置信区间宽度等高线图在线性混合效应模型样本量规划中的应用

doi:10.3724/SP.J.1041.2024.00124

心理学报 ›› 2024, Vol. 56 ›› Issue (1): 124-138.doi: 10.3724/SP.J.1041.2024.00124

置信区间宽度等高线图在线性混合效应模型样本量规划中的应用

刘玥¹, 徐雷¹, 刘红云²^,³(), 韩雨婷⁴, 游晓锋⁵, 万志林¹

¹四川师范大学脑与心理科学研究院, 成都 610066
²应用实验心理北京市重点实验室, 北京 100875
³北京师范大学心理学部, 北京 100875
⁴北京语言大学心理学院, 北京 100083
⁵南昌师范学院数学与信息科学学院, 南昌 360111

收稿日期:2023-01-04 发布日期:2023-11-23 出版日期:2024-01-25
通讯作者: 刘红云 E-mail:hyliu@bnu.edu.cn
基金资助:
*国家自然科学基金项目(32071091);*国家自然科学基金项目(32200920);四川省自然科学基金青年项目(2022NSFSC1788);四川省自然科学基金青年项目(2022NSFSC1631);四川省自然科学基金青年项目(2022NSFSC1691)

Confidence interval width contours: Sample size planning for linear mixed-effects models

LIU Yue¹, XU Lei¹, LIU Hongyun²^,³(), HAN Yuting⁴, YOU Xiaofeng⁵, WAN Zhilin¹

¹Institute of Brain and Psychological Sciences, Sichuan Normal University, Chengdu 610066, China
²Beijing Key Laboratory of Applied Experimental Psychology, Beijing Normal University, Beijing 100875, China
³Faculty of Psychology, Beijing Normal University, Beijing 100875, China
⁴School of Psychology, Beijing Language and Culture University, Beijing, 100083, China
⁵School of Mathematics and Information Science, Nanchang Normal University, Nanchang 360111, China

Received:2023-01-04 Online:2023-11-23 Published:2024-01-25
Contact: LIU Hongyun E-mail:hyliu@bnu.edu.cn

摘要/Abstract

摘要：

线性混合效应模型在分析具有嵌套结构的心理学实验数据时具有明显优势。本文提出了置信区间宽度等高线图用于该模型的样本量规划。通过等高线图, 确定同时符合检验力、效应量准确性以及置信区间宽度要求的被试量和试次数。结合关注被试内实验效应和被试变量调节效应的两类典型模型, 通过两个模拟研究, 采用基于蒙特卡洛模拟方法, 探索效应量、随机效应大小和被试变量类型对置信区间宽度等高线图及样本量规划结果的影响。

关键词: 线性混合效应模型, 多水平模型, 检验力分析, 效应量, 置信区间宽度

Abstract:

Hierarchical data, which is observed frequently in psychological experiments, is usually analyzed with the linear mixed-effects models (LMEMs), as it can account for multiple sources of random effects due to participants, items, and/or predictors simultaneously. However, it is still unclear of how to determine the sample size and number of trials in LMEMs. In history, sample size planning was conducted based purely on power analysis. Later, the influential article of Maxwell et al. (2008) has made clear that sample size planning should consider statistical power and accuracy in parameter estimation (AIPE) simultaneously. In this paper, we derive a confidence interval width contours plot with the codes to generate it, providing power and AIPE information simultaneously. With this plot, sample size requirements in LMEMs based on power and AIPE criteria can be decided. We also demonstrated how to run sensitivity analysis to assess the impact of the magnitude of experiment effect size and the magnitude of random slope variance on statistical power, AIPE and the results of sample size planning.

There were two sets of sensitivity analysis based on different LMEMs. Sensitivity analysis Ⅰ investigated how the experiment effect size influenced power, AIPE and the requirement of sample size for within-subject experiment design, while sensitivity analysis Ⅱ investigated the impact of random slope variance on optimal sample size based on power and AIPE analysis for the cross-level interaction effect. The results for binary and continuous between-subject variables were compared. In these sensitivity analysis, two factors regarding sample size varied: number of subjects (I= 10, 30, 50, 70, 100, 200, 400, 600, 800), number of trials (J= 10, 20, 30, 50, 70, 100, 150, 200, 250, 300). The additional manipulated factor was the effect size of experiment effect (standard coefficient of experiment condition = 0.2, 0.5, 0.8, in sensitivity analysis I) and the magnitude of random slope variance (0.01, 0.09 and 0.25, in sensitivity analysis Ⅱ). A random slope model was used in sensitivity analysis Ⅰ, while a random slope model with level-2 independent variable was used in sensitivity analysis II. Data-generating model and fitted model were the same. Estimation performance was evaluated in terms of convergence rate, power, AIPE for the fixed effect, AIPE for the standard error of the fixed effect, and AIPE for the random effect.

The results are as following. First, there were no convergence problems under all the conditions, except that when the variance of random slope was small and a maximal model was used to fit the data. Second, power increased as sample size, number of trials or effect size increased. However, the number of trials played a key role for the power of within-subject effect, while sample size was more important for the power of cross-level effect. Power was larger for continuous between-subject variable than for binary between-subject variable. Third, although the fixed effect was accurately estimated under all the simulation conditions, the width 95% confidence interval (95% width) was extremely large under some conditions. Lastly, AIPE for the random effect increased as sample size and/or number of trials increased. The variance of residual was estimated accurately. As the variance of random slope increased, the accuracy of the estimates of variances of random intercept decreased, and the accuracy of the estimates of random slope increased.

In conclusion, if sample size planning was conducted solely based on power analysis, the chosen sample size might not be large enough to obtain accurate estimates of effects size. Therefore, the rational for considering statistical power and AIPE during sample size planning was adopted. To shed light on this issue, this article provided a standard procedure based on a confidence interval width contours plot to recommend sample size and number of trials for using LMEMs. This plot visualizes the combined effect of sample size and number of trials per participant on 95% width, power and AIPE for random effects. Based on this tool and other empirical considerations, practitioners can make informed choices about how many participants to test, and how many trials to test each one for.

Key words: linear mixed-effects models, multilevel models, power analysis, effect size, confidence interval width

中图分类号:

B841

刘玥, 徐雷, 刘红云, 韩雨婷, 游晓锋, 万志林. (2024). 置信区间宽度等高线图在线性混合效应模型样本量规划中的应用. 心理学报, 56(1), 124-138.

LIU Yue, XU Lei, LIU Hongyun, HAN Yuting, YOU Xiaofeng, WAN Zhilin. (2024). Confidence interval width contours: Sample size planning for linear mixed-effects models. Acta Psychologica Sinica, 56(1), 124-138.

ES	I	J
ES	I	10	20	30	50	70	100	150	200	250	300
0.2	10	0.107	0.112	0.131	0.168	0.181	0.224	0.279	0.312	0.369	0.379
	30	0.118	0.152	0.202	0.266	0.335	0.446	0.585	0.677	0.738	0.802
	50	0.170	0.175	0.224	0.278	0.409	0.490	0.677	0.756	0.832	0.888
	70	0.125	0.171	0.218	0.311	0.412	0.543	0.683	0.791	0.866	0.930
	100	0.133	0.169	0.233	0.335	0.420	0.535	0.701	0.816	0.893	0.935
	200	0.147	0.188	0.234	0.344	0.455	0.586	0.745	0.845	0.913	0.951
	400	0.115	0.194	0.232	0.345	0.433	0.574	0.766	0.852	0.918	0.958
	600	0.123	0.193	0.236	0.376	0.447	0.606	0.740	0.878	0.931	0.965
	800	0.147	0.202	0.245	0.377	0.480	0.549	0.764	0.909	0.948	0.969
0.5	10	0.298	0.481	0.626	0.804	0.891	0.975	0.994	0.999	1.000	1.000
	30	0.383	0.631	0.782	0.927	0.986	0.997	1.000	1.000	1.000	1.000
	50	0.438	0.659	0.810	0.959	0.992	1.000	1.000	1.000	1.000	1.000
	70	0.430	0.651	0.822	0.963	0.992	0.999	1.000	1.000	1.000	1.000
	100	0.453	0.659	0.845	0.967	0.996	0.999	1.000	1.000	1.000	1.000
	200	0.451	0.679	0.846	0.968	0.999	0.998	1.000	1.000	1.000	1.000
	400	0.453	0.714	0.856	0.976	0.997	1.000	1.000	1.000	1.000	1.000
	600	0.416	0.695	0.849	0.972	0.994	1.000	1.000	1.000	1.000	1.000
	800	0.464	0.715	0.850	0.972	0.999	1.000	1.000	1.000	1.000	1.000
0.8	10	0.626	0.876	0.959	0.997	1.000	1.000	1.000	1.000	1.000	1.000
	30	0.715	0.952	0.995	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	50	0.747	0.956	0.993	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	70	0.773	0.958	0.999	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	100	0.766	0.968	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	200	0.766	0.977	0.995	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	400	0.799	0.970	0.999	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	600	0.783	0.976	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	800	0.805	0.973	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000

ES	I	J
ES	I	10	20	30	50	70	100	150	200	250	300
0.2	10	0.107	0.112	0.131	0.168	0.181	0.224	0.279	0.312	0.369	0.379
	30	0.118	0.152	0.202	0.266	0.335	0.446	0.585	0.677	0.738	0.802
	50	0.170	0.175	0.224	0.278	0.409	0.490	0.677	0.756	0.832	0.888
	70	0.125	0.171	0.218	0.311	0.412	0.543	0.683	0.791	0.866	0.930
	100	0.133	0.169	0.233	0.335	0.420	0.535	0.701	0.816	0.893	0.935
	200	0.147	0.188	0.234	0.344	0.455	0.586	0.745	0.845	0.913	0.951
	400	0.115	0.194	0.232	0.345	0.433	0.574	0.766	0.852	0.918	0.958
	600	0.123	0.193	0.236	0.376	0.447	0.606	0.740	0.878	0.931	0.965
	800	0.147	0.202	0.245	0.377	0.480	0.549	0.764	0.909	0.948	0.969
0.5	10	0.298	0.481	0.626	0.804	0.891	0.975	0.994	0.999	1.000	1.000
	30	0.383	0.631	0.782	0.927	0.986	0.997	1.000	1.000	1.000	1.000
	50	0.438	0.659	0.810	0.959	0.992	1.000	1.000	1.000	1.000	1.000
	70	0.430	0.651	0.822	0.963	0.992	0.999	1.000	1.000	1.000	1.000
	100	0.453	0.659	0.845	0.967	0.996	0.999	1.000	1.000	1.000	1.000
	200	0.451	0.679	0.846	0.968	0.999	0.998	1.000	1.000	1.000	1.000
	400	0.453	0.714	0.856	0.976	0.997	1.000	1.000	1.000	1.000	1.000
	600	0.416	0.695	0.849	0.972	0.994	1.000	1.000	1.000	1.000	1.000
	800	0.464	0.715	0.850	0.972	0.999	1.000	1.000	1.000	1.000	1.000
0.8	10	0.626	0.876	0.959	0.997	1.000	1.000	1.000	1.000	1.000	1.000
	30	0.715	0.952	0.995	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	50	0.747	0.956	0.993	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	70	0.773	0.958	0.999	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	100	0.766	0.968	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	200	0.766	0.977	0.995	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	400	0.799	0.970	0.999	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	600	0.783	0.976	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	800	0.805	0.973	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000

criteria	I	J
criteria	I	10	20	30	50	70	100	150	200	250	300
rbias	10	0.008	0.009	0.001	0.007	0.001	0.012	0.001	0.001	0.001	0.000
	30	0.007	0.002	0.006	0.017	0.003	0.003	0.007	0.001	0.001	0.003
	50	0.024	0.009	0.004	0.011	0.003	0.002	0.011	0.005	0.013	0.002
	70	0.003	0.004	0.019	0.001	0.017	0.001	0.001	0.001	0.004	0.005
	100	0.019	0.005	0.004	0.000	0.004	0.002	0.004	0.003	0.002	0.004
	200	0.013	0.007	0.015	0.007	0.014	0.010	0.001	0.001	0.002	0.003
	400	0.026	0.025	0.008	0.003	0.008	0.004	0.001	0.003	0.000	0.002
	600	0.016	0.011	0.007	0.003	0.005	0.005	0.004	0.002	0.003	0.006
	800	0.005	0.010	0.010	0.004	0.001	0.013	0.003	0.005	0.000	0.001
width	10	1.411	1.036	0.861	0.709	0.633	0.565	0.506	0.476	0.458	0.444
	30	1.197	0.860	0.713	0.573	0.498	0.434	0.376	0.343	0.321	0.306
	50	1.151	0.827	0.685	0.542	0.468	0.403	0.343	0.309	0.286	0.270
	70	1.125	0.817	0.669	0.530	0.453	0.389	0.328	0.293	0.269	0.252
	100	1.122	0.798	0.665	0.519	0.443	0.377	0.316	0.280	0.256	0.238
	200	1.091	0.786	0.649	0.505	0.431	0.362	0.301	0.265	0.240	0.221
	400	1.096	0.782	0.644	0.501	0.424	0.355	0.294	0.256	0.230	0.212
	600	1.086	0.778	0.643	0.497	0.422	0.353	0.290	0.254	0.227	0.209
	800	1.076	0.778	0.638	0.497	0.423	0.354	0.290	0.252	0.226	0.207
SE-SD bias	10	0.006	0.024	0.023	0.027	0.025	0.032	0.035	0.046	0.048	0.047
	30	−0.006	0.005	0.007	0.008	0.013	0.015	0.017	0.020	0.019	0.022
	50	−0.004	0.004	0.002	0.007	0.009	0.012	0.011	0.010	0.014	0.014
	70	0.004	−0.006	0.006	0.003	0.006	0.006	0.010	0.009	0.012	0.011
	100	−0.006	−0.007	0.002	0.004	0.003	0.004	0.008	0.005	0.007	0.008
	200	−0.002	0.006	0.000	0.000	0.004	0.005	0.002	0.002	0.003	0.004
	400	0.000	−0.011	0.007	0.002	0.001	0.001	0.002	0.004	0.004	0.002
	600	0.000	−0.002	0.000	−0.007	−0.004	0.001	−0.002	0.001	0.002	0.003
	800	−0.008	0.003	0.000	0.000	0.000	0.004	0.002	0.002	0.000	0.001

criteria	I	J
criteria	I	10	20	30	50	70	100	150	200	250	300
rbias	10	0.008	0.009	0.001	0.007	0.001	0.012	0.001	0.001	0.001	0.000
	30	0.007	0.002	0.006	0.017	0.003	0.003	0.007	0.001	0.001	0.003
	50	0.024	0.009	0.004	0.011	0.003	0.002	0.011	0.005	0.013	0.002
	70	0.003	0.004	0.019	0.001	0.017	0.001	0.001	0.001	0.004	0.005
	100	0.019	0.005	0.004	0.000	0.004	0.002	0.004	0.003	0.002	0.004
	200	0.013	0.007	0.015	0.007	0.014	0.010	0.001	0.001	0.002	0.003
	400	0.026	0.025	0.008	0.003	0.008	0.004	0.001	0.003	0.000	0.002
	600	0.016	0.011	0.007	0.003	0.005	0.005	0.004	0.002	0.003	0.006
	800	0.005	0.010	0.010	0.004	0.001	0.013	0.003	0.005	0.000	0.001
width	10	1.411	1.036	0.861	0.709	0.633	0.565	0.506	0.476	0.458	0.444
	30	1.197	0.860	0.713	0.573	0.498	0.434	0.376	0.343	0.321	0.306
	50	1.151	0.827	0.685	0.542	0.468	0.403	0.343	0.309	0.286	0.270
	70	1.125	0.817	0.669	0.530	0.453	0.389	0.328	0.293	0.269	0.252
	100	1.122	0.798	0.665	0.519	0.443	0.377	0.316	0.280	0.256	0.238
	200	1.091	0.786	0.649	0.505	0.431	0.362	0.301	0.265	0.240	0.221
	400	1.096	0.782	0.644	0.501	0.424	0.355	0.294	0.256	0.230	0.212
	600	1.086	0.778	0.643	0.497	0.422	0.353	0.290	0.254	0.227	0.209
	800	1.076	0.778	0.638	0.497	0.423	0.354	0.290	0.252	0.226	0.207
SE-SD bias	10	0.006	0.024	0.023	0.027	0.025	0.032	0.035	0.046	0.048	0.047
	30	−0.006	0.005	0.007	0.008	0.013	0.015	0.017	0.020	0.019	0.022
	50	−0.004	0.004	0.002	0.007	0.009	0.012	0.011	0.010	0.014	0.014
	70	0.004	−0.006	0.006	0.003	0.006	0.006	0.010	0.009	0.012	0.011
	100	−0.006	−0.007	0.002	0.004	0.003	0.004	0.008	0.005	0.007	0.008
	200	−0.002	0.006	0.000	0.000	0.004	0.005	0.002	0.002	0.003	0.004
	400	0.000	−0.011	0.007	0.002	0.001	0.001	0.002	0.004	0.004	0.002
	600	0.000	−0.002	0.000	−0.007	−0.004	0.001	−0.002	0.001	0.002	0.003
	800	−0.008	0.003	0.000	0.000	0.000	0.004	0.002	0.002	0.000	0.001

ES	I	J
ES	I	10	20	30	50	70	100	150	200	250	300
0.2	10	0.985	0.992	0.993	0.993	0.994	0.993	0.996	0.991	0.981	0.980
	30	0.989	0.994	0.994	0.999	0.998	0.994	0.995	0.986	0.985	0.974
	50	0.990	0.996	0.997	0.994	0.995	0.983	0.982	0.977	0.964	0.961
	70	0.986	0.991	0.996	0.996	0.992	0.987	0.982	0.965	0.962	0.938
	100	0.983	0.986	0.995	0.991	0.988	0.993	0.971	0.964	0.948	0.932
	200	0.973	0.967	0.976	0.985	0.979	0.960	0.953	0.936	0.913	0.885
	400	0.962	0.949	0.957	0.965	0.949	0.929	0.907	0.894	0.880	0.878
	600	0.934	0.928	0.939	0.947	0.910	0.907	0.881	0.849	0.826	0.835
	800	0.918	0.991	0.912	0.897	0.863	0.864	0.846	0.820	0.791	0.772
0.5	10	0.986	0.992	0.992	0.996	0.995	0.996	0.997	0.986	0.991	0.985
	30	0.986	0.998	0.995	0.998	0.994	0.993	0.989	0.994	0.984	0.976
	50	0.983	0.993	0.996	0.993	0.993	0.987	0.983	0.971	0.960	0.959
	70	0.989	0.985	0.998	0.996	0.986	0.985	0.986	0.973	0.940	0.939
	100	0.984	0.965	0.993	0.994	0.989	0.990	0.982	0.962	0.937	0.919
	200	0.977	0.942	0.986	0.981	0.981	0.975	0.951	0.938	0.915	0.902
	400	0.962	0.927	0.962	0.961	0.955	0.946	0.913	0.891	0.871	0.864
	600	0.947	0.992	0.933	0.921	0.907	0.916	0.877	0.836	0.849	0.848
	800	0.925	0.994	0.929	0.900	0.879	0.865	0.874	0.818	0.784	0.760
0.8	10	0.989	0.988	0.991	0.994	0.994	0.994	0.995	0.988	0.982	0.974
	30	0.991	0.99	0.994	0.998	0.995	0.996	0.988	0.984	0.986	0.980
	50	0.984	0.987	1.000	0.998	0.986	0.985	0.977	0.973	0.959	0.942
	70	0.984	0.973	0.998	0.992	0.993	0.974	0.978	0.959	0.958	0.952
	100	0.981	0.942	0.992	0.995	0.994	0.991	0.980	0.960	0.941	0.928
	200	0.967	0.942	0.984	0.984	0.978	0.970	0.947	0.933	0.915	0.897
	400	0.959	0.992	0.947	0.951	0.947	0.940	0.903	0.887	0.886	0.871
	600	0.929	0.994	0.937	0.927	0.916	0.915	0.903	0.866	0.827	0.826
	800	0.929	0.994	0.913	0.910	0.885	0.876	0.859	0.823	0.799	0.770

置信区间宽度等高线图在线性混合效应模型样本量规划中的应用

Confidence interval width contours: Sample size planning for linear mixed-effects models

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图/表 40

I	J
I	10	20	30	50	70	100	150	200	250	300
10	0.211	0.307	0.446	0.653	0.796	0.914	0.972	0.995	0.995	1.000
30	0.296	0.472	0.596	0.835	0.913	0.974	1.000	1.000	1.000	1.000
50	0.316	0.464	0.640	0.830	0.940	0.983	0.999	1.000	1.000	1.000
70	0.320	0.487	0.642	0.850	0.942	0.991	1.000	1.000	1.000	1.000
100	0.297	0.506	0.660	0.867	0.951	0.987	0.999	1.000	1.000	1.000
200	0.329	0.517	0.658	0.877	0.952	0.989	1.000	1.000	1.000	1.000
400	0.353	0.527	0.660	0.876	0.964	0.992	0.999	1.000	1.000	1.000
600	0.315	0.547	0.660	0.875	0.958	0.995	1.000	1.000	1.000	1.000
800	0.308	0.529	0.686	0.895	0.963	0.992	1.000	1.000	1.000	1.000

criteria	I	J
criteria	I	10	20	30	50	70	100	150	200	250	300
bias	10	-0.004	0.005	-0.001	-0.004	-0.001	0.006	-0.001	0.000	0.000	0.000
	30	0.004	0.001	-0.003	-0.008	-0.002	-0.002	-0.003	-0.001	0.001	0.001
	50	0.012	0.004	0.002	0.005	-0.001	-0.001	0.006	-0.002	-0.007	-0.001
	70	0.001	-0.002	-0.010	-0.001	-0.009	-0.001	-0.001	0.001	0.002	0.002
	100	0.009	-0.002	0.002	0.000	-0.002	0.001	-0.002	0.002	0.001	-0.002
	200	0.006	-0.003	-0.007	-0.004	0.007	-0.005	0.000	-0.001	0.001	-0.001
	400	0.013	0.013	-0.004	-0.002	0.004	-0.002	-0.001	-0.001	0.000	0.001
	600	-0.008	-0.005	-0.003	0.002	0.002	-0.002	-0.002	-0.001	0.001	0.003
	800	-0.002	0.005	-0.005	0.002	0.000	-0.007	0.001	-0.002	0.000	0.001
RMSE	10	0.354	0.240	0.196	0.154	0.136	0.112	0.094	0.076	0.069	0.066
	30	0.312	0.215	0.175	0.139	0.114	0.096	0.079	0.067	0.063	0.056
	50	0.297	0.207	0.173	0.132	0.110	0.091	0.077	0.069	0.059	0.055
	70	0.283	0.215	0.165	0.132	0.110	0.093	0.074	0.066	0.057	0.053
	100	0.292	0.211	0.167	0.128	0.110	0.092	0.073	0.066	0.058	0.053
	200	0.280	0.195	0.166	0.129	0.106	0.088	0.075	0.066	0.058	0.052
	400	0.280	0.211	0.158	0.125	0.107	0.090	0.073	0.061	0.054	0.052
	600	0.277	0.201	0.164	0.134	0.111	0.089	0.076	0.064	0.056	0.050
	800	0.282	0.195	0.162	0.127	0.108	0.087	0.072	0.063	0.058	0.052
CP	10	0.927	0.958	0.965	0.972	0.984	0.990	0.986	0.997	0.997	0.999
	30	0.921	0.935	0.952	0.957	0.967	0.972	0.979	0.989	0.992	0.994
	50	0.923	0.951	0.949	0.955	0.956	0.968	0.973	0.972	0.983	0.989
	70	0.927	0.943	0.942	0.953	0.957	0.971	0.973	0.976	0.983	0.983
	100	0.919	0.931	0.952	0.962	0.950	0.964	0.965	0.962	0.978	0.978
	200	0.916	0.947	0.943	0.941	0.953	0.958	0.960	0.950	0.963	0.968
	400	0.929	0.920	0.948	0.951	0.949	0.952	0.957	0.955	0.960	0.956
	600	0.936	0.937	0.951	0.932	0.937	0.953	0.940	0.956	0.954	0.963
	800	0.909	0.949	0.935	0.947	0.940	0.956	0.953	0.961	0.953	0.957

criteria	I	J
criteria	I	10	20	30	50	70	100	150	200	250	300
bias	10	-0.012	-0.009	0.000	0.001	0.004	0.003	0.004	0.001	-0.004	-0.002
	30	0.001	0.008	-0.004	0.004	0.000	0.001	0.002	-0.001	0.005	-0.002
	50	-0.001	-0.004	-0.001	-0.003	0.002	0.000	-0.002	0.001	0.001	-0.003
	70	0.007	-0.003	-0.005	-0.002	0.002	0.000	0.003	0.004	0.000	-0.002
	100	-0.005	-0.003	-0.002	-0.002	0.002	-0.005	-0.005	0.001	0.002	0.003
	200	0.005	0.000	-0.008	-0.004	-0.003	-0.002	0.002	-0.003	0.003	0.001
	400	0.026	-0.010	0.002	0.001	0.002	0.000	0.001	-0.005	0.002	-0.001
	600	0.010	0.010	-0.006	0.000	-0.003	0.003	0.000	0.000	-0.001	0.000
	800	-0.016	0.003	0.001	0.003	0.000	0.003	0.005	-0.002	0.000	-0.005
rbias	10	0.024	0.018	0.000	0.002	0.009	0.006	0.009	0.002	0.008	0.003
	30	0.001	0.015	0.008	0.009	0.000	0.003	0.005	0.002	0.009	0.003
	50	0.002	0.008	0.001	0.007	0.004	0.001	0.003	0.003	0.001	0.006
	70	0.014	0.005	0.009	0.004	0.003	0.001	0.007	0.008	0.001	0.004
	100	0.011	0.007	0.004	0.004	0.003	0.010	0.010	0.003	0.004	0.005
	200	0.011	0.001	0.017	0.007	0.007	0.003	0.004	0.006	0.005	0.001
	400	0.052	0.019	0.004	0.002	0.003	0.000	0.002	0.010	0.003	0.002
	600	0.020	0.019	0.013	0.001	0.006	0.005	0.000	0.000	0.002	0.001
	800	0.033	0.007	0.001	0.006	0.001	0.006	0.009	0.005	0.001	0.009
RMSE	10	0.424	0.311	0.248	0.180	0.162	0.134	0.113	0.097	0.087	0.078
	30	0.390	0.269	0.221	0.173	0.146	0.121	0.095	0.071	0.075	0.069
	50	0.375	0.259	0.205	0.167	0.139	0.117	0.096	0.084	0.072	0.066
	70	0.363	0.255	0.213	0.164	0.137	0.111	0.097	0.079	0.072	0.065
	100	0.344	0.251	0.205	0.161	0.134	0.116	0.092	0.080	0.072	0.068
	200	0.358	0.254	0.212	0.153	0.136	0.113	0.094	0.077	0.072	0.061
	400	0.366	0.254	0.204	0.158	0.133	0.114	0.093	0.081	0.070	0.066
	600	0.355	0.254	0.208	0.162	0.138	0.110	0.091	0.079	0.068	0.066
	800	0.355	0.245	0.208	0.154	0.132	0.111	0.086	0.077	0.069	0.064
width	10	1.736	1.272	1.056	0.846	0.740	0.649	0.575	0.525	0.501	0.480
	30	1.488	1.071	0.879	0.698	0.604	0.520	0.440	0.347	0.368	0.347
	50	1.395	1.027	0.849	0.668	0.573	0.487	0.410	0.365	0.335	0.313
	70	1.402	1.008	0.834	0.654	0.557	0.474	0.396	0.349	0.319	0.297
	100	1.393	0.990	0.822	0.643	0.548	0.464	0.384	0.337	0.307	0.284
	200	1.352	0.993	0.804	0.629	0.534	0.451	0.369	0.324	0.292	0.270
	400	1.363	0.969	0.807	0.620	0.525	0.442	0.364	0.317	0.284	0.261
	600	1.364	0.975	0.796	0.623	0.527	0.441	0.363	0.314	0.282	0.258
	800	1.357	0.975	0.793	0.622	0.524	0.442	0.361	0.313	0.280	0.256
CP	10	0.944	0.947	0.961	0.979	0.972	0.977	0.984	0.989	0.993	1.000
	30	0.917	0.944	0.947	0.957	0.963	0.971	0.979	0.985	0.988	0.988
	50	0.909	0.948	0.951	0.953	0.957	0.962	0.965	0.968	0.980	0.979
	70	0.915	0.938	0.940	0.955	0.960	0.973	0.963	0.974	0.974	0.976
	100	0.927	0.936	0.951	0.946	0.952	0.948	0.965	0.959	0.970	0.964
	200	0.917	0.941	0.925	0.953	0.954	0.949	0.943	0.969	0.949	0.973
	400	0.921	0.937	0.947	0.941	0.956	0.947	0.949	0.947	0.951	0.952
	600	0.914	0.928	0.941	0.931	0.938	0.955	0.949	0.946	0.950	0.942
	800	0.904	0.932	0.930	0.952	0.946	0.953	0.965	0.956	0.950	0.954
SE-SD bias	10	0.019	0.014	0.021	0.036	0.027	0.032	0.034	0.036	0.041	0.045
	30	-0.011	0.004	0.003	0.005	0.008	0.011	0.017	0.018	0.019	0.020
	50	-0.020	0.003	0.012	0.003	0.007	0.007	0.009	0.009	0.014	0.014
	70	-0.006	0.002	-0.001	0.002	0.005	0.010	0.004	0.010	0.009	0.010
	100	0.012	0.002	0.004	0.003	0.005	0.002	0.006	0.006	0.007	0.005
	200	-0.013	-0.001	-0.007	0.008	0.001	0.002	0.001	0.005	0.002	0.008
	400	-0.017	-0.007	0.001	0.000	0.001	-0.001	-0.001	0.000	0.002	0.001
	600	-0.007	-0.006	-0.004	-0.003	-0.003	0.002	0.002	0.001	0.004	0.000
	800	-0.008	0.003	-0.006	0.005	0.001	0.002	0.006	0.003	0.003	0.001

Parameter	I	J
Parameter	I	10	20	30	50	70	100	150	200	250	300
${{\sigma }^{2}}$	10	0.063	0.036	0.028	0.019	0.016	0.013	0.010	0.009	0.008	0.007
	30	0.033	0.017	0.012	0.008	0.006	0.005	0.004	0.003	0.003	0.003
	50	0.024	0.013	0.009	0.006	0.004	0.003	0.002	0.002	0.002	0.002
	70	0.021	0.011	0.007	0.005	0.003	0.003	0.002	0.001	0.001	0.001
	100	0.017	0.008	0.006	0.004	0.003	0.002	0.001	0.001	0.001	0.001
	200	0.013	0.006	0.004	0.002	0.002	0.001	0.001	0.001	0.001	0.000
	400	0.009	0.004	0.003	0.002	0.001	0.001	0.001	0.000	0.000	0.000
	600	0.008	0.004	0.002	0.001	0.001	0.001	0.000	0.000	0.000	0.000
	800	0.006	0.003	0.002	0.001	0.001	0.001	0.000	0.000	0.000	0.000
$\tau _{00}^{2}$	10	0.405	0.283	0.212	0.174	0.149	0.129	0.111	0.092	0.086	0.081
	30	0.227	0.156	0.130	0.107	0.089	0.078	0.067	0.065	0.059	0.059
	50	0.175	0.130	0.102	0.086	0.072	0.066	0.063	0.057	0.056	0.055
	70	0.154	0.112	0.094	0.076	0.068	0.064	0.057	0.055	0.055	0.054
	100	0.129	0.095	0.079	0.068	0.061	0.058	0.053	0.051	0.052	0.053
	200	0.101	0.076	0.063	0.056	0.055	0.055	0.053	0.054	0.052	0.053
	400	0.074	0.063	0.058	0.052	0.054	0.052	0.052	0.053	0.051	0.052
	600	0.067	0.058	0.055	0.054	0.053	0.053	0.053	0.052	0.052	0.052
	800	0.062	0.057	0.056	0.053	0.052	0.053	0.052	0.052	0.053	0.053
$\tau _{11}^{2}$	10	1.444	0.961	0.745	0.593	0.469	0.387	0.328	0.265	0.233	0.212
	30	0.967	0.612	0.480	0.332	0.274	0.222	0.176	0.151	0.130	0.121
	50	0.719	0.495	0.384	0.261	0.211	0.170	0.135	0.114	0.101	0.090
	70	0.689	0.417	0.314	0.215	0.168	0.146	0.112	0.095	0.086	0.075
	100	0.598	0.340	0.267	0.182	0.146	0.116	0.095	0.082	0.068	0.065
	200	0.431	0.251	0.188	0.126	0.106	0.086	0.068	0.057	0.050	0.041
	400	0.309	0.176	0.132	0.090	0.073	0.059	0.046	0.041	0.035	0.031
	600	0.258	0.145	0.109	0.077	0.059	0.047	0.039	0.031	0.029	0.026
	800	0.211	0.122	0.090	0.066	0.052	0.042	0.034	0.029	0.025	0.023
$\omega _{00}^{2}$	10	0.522	0.326	0.252	0.189	0.159	0.128	0.105	0.089	0.084	0.078
	30	0.286	0.171	0.139	0.104	0.083	0.070	0.056	0.049	0.043	0.039
	50	0.232	0.139	0.107	0.080	0.062	0.053	0.042	0.037	0.032	0.029
	70	0.206	0.121	0.092	0.067	0.056	0.046	0.037	0.032	0.028	0.026
	100	0.176	0.109	0.080	0.059	0.047	0.039	0.031	0.027	0.024	0.021
	200	0.151	0.083	0.059	0.045	0.036	0.029	0.022	0.018	0.017	0.015
	400	0.129	0.065	0.050	0.032	0.026	0.021	0.016	0.014	0.013	0.011
	600	0.125	0.065	0.046	0.030	0.022	0.019	0.014	0.011	0.010	0.009
	800	0.123	0.059	0.043	0.028	0.021	0.016	0.013	0.010	0.009	0.008

L2_IV	${{\tau }_{11.std}}^{2}$	I	J
L2_IV	${{\tau }_{11.std}}^{2}$	I	10	20	30	50	70	100	150	200	250	300
分类	0.01	10	0.982	0.994	0.992	0.987	0.989	0.983	0.984	0.984	0.976	0.972
		30	0.983	0.985	0.978	0.977	0.976	0.970	0.966	0.961	0.967	0.968
		50	0.985	0.977	0.970	0.970	0.966	0.962	0.958	0.932	0.941	0.947
		70	0.973	0.970	0.962	0.972	0.946	0.937	0.923	0.920	0.901	0.905
		100	0.969	0.965	0.951	0.935	0.922	0.928	0.882	0.863	0.879	0.857
		200	0.942	0.948	0.907	0.911	0.861	0.801	0.791	0.765	0.743	0.733
		400	0.898	0.883	0.878	0.844	0.790	0.775	0.677	0.638	0.604	0.606
		600	0.876	0.846	0.842	0.805	0.736	0.655	0.613	0.575	0.553	0.527
		800	0.845	0.835	0.823	0.729	0.712	0.660	0.562	0.533	0.489	0.486
	0.09	10	0.988	0.994	0.997	0.997	0.995	0.996	0.994	0.991	0.982	0.990
		30	0.992	0.990	0.998	1.000	0.996	0.995	0.991	0.989	0.989	0.982
		50	0.982	0.988	0.997	0.994	0.990	0.992	0.987	0.976	0.966	0.954
		70	0.987	0.997	0.995	0.996	0.992	0.981	0.971	0.972	0.945	0.919
		100	0.986	0.994	0.996	0.992	0.993	0.985	0.967	0.949	0.943	0.937
		200	0.979	0.991	0.984	0.987	0.967	0.975	0.954	0.941	0.933	0.906
		400	0.970	0.964	0.956	0.961	0.954	0.943	0.932	0.921	0.903	0.886
		600	0.940	0.944	0.926	0.926	0.914	0.919	0.899	0.852	0.829	0.845
		800	0.923	0.916	0.933	0.925	0.877	0.890	0.835	0.812	0.790	0.774
	0.25	10	0.985	0.992	0.998	0.998	0.994	0.992	0.990	0.988	0.985	0.975
		30	0.995	0.999	0.997	0.995	0.995	0.993	0.993	0.989	0.977	0.963
		50	0.993	0.999	0.999	0.997	0.988	0.988	0.987	0.980	0.965	0.970
		70	0.999	0.998	0.998	0.996	0.990	0.988	0.989	0.978	0.960	0.956
		100	0.995	0.998	0.995	0.995	0.993	0.981	0.983	0.977	0.963	0.961
		200	0.996	0.987	0.989	0.979	0.969	0.974	0.969	0.963	0.967	0.957
		400	0.978	0.972	0.960	0.960	0.937	0.936	0.926	0.935	0.920	0.932
		600	0.959	0.953	0.936	0.939	0.910	0.883	0.899	0.877	0.870	0.871
		800	0.934	0.917	0.929	0.935	0.890	0.875	0.875	0.794	0.810	0.833
连续	0.09	10	0.992	0.996	0.992	0.994	0.994	0.995	0.996	0.990	0.980	0.980
		30	0.989	0.992	0.994	0.998	0.996	0.992	0.993	0.992	0.982	0.965
		50	0.986	0.994	0.996	0.997	0.994	0.986	0.977	0.972	0.969	0.965
		70	0.990	0.995	0.996	0.996	0.995	0.985	0.981	0.952	0.947	0.935
		100	0.988	0.998	0.991	0.992	0.995	0.990	0.979	0.969	0.952	0.934
		200	0.985	0.984	0.979	0.986	0.980	0.971	0.961	0.939	0.950	0.924
		400	0.984	0.976	0.968	0.958	0.937	0.948	0.943	0.915	0.881	0.878
		600	0.979	0.949	0.948	0.939	0.918	0.883	0.897	0.881	0.839	0.859
		800	0.961	0.939	0.920	0.899	0.885	0.866	0.869	0.840	0.786	0.817

L2_IV	$τ_{11.std}^{2}$	I	J
L2_IV	$τ_{11.std}^{2}$	I	10	20	30	50	70	100	150	200	250	300
分类	0.01	10	0.042	0.040	0.042	0.037	0.048	0.055	0.076	0.084	0.081	0.081
		30	0.046	0.065	0.058	0.082	0.074	0.104	0.101	0.151	0.155	0.165
		50	0.053	0.047	0.074	0.087	0.102	0.146	0.147	0.176	0.173	0.201
		70	0.045	0.073	0.074	0.101	0.118	0.137	0.177	0.248	0.273	0.267
		100	0.059	0.086	0.095	0.134	0.150	0.182	0.268	0.287	0.382	0.396
		200	0.086	0.100	0.138	0.213	0.266	0.351	0.483	0.535	0.611	0.630
		400	0.134	0.196	0.265	0.391	0.463	0.628	0.742	0.839	0.843	0.884
		600	0.166	0.259	0.357	0.550	0.645	0.768	0.896	0.946	0.973	0.983
		800	0.169	0.320	0.430	0.676	0.764	0.888	0.954	0.983	0.998	0.998
	0.09	10	0.061	0.075	0.096	0.112	0.119	0.122	0.132	0.148	0.131	0.138
		30	0.097	0.129	0.148	0.179	0.212	0.205	0.215	0.236	0.249	0.247
		50	0.129	0.160	0.197	0.246	0.294	0.321	0.332	0.373	0.362	0.389
		70	0.148	0.203	0.258	0.340	0.402	0.405	0.438	0.477	0.468	0.513
		100	0.182	0.277	0.322	0.455	0.494	0.528	0.590	0.646	0.617	0.644
		200	0.314	0.497	0.613	0.732	0.804	0.844	0.882	0.879	0.921	0.919
		400	0.533	0.797	0.881	0.947	0.974	0.989	0.992	0.996	0.994	0.995
		600	0.747	0.924	0.975	0.995	0.998	0.999	1.000	1.000	1.000	1.000
		800	0.859	0.976	0.995	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.25	10	0.097	0.093	0.121	0.129	0.128	0.123	0.134	0.133	0.124	0.145
		30	0.124	0.166	0.188	0.227	0.223	0.246	0.241	0.246	0.240	0.253
		50	0.199	0.257	0.301	0.359	0.353	0.389	0.383	0.403	0.378	0.415
		70	0.262	0.343	0.395	0.443	0.473	0.496	0.476	0.494	0.509	0.515
		100	0.339	0.455	0.509	0.576	0.615	0.598	0.649	0.664	0.651	0.682
		200	0.597	0.753	0.792	0.864	0.889	0.916	0.918	0.924	0.921	0.936
		400	0.874	0.962	0.977	0.989	0.994	0.998	1.000	0.999	1.000	0.999
		600	0.969	0.998	0.998	1.000	1.000	1.000	1.000	1.000	1.000	1.000
		800	0.993	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
连续	0.09	10	0.075	0.090	0.099	0.137	0.133	0.145	0.158	0.153	0.148	0.157
		30	0.117	0.138	0.181	0.242	0.239	0.257	0.310	0.298	0.324	0.337
		50	0.151	0.232	0.243	0.357	0.374	0.434	0.451	0.459	0.490	0.515
		70	0.191	0.307	0.364	0.420	0.481	0.560	0.581	0.599	0.634	0.645
		100	0.241	0.410	0.461	0.602	0.644	0.687	0.754	0.757	0.794	0.803
		200	0.451	0.680	0.752	0.872	0.911	0.921	0.966	0.966	0.983	0.976
		400	0.730	0.931	0.972	0.991	0.995	0.998	1.000	1.000	1.000	1.000
		600	0.890	0.984	0.993	0.999	1.000	1.000	1.000	1.000	1.000	1.000
		800	0.963	0.999	0.998	1.000	1.000	1.000	1.000	1.000	1.000	1.000

criteria	I	J
criteria	I	10	20	30	50	70	100	150	200	250	300
bias	10	0.016	0.000	0.002	-0.003	-0.006	0.002	0.000	-0.003	-0.003	-0.003
	30	-0.002	-0.005	-0.003	0.009	-0.003	0.001	-0.002	0.000	0.001	0.002
	50	0.000	0.003	-0.002	0.001	-0.001	-0.001	-0.003	0.000	0.000	0.000
	70	0.000	0.003	-0.002	0.000	-0.002	-0.002	-0.002	0.001	0.000	-0.002
	100	-0.005	-0.004	0.001	-0.001	0.001	-0.002	0.001	-0.001	0.002	0.000
	200	0.000	-0.002	-0.001	-0.001	-0.001	-0.001	0.001	0.000	0.000	0.000
	400	0.005	0.001	0.001	0.000	-0.001	0.001	-0.001	0.001	-0.001	0.000
	600	0.000	-0.002	-0.001	0.000	0.000	-0.001	0.000	0.000	0.000	-0.001
	800	-0.001	-0.001	-0.002	0.001	0.000	0.001	0.001	0.001	0.000	0.000
rbias	10	0.032	0.000	0.005	0.007	0.013	0.005	0.001	0.006	0.006	0.005
	30	0.003	0.011	0.006	0.017	0.006	0.003	0.004	0.001	0.001	0.004
	50	0.000	0.007	0.004	0.002	0.003	0.002	0.005	0.001	0.001	0.001
	70	0.001	0.005	0.005	0.000	0.005	0.004	0.005	0.002	0.000	0.004
	100	0.010	0.008	0.002	0.003	0.001	0.003	0.002	0.003	0.004	0.000
	200	0.001	0.004	0.003	0.002	0.002	0.002	0.002	0.001	0.001	0.000
	400	0.009	0.002	0.001	0.001	0.002	0.002	0.002	0.001	0.002	0.000
	600	0.001	0.004	0.002	0.001	0.001	0.002	0.001	0.001	0.000	0.001
	800	0.002	0.003	0.004	0.002	0.001	0.002	0.001	0.003	0.001	0.001
RMSE	10	0.398	0.284	0.236	0.189	0.162	0.141	0.121	0.111	0.098	0.094
	30	0.235	0.176	0.140	0.109	0.094	0.082	0.068	0.065	0.060	0.054
	50	0.181	0.128	0.106	0.085	0.073	0.066	0.055	0.047	0.043	0.043
	70	0.149	0.108	0.091	0.073	0.062	0.052	0.046	0.042	0.039	0.037
	100	0.133	0.093	0.076	0.062	0.052	0.045	0.039	0.034	0.032	0.031
	200	0.092	0.064	0.054	0.042	0.035	0.033	0.027	0.026	0.023	0.022
	400	0.064	0.045	0.038	0.032	0.028	0.022	0.018	0.017	0.017	0.016
	600	0.054	0.038	0.031	0.025	0.022	0.018	0.015	0.014	0.013	0.012
	800	0.045	0.032	0.027	0.022	0.019	0.016	0.014	0.012	0.012	0.011
width	10	1.818	1.353	1.187	1.007	0.939	0.880	0.836	0.816	0.798	0.789
	30	1.020	0.771	0.667	0.585	0.548	0.512	0.485	0.474	0.465	0.458
	50	0.783	0.595	0.522	0.454	0.423	0.397	0.378	0.367	0.361	0.355
	70	0.660	0.503	0.442	0.386	0.358	0.338	0.319	0.310	0.305	0.301
	100	0.549	0.421	0.368	0.323	0.300	0.282	0.267	0.260	0.255	0.252
	200	0.388	0.299	0.262	0.228	0.212	0.200	0.189	0.184	0.181	0.178
	400	0.274	0.211	0.185	0.161	0.150	0.141	0.134	0.130	0.128	0.126
	600	0.224	0.172	0.152	0.132	0.123	0.115	0.109	0.106	0.104	0.103
	800	0.194	0.149	0.131	0.114	0.106	0.100	0.095	0.092	0.090	0.089
CP	10	0.957	0.963	0.959	0.957	0.956	0.963	0.949	0.935	0.951	0.936
	30	0.963	0.949	0.944	0.959	0.955	0.953	0.957	0.948	0.938	0.946
	50	0.949	0.960	0.953	0.954	0.957	0.940	0.943	0.964	0.953	0.943
	70	0.957	0.959	0.947	0.948	0.947	0.964	0.943	0.939	0.938	0.939
	100	0.953	0.941	0.963	0.933	0.956	0.945	0.948	0.956	0.958	0.949
	200	0.949	0.962	0.950	0.951	0.955	0.940	0.946	0.937	0.952	0.960
	400	0.951	0.950	0.945	0.942	0.932	0.957	0.957	0.962	0.944	0.937
	600	0.946	0.963	0.956	0.950	0.933	0.953	0.964	0.953	0.957	0.958
	800	0.948	0.962	0.950	0.945	0.944	0.929	0.923	0.951	0.945	0.953
SE-SD bias	10	0.049	0.033	0.026	0.015	0.013	0.006	0.005	0.000	0.005	0.002
	30	0.009	-0.002	0.004	0.004	0.003	0.000	0.002	-0.002	-0.002	0.001
	50	0.006	0.006	0.004	0.002	0.002	-0.003	-0.001	0.003	0.002	0.000
	70	0.009	0.004	0.002	0.000	0.000	0.002	0.000	-0.001	-0.001	0.000
	100	-0.002	0.000	0.001	-0.002	0.001	0.000	0.000	0.001	0.000	0.000
	200	0.000	0.002	0.000	0.000	0.002	-0.001	0.000	-0.001	0.000	0.000
	400	0.001	0.001	0.000	-0.002	-0.002	0.001	0.001	0.000	0.000	-0.001
	600	-0.001	0.000	0.000	0.000	-0.001	0.000	0.001	0.000	0.000	0.000
	800	0.001	0.001	0.000	0.000	0.000	0.000	-0.001	0.000	0.000	0.000