ESTIMATION OF AVERAGE DOMINANCE OF GENES 513 



than one even though the true value exceeds one. F becomes about .75 if the 

 data are doubled, between .85 and .9 if the data are tripled, and about .95 if 

 the data are quadrupled. With the degrees of freedom indicated, the proba- 

 bility of an estimate less than 1.0 for a is in all cases close to .05, and that of 

 an estimate significantly less than one is much smaller. This is an important 

 point since it means a very small chance of erroneously concluding that a is 

 less than one if its real value is greater thaji one by any very important 

 amount. 



The general point to note is that the amounts of data indicated in Table 

 30.9 are moderate for any combination of c ^ 1.0 and a ^ 1.4. In addition, 

 it is not prohibitive when both a and c are (within the ranges considered) 

 either large or small. Actually, as indicated by earlier references, estimates 

 of c for corn yield from data collected to date at the North Carolina Experi- 

 ment Station have been somewhat less than .50. 



An exact F test of the hypothesis that o ^ 1.0 is not provided in the vari- 

 ance analysis of either Experiment I or II. In both instances there is a func- 

 tion {R) of three mean squares that provides an approximate F test. They 

 are given below. Remember for Experiment II that we are assuming m = n 



Experiment R 



I R^ = {2n+i)Mn/{iMn+2nMn) 



II Ri = {2n+\)M^/{Mio+2nM<,i) 



and using M20 to symbolize the mean of M 21 and ^22- As was true for the test 

 ratio of Experiment III, the expectations of numerator and denominator are 

 equal in both of these ratios w-hen a — 1.0, but when a > 1.0 the expectation 

 of the numerator exceeds that of the denominator. Also, the estimate of a is 

 greater than one only when the test ratio is greater than one. Values of (f> for 

 Experiments I and II in Table 30.8 are the ratios of expectations of numera- 

 tor and denominator in these test ratios. As suggested by relative sizes of 4> 

 for the three experiments, more data are required in Experiment II than in 

 III, and still more are required in I. However, the degrees of freedom sup- 

 plied are greater relative to numbers of plots used than in III so differences 

 in data required cannot be judged properly in terms of the 0's. 



The data requirement cannot be determined as accurately as for Experi- 

 ment III, primarily because degrees of freedom that should properly be as- 

 signed to the denominators of the test ratios cannot be known exactly 

 though they can be approximated by the method of Satterthw-aite (1946). 

 For the same reason, determination of the approximate data requirement is 

 more time-consuming. Attention will therefore be confined to the three situa- 

 tions indicated below. Degrees of freedom for Experiment I refer to the mean 

 square, Mn, and for Experiment II to M23. In both cases, n was assumed to 

 be 4.0. Thus in II, progenies per set would be 16 as was assumed for Experi- 

 ment III. This would make degrees of freedom for M 23 be 9/32 of the number 



