304 I The Process of Evolution 



lation coefficients as a measure of the similarity of each individual 

 to every other individual.^ These coefficients are conveniently dis- 

 played in an array called a Q matrix (Table 13.1). Q matrices show 

 the estimates of similarities among individuals. One may wish to 

 know the patterns in which characters are associated, for instance, 

 whether or not individuals with long forewings also tend to have a 

 wide spot in the discal cell of the forewing. In this case, coefficients 

 may be calculated that show the relationships of the characters with 

 each other; these may be arrayed in what is known as an R matrix. 

 In our example, each of the 13 individual butterflies is compared with 

 all 12 others, giving 78 (13X12/2) comparisons. The coefficients 

 comparing each of the 75 characters with all 74 others were com- 

 puted, but the R matrix containing the 2,775 (75X74/2) coeffi- 

 cients is too large to reproduce conveniently. The R matrix is 

 especially useful in providing clues as to whether two characters are 

 actually measurements of the same thing. ( For instance, in a human 

 being one would not ordinarily consider the color of the left eye 

 and the color of the right eye as two separate characters, since in 

 most samples they would be perfectly positively correlated.) 



It is helpful in visualizing patterns of relationship to express the 

 structure in a Q matrix in the form of a dendrogram (treelike dia- 

 gram). One way is to form the nucleus of clusters with pairs of 

 highly correlated entities and to add individuals to the clusters ac- 

 cording to a specified procedure. The relationships of the 13 Eu- 

 phydryas specimens as obtained by one method [weighted variable 

 group ( WVG ) ] of expressing structure in a matrix are shown in Fig. 

 13.1. The only significant feature of this diagram of relationships is 

 the level at which stems join. The level at which two individual 

 stems join may be read at the ordinate as the correlation coefficient 

 of the two entities. When stems of groups join, the level is an aver- 

 age correlation of the two groups. Note that, while the dendrogram 

 makes it much easier to grasp the basic patterns of relationship in 

 the Q matrix, considerable information is lost in the process of ex- 

 pressing the structure. Thus, although the relationships of the Kings 



* The product-moment correlation coefficient may take values from — 1 to -|- 1 . 

 Perfect positive correlation (maximum similarity— the correlation of an indi- 

 vidual with itself) is +1, perfect negative correlation (maximum dissimilarity 

 in every character) is —1. A correlation coefficient of zero indicates no correla- 

 tion; the individuals are neither more nor less similar than one would expect if 

 the character values for each were assigned at random. In the matrix of 

 coefficients given, those with a value larger than .25 are significantly different 

 from zero ( P < .01 ) . The product-moment correlation coefficient is only one 

 of several coefficients which may be used to assay similarity or difference. 



