Giles: Multivariate Analysis of Pleistocene and Recent Coyotes 381 



Horizontal-ramus breadth (HRB). — Measured at the same place as the hori- 

 zontal-ramus depth, and perpendicular to the vertical axis of the horizontal ramus. 



STATISTICAL PROCEDURE 



The method and theory of calculating the D 2 statistic and the discriminant func- 

 tion as used in this study are fully set forth in Mahalanobis, Majumdar, and Rao 

 (1949), Rao (1948, 1952), Bronowski and Long (1952), and Mukherjee, Rao, and 

 Trevor (1955). 



Z> 2 is better adapted to morphological studies than certain other methods that 

 might be used (e.g., Pearson's Coefficient of Racial Likeness) in that a simple 

 theoretical determination of the distance between groups is possible, only the 

 fraction of a measurement uncorrelated with other measurements is utilized in 

 discrimination, and sample sizes do not affect the calculations. 



Although the technique seems complex, it is rather well fitted to computational 

 schemes and practical application. The laboriousness of the calculations depends 

 on the equipment at hand and the number of variables considered. In many cases 

 three or four rather than the eight measurements used in my analysis of the 

 cranium might suffice — in fact, only five measurements of the lower jaw were 

 enough. "Whether the results are considered to be worth the time involved will, of 

 course, depend upon the data tested and the analyst's opinion of the value of sta- 

 tistically based conclusions. 



D 2 Analysis 



If N groups are to be studied, with p characters measured on each individual, the 

 mean for each of the characters is found for each group (table 3) and then put 

 in standard form. The standard deviation for all of the characters and the co- 

 efficient of correlation (r) between each character and the others is found for 

 all groups, as shown in table 4. 



The next step is making a transformation to a set of uncorrelated variables. 

 The method by which this is done is described in detail in the references. This 

 transformation results in a series of equations into which the standardized mean 

 values for each character are substituted. The If between each group and the 

 others is found by summing the squares of the differences between each of the 

 transformed values. 



When reasonably large samples are used, a cTit-square test of significance can 

 be applied, using p degrees of freedom (Mahalanobis, Majumdar, and Rao, 1949). 

 The distances between groups can be shown by representing the D values in a 

 diagram. With more than three groups, however, this representation is only ap- 

 proximate. 



Discriminant-function Analysis 



Burma (1949) used multivariate analysis to test the significance of differences in 

 two invertebrate populations, giving the step-by-step procedure. The D 2 analysis 

 and the discriminant function described below differ considerably in detail from 

 those used in Burma's application. Burma was interested in testing the significance 

 of differences between two groups; my application of the D" method defines the 

 relative affinities of a number of groups geographically as well as giving a sig- 



