THE STUDY OF MORPHOLOGICAL INTEGRATION

The book of Olson and Miller (1958), despite certain inadequacies, is in my opinion a landmark in our understanding of the interrelations of the parts of organisms. It includes the first demonstration that functionally related characters tend to be more highly correlated than others, and is the first study of several other aspects of integration. Because studies in this field are continuing by several investigators, I wish to discuss some points bearing on the methodology and interpretation. No important new evidence on integration is given in the present note; the main purpose of this note is the presentation of a simple and adequate method for the study of integration. The statistical criticisms of Olson and Miller's method given by Bock (1960) are in my opinion correct but not sufficiently serious to invalidate any general conclusions so far published. One of these criticisms is that the estimated value of the correlation coefficient r should be used rather than its lower confidence limit. These two statistics have the same ranking with equal sample sizes, and with different sample sizes the confidence limits reflect this difference as well as the correlation. If it is desired to exclude non-significant correlations, this can be done by simply ignoring all correlations below some common level at which some are not significantly different from zero, although it should be noted that the region of non-significance applies to each correlation individually and not to all considered together, and that an unknown number of real correlations will also fall in this region. The other criticism given by Bock is that individual correlations are frequently not significantly different from each other and that this fact invalidates the choosing of one pair of characters for special treatment rather than some other pair. This defect probably has relatively little influence on the general patterns found, as noted by Miller and Olson (1960). Olson and Miller (1958) discarded certain correlations (by a reproducible method) in the formation of their groups of integrated characters, and used as primary groups only those in which all members were mutually correlated above some fixed level. I regard this as unfortunate, as they did (p. 147), because the discarded correlations are as real as those retained, sampling error will form or eliminate a moderate number of groups, and part of the information in the value of the correlations is lost. The reason they used this procedure is that the complex pattern of intersection among groups was thereby much simplified. An alternative treatment, using all correlations, is the following. Consider the matrix of estimated absolute correlations, Irl. Normalize each r by the z-transformation (Fisher, 1958), corrected if necessary for sample size. Find the pair of characters with the highest lzl These two characters will now be treated as a single character and the means of their signed z correlations with each other character are used as the correlations of this combined character. Repeat this process as long as desired, when combining the correlations of a group using the original signed z correlations rather than those of the group, until sufficiently large groups have been formed or sufficiently low correlations are present between the groups or any characters still separate. At any level the groups are the mutually exclusive ones that maximize the mean of all the correlations among the components of these groups (more precisely, that minimize the mean