Moderated Mediation Analyses of Intensive Longitudinal Data

doi:10.3724/SP.J.1041.2025.0915

Abstract

Abstract:

Intensive longitudinal data (ILD) is increasing in fields such as psychology and management, yet research on analytical methods for ILD remains relatively scant. Traditionally, the ILD is statistically modeled as a two-level structure, with Level 1 being the time and Level 2 being individuals. Existing analytical methods treat longitudinal moderated mediation as multilevel moderated mediation, without considering the lagged relationship between variables. This paper describes in detail how to construct four intensive longitudinal moderated mediation (ILMM) models with dynamic structural equation model (DSEM). A simulation study is conducted to examine the estimation accuracy of the 1-1-1 intensive longitudinal mediation model moderated by a level 2 moderator. An example is employed to demonstrate how to conduct ILMM analysis with DSEM by Mplus. Mplus codes for analyzing all these ILMM models are provided (The complete dataset, Mplus syntax files, and analysis outputs can be downloaded at https://osf.io/e273c/).

Key words: intensive longitudinal data, moderated mediation effect, dynamic structural equation model

FANG Jie, WEN Zhonglin, WANG Huihui, GU Honglei. (2025). Moderated Mediation Analyses of Intensive Longitudinal Data. Acta Psychologica Sinica, 57(5), 915-928.

Figures/Tables 6

Figure 1. 1-1-1 intensive longitudinal moderated mediation model based on DSEM (Adapted from Fang et al., 2024)

Figure 2. 2-1-1 intensive longitudinal moderated mediation model based on DSEM.

Figure 3. 2-2-1 intensive longitudinal moderated mediation model based on DSEM.

Table 1 Summary of the intensive longitudinal moderated mediation model

moderator	Mediation model	Moderted path
moderator	Mediation model	first-stage, second-stage and direct paths	first-stage and direct paths	second-stage and direct paths
Z_j	1-1-1	Equation (1)-(7)	kZ_j in Equation 5 has to be removed	dZ_j in Equation 4 and gZ_j in Equation 5 has to be removed
	2-1-1	Equation (13)-(19)	kl_MjZ_j in Equation 16 has to be removed	dZ_j and gX_jZ_j in Equation 15 has to be removed
	2-2-1	Equation (20)-(24)	kM_jZ_j in Equation 21 has to be removed	dZ_j and gX_jZ_j in Equation 24 has to be removed
Z_tj	1-1-1	Equation (25)-(26), and (29)-(37)	$k j i n t m z t ? 1 j c$ in Equation 36 has to be removed	$d j Z (t ? 1) j c$ and $d j Z (t ? 1) j c$ in Equation 34 has to be removed

Table 1 Summary of the intensive longitudinal moderated mediation model

moderator	Mediation model	Moderted path
moderator	Mediation model	first-stage, second-stage and direct paths	first-stage and direct paths	second-stage and direct paths
Z_j	1-1-1	Equation (1)-(7)	kZ_j in Equation 5 has to be removed	dZ_j in Equation 4 and gZ_j in Equation 5 has to be removed
	2-1-1	Equation (13)-(19)	kl_MjZ_j in Equation 16 has to be removed	dZ_j and gX_jZ_j in Equation 15 has to be removed
	2-2-1	Equation (20)-(24)	kM_jZ_j in Equation 21 has to be removed	dZ_j and gX_jZ_j in Equation 24 has to be removed
Z_tj	1-1-1	Equation (25)-(26), and (29)-(37)	$k j i n t m z t ? 1 j c$ in Equation 36 has to be removed	$d j Z (t ? 1) j c$ and $d j Z (t ? 1) j c$ in Equation 34 has to be removed

Table 2 Relative Bias (%) and Mean Squared Error (×100) for Coefficient, Variance, and Covariance Estimates

parameter	N×T	100×10		100×20		100×50		200×10		200×20		200×50
parameter	True value	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE
a	0.2	1.2	0.3	0.5	0.2	2.7	0.2	0.9	0.2	0.4	0.1	1.7	0.1
b	0.2	2.9	0.3	0.6	0.2	1.5	0.2	2.4	0.2	?0.3	0.1	0.4	0.1
g	0.2	?0.1	0.3	?0.5	0.2	?0.7	0.2	0.9	0.1	?0.4	0.1	?0.3	0.1
k	0.2	0.3	0.3	?1.7	0.2	0.4	0.2	1.4	0.2	?1.2	0.1	?0.3	0.1
E(a_jb_j)	0.273	4.1	0.4	3.1	0.3	4.3	0.3	3.5	0.2	1.7	0.2	2.2	0.1
diff	0.16	0.9	0.2	?1.5	0.1	0.7	0.1	2.3	0.1	?1.1	0.1	0.2	0.1
$σ a j 2$	0.16	11.1	0.2	7.6	0.1	5.0	0.1	4.4	0.1	3.3	0.1	2.5	0.0
$σ b j 2$	0.16	16.5	0.3	9.4	0.1	8.5	0.1	10.9	0.1	4.8	0.1	4.6	0.0
$σ a j b j$	0.113	5.0	0.1	6.4	0.1	5.8	0.1	3.3	0.1	3.8	0.0	3.2	0.0
a	0.0	0.3	0.3	0.3	0.2	0.5	0.2	0.2	0.2	0.1	0.1	0.4	0.1
b	0.0	?0.1	0.3	0.3	0.2	?0.3	0.2	0.1	0.2	?0.1	0.1	?0.4	0.1
g	0.0	0.1	0.3	?0.1	0.2	?0.1	0.2	0.3	0.1	0.0	0.1	0.1	0.1
k	0.0	?0.6	0.3	?0.6	0.2	0.1	0.2	?0.6	0.2	?0.6	0.1	?0.1	0.1
E(a_jb_j)	0.0	?0.1	0.1	0.0	0.1	0.1	0.1	?0.1	0.1	0.0	0.0	0.0	0.0
$σ a j 2$	0.16	10.9	0.2	7.9	0.1	4.0	0.1	5.4	0.1	3.9	0.1	2.5	0.0
$σ b j 2$	0.16	15.6	0.3	7.9	0.1	8.2	0.1	9.7	0.1	2.6	0.1	4.4	0.0
$σ a j b j$	0.0	?0.1	0.1	0.0	0.1	0.1	0.0	?0.1	0.1	0.0	0.0	0.0	0.0

Table 2 Relative Bias (%) and Mean Squared Error (×100) for Coefficient, Variance, and Covariance Estimates

parameter	N×T	100×10		100×20		100×50		200×10		200×20		200×50
parameter	True value	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE	RB	MSE
a	0.2	1.2	0.3	0.5	0.2	2.7	0.2	0.9	0.2	0.4	0.1	1.7	0.1
b	0.2	2.9	0.3	0.6	0.2	1.5	0.2	2.4	0.2	?0.3	0.1	0.4	0.1
g	0.2	?0.1	0.3	?0.5	0.2	?0.7	0.2	0.9	0.1	?0.4	0.1	?0.3	0.1
k	0.2	0.3	0.3	?1.7	0.2	0.4	0.2	1.4	0.2	?1.2	0.1	?0.3	0.1
E(a_jb_j)	0.273	4.1	0.4	3.1	0.3	4.3	0.3	3.5	0.2	1.7	0.2	2.2	0.1
diff	0.16	0.9	0.2	?1.5	0.1	0.7	0.1	2.3	0.1	?1.1	0.1	0.2	0.1
$σ a j 2$	0.16	11.1	0.2	7.6	0.1	5.0	0.1	4.4	0.1	3.3	0.1	2.5	0.0
$σ b j 2$	0.16	16.5	0.3	9.4	0.1	8.5	0.1	10.9	0.1	4.8	0.1	4.6	0.0
$σ a j b j$	0.113	5.0	0.1	6.4	0.1	5.8	0.1	3.3	0.1	3.8	0.0	3.2	0.0
a	0.0	0.3	0.3	0.3	0.2	0.5	0.2	0.2	0.2	0.1	0.1	0.4	0.1
b	0.0	?0.1	0.3	0.3	0.2	?0.3	0.2	0.1	0.2	?0.1	0.1	?0.4	0.1
g	0.0	0.1	0.3	?0.1	0.2	?0.1	0.2	0.3	0.1	0.0	0.1	0.1	0.1
k	0.0	?0.6	0.3	?0.6	0.2	0.1	0.2	?0.6	0.2	?0.6	0.1	?0.1	0.1
E(a_jb_j)	0.0	?0.1	0.1	0.0	0.1	0.1	0.1	?0.1	0.1	0.0	0.0	0.0	0.0
$σ a j 2$	0.16	10.9	0.2	7.9	0.1	4.0	0.1	5.4	0.1	3.9	0.1	2.5	0.0
$σ b j 2$	0.16	15.6	0.3	7.9	0.1	8.2	0.1	9.7	0.1	2.6	0.1	4.4	0.0
$σ a j b j$	0.0	?0.1	0.1	0.0	0.1	0.1	0.0	?0.1	0.1	0.0	0.0	0.0	0.0

Table 3 95% Credible Intervals Coverage Rates, Power, and Type I Error Rate for Coefficient Estimates

Coefficient	N×T	100×10		100×20		100×50		200×10		200×20		200×50
Coefficient	True value	Cov	Pow	Cov	Pow	Cov	Pow	Cov	Pow	Cov	Pow	Cov	Pow
a	0.2	0.945	0.925	0.96	0.99	0.965	0.995	0.945	1	0.94	1	0.95	1
b	0.2	0.975	0.950	0.95	0.99	0.955	1	0.94	1	0.945	1	0.95	1
g	0.2	0.950	0.945	0.945	0.975	0.97	1	0.965	1	0.96	1	0.965	1
k	0.2	0.96	0.94	0.93	0.96	0.965	0.995	0.955	1	0.94	1	0.95	1
E(a_jb_j)	0.273	0.955	1	0.955	1	0.97	1	0.96	1	0.93	1	0.95	1
diff	0.16	0.935	0.945	0.945	0.985	0.97	1	0.975	1	0.935	1	0.955	1
Coefficient	N×T True value	100×10		100×20		100×50		200×10		200×20		200×50
Coefficient	N×T True value	Cov	Error	Cov	Error	Cov	Error	Cov	Error	Cov	Error	Cov	Error
a	0.0	0.949	0.051	0.95	0.05	0.965	0.035	0.939	0.061	0.935	0.065	0.965	0.045
b	0.0	0.96	0.04	0.94	0.06	0.975	0.025	0.949	0.051	0.96	0.04	0.955	0.035
g	0.0	0.949	0.051	0.945	0.055	0.97	0.03	0.965	0.035	0.95	0.05	0.955	0.045
k	0.0	0.934	0.066	0.955	0.045	0.96	0.04	0.955	0.045	0.965	0.035	0.945	0.055
E(a_jb_j)	0.0	0.97	0.03	0.95	0.05	0.965	0.035	0.944	0.056	0.945	0.055	0.945	0.055

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