Canonical Discriminant Function Analysis Research Articles

The use of multivariate morphometric methods in the separation of alate morphs of the rose‐grain aphid, Metopolophium dirhodum

SUMMARYA preliminary morphometric analysis of alate Metopolophium dirhodum was carried out. The alate morphs studied were: gynoparae and alate emigrants (collected from Rosa), summer exules (collected from maize in the field and from wheat in the laboratory) and winter exules (collected from grasses). Comparing the different morphs using canonical discriminant function analysis gave good separations (always in excess of 80% and often 100%). If this result is found to hold for specimens from a wider temporal and geographical range, it may be used to determine the sources of flying alatae, particularly spring colonists.

Nonpharmacologic treatment of chronic headache: prediction of outcome.

We studied the ability of headache history, a 4-week headache diary, standard psychological tests, and laboratory measures of psychophysiologic responses to stress to predict the outcome of relaxation therapy and biofeedback for three types of chronic headache. Using canonical discriminant function analyses, each potential predictor set was tested separately, and all four were tested together. Information from the headache history alone correctly classified 89 to 95% of patients as improved or unimproved. No other single predictor set was consistently better than headache history. When all four predictor sets were combined, prediction improved; 93 to 100% of patients were correctly classified.

The Study of Biological Versus Statistical Variation in Multivariate Morphometrics: The Descriptive Use of Multiple Regression Analysis

Albrecht, G. H. (Department of Anatomy, School of Medicine, University of Southern California, Los Angeles, California 90033) 1979. The study of biological versus statistical variation in multivariate morphometrics: The descriptive use of multiple regression analysis. Syst. Zool. 28:338-344.-Multivariate statistical techniques (such as canonical variate and principal component analyses) are often used to ordinate or summarize morphometric data to facilitate biological interpretation of the morphological relationships under study. While the major axes of statistical variation which are derived in such analyses may have direct biological significance, there is no a priori reason that the biological and statistical determinants of morphological variation necessarily be concordant. Multiple regression provides a simple means of identifying and describing the maximum degree of relationship between (1) a variable, such as size or latitude, which is thought to have some biological relevance to the problem at hand, and (2) the set of uncorrelated variables, such as canonical or principal component variates, which represent the major axes of statistical variation and which may be thought of as a convenient, analytically efficient system of reference axes describing the multivariate data space. Of particular significance is the ability to examine the full multidimensional space and detect biological information having an angular relationship to the major axes of statistical variation. [Multiple regression analysis; multivariate analysis; morphometrics; statistical and biological variation.] Multivariate statistical techniques are often used to ordinate morphometric data so that biological parameters underlying morphological relationships among individuals or groups may be more readily discovered. Commonly used techniques include canonical variate, discriminant function, principal component, and principal coordinate analyses (see Blackith and Reyment, 1971, for definitions and examples). All have in common a primary purpose of summarizing multivariate data in a relatively few dimensions that retain the majority of information formerly dispersed among the larger array of original variables. An additional advantage shared by all is the lack of statistical correlation among the transformed variates as compared to the complex of statistical dependencies usually found among the original variables. Such a reduction in both dimensionality and correlation results in a greater probability that the investigator will be able to make biologicallv relevant statements about the morphometric relationships under study. Populations (or individuals) are often found to be ordered on the first few transformed variates of a multivariate analysis according to morphological gradients suggestive of differences in size, shape, time, function, behavior, or ecology. For example, Oxnard (1967; nine measurements of the scapula of 27 genera of primates) interpreted the first canonical variate as reflecting the extent to which the shoulder is subjected to compressive or tensile forces, the second canonical variate as reflecting relative degrees of arboreality or terrestriality, and the third canonical variate as reflecting the uniqueness of the human condition. Johnston and Selander (1971; 16 measurements of the skeleton of 33 populations of North American house sparrows) interpreted the first and second principal component variates, after regressiot analyses involving 15 environmental variables, as reflecting classic examples

Recent Trends of Research Work in Multivariate Analysis

In the issue of Biometrics (Vol. 20, part 2) dedicated to the memory of Ronald Aylmer Fisher I reviewed the contributions by Fisher and some of the salient features of research work done in multivariate analysis up to 1964. Fisher's contributions as well as the related methodology developed by Wilks ([1932; 1946]-likelihood ratio criteria for testing multivariate hypotheses), Bartlett ([1947; 1951]-decomposition of Wilks's criteria for testing different aspects of a null hypothesis), Rao ([1946; 1948]-analysis of dispersion' as a generalization of the univariate analysis of variance, and tests for additional information supplied by a subset or functions of measurements), and Williams ([1952a; 19591-residual canonical correlations in testing goodness of fit of specified discriminants) may be described as a study of association between two sets of variables. One set is called predictor and another set, criterion variables. Some of the variables may be hypothetical (unobservable), and others may have values on a dummy or interval scale. I have indicated in Rao [1960] how various multivariate methods such as regression, canonical correlations, analysis of dispersion and canonical analysis, discriminant function, factor and latent structure analyses, treatment of contingency tables etc., can be classified by the nature of predictor and criterion variables. However, the classification provides only a suitable framework for the discussion of different problems but does not imply that the statistical methods appropriate for one problem can be simply deduced from those of another. Since 1964 considerable progress has been made in several directions of the work initiated by Fisher. Exact distributions have been found of several likelihood ratio criteria and of roots of determinantal equations involving random matrices which arise in multivariate statistical analysis and in some problems of physics. Further progress has been made in testing goodness of fit of assigned discriminant functions. The theory and construction of discriminant functions for deciding between composite hypotheses (involving nuisance parameters) has been developed. A satisfactory