MORPHOLOGICAL INTEGRATION: A DISCUSSION

pairs of measures involving one member of the basic pair by a difference in "r" of less than .05 and some by a difference in "r" of only .01, differences that are certainly non-significant. Rough calculations show that for samples in the range of 20 to 40 individuals, the "r's" must differ by at least .10 to .20 to be considered significantly different when they are in the range of .80 to .95. Usually the difference in "r" must be even higher, in the range of .30 to .50, before the "r's" between 0 and .80 are significantly different. Thus in studies of morphological integration, the "r" value of most basic pairs does not differ significantly from the "r" of many other pairs of measures involving one member of the basic pair. Consequently, one cannot, with any assurance, choose one pair of measures from a group of pairs whose "r" values are not significantly different and give it a special meaning -the basic pair. A second problem is that the basic pairs may not be the same if one repeats a study of morphological integration using a different sample. Therefore, I can only conclude that the definition of the basic pair does not have a sound statistical foundation. The consequences are obvious since the choice of basic pairs is an essential step in the pF-model. Conclusion. The two points in the statistical model discussed above are sufficient to invalidate the present pF-model. First, instead of dealing with "high" levels of correlation, the levels are rather low; and second, the correlation coefficients can be distinguished from each other with a very low degree of precision, or conversely, the "r's" must differ by at least .10 to .20 before one can distinguish with any certainty between populations from which they were drawn. Since the entire argument of "Morphological integration" is built on the pF-model, it can thus be no better than the model. Unless the authors can correct these points and present a statistically sound pF-model, it can only be concluded that the method of "Morphological integration" is statically unsound and of little use in evolutionary studies. Acknowledgments. I wish to thank the many people who have helped me during the course of this study. Special thanks must go to Dr. W. G. Cochran of the Statistics Department, Harvard University, who very kindly checked my statistics and to Dr. G. G. Simpson with whom I discussed the entire problem. Drs. Ernst Mayr, Carl Gans and Professor Bryan Patterson must be thanked for their aid in checking the statistics and reading the manuscript. These workers, of course, do not necessarily agree with all of the remarks contained in this paper and I am solely responsible for all errors in the statistics and the interpretation of "Morphological integration."