XXIX — XXX] Introduction liii 



Considering only the equation as far as Everitt's Tables extend, we have 

 ( r ) = - -16079 + 13557c + -01014r 2 + 01585r»+ , 0068lr* + -00470?- 5 + -00507?'" = 0. 

 This leads to r = 93G5, but the series indicates that the terms are far from 

 converging rapidly. 



The first 12 tetrachoric functions were then used, the last six being found 

 by the table of p n and q n above, and the value of r was found to be - 9152. 

 Then 18 functions were used and gave r = -9114. 



Lastly 24 tetrachoric functions were used, and the equation below obtained, 

 which led to r = '9105. 



$ ( r ) = - -16079 + -13557r + 01014r 2 + -01585^ + -00681?- 4 + -00470c 5 

 + OOSO?;- 11 + -00167r + -00395^ + O0055r« + 00315c 10 

 + OOOlOr" + -00255c 12 - -00009c 13 + -00212-c" - -00015 c 13 

 + 00174c 111 - 00014c 17 + -00143r 18 - -00011?- 19 + -00118?- 20 

 - -00057r 21 + -00096c 5 - - -00003c- 3 + -00079c 24 . 

 It will be seen that even with this very large amount of labour we cannot be 

 sure of having reached a final result*. To obviate this the following table was 

 constructed by Everitt, and there is no doubt that the extension of this table to 

 the whole range of correlation would much simplify the discovery of tetrachoric r t . 

 At present the calculation of high values of r t , for negative correlations is in hand. 



Table XXX (pp. 52—27) 



Supplementary Tables for determining High Correlations from Tetrachoric 

 Groupings. (P. F. Everitt, Biometrika, Vol. vm. pp. 385 — 395.) 



Using the notation of p. 1, 



d 1 /•»/•=>= - j —1— (a;2 + j, 2 _ 2raj ,) 



"= == e 'i-H v * »' dxdy (xl) 



in the case of a tetrachoric table, or 



d _ J_ /""--to 1 



N 



= -Lf e'^Ydy 

 \l-lirh y 



x. rr 1 f -hXdX ., , h-yr 

 where Y= — = e 2 , if t «- 

 ^2-rrJt 



.(xli). 



Vl -r 2 



* Mr H. E. Soper working out this example draws my attention to the fact that convergence is closely 

 given by a form : r n =r„ (1 + n .c"), where n is the number of terms used and a and c are constants. 

 Hence (r„ - r„ ) (r n+2m - r„ ) = (r n+m - r K ) 2 , 



r n + r n+2m _ 2)-„ t m 



In our ease take n = 6, m = 6, and we find 



°° r r , + r M -2r n 

 The value j- 24 is -9105. In this case « = -1567 and c = -7574, but we cannot assert that these would 

 be constants for all tables. If we use r V i, l"u and fjj, we find )•„ = -9102. 



