A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models

Jules L. Ellis () and Klaas Sijtsma
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Jules L. Ellis: Behavioural Science Institute, Radboud University Nijmegen
Klaas Sijtsma: Tilburg University

Psychometrika, 2023, vol. 88, issue 2, No 2, 387-412

Abstract: Abstract The goodness-of-fit of the unidimensional monotone latent variable model can be assessed using the empirical conditions of nonnegative correlations (Mokken in A theory and procedure of scale-analysis, Mouton, The Hague, 1971), manifest monotonicity (Junker in Ann Stat 21:1359–1378, 1993), multivariate total positivity of order 2 (Bartolucci and Forcina in Ann Stat 28:1206–1218, 2000), and nonnegative partial correlations (Ellis in Psychometrika 79:303–316, 2014). We show that multidimensional monotone factor models with independent factors also imply these empirical conditions; therefore, the conditions are insensitive to multidimensionality. Conditional association (Rosenbaum in Psychometrika 49(3):425–435, 1984) can detect multidimensionality, but tests of it (De Gooijer and Yuan in Comput Stat Data Anal 55:34–44, 2011) are usually not feasible for realistic numbers of items. The only existing feasible test procedures that can reveal multidimensionality are Rosenbaum’s (Psychometrika 49(3):425–435, 1984) Case 2 and Case 5, which test the covariance of two items or two subtests conditionally on the unweighted sum of the other items. We improve this procedure by conditioning on a weighted sum of the other items. The weights are estimated in a training sample from a linear regression analysis. Simulations show that the Type I error rate is under control and that, for large samples, the power is higher if one dimension is more important than the other or if there is a third dimension. In small samples and with two equally important dimensions, using the unweighted sum yields greater power.

Keywords: unidimensional measurement; multidimensional measurement; monotone latent variable model; monotone homogeneity model; conditional association (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11336-023-09905-w

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