For testing marginal homogeneity of a square contingency table under the restricted alternative that either is stochastically larger than the other, approximately somewhere most powerful tests are studied. The Mann-Whitney test is included in them. Among the approximately somewhere most powerful tests, an optimal one is derived by the most stringent principle and compared with the Mann-Whitney test and the Stuart's test with respect to their asymptotic powers. Simulation study indicates that the Mann-Whitney and most stringent tests both have consistently higher powers, compared with the Stuart's test for detecting the locational shifts when a bivariate continuous distribution is assumed as an underlying distribution. It is indicated that the Mann-Whitney test competes well with the approximately most stringent somewhere most powerful test, but the latter is more reliable than the former.
A model, in which the means and the variance-covariance matrix of observed variables change with an external variable, is proposed. This is an extension of the analysis of covariance structures in several populations. Assuming that the observed variables, given the value of an external variable, have a multivariate normal distribution, the maximum likelihood estimates of the parameters in the model can be obtained by the Fisher's scoring method. The model with a constant variance-covariance matrix, the model with constant correlations, the model of a single common factor and the model of oblique multiple factors with constant factor loadings are disucussed for the model of the variance-covariance matrix. Finally, examples of intelligence test scores are provided, where the external variable is age.
A sensitivity analysis method for eigenvalue problems in multivariate analysis was proposed by De Sarbo, et al. (1982) and developed by Ueda (1988). Metric multidimensional scaling (MDS) is a spectral decomposition of symmetric matrices, where eigenvalues and eigenvectors play important roles. It is shown that the ε-neighborhood vectors defined in Ueda (1988) give the 2ε-neighborhood vectors for the goodness-of-fit index in a spectral decomposition. Considering this fact and using the same method as Ueda (1988), sensitivity for metric MDS is discussed.
The concept of stability has been developed in structural stabilty theory of mathematics. This concept plays an important role in characterizing the nature of a scientific law. Yoshino (1989a) introduced it into axiomatic measurement theory developed by mathematical psychologists, and analyzed some experimental data concerning psychophysical laws. The main objective of the present paper is to develop a concept of “degree of stability” for the further study of the nature of a scientific law, especially, a psychophysical law. Some experimental data on the Stevens' power law are analyzed from the standpoint of degree of stability. Finally, some possible experiments are suggested for the future development of this research.