Giles: Multivariate Analysis of Pleistocene and Recent Coyotes 371 



or perhaps show that one or more groups were not definable on the basis of such 

 measurements. 



The simplest method would be to choose one character — for example, skull 

 length — which differs among the groups, measure it for all specimens, find the 

 mean for each group, and express the difference among the groups as the difference 

 between each group's mean and the means of the others. This would give numerical 

 values for the symbols D 12 , D 13 , and so on. The group whose mean was least like 

 that of another group would thus be placed farthest from it. The modified exten- 

 sion of this method that I have used is not intended to sort out populations from a 

 heterogeneous assemblage. The populations must first be segregated by an external 

 criterion such as location, stratigraphic position, or age. 



The complicated morphology of vertebrates suggests that the distance between 

 groups should be determined not just from the differences between the means of 

 one measurement, but from the sum of the differences of the means of a theoreti- 

 cally endless series of measurements representing all possible dimensions of the 

 animal. This is not only impossible; it is also unnecessary, for two reasons. (1) 

 Many measurements are practically identical or are merely part of a previously 

 measured dimension, and thus highly correlated with each other, so that each 

 additional measurement adds little to the total picture. (2) The attainable sharp- 

 ness of discrimination between any two groups is limited by intragroup variability. 



If the distance between groups is to be determined by the sum of the differences 

 between the means of a number of measurements or variables (no more than eight 

 or nine are generally feasible), two corrections are necessary to make the calcula- 

 tions reliable. First, the disparity between two mean-differences which is due only 

 to a difference in gross size of the characters measured must be removed: a differ- 

 ence of a centimeter between two means for head length, for example, is not likely 

 to be ten times as important as a mean-difference of one millimeter in molar length. 

 Second, the possibility of error resulting from the correlation between two vari- 

 ables must be removed. If, for example, a specimen with a short head is compared 

 with a specimen with a long one, its head will probably also be narrower than that 

 of the second specimen. Part of this difference in head breadth may be related to a 

 difference in the over-all size of the two specimens, but part may be attributable to 

 the fact that the second belongs to a group with distinctive, characteristically 

 broad heads. The part that is correlated with the difference in over-all size must be 

 excluded before the true value of the breadth measurement can be determined. 



Making allowance for correlated measurements is important for another reason. 

 As mentioned earlier, successive measurements tend to become more and more 

 closely correlated, and may finally completely overlap each other. If correlation 

 is not taken into account, the mean-differences of several measurements whose 

 correlation may be as much as 100 per cent will be added to the total sum. This 

 difficulty can be avoided only if the mean-difference added by each new measure- 

 ment is adjusted in proportion to its correlation with all the measurements coming 

 before it. 



Multivariate analysis may also be applied to the problem of allocating a speci- 

 men to either of two groups, or possibly excluding it from both. The procedure has 

 two parts: (1) determining whether the specimen could belong to group A, and 



