Quantitative characters and dominance

Use of third degree statistics with this problem has been illustrated by Fisher, Immer, and Tedin (Genetics 17:107, 1932).

The less powerful but more ready attack with means does not require so extensive nor intricate data. Essentially the method is to test for departure from the additive scheme except for dominance by comparing F2 mean with the mid-point of F1 and parents, and backcross mean with mid-point of F1 and parent. Some extension of the method is proposed and illustrated below.

Denote: n - number gene pairs heterozygous in cross; n1 - plus pairs parent farther from F1, n2 - pairs in near parent; n1 + n2 = n; α - aA affect minus aa effect; k - dominance factor, (AA-aA)/(aA-aa); R - minimum phenotype summing effects of pairs aa or AA in both parents and aa effects of n pairs; FP - parent farther from F1; NP - near parent; etc.

For the additive scheme with pure parents;

FP = n1α + n1 k a + R (1)
NP = n2α + n2 kα + R (2)
F1 = nα      + R (3)
F2 = 3/4 nα + 1/4 nkα + R (4)
FB = 1/2 nα + 1/2 (1 + k) n1α + R (5)
NB = 1/2 nα + 1/2 (1 + k) n2α + R (6)


Eliminating R from (1) to (6) and combining n1 and n2 provides seven not entirely independent estimates of (1-k) nα and an eighth comparison (2F2-B) = 0. Take: P - sum of parents; F - sum of F1 and F2; and sum of backcrosses. For Lindstrom's data on relative yields of three inbred lines of maize and their hybrids (Proc. 7th. Int. Gen. Cong.):


  (l-k) nα d
4(F1 - F2) 136.8% F1 15.1
4/3(F-P) 124.5 2.8
2(B-P) 142.0 20.3
(2F1-P) 127.6 5.9
2(2F2-P) 118.4 -3.3
4(F-B) 89.6 -32.1
2(2F1-B) 113.2 -8.5
Mean 121.7 (2F2-B) = 11.8;
should be 0.


Lindstrom's data probably are a fair representation of the usual result - see Neal, J. Am. Soc. Agron. 27: 666.

The seven estimates of (1-k) nα are expected to be homogeneous and (2F2 = B) on the additive scheme, with no restrictions as to linkage, or as to degree, direction or other variation of dominance, or variation of Alpha.

In the event of no significant departure from the additive scheme the mean estimate of (1-k) nα may be of value to the breeder without further resolution into its factors. The quantity (1 + k) nα or (nα + nkα) estimates total range of genetic variation for the specific cross with free assortment. Distance from the lower extremity to F1 is nα; from F1 to upper extremity is nkα. The two are equal with no dominance. With dominance their difference is (1 - k) nα. Total depression by inbreeding is 1/2 (1 - k) nα; depression from F1 to F2 is 1/4 (1 - k) nα.

Taking the present case as additive, k = (-121.7/nα) + 1. Then, nα must be as great as 121.7% F1 if the conclusion of negative k is to be avoided. The factor k varies from unity for no dominance, through zero for complete dominance to negative values for over-dominance, "super-dominance," or "diverse alleles". With the conclusion of "complete" dominance (k = 0) nα must be taken 121.7 and the minimum phenotype minus 21.7. Taking the minimum at zero, nα is 100% and k is minus 21.7. The correct explanation of heterosis for yield in maize may lie somewhere between these somewhat arbitrary limits, involving both negative R and negative k. Note that on the additive scheme F1 will not exceed the sum of parents without negative R or negative k, yet most maize inbred yields are less than one-half of F1 yield. Tf k be negative, selection for increased inbred yield will tend towards lower F1 yield.

The obtained value of 121.7 places expected yield of a homozygote from these crosses at 39.2% F1, which is higher than usually obtained. The example may not be strictly additive. For further illustration of method, four of the seven estimates of (1 - k) nα involve (-P) with average deviation plus 6.4, indicating that slightly higher inbred yields may be expected with the present hypothesis.

Cross of heterozygous maize varieties - Tuxpan × Golden Cross Bantam

  F2 Backcross
to G.C.B.
  O C* O C** (1-k) nα F2 sk
Number leaves 13.7 13.9 12.0 11.6 +1.7 1.47 -3
Height, feet 7.5 7.3 5.4 5.9 +2.7 1.00 -2
Days to silking 73.9 70.9 66.6 65.2 -6.9 4.47 +3
Tassel length, ins. 17.4 16.6 14.6 14.7 +3.2 2.28 -1
Silking shoots 4.7 4.8 6.5 5.5 +2.8 2.30 +4
Ear diameter, cm. 4.4 4.6 4.2 4.3 +0-1 0.37 0
Cob diameter, cm. 2.5 2.5 2.3 2.3 -0.1 0.26 0
Husk length, cm. 24.0 24.6 21.1 22.8 -3.2 2.96 0
Ear length, cm. 19.1 19.6 18.1 18.6 +5-3 2.84 -1?
Husk extension, cm. 5.0 5.2 3.0 4.3 -8.0 3.33 +3
Number tillers .9 .97 1.1 1.3 +0.9 0.91 +5
No. kernel rows 13.4 13.3 11.6 12.0 +0.4 2.27 +1


*   Mid-point between F1 and mean of parents.
**   Mid-point between F1 and Golden Cross Bantam.
sk   Inspection grade of skewness: grade 5 as 1/2 of a normal distribution.


Although these records are from heterozygous parents they show generally good agreement with the additive hypothesis. Interpretation for any character will involve first the comparison of F2 and backcross means. Where agreement seems good, (1 - k) nα is next compared with skewness as to magnitude and direction. Finally, (1 - k) nα as a measure of dominance bias is considered with some measure of variation. Number of silking shoots and number of tillers have apparent skewness opposed in direction to the dominance bias. For tillers the explanation seems to lie in a piling up of nearly half of the frequency in the zero class; the character is not to the left of or below zero. No explanation for silking shoots is apparent.

It is indicated that continued inbreeding would increase total husk length 1.6cm., while ear length would be shortened 2.6 cm. Husk extension would then increase about 4.0 cm. with inbreeding and decrease with crossbreeding of inbreds.

Powers (J. Agr. Res. 63: 161) presents records on plant height in centimeters for four tomato crosses. Mean estimates of (1 - k) nα are:


  Danmark Johannisfeur
Red Currant 25.3 10.8
Johannisfeur 13.8, 7.1*  
Bonny Best   1.1

* Records for two years


The seven individual estimates on which each of the above is based do not show marked heterogeneity within any set in the writer's judgment. Variance of (1 - k) nα is apparently much greater between than within these crosses. Deviation from O of (2F2 - B) is slight in each case. The cross Johannisfeur × Bonny Best was discussed separately by Powers. He found departure of F1 from mid-point of the parents not significant. F1 and F2 seem almost identical. Yet the seven estimates of (1 - k) nα are, 0.24, 1.31, 1.28, 1.04, 1.84, 1.36, 0.80; all positive, suggesting the expected mean may be some small positive value due to some degree of dominance bias, geometric interaction, or non-linear scale of environmental effects. Since (2F2 - B) = 0.28, dominance may be the favored conclusion.

For the cross Danmark × Red Currant the far parent is 16.6 cm. from F1. Since (1 - k) nα = 25.3 cm. the minimum phenotype, R, is 25.3 cm. farther from F1 than is the maximum. It would appear that the far parent Red Currant has plus gene values not found in the other parent sufficient to explain the excess of F1 over the taller parent. The writer sees no suggestion of negative k or negative R in the three crosses which have F1s taller than the taller parent.

Powers has noted that dominance bias may be affected by environment, which view is supported by the two records for separate years on one cross. Extensive analysis by higher order statistics, not being easily repeated, might be of doubtful value, if confined to one season or location.

There in of course, no new principle involved in the analysis by comparison of means suggested here. More efficient statistics for judging significance in some of the comparisons may be developed perhaps. If values of k, n, and alpha could be resolved by extensive analysis the quantity (1 - k) nα would still be of prime interest as a measure of dominance depression of efficiency of selection. Progress in breeding towards an objective involving several quantitative characters may sometimes be hastened by an efficient balancing of backcross and selection pressures. Those characters which have strong dominance depression away from the objective will be more difficult to recover from crosses by selection. Insofar as possible such characters should be collected in the recurrent parent, and thus largely recovered by backcrossing.

In the event of negative k (aA increment exceeds AA increment), regression of phenotype on number of plus genes (A) will rise to a point beyond which the mean effect of an (A - a) substitution is negative because of increasing homozygosity. From this point the A distribution of phenotype will be doubled back on itself with respect to gene number values. Analysis by comparison of means will not be distorted. Presumably, analysis by higher order statistics may also not be distorted but that must be investigated.

Analysis by comparison of means would seem to be a ready method where more extensive analyses cannot be employed or a reasonable preliminary to more powerful methods.

Fred B. Hull