Exploring Factor Model Parameters across Continuous Variables with Local Structural Equation Models

ABSTRACT

Using an empirical data set, we investigated variation in factor model parameters across a continuous moderator variable and demonstrated three modeling approaches: multiple-group mean and covariance structure (MGMCS) analyses, local structural equation modeling (LSEM), and moderated factor analysis (MFA). We focused on how to study variation in factor model parameters as a function of continuous variables such as age, socioeconomic status, ability levels, acculturation, and so forth. Specifically, we formalized the LSEM approach in detail as compared with previous work and investigated its statistical properties with an analytical derivation and a simulation study. We also provide code for the easy implementation of LSEM. The illustration of methods was based on cross-sectional cognitive ability data from individuals ranging in age from 4 to 23 years. Variations in factor loadings across age were examined with regard to the age differentiation hypothesis. LSEM and MFA converged with respect to the conclusions. When there was a broad age range within groups and varying relations between the indicator variables and the common factor across age, MGMCS produced distorted parameter estimates. We discuss the pros of LSEM compared with MFA and recommend using the two tools as complementary approaches for investigating moderation in factor model parameters.

KEYWORDS:

• Local structural equation model
• moderated factor analysis
• multiple-group mean and covariance structures
• age differentiation of cognitive abilities
• WJ-III tests of cognitive abilities

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

No funding from a third party was available to complete this work.

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 thank the Woodcock-Munõz Foundation for access to the normative data from the Woodcock-Johnson III Tests of Cognitive Abilities and Kevin McGrew for help with the data acquisition procedure. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institutions is not intended and should not be inferred.

Notes

1 Note that different ways of identifying the common factor are possible—for example, by setting the loading of one indicator to 1 and specifying the intercept to be 0 (Bollen, 1989).

2 The reason for this is that as the bandwidth increases, the overlap between the highly weighted observations increases. In each model, at a focal point, the overlap between the sample and neighboring samples becomes larger, and as a consequence, the parameter estimates of the different models become more similar. If the bandwidth approximated infinity, then all observations would be fully weighted in every model. All parameter plots would consequently show straight lines because the parameter estimates would be identical in every model. By contrast, a bandwidth that approximates 0 would use a weighting function that gives full weight to observations at that particular focal point while giving all other observations a weight of 0.

3 First, from the output of the MCMC chain, a k-dimensional estimate and its covariance matrix V can be calculated. For large sample sizes and noninformative prior distributions, statistical inference that is based on the posterior distribution (obtained from the MCMC output) approximates maximum likelihood inference (Gelman et al., 2014; see also Walker, 1969). Using the asymptotic multivariate normality of , the chi-square statistic χ² = δ^TV⁻¹δ of the Wald test for the null hypothesis δ = 0 is formed; it is asymptotically χ² distributed with k degrees of freedom.