Grade of membership models in low dimensions

Abstract

Grade of Membership (GoM) Models have always been presented by their inventors as statistical applications of fuzzy set theory. This paper develops an alternative formulation, recasting GoM as a geometric dimensionality-reduction technique in terms of an underlying family of metrics, exposing a close relationship with Principal Components. The geometric viewpoint facilitates intuitive understanding and guides an investigation into the robustness of GoM estimates to violations of assumption, with test cases drawn from the National Survey of Families and Households. Analysis is restricted to visualizable, low-dimensional cases with two pure types and 3 to 9 dichotomous variables, and to “conditional” GoM, the version more commonly used but less commonly studied. In these low-dimensional settings, I find GoM to be a successful technique for recovering an underlying gradient among individuals when such a gradient is actually present. GoM is only moderately sensitive to moderate violations in its assumptions. In the cases studied, GoM solutions turn out to be remarkably close to Principal Component solutions.