A Two-level Moderated Latent Variable Model with Single Level Data

Abstract

With single-level data, Yuan, Cheng and Maxwell developed a two-level regression model for more accurate moderation analysis. This article extends the two-level regression model to a two-level moderated latent variable (2MLV) model, and uses a Bayesian approach to estimate and test the moderation effects. Monte Carlo results indicate that: 1) the new method yields more accurate estimate of the interaction effect than those via the product-indicator (PI) approach and latent variable interaction (LVI) with single-level model, both are also estimated via Bayesian method; 2) the coverage rates of the credibility interval following the 2MLV model are closer to the nominal 95% than those following the other methods; 3) the test for the existence of the moderation effect is more reliable in controlling Type I errors than both PI and LVI, especially under heteroscedasticity conditions. Moreover, a more interpretable measure of effect size is developed based on the 2MLV model, which directly answers the question as to what extent a moderator can account for the change of the coefficient between the predictor and the outcome variable. A real data example illustrates the application of the new method.

Keywords:

• Moderation effect
• latent variables
• effect size
• Bayesian estimation

Article information

Conflict of interest disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant 31971029 and 31571152 from the Natural Science Foundation of China, and by Grant GJK2017015 from the National Education Examinations Authority (NEEA) under the Chinese Ministry of Education.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The authors would like to thank the Action Editor Dr. Sarah Depaoli and two anonymous reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions or the NSFC is not intended and should not be inferred.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Note that R²=1 when ${\tau }_{11}=0.$ This implies that $b_{1i}$ is not affected by any other variable after controlling Z_i, or Z_i completely explains all of the between-person variance of $b_{1i}.$ When R²=1 and ${\gamma }_{11}$ is small, it may imply that there does not exist much variation in $b_{1i},$ although the variation is completely accounted for by Z_i. A small value of ${\gamma }_{11}$ can also be due to the scale of Z_i.

2 Note that frequentist methods can provide standard errors for parameter estimates and confidence intervals for the unknown parameters. These are parallel to posterior standard deviation and credible interval of the Bayesian approach but with a different perspective.

3 As we will see, the 2MLV model performs about the same as the PI or LVI approach to moderation analysis even when both ${\gamma }_{11}=0$ and ${\tau }_{11}=0,$ which are unlikely to hold when moderation analysis is conducted empirically.

4 We run multiple tests for the number of burn-in iterations required for achieving convergence. Results indicated that 10,000 burn-in iterations are sufficient, and the results are essentially the same when doubled the number of iterations. Hence, our Bayesian results were obtained from 10,000 simulated observations after 10,000 burn-in iterations.

5 In our initial study, we thinned the chain by taking every 5^th, 10^th, and 15^th observations, respectively, and essentially the same results were obtained.

6 We also used maximum likelihood (ML) to estimate the latent interaction effect with the PI and LVI approaches, and the results indicated that ML and the Bayesian estimators perform about the same.

7 We have also studied conditions with different number of indicators, and the conclusion regarding the performances of the different methods is essentially the same.

8 While the methodology development for 2MLV, PI and LVI is based on continuous data, researchers in psychology and other social sciences tend to have data in Likert scale. Fortunately, with more than 4 categories, the bias by treating ordinal data as continuous is minimal.

9 In this article, for the purpose of illustrating the application of the 2LMV model, the five PVs are treated as observed indicators of the latent variable mathematics achievement, and the imputation errors will be combined with the measurement errors. However, it is worth noting that if the purpose is to conduct an empirical study, the suggestion of PISA data analysis manual should be followed (see OECD, 2009).

10 Currently, the Deviance Information Criterion is not available in Mplus 8.3 (See Asparouhov & Muthén, 2019).