512 



R. E. COMSTOCK AND H. F. ROBINSON 



All values of ^ listed in the table are for r = 2. However, the effect of mul- 

 tiplying r by any constant is the same as dividing c by the same constant. 

 Hence, <^ for c = 1 and r = 8 is the same as for c = .25 and r — 2; for 

 c = 4 and r = 4 is the same as for c — 2 and r = 2; etc. 



Table 30.9 lists the approximate degrees of freedom required for Msi and 

 Mzi if F.05 is to equal so that P will be .50. As an example to clarify the 

 significance of this table, assume that c = 1.0, a = 1.4, and r = 2. Then if 

 the data provide 142 degrees of freedom for both Mzi and M32, the probabili- 



TABLE 30.9 



.\PPROXIAJ.ATE DEGREES OF FREEDOM* RE- 

 QUIRED TO MAKE P = .50 IN 

 EXPERIMENT III 



♦Obtained assuming normal distribution of Fisher's z and em- 

 ploying the facts that <tv = i(l/f\ + I (fi) (where h and J2 are de- 

 grees of freedom for the two mean squares) and F = e-'. 



ty of the estimate of a being significantly greater (at the 5 per cent point) 

 than one is one-half. Degrees of freedom can be related to amount of data as 

 follows. Suppose that n, the number of progeny pairs per set, is 8. Then de- 

 grees of freedom will be 7/8 the number of progeny pairs, and assuming two 

 replications, r = 2, degrees of freedom will be 7/32 the number of plots in the 

 experiment. The 142 degrees of freedom indicated in the specific instance 

 singled out above would require data on a total of about 650 plots. 



An obvious question is whether increasing replications is as effective as in- 

 creasing the number of progeny pairs. Consider the case where c = 4.0 and 

 a — 1.6. Degrees of freedom required are 450 when r = 2. But remembering 

 that multiplying r by a constant has the same effect on as division of c by 

 the same constant, we see that with four replications degrees of freedom re- 

 quired would be only 150. Thus with two replications a total of about 2056 

 plots would berequired, whereas with four replications only about 1370 would 

 be needed. The same is not true for the entire area of the table. Careful in- 

 spection will show that when c is 1.00 or less, doubling the number of progeny 

 pairs is more effective than increasing replications from two to four. But 

 when c is 2.0 or greater, the opposite is true. 



Also pertinent are (1) the effect on P of increasing data above amounts in- 

 dicated in Table 30.9, and (2) the probability of an estimate of a that is less 



