Latent Variable Interactions With Ordered-Categorical Indicators: Comparisons of Unconstrained Product Indicator and Latent Moderated Structural Equations Approaches

Oct 15, 2026

Methods to handle ordered-categorical indicators in latent variable interactions have been developed, yet they have not been widely applied. This article compares two latent variable interaction modeling approaches—unconstrained product indicator (UPI) and latent moderated structural equations (LMS)—in the presence of ordered-categorical indicators. A simulation study varies sample size, indicator distributions, and category conditions, and evaluates four UPI product-indicator creation strategies. Results favor parceling for UPI or LMS under symmetric categorical distributions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages31

Reading Time1 h 18 min

AccessFree to download

Abstract

Introduction and Motivation

Prevalence in educational and psychological research

Handling categorical indicators in latent interactions

Literature Review of UPI and LMS Usage

Methodological Framework

Latent interaction structural model

Factor model for latent variables

Simulation Study Design and Strategies

Latent Variable Interactions With Ordered-Categorical Indicators: Comparisons of Unconstrained Product Indicator and Latent Moderated Structural Equations Approaches Ezgi Ayt¨urk 1, Heining Cham 1, Patricia A. Jennings2 and Joshua L. Brown 1 Educational and Psychological Measurement 2020, Vol. 80(2) 262–292 􀀂 The Author(s) 2019 Article reuse guidelines: [sagepub.com/journals-permissions](sagepub.com/journals-permissions)[ ](sagepub.com/journals-permissions)[DOI: 10.1177/0013164419865017](DOI: 10.1177/0013164419865017)[ ](DOI: 10.1177/0013164419865017)[journals.sagepub.com/home/epm](journals.sagepub.com/home/epm) Abstract Methods to handle ordered-categorical indicators in latent variable interactions have been developed, yet they have not been widely applied. This article compares the performance of two popular latent variable interaction modeling approaches in handling ordered-categorical indicators: unconstrained product indicator (UPI) and latent moderated structural equations (LMS). We conducted a simulation study across sample sizes, indicators’ distributions and category conditions. We also studied four strategies to create sets of product indicators for UPI. Results supported using a parceling strategy to create product indicators in the UPI approach or using the LMS approach when the categorical indicators are symmetrically distributed. We applied these models to study the interaction effect between third-to fifth-grade students’social skills improvement and teacher–student closeness on their state English language arts test scores. Keywords latent interaction, product indicator, categorical data, parceling 1 Fordham University, Bronx, NY, USA 2University of Virginia, Charlottesville, VA, USA Corresponding Author: Ezgi Aytrk, Department of Psychology, Fordham University, 441 East Fordham Road, 226 Dealy Hall, Bronx, NY 10458, USA. Email: [eayturk@fordham.edu](eayturk@fordham.edu) Interaction (moderation) effects, when the effects of predictors on outcomes are dependent on moderators, are commonly studied in educational and psychological research. For instance, West, Aiken, Cham, and Liu (2013) found that more than 60% of the published articles in two abnormal and clinical psychology journals have tested at least one interaction effect. Latent variable interaction modeling approaches have been developed when the predictor or moderator are best represented by confirmatory factor models. Among them, the unconstrained product indicator (UPI; Marsh, Wen, & Hau, 2004) and latent moderated structural equations (LMS; Klein & Moosbrugger, 2000; Schermelleh-Engel, Klein, & Moosbrugger, 1998) are popular most likely because of their availability in software packages. However, these approaches assume that the indicators of the predictor and the moderator are multivariate normally distributed and are measured on continuous scales. In practice, researchers often work with ordered-categorical variables measured on Likert-type scales (e.g., strongly disagree, disagree, agree, strongly agree) or binary responses (e.g., true/false) in psychological or educational measures. Although there are methods developed specifically to handle ordered-categorical variables in latent variable interaction modeling (e.g., Lee, Song, & Cai, 2010; Rizopoulos & Moustaki, 2008; Song & Lee, 2005), UPI and LMS still appear to be the most popular methods. According to Google Scholar, the original articles of the UPI (Marsh et al., 2004) and LMS (Klein & Moosbrugger, 2000) had 255 and 226 citations between January 1, 2017 and February 16, 2019, while the articles of methods that are developed to handle ordered-categorical variables had less than 35 citations each since their publication. We conducted a literature review to further understand how applied researchers used UPI and LMS. We used Google Scholar to search for research articles which cited the original articles of the UPI (Marsh et al., 2004) and LMS (Klein & Moosbru

FAQs

Why are UPI and LMS challenging with ordered-categorical indicators?

They rely on distributional assumptions and continuous measurement, while many real applications use ordered-categorical (Likert-type) or binary responses. The study focuses on how well these approaches perform when those assumptions are violated.

What does the article compare between UPI and LMS?

It compares the performance of unconstrained product indicator (UPI) and latent moderated structural equations (LMS) for modeling latent variable interactions when indicators are ordered-categorical, including the arrangement of interaction terms for UPI.

What simulation results does the paper report?

Findings support using a parceling strategy to create product indicators for UPI, and using the LMS approach when categorical indicators are symmetrically distributed.

Oct 15, 2026

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