Latent Moderation Analysis: A Factor Score Approach

Abstract

Moderation analysis with latent variables is an important topic in social science research. Although different methods have been proposed for latent moderation analysis in the past three decades, these methods have weaknesses in certain circumstances. We therefore propose the factor score approach as a straightforward and implementation-friendly alternative in latent moderation analysis. This approach has several advantages over other existing methods, such as being able to test higher-order interaction models and interaction-as-outcome models. We compared the empirical performance of the factor score approach and other commonly used methods, namely the unconstrained product indicator approach and the latent moderated structural equation approach, by conducting a simulation study. Results indicated that the factor score approach worked satisfactorily under a range of model conditions. Using these results, we can offer applied researchers some practical guidelines of use for the factor score approach with regard to the subject variable (N/P) ratio and reliability level.

Keywords:

• Moderation
• interaction
• factor score

Notes

¹ In the current simulation study, we calculated the average computing time that the LMS took to test a simple latent moderation model. On average, the LMS took 45.997 seconds on an Intel Core i5-4570 computer (3.20 GHz) to estimate the effects. For comparison, we also calculated the average computing time for the FS. On average, the FS took 0.102 seconds.

² Considering a simple moderation model ($X\stackrel{\begin{array}{c}W\\ \downarrow \end{array}}{\to }Y$) in a cross-lagged panel design across two timepoints ($t1$ and $t2$), one may be interested in examining the longitudinal moderation effect across two timepoints: $Y^{t2}={\beta }_1^{t1}{X}^{t1}+{\beta }_2^{t1}{W}^{t1}+{\beta }_3^{t1}{X}^{t1}{W}^{t1}+e_{y^{t2}}$, or examining the cross-sectional moderation effect at each time point: $Y^{t1}={\beta }_1^{t1}{X}^{t1}+{\beta }_2^{t1}{W}^{t1}+{\beta }_3^{t1}{X}^{t1}{W}^{t1}+e_{y^{t1}}$,$Y^{t2}={\beta }_1^{t2}{X}^{t2}+{\beta }_2^{t2}{W}^{t2}+{\beta }_3^{t2}{X}^{t2}{W}^{t2}+e_{y^{t2}}$.

In both scenarios, the autoregressive effects must be specified for all first-order variables in a cross-lagged panel design: $W^{t2}={\alpha }_1{W}^{t1}+e_{W^{t2}}$, $X^{t2}={\alpha }_2{X}^{t1}+e_{X^{t2}}$, $Y^{t2}={\alpha }_3{Y}^{t1}+e_{Y^{t2}}$. When the autoregressive effects for the first-order variables are specified, the autoregressive effect for the product term is a natural consequence and should be specified in the model. In this specification, the product term of $X^{t2}{W}^{t2}$ is specified as an endogenous variable: $X^{t2}{W}^{t2}=({\alpha }_1{X}^{t1}+e_{W^{t1}})+({\alpha }_2{W}^{t1}+e_{X^{t2}})$, $X^{t2}{W}^{t2}={\alpha }_1{\alpha }_2{X}^{t1}{W}^{t1}+({\alpha }_1{X}^{t1}{e}_{X^{t1}}+{\alpha }_2{X}^{t1}{e}_{X^{t1}}+e_{W^{t2}}{e}_{X^{t2}}).$

Additional information

Funding

This work was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, China (Project No.: CUHK 14614719).