A Multivariate Procedure for Comparing Mean Vectors for Populations with Unequal Regression Coefficient and Residual Covariance Matrices

In fisheries research there is a need to compare vectors of means of continuous random response variables adjusted for concomitant variations of covariables for populations that have unequal regression coefficient and residual covariance matrices. A multivariate procedure that provides an extended comparison of vectors of adjusted means is presented. An example is presented using a real data set. The procedure is quite general and applicable to many other fields of research.     

Comparing Vectors of Adjusted Means when Covariance Matrices are Unequal

The ability to distinguish between stocks of fish is of vital importance in fisheries management programs. Morphometric measurements are commonly employed to separate biological populations. Unique role of multivariate analysis of covariance in separating fish stocks has been recognized recently. However, fish populations frequently differ in their covariance matrices and sometimes also in vectors of slope (on the covariate) coefficients. In this article the following are presented: (a) a procedure for comparing adjusted (for variations in body sizes of specimens) mean vectors of populations with unequal covariance matrices and slopes, and (b) a study to distinguish between beaked redfishes based on the analysis of a real data set by this procedure. This will also serve as a numerical example.     

A Multivariate Analysis of Hybridization when Covariance Matrices are Unequal

Numerous occurrences of natural hybridization have been known in many groups of animals. It has a bearing on growth, nomenclature, speciation, genetics and wildlife management. It is well recognised that demonstration of intermediacy based on several characters makes the identification of hybrids more certain than that based on just a single character, and that differences among the hybrids and the parental populations should be analysed for variations due to the additive genetic (A) and the non-additive genetic (NA) factors separately. In the present paper (i) it is pointed out that sometimes the assumption that covariance matrices of the hybrids and the parental populations are equal, may not be valid, and (ii) a multivariate method of testing hypotheses analysing differences between the hybrids and the parents, qualified by the A and the NA factors, when covariance matrices are not equal, is submitted.