Standardized estimates for latent interaction effects: Method comparison and selection strategy

doi:10.3724/SP.J.1041.2022.00091

Abstract

Abstract:

Standardized estimation plays an important role in model interpretation and effect size comparison. The appropriate standardized estimation of latent interaction effects, appeared over 10 years ago, has been widely adopted. However, there is no systematic comparison of different methods in their appropriate standardized estimation. Through a simulation study in different conditions, we compared the latent interaction effects of three methods: the product indicator approach, Latent Moderated Structural Equations (LMS), Bayesian methods with or without prior information. The results showed that, when normality assumption was satisfied, LMS and the informative Bayesian method performed best in standardized estimation. When normality assumption was violated, however, the product indicator approach was more robust than the other methods, but a large sample size (not less than 500) was necessary. For small samples, when the correlation between exogenous latent variables was close to zero, the noninformative Bayesian method could be used.

Key words: latent variable, interaction effect, product indicator approach, Latent Moderated Structural Equations, Bayesian estimation, standardized estimation

WEN Zhonglin, OUYANG Jinying, FANG Junyan, LIU Xiqin. (2022). Standardized estimates for latent interaction effects: Method comparison and selection strategy. Acta Psychologica Sinica, 54(91), 91-107.

Figures/Tables 6

References 64

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N	Method	Proper solution%	M	SD	SE	SE Bias%	Type I error rate
ϕ₁₂ = 0, ξ₁ (0.001, -0.004), ξ₂ (-0.008, -0.007), ξ₁ξ₂ (-0.024, 6.163)
100	PI	72.8	-0.009	0.239	0.180	-24.5	0.069
	LMS	100.0	-0.003	0.148	0.147	-0.8	0.082
	BN	100.0	-0.005	0.146	0.147	0.6	0.060
	BI	100.0	-0.003	0.047	0.083	75.8	0.000
200	PI	92.2	0.001	0.137	0.116	-15.3	0.043
	LMS	100.0	0.005	0.105	0.100	-4.8	0.080
	BN	100.0	0.019	0.107	0.098	-8.5	0.088
	BI	100.0	0.011	0.053	0.068	28.3	0.010
500	PI	100.0	0.000	0.077	0.072	-6.6	0.052
	LMS	100.0	0.002	0.064	0.063	-1.6	0.060
	BN	100.0	0.001	0.065	0.063	-3.1	0.058
	BI	100.0	0.002	0.046	0.053	14.1	0.030
ϕ₁₂ = 0.3, ξ₁ (-0.001, 0.000), ξ₂ (-0.001, 0.001), ξ₁ξ₂ (1.619, 7.628)
100	PI	77.4	0.006	0.207	0.163	-21.4	0.059
	LMS	100.0	0.006	0.136	0.129	-4.8	0.082
	BN	100.0	0.008	0.133	0.135	1.1	0.044
	BI	100.0	0.003	0.049	0.081	65.3	0.000
200	PI	92.4	0.009	0.121	0.105	-13.1	0.045
	LMS	100.0	0.003	0.092	0.089	-3.4	0.070
	BN	100.0	0.016	0.093	0.088	-5.7	0.068
	BI	100.0	0.010	0.051	0.064	24.8	0.008
500	PI	99.6	0.004	0.064	0.063	-0.4	0.034
	LMS	100.0	0.001	0.054	0.055	1.2	0.048
	BN	100.0	0.001	0.056	0.055	-1.5	0.054
	BI	100.0	0.001	0.042	0.048	14.4	0.020
ϕ₁₂ = 0.7, ξ₁ (-0.005, 0.002), ξ₂ (-0.005, -0.002), ξ₁ξ₂ (2.673, 11.073)
100	PI	84.2	0.009	0.153	0.120	-21.9	0.076
	LMS	99.8	0.008	0.099	0.094	-4.5	0.078
	BN	96.2	0.011	0.100	0.103	3.5	0.037
	BI	100.0	0.006	0.048	0.072	49.1	0.006
200	PI	97.2	0.008	0.089	0.079	-10.6	0.045
	LMS	100.0	0.003	0.065	0.065	0.2	0.054
	BN	100.0	0.013	0.067	0.065	-2.1	0.062
	BI	100.0	0.012	0.045	0.054	20.5	0.034
500	PI	100.0	0.004	0.049	0.048	-2.3	0.046
	LMS	100.0	0.002	0.040	0.041	0.9	0.048
	BN	100.0	0.001	0.041	0.040	-2.3	0.050
	BI	100.0	0.000	0.035	0.037	7.0	0.032

N	Method	Proper solution%	M	SD	SE	SE Bias%	Type I error rate
ϕ₁₂ = 0, ξ₁ (0.001, -0.004), ξ₂ (-0.008, -0.007), ξ₁ξ₂ (-0.024, 6.163)
100	PI	72.8	-0.009	0.239	0.180	-24.5	0.069
	LMS	100.0	-0.003	0.148	0.147	-0.8	0.082
	BN	100.0	-0.005	0.146	0.147	0.6	0.060
	BI	100.0	-0.003	0.047	0.083	75.8	0.000
200	PI	92.2	0.001	0.137	0.116	-15.3	0.043
	LMS	100.0	0.005	0.105	0.100	-4.8	0.080
	BN	100.0	0.019	0.107	0.098	-8.5	0.088
	BI	100.0	0.011	0.053	0.068	28.3	0.010
500	PI	100.0	0.000	0.077	0.072	-6.6	0.052
	LMS	100.0	0.002	0.064	0.063	-1.6	0.060
	BN	100.0	0.001	0.065	0.063	-3.1	0.058
	BI	100.0	0.002	0.046	0.053	14.1	0.030
ϕ₁₂ = 0.3, ξ₁ (-0.001, 0.000), ξ₂ (-0.001, 0.001), ξ₁ξ₂ (1.619, 7.628)
100	PI	77.4	0.006	0.207	0.163	-21.4	0.059
	LMS	100.0	0.006	0.136	0.129	-4.8	0.082
	BN	100.0	0.008	0.133	0.135	1.1	0.044
	BI	100.0	0.003	0.049	0.081	65.3	0.000
200	PI	92.4	0.009	0.121	0.105	-13.1	0.045
	LMS	100.0	0.003	0.092	0.089	-3.4	0.070
	BN	100.0	0.016	0.093	0.088	-5.7	0.068
	BI	100.0	0.010	0.051	0.064	24.8	0.008
500	PI	99.6	0.004	0.064	0.063	-0.4	0.034
	LMS	100.0	0.001	0.054	0.055	1.2	0.048
	BN	100.0	0.001	0.056	0.055	-1.5	0.054
	BI	100.0	0.001	0.042	0.048	14.4	0.020
ϕ₁₂ = 0.7, ξ₁ (-0.005, 0.002), ξ₂ (-0.005, -0.002), ξ₁ξ₂ (2.673, 11.073)
100	PI	84.2	0.009	0.153	0.120	-21.9	0.076
	LMS	99.8	0.008	0.099	0.094	-4.5	0.078
	BN	96.2	0.011	0.100	0.103	3.5	0.037
	BI	100.0	0.006	0.048	0.072	49.1	0.006
200	PI	97.2	0.008	0.089	0.079	-10.6	0.045
	LMS	100.0	0.003	0.065	0.065	0.2	0.054
	BN	100.0	0.013	0.067	0.065	-2.1	0.062
	BI	100.0	0.012	0.045	0.054	20.5	0.034
500	PI	100.0	0.004	0.049	0.048	-2.3	0.046
	LMS	100.0	0.002	0.040	0.041	0.9	0.048
	BN	100.0	0.001	0.041	0.040	-2.3	0.050
	BI	100.0	0.000	0.035	0.037	7.0	0.032

N	Method	Proper solution%	M	SD	SE	SE Bias%	Type I error rate
ϕ₁₂ = 0, ξ₁ (1.141, 1.962), ξ₂ (1.152, 2.002), ξ₁ξ₂ (1.278, 20.126)
100	PI	64.6	0.031	0.239	0.198	-17.0	0.043
	LMS	99.4	0.015	0.155	0.158	2.1	0.078
	BN	100.0	0.020	0.153	0.153	-0.3	0.064
	BI	100.0	0.005	0.047	0.083	77.1	0.004
200	PI	83.2	0.009	0.157	0.130	-17.2	0.060
	LMS	100.0	0.013	0.112	0.108	-3.9	0.078
	BN	100.0	0.030	0.114	0.102	-10.1	0.096
	BI	100.0	0.017	0.054	0.068	26.5	0.034
500	PI	97.8	0.001	0.087	0.077	-11.5	0.059
	LMS	100.0	0.016	0.067	0.067	-0.9	0.092
	BN	100.0	0.017	0.069	0.064	-7.3	0.088
	BI	100.0	0.013	0.047	0.053	12.9	0.034
ϕ₁₂ = 0.3, ξ₁ (0.855, 1.121), ξ₂ (0.858, 1.161), ξ₁ξ₂ (3.200, 32.430)
100	PI	75.8	0.008	0.194	0.155	-20.0	0.069
	LMS	100.0	0.049	0.135	0.132	-2.1	0.088
	BN	100.0	0.058	0.135	0.137	1.6	0.060
	BI	100.0	0.021	0.049	0.081	64.7	0.010
200	PI	91.6	0.005	0.113	0.104	-8.2	0.057
	LMS	100.0	0.044	0.090	0.090	-0.2	0.068
	BN	100.0	0.061	0.092	0.088	-4.3	0.122
	BI	100.0	0.035	0.051	0.063	25.4	0.046
500	PI	100.0	-0.002	0.069	0.064	-7.8	0.056
	LMS	100.0	0.044	0.056	0.055	-1.3	0.122
	BN	100.0	0.046	0.057	0.055	-4.5	0.136
	BI	100.0	0.034	0.043	0.047	11.4	0.088
ϕ₁₂ = 0.7, ξ₁ (0.850, 1.075), ξ₂ (0.856, 1.110), ξ₁ξ₂ (6.275, 86.323)
100	PI	95.0	0.002	0.137	0.118	-14.0	0.072
	LMS	100.0	0.064	0.100	0.098	-2.4	0.106
	BN	97.2	0.074	0.101	0.107	5.8	0.080
	BI	100.0	0.035	0.048	0.071	48.2	0.032
200	PI	100.0	0.000	0.081	0.074	-8.8	0.058
	LMS	100.0	0.059	0.067	0.064	-3.8	0.136
	BN	99.6	0.076	0.069	0.067	-3.4	0.177
	BI	100.0	0.053	0.045	0.053	17.0	0.118
500	PI	100.0	-0.001	0.046	0.043	-6.4	0.066
	LMS	100.0	0.056	0.041	0.040	-2.5	0.272
	BN	100.0	0.057	0.042	0.040	-5.8	0.292
	BI	100.0	0.048	0.035	0.036	2.7	0.248

N	Method	Proper solution%	M	SD	SE	SE Bias%	Type I error rate
ϕ₁₂ = 0, ξ₁ (1.141, 1.962), ξ₂ (1.152, 2.002), ξ₁ξ₂ (1.278, 20.126)
100	PI	64.6	0.031	0.239	0.198	-17.0	0.043
	LMS	99.4	0.015	0.155	0.158	2.1	0.078
	BN	100.0	0.020	0.153	0.153	-0.3	0.064
	BI	100.0	0.005	0.047	0.083	77.1	0.004
200	PI	83.2	0.009	0.157	0.130	-17.2	0.060
	LMS	100.0	0.013	0.112	0.108	-3.9	0.078
	BN	100.0	0.030	0.114	0.102	-10.1	0.096
	BI	100.0	0.017	0.054	0.068	26.5	0.034
500	PI	97.8	0.001	0.087	0.077	-11.5	0.059
	LMS	100.0	0.016	0.067	0.067	-0.9	0.092
	BN	100.0	0.017	0.069	0.064	-7.3	0.088
	BI	100.0	0.013	0.047	0.053	12.9	0.034
ϕ₁₂ = 0.3, ξ₁ (0.855, 1.121), ξ₂ (0.858, 1.161), ξ₁ξ₂ (3.200, 32.430)
100	PI	75.8	0.008	0.194	0.155	-20.0	0.069
	LMS	100.0	0.049	0.135	0.132	-2.1	0.088
	BN	100.0	0.058	0.135	0.137	1.6	0.060
	BI	100.0	0.021	0.049	0.081	64.7	0.010
200	PI	91.6	0.005	0.113	0.104	-8.2	0.057
	LMS	100.0	0.044	0.090	0.090	-0.2	0.068
	BN	100.0	0.061	0.092	0.088	-4.3	0.122
	BI	100.0	0.035	0.051	0.063	25.4	0.046
500	PI	100.0	-0.002	0.069	0.064	-7.8	0.056
	LMS	100.0	0.044	0.056	0.055	-1.3	0.122
	BN	100.0	0.046	0.057	0.055	-4.5	0.136
	BI	100.0	0.034	0.043	0.047	11.4	0.088
ϕ₁₂ = 0.7, ξ₁ (0.850, 1.075), ξ₂ (0.856, 1.110), ξ₁ξ₂ (6.275, 86.323)
100	PI	95.0	0.002	0.137	0.118	-14.0	0.072
	LMS	100.0	0.064	0.100	0.098	-2.4	0.106
	BN	97.2	0.074	0.101	0.107	5.8	0.080
	BI	100.0	0.035	0.048	0.071	48.2	0.032
200	PI	100.0	0.000	0.081	0.074	-8.8	0.058
	LMS	100.0	0.059	0.067	0.064	-3.8	0.136
	BN	99.6	0.076	0.069	0.067	-3.4	0.177
	BI	100.0	0.053	0.045	0.053	17.0	0.118
500	PI	100.0	-0.001	0.046	0.043	-6.4	0.066
	LMS	100.0	0.056	0.041	0.040	-2.5	0.272
	BN	100.0	0.057	0.042	0.040	-5.8	0.292
	BI	100.0	0.048	0.035	0.036	2.7	0.248

N	Method	Proper solution%	M	M Bias%	SD	SE	SE Bias%	Statistical power
ϕ₁₂ = 0, ξ₁ (0.001, -0.004), ξ₂ (-0.008, -0.007), ξ₁ξ₂ (-0.024, 6.163)
100	PI	73.8	0.217	8.3	0.216	0.183	-15.6	0.236
	LMS	100.0	0.189	-5.7	0.148	0.144	-2.6	0.300
	BN	100.0	0.183	-8.3	0.146	0.145	-0.2	0.268
	BI	100.0	0.195	-2.6	0.051	0.084	65.0	0.766
200	PI	95.2	0.213	6.7	0.141	0.129	-8.8	0.416
	LMS	100.0	0.199	-0.5	0.104	0.098	-6.3	0.566
	BN	100.0	0.215	7.3	0.106	0.095	-10.4	0.638
	BI	100.0	0.208	4.1	0.054	0.067	24.1	0.924
500	PI	100.0	0.204	2.1	0.081	0.079	-2.4	0.774
	LMS	100.0	0.201	0.3	0.061	0.061	0.0	0.878
	BN	100.0	0.202	1.2	0.062	0.061	-1.7	0.888
	BI	100.0	0.201	0.7	0.045	0.051	13.2	0.986
ϕ₁₂ = 0.3, ξ₁ (-0.001, 0.000), ξ₂ (-0.001, 0.001), ξ₁ξ₂ (1.619, 7.628)
100	PI	81.2	0.230	15.2	0.210	0.176	-15.9	0.273
	LMS	100.0	0.198	-1.1	0.133	0.124	-7.0	0.420
	BN	100.0	0.200	-0.2	0.130	0.131	1.1	0.374
	BI	100.0	0.200	-0.2	0.051	0.081	58.5	0.778
200	PI	95.8	0.226	12.9	0.142	0.120	-15.3	0.476
	LMS	100.0	0.201	0.7	0.089	0.086	-3.4	0.626
	BN	100.0	0.215	7.6	0.089	0.085	-4.7	0.674
	BI	100.0	0.209	4.6	0.051	0.063	21.9	0.962
500	PI	99.6	0.207	3.7	0.072	0.071	-1.4	0.871
	LMS	100.0	0.201	0.7	0.054	0.054	0.1	0.954
	BN	100.0	0.204	2.1	0.055	0.053	-3.5	0.964
	BI	100.0	0.202	1.0	0.042	0.046	9.3	0.998
ϕ₁₂ = 0.7, ξ₁ (-0.005, 0.002), ξ₂ (-0.005, -0.002), ξ₁ξ₂ (2.673, 11.073)
100	PI	89.6	0.232	15.9	0.165	0.134	-18.7	0.446
	LMS	99.8	0.203	1.4	0.095	0.090	-5.4	0.623
	BN	97.0	0.206	3.2	0.094	0.099	5.4	0.555
	BI	100.0	0.202	0.9	0.048	0.070	46.2	0.906
200	PI	98.2	0.218	8.8	0.104	0.089	-14.0	0.735
	LMS	100.0	0.202	0.9	0.063	0.063	-0.7	0.878
	BN	100.0	0.214	7.1	0.063	0.063	-0.9	0.922
	BI	100.0	0.212	5.8	0.045	0.052	15.8	0.992
500	PI	100.0	0.206	3.0	0.058	0.054	-5.7	0.982
	LMS	100.0	0.201	0.6	0.039	0.039	-0.1	0.998
	BN	100.0	0.203	1.4	0.040	0.038	-5.7	1.000
	BI	100.0	0.202	0.8	0.034	0.035	3.0	1.000