Abstract:

At present, the analysis of moderating effect is mainly based on cross sectional data. This article discusses how to analyze the moderating effect with longitudinal data. If the independent variable X and the dependent variable Y are longitudinal data, longitudinal moderation models can be divided into three categories according to the type of moderator: time-invariant moderator, time-variant moderator, and moderator generated from X or Y. For example, X_tj is divided into two parts, time-varying intra-individual differences $X_{t j}-\bar{X}_{\bullet} j$ and time-invariant inter-individual differences$\bar{X}_{\boldsymbol{\bullet} j}$, and then the moderating effect of $\bar{X}_{\boldsymbol{\bullet} j}$ on the relationship between $(X_{t j}-\bar{X}_{\bullet} j)$ and Y_tj can be analyzed. In that case, there will be no new moderator Z, which is characteristic of moderation research on longitudinal data in contrast to research on cross-sectional data.

Four types of longitudinal moderation analysis approaches are summarized: 1) Multilevel model (MLM); 2) Multilevel structural equation model (MSEM); 3) Cross-lagged model (CLM); 4) Latent growth model (LGM). It is found that the decomposition of the moderating effect and the use of the latent moderating structural equation (LMS) method are the two characteristics of the moderation analysis for longitudinal data. Specifically, MLM, MSEM, and CLM divide the moderating effect of longitudinal data into three parts: the time-varying intra-individual part, time-invariant inter-individual part, and the cross-level part. In addition, the moderating effect of longitudinal data can be decomposed into the moderating effect of initial level and rate of change by LGM.

In the present study, we propose a procedure to analyze longitudinal mediation analysis. The first step is to decide whether it is necessary to make a causal inference. If the aim of research is to make a causal inference, CLM should be adopted to analyze longitudinal moderation. Otherwise, proceed with the second step. The second step is to decide whether it is necessary to treat longitudinal data as multilevel data. If longitudinal data is treated as multilevel data, MSEM should be adopted to analyze longitudinal moderation, because MSEM and MLM are more suitable for describing individual differences. Otherwise, LGM should be adopted to analyze longitudinal moderation, because only an LGM can simultaneously examine the effect of some variables on change and how the change affects other variables. The third step is to decide whether MSEM converges. If MSEM converges, the result of MSEM should be reported. Otherwise, MLM should be adopted to analyze longitudinal moderation. Compared with MLM, MSEM takes sampling error into account when the group mean is calculated, but the convergence of the MSEM is more difficult. Therefore, the MSEM with sampling error taken into account is preferred. If convergence fails, MLM will be considered.

This paper exemplifies how to conduct the proposed procedure by using Mplus. Directions for future research on moderation analysis of longitudinal data are discussed, such as the moderation analysis for intensive longitudinal data based on the dynamic structural equation model.

Key words: longitudinal data, moderation effect, multilevel model, multilevel structural equation model, cross-lagged model, latent growth model