A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models.

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.