294 



Miss M. Beeton and Prof. Karl Pearson. 



20 years of age. Evidence of this selection is to be found in the 

 facts that (1) fathers have a mean age 5 to 7 years greater than 

 that of their sons ; (2) the variability of their age at death is very 

 sensibly less than the variability of their sons' age, i.e., as 2'9 to 3*5, 

 or (3) by noticing for example that in our first table 82 sons as 

 against 20 fathers die before 32*5 years of age, and that in our second 

 table some 100 sons as against 20 fathers die before 35 years of age. 

 Clearly the group " son " is a much weaker type than the group 

 " father." As will be shown in a memoir on the effect of selection on 

 correlation, this want of likeness in fathers and sons itself tends to 

 modify the correlation between them.* 



While this selection occurs only in the case of fathers and sons and 

 not in the case of brethren, still the general character of the correla- 

 tion surface is alike in both. It is known that the curve of frequency 

 of death at different agesf is by no means normal. It is probably 

 compound, and only approximates to normality round three score 

 years and ten. It would hardly, however, fulfil a useful purpose to 

 deal only with the correlation of ages of death of relatives both dying 

 under the old age mortality group, even if on the sunny side of 70 we 

 could distinguish old age from middle age mortality. But in dealing 

 with correlation and regression in such cases as this, we must throw 

 entirely on one side any notion of normal surface and curves of error, 

 and go simply to the kernel of the affair. 



What we want is the law connecting the mean age at death of one 

 relative when another relative has died at a given age. When the 

 given age of the latter and the mean age of the former are plotted to 

 form a curve, this curve is the regression curve whatever be the form 

 of the frequency surface. The line of closest fit to this curve is the 

 regression line, and Yule's theorem^ tells us that the slope of this line 

 is found in exactly the same way as if the frequency surface were a 

 normal distribution. The slope of this line has nothing whatever to 

 do with the particular form of surface, and may be found even if we 

 cut off a portion of the surface parallel to one axis, e.g., if we take the 

 regression line for fathers or sons we get the best fitting lines in pre- 

 cisely the same manner whether we take all sons dying from infancy 

 to old age, or only those from 20 years onwards. If, of course, the 

 regression curve is sensibly linear, then the regression line is the true 

 curve of regression. Everything proved in the memoir, " On the Law 

 of Ancestral Heredity "§ holds for such linear regression equally 

 well ; we need not suppose normal correlation. Now the reader has 



* Not of course very largely, still with the values given in the first series of 

 fathers and sons, the correlation would be reduced about - 86 to - 9 of its value by 

 the selection of fathers 



f ' Phil. Trans.,' A, vol. 186, p. 406 and plate 16. 



% ' Roy. Soc. Proc.,' vol. 60, p. 480. 



§ ' Roy. Soc. Proc.,' vol. 62, p. 336. 



