Standard and Nonstandard Log-Linear Symmetry Models for Measuring Change in Categorical Variables

Abstract

Abstract This article exemplifies how to recast well-known statistical tests in terms of more general statistical models. The example used is the Bowker test, a popular test for assessing axial symmetry in square cross-tabulations. Researchers use this test to assess patterns of change in categorical variables across two occasions. This article reviews how the Bowker test can be equivalently expressed in terms of log-linear models with side constraints that specify (1) those pairs of cells that are supposed to be symmetrical with regard to the main diagonal and (2) that the main diagonal cells remain untouched when estimating expected cell frequencies. This article proposes expressing the model of axial symmetry through vectors in the design matrix. In addition, this article recasts quasi-symmetry models in terms of nonstandard log-linear models. One benefit from this new formulation is that estimation of parameters and expected cell frequencies can be performed without unfolding the cross-tabulation. The article also introduces multiple group models. In addition, it is shown how shift patterns can be made part of models. Benefits from recasting well-known tests in terms of more general statistical models include that parameters can be interpreted and that tests can be parts of more elaborate research designs.