A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models

Area of Science:

• Psychometrics
• Statistical modeling
• Latent variable analysis

Background:

• Unidimensional monotone latent variable models rely on empirical conditions like nonnegative correlations for goodness-of-fit assessment.
• Existing methods for detecting multidimensionality, such as conditional association tests, are often computationally infeasible for large item sets.

Purpose of the Study:

• To demonstrate that common goodness-of-fit conditions for unidimensional models are also met by multidimensional models with independent factors, rendering them insensitive to multidimensionality.
• To propose an improved feasible test for detecting multidimensionality by conditioning on a weighted sum of items, rather than an unweighted sum.

Main Methods:

• Theoretical analysis showing that multidimensional monotone factor models imply existing goodness-of-fit conditions.
• Development of a novel conditional association test by estimating weights from a linear regression in a training sample.
• Simulation studies to evaluate the Type I error rate and power of the proposed test under various multidimensional scenarios.

Main Results:

• The empirical conditions used for unidimensional models are insensitive to multidimensionality when factors are independent.
• The proposed weighted sum conditioning improves upon existing feasible tests (Rosenbaum's Case 2 and Case 5) for detecting multidimensionality.
• Simulations indicate controlled Type I error rates and enhanced power, particularly when dimensions have unequal importance or a third dimension exists.

Conclusions:

• Standard goodness-of-fit tests are insufficient for detecting multidimensionality in certain latent variable models.
• The improved conditional association test offers a feasible and more powerful method for identifying multidimensionality.
• The effectiveness of the weighted sum approach depends on sample size and the relative importance of the latent dimensions.