60 



THE AMERICAN NATURALIST 



[Vol. L 



of both characters being measurable on a quantitative scale, or 

 one character recorded in a number of categories and the other 

 measurable on a quantitative scale, both characters are not quan- 

 titatively measurable, but describable in a number of classes only, 

 neither the correlation coefficient nor the correlation ratio can be 

 used. In such cases, which in practical work are very frequent, 

 Pearson's contingency methods 30 must be used. These have been 

 too long in use to require discussion or illustration here. Certain 

 corrections to be applied will be considered at another time. 



The probable error of the contingency coefficient presents con- 

 siderable difficulty. Those who have to deal with it should con- 

 sult papers by Blakeman and Pearson 31 and by Pearson. 32 



One Variate Classified in Alternative, the Other in Multiple 

 Categories. — Consider a contingency table reduced to a two-fold 

 grouping for one of the characters, but retaining the multiple 

 division for the other. Such a table is comparable with the con- 

 densation of the correlation ratio table discussed above. It must 

 be analyzed by a special method. 33 



The formula has not as yet had extensive practical application. 

 It has been used to determine the relationship between alcoholism 

 as an alternative character and type of crime classed in multiple 

 categories, and between alcoholism in the parent and health of 

 the children. It may prove especially valuable in dealing with 

 the interrelationship of various teratologieal conditions in mor- 

 phological work. 



Both Characters Classified in Alternative Categories Only.— As 

 the extreme case we may think of a contingency table reduced 

 to a two-fold grouping for each of the characters. This is then 

 the four-fold table for alternative characters, i. e., (A) and (not 

 -A), (B) and (not -B). 



In the past, two methods have been chiefly employed for obtain- 

 ing constants from such tables, Pearson's four-fold correlation 

 coefficient and Yule's coefficient of association. 



30 Pearson, K., "On the Theory of Contingency and its Relation to Asso- 

 ciation and Normal Correlation," Drapers' Co. Res Mem Biom Ser I 

 Dulan & Co., 1904. 



31 Blakeman, John, and K. Pearson, "On the Probable Error of Mean 

 Square Contingency/' Biometrika, 5: 191-197, 1906. 



32 Pearson, k., "On the Probable Error of a Coefficient of Mean Square 

 Contingency," BiometriTca, 10: 570-573, 1915. 



33 Pearson, K., "On a New Method of Determining Correlation when One 



