab - (T'a + ab) - (T'b + ab) + (2/n-1)(T' + ab) = 0 (2)
ab = (n - 1) (T'a + T'b) - 2T' (3)

n2 ā€‘ 5n + 6

This is the same solution one obtains by minimizing the total variation for specific combining ability.

Table 1 presents the yield data of seven inbred lines in all possible single cross combinations. Inbreds Mp 309 and Mp 311 were both developed from the open pollinated variety Whatley. The relatively low yield of the single cross Mp 309 X Mp 311 (93 bushels per acre) compared with the high yields obtained when Mp 309 and Mp 311 were crossed with inbreds developed from other varieties, suggest that the two inbreds are somewhat related. In order to evaluate the general combining ability of Mp 309 and Mp 311 properly, the yield value of the single cross Mp 309 X Mp 311 will be assumed missing, and substituting in (3) we obtain

Mp 309 x Mp 311 =
(n-1) (T'a + T'b) - 2T'

n2 - 5n + 6
= 6(581 + 572) - 4604

= 116

The substitution of the predicted value of 116 bushels per acre for Mp 309 x Mp 311 is probably a much better figure to use in computing general combining values in the two-way table, than the actual yield of 93 bushels.

When more than one single cross is missing, "guess" values can be inserted for all but one of the missing single crosses. As each single cross value is solved, it can be placed in the table and used in solving another missing single cross. When all "guess" values have been replaced by estimated values, the procedure can be repeated. The second estimations should be accurate enough for all practical purposes.

It has been suggested that predicting single-cross yields will then enable one to predict double-cross yields (based on the nonexistent yields of the single crosses) and reduce corn breeding to an indoor sport with a calculating machine. The unpleasant facts remain that inbreds still are essential and that actual yields of single crosses are usually better criteria of the value of the lines than predicted ones.

Table 1.—Yields in bushels per acre for 21 single crosses of corn compared at State College, Miss, in 1950.

Parent lines Mp 305 Mp 307 Mp 309 MP 311 T101 T61 Tx61M Totals
Mp 305 133 127 123 113 128 115 739
Mp 307 133 120 115 104 114 113 699
Mp 309 127 120 x* 112 124 98 581=T'a
MP 311 123 115 x* 111 130 93 572=T'b
T101 113 104 112 111 114 105 659
T61 128 114 124 130 114 110 720
Tx61M 115 113 98 93 105 110 634
                4604 =2T'

*Actual yield 93; predicted yield 116.