Model Selection of Nested and Non-Nested Item Response Models Using Vuong Tests

Abstract

In this paper, we apply Vuong’s general approach of model selection to the comparison of nested and non-nested unidimensional and multidimensional item response theory (IRT) models. Vuong’s approach of model selection is useful because it allows for formal statistical tests of both nested and non-nested models. However, only the test of non-nested models has been applied in the context of IRT models to date. After summarizing the statistical theory underlying the tests, we investigate the performance of all three distinct Vuong tests in the context of IRT models using simulation studies and real data. In the non-nested case we observed that the tests can reliably distinguish between the graded response model and the generalized partial credit model. In the nested case, we observed that the tests typically perform as well as or sometimes better than the traditional likelihood ratio test. Based on these results, we argue that Vuong’s approach provides a useful set of tools for researchers and practitioners to effectively compare competing nested and non-nested IRT models.     

Multi‐Criteria Ratios: What is the Unit?

When Analytic Hierarchy Process ratios are normalized to sum to unity, the unit of measure becomes obscure. This paper investigates this obscurity and whether ratio measurement is possible when there is no prior knowledge of the measurement unit. Initially, we look at ratio scales of tangible attributes of objects with well‐known measures. Then, ratio scales of unknown intangible attributes of objects are analysed. We discover that natural zero and a specific unit of measure are not necessarily used explicitly in deriving ratio scale measures. Nevertheless, the derived scale does have a derived unit of measurement. We conclude that although composite multi‐criteria answers are possible in ratio form, it is important to know that a unit of measure exists if ambiguities are to be avoided. Copyright © 2011 John Wiley & Sons, Ltd.