Rank-deficiencies in a reduced information latent variable model

TL;DR

This paper investigates rank deficiencies in a specific latent variable model, demonstrating their existence, effects on convergence, and implications for estimation stability and practice.

Contribution

It explicitly characterizes the null space of rank deficiencies in the reduced-group split-ballot multitrait-multimethod model and discusses their impact on convergence and estimation.

Findings

Rank deficiencies exist and are explicitly characterized.

Sample size and distance from deficiency interact affecting convergence.

Latent variable correlations remain unaffected by rank deficiencies.

Abstract

Latent variable models are well-known to suffer from rank deficiencies, causing problems with convergence and stability. Such problems are compounded in the "reduced-group split-ballot multitrait-multimethod model", which omits a set of moments from the estimation through a planned missing data design. This paper demonstrates the existence of rank deficiencies in this model and give the explicit null space. It also demonstrates that sample size and distance from the rank-deficient point interact in their effects on convergence, causing convergence to improve or worsen depending on both factors simultaneously. Furthermore, it notes that the latent variable correlations in the uncorrelated methods SB-MTMM model remain unaffected by the rank deficiency. I conclude that methodological experiments should be careful to manipulate both distance to known rank-deficiencies and sample size, and report all results, not only the apparently converged ones. Practitioners may consider that, even in the presence of nonconvergence or so-called "inadmissible" estimates, a subset of parameter estimates may still be consistent and stable.