Page:Proceedings of the Royal Society of London Vol 60.djvu/525

490 exist correlation between u and v. Thus a real danger arises when a statistical biologist attribntes the correlation between two functions like uand vto organic relationship. The particular case that if likely to occur is when uand v are indices with the same denominator for the correlation of indices seems at first sight a very plausible measure of organic correlation.

The difficulty and dang’er which arise from the use of indices was brought home to me recently in an endeavour to deal with a considerable series of personal equation data. In this case it was convenient to divide the errors made by three observers in estimating a variable quantity by the actual value of the quantity. As a result there appeared a high degree of correlation between three series of absolutely independent judgments. It was some time before I realised that this correlation had nothing to do with the manner of judging, but was a special case of the above principle due to the use of indices.

A further illustration is of the following kind. Select three numbers within certain ranges at random, say x, y, z, these will be pair and pair uncorrelated. Form the proper fractions xjy and zjy for each triplet, and correlation will be found between these indices.

The application of this idea to biology seems of considerable importance. For example, a quantity of bones are taken from an ossuarium, and are put together in groups, which are asserted to be those of individual skeletons. To test this a biologist takes the triplet femur, tibia, humerus, and seeks the correlation between the indices femur / humerus and tibia / humerus. He might reasonably conclude that this correlation marked organic relationship, and believe that the bones had really been put together substantially in their individual grouping. As a matter of fact, since the coefficients of variation for femur, tibia, and humerus are approximately equal, there would be, as we shall see later, a correlation of about 0'4 to 0'*> between these indices had the bones been sorted absolutely at random. I term this a spurious organic correlation, or simply a spurious correlation. I understand by this phrase the amount of correlation which would still exist between the indices, were the absolute lengths on which they depend distributed at random.

It has hitherto been usual to measure the organic correlation of the organs of shrimps, prawns, crabs, Ac., by the correlation of indices in which the denominator represents the total body length or total carapace length. Now suppose a table formed of the absolute lengths and the indices of, say, some thousand individuals. Let an “ imp ” (allied to the Maxwellian demon) redistribute the indices at random, they would then exhibit no correlation; if the corresponding absolute lengths followed along with the indices in the redistribution, they also would exhibit no correlation. Now let us suppose the indices not to have been calculated, but the imp to redistribute the abso-