IF ATTRIBUTES IN STATISTICS. 



271* 



I'xrtial Associations and Associations between Groups of Attribute*. 



29. In the value of Q, as written in equation (1), p. 272, the " Universe of 

 Discourse " is understood, not expressed. If " A " represent, say, deafness, and " B " 

 blindness, we are probably dealing with the association of these infirmities within at 

 most one nation, <.</., English, or even one sex of tin- nation, e.g., English men. 

 Letters are not given to represent that the universe is so limited, it l>eing generally 

 obvious from the context, but if we take D = English, E = men, we can write Q 



(ABDE)(DE) - 

 (ABDE)(aDE) + 



BDE)(A/9DE) 



xB1>K)(A/3I>Ki 



(11), 



adding the letters DE to every group. Such a coefficient of association will be 

 termed a 2>artial coefficient, as distinguished from the total coefficient of equation (1). 

 We may speak of partial coefficients of the 1st, 2nd, .... nth orders, according as 

 the universe is limited by the specification of 1, 2, :i . . . . n attributes. These 

 partial and total coefficients of association correspond roughly in their nature to 

 partial and total coefficients of correlation. In the latter cii.se, however, we limit the 

 universe by specifying that in all members of the universe variable x shall have the 

 fixed magnitude // ; in the former case we only specify that x shall exceed h or be less 

 than h. 



The following notation for coefficients of association seems concise and convenient. 

 The total association between A and B we shall denote by AB between two vertical 

 lines thus |AB|. The partial association in the universe of C's, CD's, OS's, or 

 OSes we shall denote by |AB|C|, |AB CD|, |AB|C8|, JABICSej. 



30. The number of possible partial coefficients becomes very high as soon as we 

 go beyond four or five variables. Supposing m attributes are given, we can form 



(t - 2)(m - ). . . (m - 4 1) 

 L2 



partial coefficients of the nth order (n < m 2) between any one pair of attributes. 

 For we can form 2" different universes with n attributes, and choose n attributes out 

 of (m 2), in 



(m - 2) (m - 3) ... (m - n 4 1) 



different ways. But the number of possible pairs of attributes (AB, AC, BC, &c.) is 

 w (m 1), and therefore the total number of possible partial coefficients of the 



uth order, 



_, -M(M- l)(t- 2) . . . (m-n-l) 



li 

 These expressions give the following figures : 



