---
author:
- DL Oberski
bibliography:
- 'identification.bib'
title: 'Rank-deficiencies in a reduced information latent variable model'
---

In a wide variety of fields, different data sources inform on the same phenomenon, the problem being to determine how these different sources should be combined, and how validly each measures the phenomenon of interest. For example, in official statistics, contradictory administrative registers and surveys may be available on citizens’ employment contracts [@oberski_evaluating_2017; @pankowska_reconciliation_2018]; in family sociology, reports from different family members may not always match up [@kenny_dyadic_2006]; and in medicine, a hospital may have data on patients’ condition from electrocardiograms, echocardiograms, radiological examinations and individual laboratory measurements simultaneously [@sammani_unravel:_2019]. In all such cases, latent variable models [@bartholomew_latent_2011] can prove powerful tools to combine different data sources measuring the same phenomenon in a principled manner [@hand_statistical_2018; @oberski2018research].

A particularly useful approach is the “multitrait-multimethod” design, which was introduced by @campbell_convergent_1959 to measure a single phenomenon (“trait”) using different data sources (“methods”), and to evaluate the sources’ validity as measures of their underlying “traits”. To analyze the resulting data, MTMM factor models were developed by @browne1984decomposition [@widaman_hierarchically_1985; @cudeck_msultiplicative_1988; @millsap_statistical_1995; @wothke1995covariance], and @eid_multitrait-multimethod_2000. Extensions to nonlinear and nonnormal latent variable models were recently developed 