ABSTRACT
Some recent measurement results and basic concepts from differential geometry are applied herein to two dimensional probabilistic latent variable models. The purpose of the chapter is to introduce a strategy for using unidimensional methods to study two and higher dimensional data sets.
Each latent variable model is associated with a manifold. Curves in a model’s manifold are approximated by fitting one dimensional models to data. This paper shows how fitted one dimensional curves can be used to select an appropriate multidimensional model.
The data sets motivating this work are formed by sampling people. Each sampled person is asked a standard set of questions. The answers to each question are dichotomous, or can be easily made dichotomous.
Vocational interest inventory data exemplify the data we seek to understand. Each year 4,200,000 young adults answer a standard set of questions about their vocational interests when they complete ACT’s UNIACT interest inventory (Swaney, 1995, p. 1). Each inventory question or item describes an activity. Each response is one of “like”, “dislike”, or “indifferent.”
Our approach is intended to be broadly applicable. Instead of vocational interest data, other data formed by eliciting a vector of qualitative responses from a large number of sampled individuals might have been used. Examples include ability or achievement measurement data, cognitive, attitudinal or personality survey data, census data, and reliability testing data.
