Data analysis and sample size planning for intensive longitudinal intervention studies using dynamic structural equation modeling

doi:10.3724/SP.J.1041.2026.0773

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LIU Yue, HE Yueling, LIU Hongyun. (2026). Data analysis and sample size planning for intensive longitudinal intervention studies using dynamic structural equation modeling. Acta Psychologica Sinica, 58(4), 773-792.

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URL: https://journal.psych.ac.cn/acps/EN/10.3724/SP.J.1041.2026.0773

https://journal.psych.ac.cn/acps/EN/Y2026/V58/I4/773

Figures/Tables 8

Table 1 Summary of Empirical Studies on Intensive Longitudinal Interventions in Psychology (2015-2025, 36 Studies)

Category	Method	Number of Studies (Percentage)
Experimental Design	Single-Arm Trial (SAT)	9(25.0%)
	Randomized Controlled Trial (RCT)	25(69.4%)
	Microrandomized Trial	2(5.6%)
Statistical Modeling	Qualitative Analysis	1(2.8%)
	Descriptive Analysis	2(5.5%)
	ANOVA, t-test, Chi-square test, Regression Analysis	10(27.8%)
	Thematic Analysis	1(2.8%)
	Linear Mixed Effects Model	20(55.6%)
	Dynamic Structural Equation Modeling (DSEM)	2(5.5%)
Sample Size Determination Method	Power Analysis	11(30.6%)
	Based on Previous or Pilot Studies	6(16.6%)
	Based on Practical Resource Considerations	1(2.8%)
	Not Mentioned	18(50.0%)

Table 1 Summary of Empirical Studies on Intensive Longitudinal Interventions in Psychology (2015-2025, 36 Studies)

Category	Method	Number of Studies (Percentage)
Experimental Design	Single-Arm Trial (SAT)	9(25.0%)
	Randomized Controlled Trial (RCT)	25(69.4%)
	Microrandomized Trial	2(5.6%)
Statistical Modeling	Qualitative Analysis	1(2.8%)
	Descriptive Analysis	2(5.5%)
	ANOVA, t-test, Chi-square test, Regression Analysis	10(27.8%)
	Thematic Analysis	1(2.8%)
	Linear Mixed Effects Model	20(55.6%)
	Dynamic Structural Equation Modeling (DSEM)	2(5.5%)
Sample Size Determination Method	Power Analysis	11(30.6%)
	Based on Previous or Pilot Studies	6(16.6%)
	Based on Practical Resource Considerations	1(2.8%)
	Not Mentioned	18(50.0%)

Table 2 Key DSEM-Based Intervention Effects

Effect dimension	Parameter	Interpretation
Mean level	$\Delta \mu $	Change in average outcome level attributable to intervention
Autoregression	$\Delta \varphi $	Change in temporal dependency (e.g., inertia)
Intra-individual variability	$\Delta \text{log(}\sigma _{i}^{2}\text{)}$	Change in within-person variability

Table 2 Key DSEM-Based Intervention Effects

Effect dimension	Parameter	Interpretation
Mean level	$\Delta \mu $	Change in average outcome level attributable to intervention
Autoregression	$\Delta \varphi $	Change in temporal dependency (e.g., inertia)
Intra-individual variability	$\Delta \text{log(}\sigma _{i}^{2}\text{)}$	Change in within-person variability

Table 3 Overview of Simulation Conditions

Component	Specification
Study design	Single-arm trial (SAT); Randomized controlled trial (RCT)
Sample size (N)	Multiple levels representing typical ILI studies
Measurement occasions (T)	Multiple levels reflecting intensive measurement
Baseline mean structure	Fixed across conditions
Intervention effect on mean	Varying effect size levels
Baseline autoregression	Low to high temporal dependency
Intervention effect on autoregression	Present vs. absent
Baseline residual variance	Fixed
Intervention effect on variablity	Decrease, no change, increase
Estimation	Bayesian DSEM

Table 3 Overview of Simulation Conditions

Component	Specification
Study design	Single-arm trial (SAT); Randomized controlled trial (RCT)
Sample size (N)	Multiple levels representing typical ILI studies
Measurement occasions (T)	Multiple levels reflecting intensive measurement
Baseline mean structure	Fixed across conditions
Intervention effect on mean	Varying effect size levels
Baseline autoregression	Low to high temporal dependency
Intervention effect on autoregression	Present vs. absent
Baseline residual variance	Fixed
Intervention effect on variablity	Decrease, no change, increase
Estimation	Bayesian DSEM

Figure 1. Contour plot of the credible interval width for ${{\gamma }_{03}}$ under a balanced single-arm design.

Figure 1. Contour plot of the credible interval width for ${{\gamma }_{03}}$ under a balanced single-arm design.

Figure 2. Contour plot of the credible interval width for ${{\gamma }_{03}}$ under a balanced randomized controlled trials design.
Note. Shaded regions indicate conditions where statistical power exceeds 0.80.

Figure 2. Contour plot of the credible interval width for ${{\gamma }_{03}}$ under a balanced randomized controlled trials design.Note. Shaded regions indicate conditions where statistical power exceeds 0.80.

Table 4 Statistical power, relative bias, and credible interval width under a balanced single-arm design

N	T	power			rbias			width
N	T	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$
30	20	0.088	0.118	0.080	?0.044	?0.202	?0.256	0.534	0.397	0.547
	40	0.178	0.250	0.180	0.024	?0.143	?0.086	0.385	0.270	0.379
	80	0.358	0.608	0.336	?0.024	?0.019	?0.048	0.271	0.191	0.270
	120	0.474	0.766	0.480	?0.001	?0.039	?0.039	0.225	0.158	0.222
	160	0.624	0.854	0.564	?0.013	?0.029	?0.037	0.197	0.141	0.196
	200	0.704	0.936	0.658	?0.012	?0.006	?0.029	0.181	0.131	0.178
60	20	0.200	0.262	0.126	?0.027	?0.163	?0.231	0.358	0.270	0.371
	40	0.400	0.552	0.328	0.026	?0.115	?0.080	0.255	0.183	0.257
	80	0.668	0.928	0.666	?0.032	?0.002	?0.017	0.180	0.130	0.183
	120	0.850	0.986	0.836	0.012	?0.018	?0.024	0.149	0.107	0.150
	160	0.922	0.996	0.912	?0.016	?0.020	?0.024	0.131	0.095	0.132
	200	0.966	1.000	0.966	?0.010	?0.010	?0.010	0.120	0.089	0.120
100	20	0.344	0.474	0.202	?0.017	?0.143	?0.186	0.271	0.205	0.283
	40	0.660	0.848	0.538	0.018	?0.077	?0.074	0.193	0.139	0.196
	80	0.888	0.996	0.882	?0.018	?0.014	0.009	0.135	0.098	0.139
	120	0.966	1.000	0.964	?0.013	?0.010	?0.004	0.113	0.082	0.114
	160	0.996	1.000	0.996	?0.002	?0.011	?0.013	0.099	0.073	0.100
	200	1.000	1.000	0.998	?0.013	?0.005	?0.010	0.091	0.067	0.091
150	20	0.560	0.652	0.364	0.011	?0.117	?0.155	0.219	0.166	0.228
	40	0.806	0.960	0.750	0.006	?0.063	?0.042	0.154	0.112	0.158
	80	0.972	0.992	0.974	?0.009	?0.024	0.007	0.109	0.079	0.112
	120	0.998	0.998	0.996	0.002	?0.020	?0.001	0.090	0.067	0.092
	160	1.000	1.000	1.000	?0.014	?0.020	0.004	0.080	0.060	0.081
	200	1.000	1.000	1.000	0.007	?0.012	0.001	0.073	0.055	0.074
200	20	0.652	0.808	0.496	0.001	?0.096	?0.125	0.188	0.142	0.196
	40	0.898	0.986	0.872	?0.010	?0.048	?0.033	0.133	0.097	0.137
	80	0.996	1.000	0.994	?0.012	?0.008	?0.007	0.094	0.069	0.096
	120	0.998	1.000	1.000	?0.006	?0.005	?0.002	0.078	0.057	0.079
	160	1.000	1.000	1.000	?0.011	?0.002	?0.010	0.069	0.051	0.070
	200	1.000	1.000	1.000	?0.011	0.000	?0.003	0.063	0.047	0.063
300	20	0.812	0.940	0.690	?0.023	?0.066	?0.103	0.153	0.116	0.160
	40	0.974	0.998	0.972	?0.013	?0.037	?0.039	0.108	0.079	0.111
	80	1.000	1.000	1.000	?0.015	0.000	0.004	0.076	0.056	0.078
	120	1.000	1.000	1.000	?0.010	?0.002	0.004	0.063	0.047	0.064
	160	1.000	1.000	1.000	?0.011	0.002	?0.010	0.056	0.042	0.057
	200	1.000	1.000	1.000	?0.006	?0.002	?0.005	0.051	0.038	0.052
400	20	0.926	0.982	0.846	?0.013	?0.051	?0.079	0.131	0.100	0.137
	40	0.998	0.998	0.992	?0.012	?0.028	?0.036	0.092	0.068	0.096
	80	1.000	1.000	1.000	?0.010	0.001	0.006	0.066	0.048	0.067
	120	1.000	1.000	1.000	?0.006	0.000	0.001	0.055	0.040	0.056
	160	1.000	1.000	1.000	?0.010	?0.001	?0.009	0.048	0.036	0.049
	200	1.000	1.000	1.000	?0.003	?0.001	?0.002	0.044	0.033	0.045

Table 4 Statistical power, relative bias, and credible interval width under a balanced single-arm design

N	T	power			rbias			width
N	T	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$	${{\gamma }_{30}}$	${{\gamma }_{40}}$	${{\omega }_{50}}$
30	20	0.088	0.118	0.080	?0.044	?0.202	?0.256	0.534	0.397	0.547
	40	0.178	0.250	0.180	0.024	?0.143	?0.086	0.385	0.270	0.379
	80	0.358	0.608	0.336	?0.024	?0.019	?0.048	0.271	0.191	0.270
	120	0.474	0.766	0.480	?0.001	?0.039	?0.039	0.225	0.158	0.222
	160	0.624	0.854	0.564	?0.013	?0.029	?0.037	0.197	0.141	0.196
	200	0.704	0.936	0.658	?0.012	?0.006	?0.029	0.181	0.131	0.178
60	20	0.200	0.262	0.126	?0.027	?0.163	?0.231	0.358	0.270	0.371
	40	0.400	0.552	0.328	0.026	?0.115	?0.080	0.255	0.183	0.257
	80	0.668	0.928	0.666	?0.032	?0.002	?0.017	0.180	0.130	0.183
	120	0.850	0.986	0.836	0.012	?0.018	?0.024	0.149	0.107	0.150
	160	0.922	0.996	0.912	?0.016	?0.020	?0.024	0.131	0.095	0.132
	200	0.966	1.000	0.966	?0.010	?0.010	?0.010	0.120	0.089	0.120
100	20	0.344	0.474	0.202	?0.017	?0.143	?0.186	0.271	0.205	0.283
	40	0.660	0.848	0.538	0.018	?0.077	?0.074	0.193	0.139	0.196
	80	0.888	0.996	0.882	?0.018	?0.014	0.009	0.135	0.098	0.139
	120	0.966	1.000	0.964	?0.013	?0.010	?0.004	0.113	0.082	0.114
	160	0.996	1.000	0.996	?0.002	?0.011	?0.013	0.099	0.073	0.100
	200	1.000	1.000	0.998	?0.013	?0.005	?0.010	0.091	0.067	0.091
150	20	0.560	0.652	0.364	0.011	?0.117	?0.155	0.219	0.166	0.228
	40	0.806	0.960	0.750	0.006	?0.063	?0.042	0.154	0.112	0.158
	80	0.972	0.992	0.974	?0.009	?0.024	0.007	0.109	0.079	0.112
	120	0.998	0.998	0.996	0.002	?0.020	?0.001	0.090	0.067	0.092
	160	1.000	1.000	1.000	?0.014	?0.020	0.004	0.080	0.060	0.081
	200	1.000	1.000	1.000	0.007	?0.012	0.001	0.073	0.055	0.074
200	20	0.652	0.808	0.496	0.001	?0.096	?0.125	0.188	0.142	0.196
	40	0.898	0.986	0.872	?0.010	?0.048	?0.033	0.133	0.097	0.137
	80	0.996	1.000	0.994	?0.012	?0.008	?0.007	0.094	0.069	0.096
	120	0.998	1.000	1.000	?0.006	?0.005	?0.002	0.078	0.057	0.079
	160	1.000	1.000	1.000	?0.011	?0.002	?0.010	0.069	0.051	0.070
	200	1.000	1.000	1.000	?0.011	0.000	?0.003	0.063	0.047	0.063
300	20	0.812	0.940	0.690	?0.023	?0.066	?0.103	0.153	0.116	0.160
	40	0.974	0.998	0.972	?0.013	?0.037	?0.039	0.108	0.079	0.111
	80	1.000	1.000	1.000	?0.015	0.000	0.004	0.076	0.056	0.078
	120	1.000	1.000	1.000	?0.010	?0.002	0.004	0.063	0.047	0.064
	160	1.000	1.000	1.000	?0.011	0.002	?0.010	0.056	0.042	0.057
	200	1.000	1.000	1.000	?0.006	?0.002	?0.005	0.051	0.038	0.052
400	20	0.926	0.982	0.846	?0.013	?0.051	?0.079	0.131	0.100	0.137
	40	0.998	0.998	0.992	?0.012	?0.028	?0.036	0.092	0.068	0.096
	80	1.000	1.000	1.000	?0.010	0.001	0.006	0.066	0.048	0.067
	120	1.000	1.000	1.000	?0.006	0.000	0.001	0.055	0.040	0.056
	160	1.000	1.000	1.000	?0.010	?0.001	?0.009	0.048	0.036	0.049
	200	1.000	1.000	1.000	?0.003	?0.001	?0.002	0.044	0.033	0.045

Table 5 Statistical power, relative bias, and credible interval width under a balanced randomized controlled design

N	T	power			rbias			width
N	T	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$
30	20	0.038	0.070	0.048	?0.134	0.017	0.004	0.933	0.676	0.932
	40	0.080	0.172	0.108	?0.013	0.014	0.004	0.679	0.482	0.662
	80	0.150	0.302	0.164	?0.035	0.003	?0.019	0.498	0.364	0.492
	120	0.242	0.366	0.216	0.009	?0.040	?0.003	0.425	0.309	0.421
	160	0.316	0.492	0.278	0.042	0.005	?0.047	0.374	0.281	0.373
	200	0.326	0.550	0.344	0.044	?0.016	0.020	0.345	0.261	0.345
60	20	0.100	0.212	0.136	?0.103	0.054	0.053	0.598	0.441	0.598
	40	0.218	0.432	0.232	0.016	0.030	?0.010	0.435	0.319	0.433
	80	0.350	0.648	0.376	?0.047	0.007	?0.013	0.324	0.241	0.323
	120	0.511	0.781	0.513	?0.025	0.015	0.033	0.273	0.207	0.276
	160	0.625	0.826	0.582	0.026	?0.003	?0.040	0.246	0.187	0.247
	200	0.706	0.890	0.700	0.005	?0.003	0.008	0.226	0.174	0.227
100	20	0.218	0.398	0.224	?0.037	0.054	0.024	0.444	0.329	0.445
	40	0.380	0.664	0.382	0.005	0.029	0.001	0.325	0.240	0.323
	80	0.594	0.878	0.624	?0.044	0.007	?0.002	0.242	0.181	0.243
	120	0.760	0.940	0.782	?0.029	0.005	0.019	0.205	0.156	0.208
	160	0.859	0.975	0.832	0.002	0.002	?0.019	0.184	0.141	0.186
	200	0.921	0.992	0.902	?0.005	?0.009	0.013	0.170	0.131	0.171
150	20	0.308	0.584	0.344	?0.057	0.029	0.033	0.355	0.264	0.356
	40	0.516	0.850	0.562	?0.023	0.036	0.024	0.261	0.192	0.260
	80	0.788	0.974	0.798	?0.042	?0.001	0.004	0.194	0.146	0.195
	120	0.902	0.988	0.926	0.007	?0.021	0.026	0.165	0.126	0.167
	160	0.962	0.996	0.956	?0.002	0.011	?0.018	0.147	0.115	0.149
	200	0.986	1.000	0.988	0.019	?0.005	0.020	0.136	0.107	0.138
200	20	0.376	0.692	0.444	?0.042	0.000	0.050	0.304	0.226	0.305
	40	0.710	0.926	0.726	?0.012	0.031	0.029	0.223	0.165	0.224
	80	0.912	0.992	0.936	?0.032	0.002	0.017	0.166	0.125	0.168
	120	0.978	1.000	0.968	0.006	?0.016	0.023	0.141	0.109	0.144
	160	0.990	1.000	0.990	?0.001	0.006	?0.003	0.126	0.099	0.129
	200	0.998	1.000	0.998	0.010	?0.003	0.022	0.117	0.092	0.119
300	20	0.560	0.874	0.636	?0.053	0.030	0.031	0.245	0.182	0.246
	40	0.860	0.994	0.884	?0.031	0.011	0.008	0.180	0.134	0.180
	80	0.982	1.000	0.984	0.004	0.012	?0.006	0.135	0.102	0.135
	120	0.998	1.000	0.994	0.004	?0.012	0.015	0.114	0.088	0.117
	160	1.000	1.000	1.000	?0.001	?0.004	?0.001	0.102	0.080	0.104
	200	1.000	1.000	1.000	0.000	?0.001	0.003	0.095	0.075	0.096
400	20	0.688	0.942	0.778	?0.049	0.040	0.038	0.211	0.156	0.212
	40	0.944	0.996	0.962	?0.020	0.026	0.034	0.155	0.115	0.156
	80	0.994	1.000	1.000	?0.013	?0.002	0.018	0.116	0.088	0.117
	120	1.000	1.000	1.000	0.000	?0.008	0.013	0.099	0.076	0.101
	160	1.000	1.000	1.000	0.000	?0.003	0.001	0.088	0.069	0.090
	200	1.000	1.000	1.000	0.005	?0.003	0.018	0.082	0.065	0.083

Table 5 Statistical power, relative bias, and credible interval width under a balanced randomized controlled design

N	T	power			rbias			width
N	T	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$	${{\gamma }_{03}}$	${{\gamma }_{41}}$	${{\omega }_{05}}$
30	20	0.038	0.070	0.048	?0.134	0.017	0.004	0.933	0.676	0.932
	40	0.080	0.172	0.108	?0.013	0.014	0.004	0.679	0.482	0.662
	80	0.150	0.302	0.164	?0.035	0.003	?0.019	0.498	0.364	0.492
	120	0.242	0.366	0.216	0.009	?0.040	?0.003	0.425	0.309	0.421
	160	0.316	0.492	0.278	0.042	0.005	?0.047	0.374	0.281	0.373
	200	0.326	0.550	0.344	0.044	?0.016	0.020	0.345	0.261	0.345
60	20	0.100	0.212	0.136	?0.103	0.054	0.053	0.598	0.441	0.598
	40	0.218	0.432	0.232	0.016	0.030	?0.010	0.435	0.319	0.433
	80	0.350	0.648	0.376	?0.047	0.007	?0.013	0.324	0.241	0.323
	120	0.511	0.781	0.513	?0.025	0.015	0.033	0.273	0.207	0.276
	160	0.625	0.826	0.582	0.026	?0.003	?0.040	0.246	0.187	0.247
	200	0.706	0.890	0.700	0.005	?0.003	0.008	0.226	0.174	0.227
100	20	0.218	0.398	0.224	?0.037	0.054	0.024	0.444	0.329	0.445
	40	0.380	0.664	0.382	0.005	0.029	0.001	0.325	0.240	0.323
	80	0.594	0.878	0.624	?0.044	0.007	?0.002	0.242	0.181	0.243
	120	0.760	0.940	0.782	?0.029	0.005	0.019	0.205	0.156	0.208
	160	0.859	0.975	0.832	0.002	0.002	?0.019	0.184	0.141	0.186
	200	0.921	0.992	0.902	?0.005	?0.009	0.013	0.170	0.131	0.171
150	20	0.308	0.584	0.344	?0.057	0.029	0.033	0.355	0.264	0.356
	40	0.516	0.850	0.562	?0.023	0.036	0.024	0.261	0.192	0.260
	80	0.788	0.974	0.798	?0.042	?0.001	0.004	0.194	0.146	0.195
	120	0.902	0.988	0.926	0.007	?0.021	0.026	0.165	0.126	0.167
	160	0.962	0.996	0.956	?0.002	0.011	?0.018	0.147	0.115	0.149
	200	0.986	1.000	0.988	0.019	?0.005	0.020	0.136	0.107	0.138
200	20	0.376	0.692	0.444	?0.042	0.000	0.050	0.304	0.226	0.305
	40	0.710	0.926	0.726	?0.012	0.031	0.029	0.223	0.165	0.224
	80	0.912	0.992	0.936	?0.032	0.002	0.017	0.166	0.125	0.168
	120	0.978	1.000	0.968	0.006	?0.016	0.023	0.141	0.109	0.144
	160	0.990	1.000	0.990	?0.001	0.006	?0.003	0.126	0.099	0.129
	200	0.998	1.000	0.998	0.010	?0.003	0.022	0.117	0.092	0.119
300	20	0.560	0.874	0.636	?0.053	0.030	0.031	0.245	0.182	0.246
	40	0.860	0.994	0.884	?0.031	0.011	0.008	0.180	0.134	0.180
	80	0.982	1.000	0.984	0.004	0.012	?0.006	0.135	0.102	0.135
	120	0.998	1.000	0.994	0.004	?0.012	0.015	0.114	0.088	0.117
	160	1.000	1.000	1.000	?0.001	?0.004	?0.001	0.102	0.080	0.104
	200	1.000	1.000	1.000	0.000	?0.001	0.003	0.095	0.075	0.096
400	20	0.688	0.942	0.778	?0.049	0.040	0.038	0.211	0.156	0.212
	40	0.944	0.996	0.962	?0.020	0.026	0.034	0.155	0.115	0.156
	80	0.994	1.000	1.000	?0.013	?0.002	0.018	0.116	0.088	0.117
	120	1.000	1.000	1.000	0.000	?0.008	0.013	0.099	0.076	0.101
	160	1.000	1.000	1.000	0.000	?0.003	0.001	0.088	0.069	0.090
	200	1.000	1.000	1.000	0.005	?0.003	0.018	0.082	0.065	0.083

Table 6 Type I error in two designs

N	T	Effect size=0.1						Effect size=0.2
		mean		autoregression		IIV		mean		autoregression		IIV
		SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT
60(30)	40	0.024	0.022	0.030	0.022	0.032	0.034	0.044	0.022	0.050	0.022	0.050	0.034
60(30)	80	0.046	0.048	0.030	0.024	0.036	0.036	0.088	0.040	0.102	0.032	0.054	0.030
60(30)	120	0.048	0.044	0.068	0.048	0.052	0.036	0.088	0.040	0.170	0.038	0.096	0.032
60(30)	160	0.034	0.028	0.066	0.036	0.046	0.046	0.154	0.044	0.196	0.036	0.120	0.042
60(30)	200	0.076	0.040	0.086	0.068	0.072	0.058	0.152	0.050	0.216	0.040	0.148	0.038
100(50)	40	0.056	0.034	0.034	0.036	0.044	0.048	0.102	0.032	0.094	0.038	0.060	0.046
100(50)	80	0.070	0.056	0.062	0.036	0.038	0.042	0.136	0.036	0.190	0.050	0.106	0.034
100(50)	120	0.056	0.060	0.090	0.050	0.078	0.034	0.184	0.042	0.290	0.056	0.166	0.034
100(50)	160	0.070	0.042	0.132	0.058	0.074	0.058	0.244	0.036	0.306	0.038	0.214	0.040
100(50)	200	0.104	0.046	0.118	0.052	0.068	0.038	0.284	0.054	0.392	0.048	0.246	0.044
200(100)	40	0.088	0.044	0.080	0.044	0.042	0.064	0.180	0.044	0.208	0.038	0.092	0.052
200(100)	80	0.086	0.048	0.106	0.044	0.072	0.048	0.294	0.062	0.400	0.048	0.202	0.040
200(100)	120	0.114	0.064	0.178	0.046	0.092	0.060	0.380	0.050	0.518	0.048	0.326	0.054
200(100)	160	0.146	0.048	0.218	0.046	0.134	0.042	0.486	0.044	0.624	0.076	0.404	0.044
200(100)	200	0.178	0.048	0.234	0.070	0.146	0.050	0.522	0.040	0.710	0.040	0.504	0.038

Table 6 Type I error in two designs

N	T	Effect size=0.1						Effect size=0.2
		mean		autoregression		IIV		mean		autoregression		IIV
		SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT	SAT	RCT
60(30)	40	0.024	0.022	0.030	0.022	0.032	0.034	0.044	0.022	0.050	0.022	0.050	0.034
60(30)	80	0.046	0.048	0.030	0.024	0.036	0.036	0.088	0.040	0.102	0.032	0.054	0.030
60(30)	120	0.048	0.044	0.068	0.048	0.052	0.036	0.088	0.040	0.170	0.038	0.096	0.032
60(30)	160	0.034	0.028	0.066	0.036	0.046	0.046	0.154	0.044	0.196	0.036	0.120	0.042
60(30)	200	0.076	0.040	0.086	0.068	0.072	0.058	0.152	0.050	0.216	0.040	0.148	0.038
100(50)	40	0.056	0.034	0.034	0.036	0.044	0.048	0.102	0.032	0.094	0.038	0.060	0.046
100(50)	80	0.070	0.056	0.062	0.036	0.038	0.042	0.136	0.036	0.190	0.050	0.106	0.034
100(50)	120	0.056	0.060	0.090	0.050	0.078	0.034	0.184	0.042	0.290	0.056	0.166	0.034
100(50)	160	0.070	0.042	0.132	0.058	0.074	0.058	0.244	0.036	0.306	0.038	0.214	0.040
100(50)	200	0.104	0.046	0.118	0.052	0.068	0.038	0.284	0.054	0.392	0.048	0.246	0.044
200(100)	40	0.088	0.044	0.080	0.044	0.042	0.064	0.180	0.044	0.208	0.038	0.092	0.052
200(100)	80	0.086	0.048	0.106	0.044	0.072	0.048	0.294	0.062	0.400	0.048	0.202	0.040
200(100)	120	0.114	0.064	0.178	0.046	0.092	0.060	0.380	0.050	0.518	0.048	0.326	0.054
200(100)	160	0.146	0.048	0.218	0.046	0.134	0.042	0.486	0.044	0.624	0.076	0.404	0.044
200(100)	200	0.178	0.048	0.234	0.070	0.146	0.050	0.522	0.040	0.710	0.040	0.504	0.038

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