Miscellanea 
413 
concerning its yield in any subsequent year. If, on the other hand, the correlation be high, 
prediction from a few years' test may be made with great probability of certainty. 
Given a measure of the "performance" of a series of varieties during a number of years it 
would at first seem quite allowable to form symmetrical tables or to use the intra-class formulae 
of a former paper* to determine the intra- varietal correlation, and to regard this as a satisfactory 
measure of the differentiation of the varieties and of the average prediction value of a year's test. 
Such is, however, not the case, for while there may be no orderly change in yield throughout the 
period under consideration, the individual years differ greatly in their average yield for all the 
varieties. The influence of this "disorderly differentiation" upon r is admirably shown by 
A. D. Hall's t table of the yield in bushels of wheat in the Rothamsted experiments. 
Let 6= yield in bushels per acre of any one of m varieties in any one of n years, y^, be the 
"first" and the "second" years of a symmetrical intra-varietal correlation surface, v^^ v<^ be the 
"first" and "second" varieties of a symmetrical intra-annual correlation surface. Then ^ty^by^ 
will be a (spurious) measure of the (persistent) differentiation of varieties, ''t^jS^^' ^ (spurious) 
measure of the differentiation (in the yield of all the varieties) of years. Applying formulae 
(v) — (ix) of Biometrika, Vol. ix. p. 450, to these data, I find 
,S'[?i(?i-l)]==2128, 
S[{n- 1) 2 (6')] = 83122-5, S[{n - 1) 2 (6'^)] = 3483626-4, 
S [2 {h')'f = 3610204-57, -S" [2 (6'^)] = 370820-13, 
5 = ,39-0613, 0-,,^= 11 1-257328, 
The I'esult is obviously spurious, for mere inspection of the entries in the table shows that 
some varieties regularly give heavier yields than others. The source of the spurious value is to 
be seen in the fact that an intra-class coefficient has been calculated from a symmetrical surface 
formed from classes (varieties) represented by a series of yields differentiated hy annual variations 
in the grotving conditions. By correcting for this source of differentiation by expressing each 
yield as a deviation from the mean yield of all the varieties for the particular year, i.e. b" = h — by, 
where the bar denotes a mean and the subscript y that it is for all the yields of a year, I have 
found f 
Measuring the differentiation of years in terms of intra-annual correlation (intra-class correla- 
tion in which each class is defined by the year and its individuals are the yields of the different 
varieties grown), I find from Hall's taVtlo 
S[m{fa- 1)] = 4440, 
.S'[(/rt- 1) 2 (6')] = 174129-2, S[{m -1)2 (?-'^)] = 7317531 -92, 
S[2 (6')P=7586436-21, ,S'[2 (6'^)] = 370820-13, 
6 = 39-2183, 0-63=110-017719, 
»-Si-„,= -791. 
Since the varieties have been shown to be differentiated, this result must also be spurious. 
Let b" = b- 6„ where the v indicates that the mean denoted by the bar is for the yield of the 
* Biometrika, Vol. ix. pp. 446—472, 1913. 
t Hall, A. D., The Book of the Rothamsted Experiments, p. 66, 1905. 
t Science, N. S. Vol. xxxvi. pp. 318 — 320, 1912. Probably a better method of dealing with such 
cases will sometime be found. So far I have not succeeded. 
Biometrika x 53 
