XVIII— XX] 



Introduction 



xxxvu 



solely in Alternate Categories. Draper's' Company Research Memoirs, Biometric 

 Series, vm. Dulau & Co.) 



We have seen under the discussion of the previous Table how to find a measure 

 of the improbability of two variates being independent, when they are classed in 

 alternate categories. The difficulty in such cases is to appreciate the relative 

 importance of very large inverse powers of 10. The object of the present tables is 

 to enable us to deduce a tetrachoric correlation, r t , of which the improbability 

 is the same as that of the given system supposing it to arise, when the two 

 variates have the same marginal frequencies but are really independent. In order 

 to do this we have to determine <r r for the given marginal frequencies, i.e. the 

 standard deviation of r t on the assumption that r is really zero. This may be easily 

 found from Abac Diagram XXI or from Table XXIV (see below). Table XVIII 

 then gives us the value of (— log P) for each value of r t and a>. If we now turn 

 to our original table and calculate its rf, this as we have seen will correspond 

 to a given (— log P). We now make the (— log P) from our ^ 2 correspond to the 

 (— logP) from our r t and „a,., this gives us a value of r t which has the same degree 

 of improbability as our observed table. In other words, instead of trying to 

 appreciate the meaning of inverse high powers of 10, we say that a table of the 

 same marginal frequency would be as improbable if it had a tetrachoric correlation 

 r t arising from random sampling of independent variates. Thus we read our 

 improbability on a scale of tetrachoric correlation. We use our correlation merely 

 as a scale to measure probability on. 



As log% 2 provides a more satisfactory basis for interpolation, and as many 

 readers use logarithm tables and not calculators, log^ 2 will be the form in which 

 X 1 will be often presented. Table XX provides the value of r t corresponding to 

 given a r and given log % 2 . 



We will assume for the present that „cr,. can be readily found from the marginal 

 totals : see p. xli below. 



Illustration. Obtain the values of r P for the five tables given above on 

 pp. xxxiv — v. 



The values of log^ 2 and cr r are as follows: 



Of the values here recorded for log ^ 2 and <r r , those for the first and third 

 cases lie beyond the limits of our Table XX. But if the point <r r = - 0634 and 

 log x 2 = '1582 be noted on the Abac XXII, p. 34, it will be seen that, having due 



