establish the best separation of the sample. 



OVERLAP OF MULTIPLE CHARACTERS 



The preceding sections have shown that 

 the amount of overlap between two samples may 

 be computed in the same units for either counted 

 or measured characters. We now consider how 

 to combine the results of comparing several 

 characters either counted, measured, or both 

 in combination . 



This is a problem which has long plagued 

 taxonomists. Ginsburg (1939) gives a good re- 

 view of the methods used by fish taxonomists. 

 These include a great variety of sums, ratios, 

 and products . Usually the combining method 

 has been developed to fit a particular problem 

 and usually, too, the results have been difficult 

 to compare statistically. Then, after a number 

 of characters have been compared, either in- 

 dividually or in some combination, it has been 

 perplexing to interpret the results . Are two 

 correlated characters to be given twice as much 

 weight as one? Are small differences in a num- 

 ber of characters as significant as a large 

 difference in one character? No satisfactory 

 answers have been found by fish taxonomists. 

 Ginsburg concluded that it was best to rely on a 

 "principal" character and place slight weight on 

 other characters. 



One elaborate method of combining char- 

 acters was used considerably by anthropologists 

 and occasionally by other taxonomists, e.g., 

 Royce (1953) . This was called a coefficient of 

 racial likeness by Pearson (1926, 1928) who 

 originated and applied it. It is essentially an 

 average of the difference between means divided 

 by the sum of the standard errors of the means, 

 with the result adjusted to samples of a constant 

 size. The adjusted value is known as the re- 

 duced coefficient of racial likeness 



R.C.R.L.= 50 



'1 °2 



(11) 

 in which n is the mean of the sth character, 

 tf^~ the standard error of the sth character, ng 



the number of observations of the sth character, 

 n the mean number of observations of all char- 

 acters, and M the number of characters. The 

 subscripts 1 and 2 refer to the first and the 

 second samples respectively. The first paren- 

 thesis involving n is the reduction factor, pur- 

 porting to eliminate the effect ox sample size. 



The coefficient of racial likeness was 

 severely criticized by Fisher (1936a) and by other 

 statisticians, who generally agreed with Fisher. 

 It has two major deficiencies (Rao 1952) . First, 

 and probably most serious, all the characters 

 are treated as though they are independent. The 

 addition of a character highly correlated with 

 one previously considered may produce a large 

 change in the coefficient even though it adds no 

 additional statistical information. Second, even 

 the reduced coefficient is not independent of 

 sample size, which in taxonomic material is 

 rarely uniform. Associated with this is the fact 

 that the weight given various. characters depends 

 on the number of measurements rather than on 

 any biological criteria. Fisher (1936a) also 

 pointed out that adding the ratios of mean differ- 

 ences to their standard error was comparable 

 to repeating a test of significance, something 

 which has no sound logical basis. Once a test 

 of significance has been made, the conclusion 

 has been reached and repeating it is mere re- 

 iteration. Because of these difficulties the 

 coefficient of racial likeness has fallen into dis- 

 use. 



Another approach to multiple character 

 analysis has been made by psychologists in their 

 "multiple factor analysis" . This is a method of 

 reducing the number of characters by "factoring" 

 to a smaller number of parameters based on the 

 intercorrelations of the characters. According 

 to Thurstone (1947), however, this method of 

 analysis has been challenged by mathematical 

 statisticians and he admits no direct relation to 

 modern statistical theory. A recent user (Stroud 

 1953) points out that it is a method of obtaining 

 semireliable explanatory information and there 

 are no adequate methods of establishing confidence 

 in the results. 



At about the same time that he condemned 

 the coefficient of racial likeness^ Fisher (1936b) 

 proposed a discriminant function especially for 



22 



