34 WJCKSELL, THE CORRELATION PUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



d, = P» - 3r/* 03 + y^ (&0 - rfti + r 2 /? i2 - r*p oa ) + 

 (59) +> ; [ r _3(^3-4r i 5 04 )- T -^ 1 (/? 31 -2^ 22 + 3r s /? 13 -4rV 04 )]- 



+ i?»[/?„-4r i 9 M ]. 



As the range of the applicability of this curve, even in the case of moderate 

 skewness of correlation, in a high degree depends on the value of r, the greatest 

 caution must be observed whenever it is used. 



Mutatis mutandis the formulae of this paragraph may be used also to obtain 

 the regression of the characteristics of the distribution of the rjis in the |-arrays. 



(12) General conclusions. The formulae of regression have been derived for all 

 the characteristics up to the fourth order in the arrays. For the two highest orders, 

 however, they have been derived only in the case of approximation referred to as 

 case I. 



Now the terms neglected are in all the formulae of the same order of magni- 

 tude in absolute measure. Not so, however, in a relative measure. Provided r is not 

 nearly equal to unity the effect of the neglected terms in percentage of the regres- 

 sing characteristics will be about of the same magnitude in the case of the means 

 and of the moments of the second order. 



But in the case of the characteristics v 3 (§) v and Ä 4 (£)■>? the percentual effect of 

 neglected terms will be greater. The quantities to be plotted out in the curves are 



proportional to ^H — ': <?,*(£), v s (^)i, an d Ml)»?- The first two quantities are of the 



order of magnitude of unity, but the last two are of the same order as the /?y, 

 The terms neglected are in all cases of the order Pij.fipq. Thus we see that when 

 it comes to a study of the forms of the curves, not merely of the absolute values of 

 their ordinates, the curves of regression of the characteristics of the fourth and third 

 order will be less well defined than the curves giving the regression of the means 

 and dispersions. This of course holds good only so far as formulae (34* I), (40 I), 

 (46), and (53) are concerned. In the case referred to as case II formula (53) will be 

 even quite illusory. From this it follows that too much importance must not be 

 attributed to the fact that according to formulae (46) and (53) the clisy is linear 

 and the concentricity is constant. 



Some conclusions may, nevertheless, still be drawn from the formulae. Pear- 

 son has introduced the term homoscedasticity when the correlation is such that 

 o-qd) and <T|(ij) are constant in all arrays. Similarly he talks of homoclisy when Sfå) 

 and £,,(£) are constant. We shall add the term homo-synagicity when #,,(£) and 

 Et{i-j) are constant. Further Pearson talks of the variates being uncorrelated when 

 |, and rj( are constant. To this last term, so far as it gives a definition of indepen- 

 dency of the variates, we shall have to advance an objection. Clearly for the varia- 

 tes, to be independent we should require that they could be separated in the correla- 

 tion function. Thus, that 



