Mathematical Contributions to the Theory of Evolution. VII. On the Correlation of Characters not Quantitatively Measurable
Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, (1900)Zusammenfassung
In August last I presented to the Society a memoir on the inheri-
tance of coat-colour in thoroughbred horses, and of eye-colour in man.
This memoir, which was read in November of last year, presented the
novel feature of determining correlation between characters which were
not capable a priori of being quantitatively measured. The theoretical
part of that memoir was somewhat brief, but I showed by illustrations
that the method could be extended to deal with problems like the
effectiveness of vaccination and of the antitoxin treatment in diphtheria.
More recently, in studying the phenomena of reversion in Basset
Hounds, Mr. Bramley-Moore indicated to me how my method, although
correct in theory, differed sensibly in the numerical results with the pro-
cesses of interpolating employed. I then proposed a new method, and
the analytical discussion of its details was worked out in part by Mr.
Bramley-Moore himself, by Mr. L. N. Gr. Filon, M.A., and by myself*
Dr. Alice Lee also came to our assistance, and the result is the present
joint paper. On the basis of the new methods, we have already worked
out upwards of sixty coefficients of correlation, principally of herednVy.
Thus the thirty-six coefficients of heredity for coat-colour in horses and
eye-colour in man have been re-calculated, as well as twelve coefficients
for heredity in coat-colour of Basset Hounds given in a paper on the
Law of Ee version presented on December 28th, and about to appear
in the ' Proceedings.' The great growth of the theoretical investigations
has, however, compelled me to break up the old memoir* of last August
into two parts, the one (the present) dealing only with theory, and the
other with its application to inheritance in the horse and man.
The theory of the present memoir depends upon a very simple
feature of normal correlation. If 28x18x2 .... 8x n be the frequency
of a complex of characters lying between X\ and X\ + 8x1, x 2 and
%z + 8x 2 . . . ., x n and x n + 8x n , where x p is the deviation of the j?th
character from its mean, then
dz d 2 z
tv' PQ CvX-n QjXq
where r pq is the correlation of the j?th and qth organs.
This simple differential relation enables us to expand z for any
number of characters in powers of the correlation coefficients (neces-
sarily less than unity) by Maclaurin's theorem. But since we may
replace a differential with regard to a coefficient of correlation by a
double differential with regard to the corresponding organs, the coeffi-
cients of correlation may be put zero before instead of after the
differentiation. In other words, we obtain a symbolic operator which,
applied to a normal surface of frequency for w-uncorrelated organs,
converts it into a correlated surface of frequency with ^n(n-l)
coefficients of correlation of arbitrary values.
Beschreibung
Includes identity relating derivative of bivariate normal density wrt correlation to derivative wrt both spatial variables.