侯杰泰结构方程模型.ppt

KT HAU SEM p. 1,结构方程模型及其应用Structural Equation Model and Its Applications,侯杰泰 Kit-Tai Hau,香港中文大学教育心理系 Educational Psychology Dept, The Chinese University of Hong Kong,KT HAU SEM p. 2,结构方程模型及其应用Structural Equation Model and Its Applications,I 简介 I Introduction,KT HAU SEM p. 3,100个推理测验分数 21, 31, 32, 05, 06, 09, 10, 22, 29, 18, 11, 01, 39, 92, 23, 27, 93, 97, 30, 02, 96, 40, 53, 78, 04, 98, 36, 07, 08, 24, 54, 55, 77, 99, 34, 03, 86, 87, 59, 60, 15, 62, 63, 43, 52, 28, 79, 58, 65, 95, 81, 85, 57, 14, 17, 33, 16, 19, 20, 37, 25, 69, 84, 61, 64, 68, 70, 42, 45, 72, 83, 89, 44, 38, 47, 71, 00, 73, 12, 35, 82, 56, 75, 41, 46, 49, 50, 94, 66, 67, 76, 51, 88, 90, 74, 13, 26, 80, 48, 91 均值Mean=53，标准差SD(Std Dev)=15,KT HAU SEM p. 4,100名学生在9个不同学科间的相关系数(correlation coefficient matrix),KT HAU SEM p. 5,KT HAU SEM p. 6,再生/隐含矩阵 (reproduced/implied matrix),KT HAU SEM p. 7,KT HAU SEM p. 8,KT HAU SEM p. 9,检查模型的准确性(accuracy)和简洁性(parsimony) 拟合优度指数（goodness of fit index），简称为拟合指数(fit index): 、NNFI、CFI df=不重复元素non-duplicating elements, p(p+1)/2 估计参数estimated parameters 在前面例子 df =9 x 10/2 21 = 24,KT HAU SEM p. 10,依据 及指定模型 找出与 相距最小的,样本相关（或协方差）矩阵 correlation/covariance matrix一个或多个有理据的可能模型 (alternative models),输出 Output,输入 Input,SEM 程式program (e.g., LISREL),、各路径参数（因子负荷loading、 因子相关系数factor correlations等） 各种拟合指数,KT HAU SEM p. 11,结构方程模型及其应用Structural Equation Model and Its Applications,I 简介 I Introduction,KT HAU SEM p. 12,100名学生在9个不同学科间的相关系数(correlation coefficient matrix),KT HAU SEM p. 13,KT HAU SEM p. 14,KT HAU SEM p. 15,KT HAU SEM p. 16,KT HAU SEM p. 17,KT HAU SEM p. 18,KT HAU SEM p. 19,KT HAU SEM p. 20,KT HAU SEM p. 21,KT HAU SEM p. 22,KT HAU SEM p. 23,_ (no. of estimated parameters) 模型 df NNFI CFI 需要估计的参数个数,_,M1 24 40 .973 .98221 = 9 Load + 9 Uniq + 3 Corr,M2 27 503 .294 .471 18 = 9 Load + 9 Uniq,M3 26 255 .647 .745 19 = 9 Load + 9 Uniq + 1 Corr,M4 26 249 .656 .752 19 = 9 Load + 9 Uniq + 1 Corr,M5 27 263 .649.72718 = 9 Load + 9 Uniq,M6 24 422 .337 .558 21 = 9 Load + 9 Uniq + 3 Corr,M7 21 113 .826 .898 24 = 9 Load + 9 Uniq + 6 Corr,_,KT HAU SEM p. 24,模型比较 (Model Comparison),自由度(df), 拟合程度 (fit), 不能保证最好，可能存在更简洁(parsimonious) 又拟合(fit)得很好的模型 输入Input: 相关（或协方差）矩阵correlation/covariance matrix 一个或多个有理据的可能模型 (alternative models) 输出Output: 既符合某指定模型，又与 差异最小的矩阵 估计各路径参数parameter（因子负荷loading、因子相关系数factor correlations等）。 计算出各种拟合指数(goodness of fit indexes),KT HAU SEM p. 25,依据 及指定模型 找出与 相距最小的,样本相关（或协方差）矩阵 correlation/covariance matrix一个或多个有理据的可能模型 (alternative models),输出 Output,输入 Input,SEM 程式program (e.g., LISREL),、各路径参数（因子负荷loading、 因子相关系数factor correlations等） 各种拟合指数,KT HAU SEM p. 26,结构方程模型的重要性 Structural Equation Model，SEM Covariance Structure Modeling，CSM LInear Structural RELationship ， LISREL (EQS, AMOS, Mplus, etc.),KT HAU SEM p. 27,结构方程模型的结构,测量模型 (measurement model) , ,外源指标exogenous（如6项社经指标）,内生指标endogenous（如语、数、英成绩）,因子负荷矩阵 (loading),误差项 (uniqueness, measurement errors),结构模型 (structural model),KT HAU SEM p. 28,结构方程模型的优点,同时处理多个因变量(many dependent variables) 同时估计因子结构factor structure和因子关系,KT HAU SEM p. 29,容许自变量independent variable和因变量dependent variable含测量误差measurement error 传统方法（如回归regression）假设自变量 independent variable没有误差 ,_ 英文 中文 _ 观察 真 误差 观察 真 误差 得分 分数 得分 分数 observed true error observed true error score score score score X Tx e Y Ty e_ 8 7 +1 5 3 +2 5 6 -1 6 7 -1 7 5 +2 9 7 +2 9 8 +1 5 8 -3 . . . . . . X = Tx + e Y = Ty + e if r (X, Y) = 0.5 r (Tx,Ty)= 0.5 /(rXt-t) (rYt-t)1/2 = 0.71 (assume rt-t =0.7),KT HAU SEM p. 30,容许更大弹性的测量模型 估计整个模型的拟合程度model fit用以比较不同模型 SEM包括:回归分析regression、因子分析(验证性因子分析CFA、 探索性因子分析EFA)、检验t-test、方差分析ANOVA、比较各组因子均值group mean comparison、交互作用模型interaction、实验设计expt design,KT HAU SEM p. 31,结构方程模型及其应用Structural Equation Model and Its Applications,I 简介 I Introduction,KT HAU SEM p. 32,Additional notes on EFA Exploratory Factor Analyses (EFA),In the above analyses, we have a structure in mind to test, this process is called confirmatory factor analysis (CFA) It is also possible that we have no “theory” in mind to test, i.e., we have the following research questions: How many cluster of subjects are there? How do these 9 subjects relate to each of these clusters (factors)? Which of these subjects are more closely related/correlated than others?,KT HAU SEM p. 33,Using LISREL, run the following program DA NI=9 NO=100 KM 1.00 0.12 1.00 0.08 0.08 1.00 0.50 0.11 0.08 1.00 0.48 0.03 0.12 0.45 1.00 0.07 0.46 0.15 0.08 0.11 1.00 0.05 0.44 0.15 0.12 0.12 0.44 1.00 0.14 0.17 0.53 0.14 0.08 0.10 0.06 1.00 0.16 0.05 0.43 0.10 0.06 0.08 0.10 0.54 1.00 PC NC=6 OU,KT HAU SEM p. 34,The output: Principal Components Analysis Eigenvalues and Eigenvectors PC_1 PC_2 PC_3 PC_4 PC_5 PC_6 - - - - - - Eigenvalue 2.56 1.66 1.63 0.69 0.59 0.56 % Variance 28.42 18.49 18.15 7.65 6.50 6.18 Cum% Var 28.42 46.91 65.06 72.71 79.21 85.39,KT HAU SEM p. 35,3 rules to determine number of factorsa) EV(eigenvalue特征值 ) 1b) scree test碎石: greatest change in slopec) meaningful dimensions,KT HAU SEM p. 36,Assume 3 factors, we run the following program and obtain further information DA NI=9 NO=100 KM 1.00 0.12 1.00 0.08 0.08 1.00 0.50 0.11 0.08 1.00 0.48 0.03 0.12 0.45 1.00 0.07 0.46 0.15 0.08 0.11 1.00 0.05 0.44 0.15 0.12 0.12 0.44 1.00 0.14 0.17 0.53 0.14 0.08 0.10 0.06 1.0 0.16 0.05 0.43 0.10 0.06 0.08 0.10 0.54 1.0 FA NF=3 OU,KT HAU SEM p. 37,The Output: Varimax-Rotated Factor Loadings Factor 1 Factor 2 Factor 3 Unique Var - - - - VAR 1 0.10 0.73 0.04 0.46 VAR 2 0.09 0.06 0.66 0.55 VAR 3 0.63 0.05 0.12 0.58 VAR 4 0.08 0.67 0.08 0.53 VAR 5 0.04 0.65 0.07 0.57 VAR 6 0.07 0.05 0.68 0.54 VAR 7 0.05 0.07 0.65 0.57 VAR 8 0.82 0.08 0.07 0.32 VAR 9 0.65 0.09 0.04 0.56,factors are assumed to be uncorrelated正交,KT HAU SEM p. 38,Promax-Rotated Factor Loadings Factor 1 Factor 2 Factor 3 Unique Var - - - - VAR 1 0.73 -0.03 0.03 0.46 VAR 2 0.00 0.66 0.02 0.55 VAR 3 -0.02 0.06 0.64 0.58 VAR 4 0.68 0.02 0.01 0.53 VAR 5 0.66 0.01 -0.03 0.57 VAR 6 -0.01 0.68 0.00 0.54 VAR 7 0.01 0.66 -0.02 0.57 VAR 8 0.00 -0.01 0.83 0.32 VAR 9 0.03 -0.03 0.66 0.56 Factor Correlations Factor 1 Factor 2 Factor 3 - - - Factor 1 1.00 Factor 2 0.19 1.00 Factor 3 0.21 0.22 1.00,Factors allowed to be correlated斜交,KT HAU SEM p. 39,KT HAU SEM p. 40,验证性因子分析,17个题目:学习态度及取向 A、B、C、D、E4、4、3、3、3题 350个学生,Confirmatory Factor Analysis, CFA,KT HAU SEM p. 41,KT HAU SEM p. 42,Confirmatory Factor Analysis Example 1 DA NI=17 NO=350 MA=KM KM SY 1 .34 1 MO NX=17 NK=5 LX=FU,FI PH=ST TD=DI,FR PA LX 4(1 0 0 0 0) 4(0 1 0 0 0) 3(0 0 1 0 0) 3(0 0 0 1 0) 3(0 0 0 0 1) OU MI SS SC,KT HAU SEM p. 43,1 23 4,running LISREL,KT HAU SEM p. 44,1 23 4,KT HAU SEM p. 45,什么情况下固定 (fixed, FI)？ 两个变量（指标或因子）间没有关系，将元素固定为0 例如，不从属，将因子负荷(LX 1,2)固定为0。又如，因子和因子没有相关，PH 1,2 固定为0。 需要设定因子的度量单位(set metric/scale) 因子没有单位 (metric)，无法计算。 一种将所有因子的方差固定为1(或其他常数)，简称为固定方差法 (fixed variance method) 一种是在每个因子中选择一个负荷固定为1(或其他常数)，简称为固定负荷法(fixed loading) 什么情况下设定为自由(free,FR):所有需要估计的参数,KT HAU SEM p. 46,KT HAU SEM p. 47,补充例子 9个题目: 第1个因子: 第1、2、3题 第2个因子: 第4、5、6题 第3个因子: 第7、8、9题 设因子1, 2, 3互有相关,固定方差法 (fixed variance),KT HAU SEM p. 48,固定方差法 (fixed variance) MO LX=9 NK=3 LX=FU,FI PH=ST TD=DI,FR FR LX 1,1 LX 2,1 LX 3,1 LX 4,2 LX 5,2 FR LX 6,2 LX 7,3 LX 8,3 LX 9,3 固定负荷法(fixed loading) MO LX=9 NK=3 LX=FU,FI PH=SY,FR TD=DI,FR FR LX 2,1 LX 3,1 LX 5,2 LX 6,2 LX 8,3 LX 9,3 VA 1 LX 1,1 LX 4,2 LX 7,3,KT HAU SEM p. 49,设因子1和因子3无关(uncorrelated)，因子1和因子2、因子2和因子3相关(correlated) 固定方差法(fixed variance) MO LX=9 NK=3 LX=FU,FI PH=ST TD=DI,FR FR LX 1,1 LX 2,1 LX 3,1 LX 4,2 LX 5,2 FR LX 6,2 LX 7,3 LX 8,3 LX 9,3 FI PH 1,3 固定负荷法(fixed loading) MO LX=9 NK=3 LX=FU,FI PH=SY,FR TD=DI,FR FR LX 2,1 LX 3,1 LX 5,2 LX 6,2 LX 8,3 LX 9,3 VA 1 LX 1,1 LX 4,2 LX 7,3 FI PH 1,3,KT HAU SEM p. 50,Number of Input Variables 17 (读入变量个数) Number of Y - Variables 0 (Y-变量个数) Number of X - Variables 17 (X-变量个数) Number of ETA - Variables 0 (Y-因子个数) Number of KSI - Variables 5 (X-因子个数) Number of Observations 350 (样品个数),KT HAU SEM p. 51,Parameter Specifications 参数设定 LAMBDA-X KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - VAR 1 1 0 0 0 0 VAR 2 2 0 0 0 0 VAR 3 3 0 0 0 0 VAR 4 4 0 0 0 0 VAR 5 0 5 0 0 0 VAR 6 0 6 0 0 0 VAR 7 0 7 0 0 0 VAR 8 0 8 0 0 0 VAR 9 0 0 9 0 0 VAR 10 0 0 10 0 0 VAR 11 0 0 11 0 0 VAR 12 0 0 0 12 0 VAR 13 0 0 0 13 0 VAR 14 0 0 0 14 0 VAR 15 0 0 0 0 15 VAR 16 0 0 0 0 16 VAR 17 0 0 0 0 17,KT HAU SEM p. 52,PHI KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - KSI 1 0 KSI 2 18 0 KSI 3 19 20 0 KSI 4 21 22 23 0 KSI 5 24 25 26 27 0 THETA-DELTA VAR1 VAR2 VAR3 VAR4 VAR5 VAR6 VAR7 VAR8 VAR9 VAR10 28 29 30 31 32 33 34 35 36 37 VAR 11 VAR 12 VAR 13 VAR 14 VAR 15 VAR 16 VAR 17 38 39 40 41 42 43 44,KT HAU SEM p. 53,Number of Iterations = 19 LISREL Estimates (Maximum Likelihood) 参数估计 LAMBDA-X KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - VAR 1 0.59 - - - - - - - - (0.06) 9.49 VAR 2 0.58 - - - - - - - - (0.06) 9.30 VAR 3 0.62 - - - - - - - - (0.06) 9.93 VAR 4 0.05 - - - - - - - - (0.07) 0.81,参数 (parameter),Standard Error, SE,t-value = 参数 / SE 0.59 / 0.06 = 9.49,KT HAU SEM p. 54,VAR 5 - - 0.64 - - - - - - (0.06) 10.46 VAR 6 - - 0.57 - - - - - - (0.06) 9.32 VAR 7 - - 0.51 - - - - - - (0.06) 8.29 VAR 8 - - 0.28 - - - - - - (0.06) 4.41 VAR 9 - - - - 0.59 - - - - (0.06) 9.56,KT HAU SEM p. 55,VAR 10 - - - - 0.61 - - - - (0.06) 9.99 VAR 11 - - - - 0.64 - - - - (0.06) 10.47 VAR 12 - - - - - - 0.62 - - (0.06) 10.28 VAR 13 - - - - - - 0.66 - - (0.06) 10.84 VAR 14 - - - - - - 0.54 - - (0.06) 8.96 VAR 15 - - - - - - - - 0.65 (0.06) 11.14 VAR 16 - - - - - - - - 0.72 (0.06) 12.19 VAR 17 - - - - - - - - 0.55 (0.06) 9.36,KT HAU SEM p. 56,PHI KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - KSI 1 1.00 KSI 2 0.52 1.00 (0.07) 7.06 KSI 3 0.40 0.53 1.00 (0.08) (0.07) 5.21 7.24 KSI 4 0.51 0.54 0.48 1.00 (0.07) (0.07) (0.07) 6.97 7.47 6.60 KSI 5 0.42 0.50 0.44 0.50 1.00 (0.07) (0.07) (0.07) (0.07) 5.77 6.99 6.22 7.17,KT HAU SEM p. 57,THETA-DELTA VAR 1 VAR 2 VAR 3 VAR 4 VAR 5 VAR 6 - - - - - - 0.65 0.66 0.61 1.00 0.59 0.67 (0.07) (0.07) (0.07) (0.08) (0.07) (0.07) 9.63 9.85 9.02 13.19 8.82 10.21 VAR 7 VAR 8 VAR 9 VAR 10 VAR 11 VAR 12 - - - - - - 0.74 0.92 0.66 0.63 0.59 0.61 (0.07) (0.07) (0.07) (0.07) (0.07) (0.06) 11.05 12.70 9.96 9.46 8.80 9.46 VAR 13 VAR 14 VAR 15 VAR 16 VAR 17 - - - - - 0.57 0.70 0.57 0.48 0.69 (0.07) (0.07) (0.06) (0.06) (0.06) 8.70 10.75 9.13 7.49 10.91,KT HAU SEM p. 58,Goodness of Fit Statistics 拟合优度统计量 Degrees of Freedom = 109 Minimum Fit Function Chi-Square = 194.57 (P = 0.00) Normal Theory Weight Least Sq Chi-Sq = 190.15 (P = 0.00) Estimated Non-centrality Parameter (NCP) = 81.15 90 Percent Confidence Interval for NCP = (46.71 ; 123.45) Minimum Fit Function Value = 0.56 Population Discrepancy Function Value (F0) = 0.23 90 Percent Confidence Interval for F0 = (0.13 ; 0.35) Root Mean Square Error of Approximation (RMSEA) = 0.046 90 Percent Confidence Interval for RMSEA = (0.035 ; 0.057) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.71 Expected Cross-Validation Index (ECVI) = 0.80 90 Percent Confidence Interval for ECVI = (0.70 ; 0.92) ECVI for Saturated Model = 0.88 ECVI for Independence Model = 5.78,Chi df = 190.15-109,0.56 x (N-1) = chi 0.56x349=194.57,KT HAU SEM p. 59,Chi-Square for Independence Model with 136 df = 1982.04 Independence AIC = 2016.04 Model AIC = 278.15 Saturated AIC = 306.00 Independence CAIC = 2098.63 Model CAIC = 491.90 Saturated CAIC = 1049.26 Normed Fit Index (NFI) = 0.90 Non-Normed Fit Index (NNFI) = 0.94 Parsimony Normed Fit Index (PNFI) = 0.72 Comparative Fit Index (CFI) = 0.95 Incremental Fit Index (IFI) = 0.95 Relative Fit Index (RFI) = 0.88 Critical N (CN) = 263.34 Root Mean Square Residual (RMR) = 0.054 Standardized RMR = 0.054 Goodness of Fit Index (GFI) = 0.94 Adjusted Goodness of Fit Index (AGFI) = 0.92 Parsimony Goodness of Fit Index (PGFI) = 0.67,NNFI=TLI,CFI = RNI,Oldest LISREL indexes: RMR, SRMR,GFI, AGFI,KT HAU SEM p. 60,Modification Indices for LAMBDA-X 修正指数 KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - VAR 1 - - 0.06 0.66 0.09 2.53 VAR 2 - - 0.38 0.53 0.23 0.11 VAR 3 - - 0.72 0.01 0.03 1.49 VAR 4 - - 0.00 0.03 0.01 0.03 VAR 5 7.73 - - 9.62 9.23 1.50 VAR 6 0.01 - - 3.29 1.07 1.50 VAR 7 0.12 - - 0.25 0.12 2.26 VAR 8 41.35 - - 3.66 22.02 4.78 VAR 9 0.40 0.02 - - 2.19 0.22 VAR 10 0.03 0.10 - - 0.30 0.22 Maximum Modification Index is 41.35 for Element ( 8,1)LX 修正指数:该参数由固定改为自由估计， 会减少的数值,KT HAU SEM p. 61,Completely Standardized Solution LAMBDA-X KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - VAR 1 0.59 - - - - - - - - VAR 2 0.58 - - - - - - - - VAR 3 0.62 - - - - - - - - VAR 4 0.05 - - - - - - - - VAR 5 - - 0.64 - - - - - - VAR 6 - - 0.57 - - - - - - VAR 7 - - 0.51 - - - - - - VAR 8 - - 0.28 - - - - - - VAR 9 - - - - 0.59 - - - - VAR 10 - - - - 0.61 - - - - VAR 11 - - - - 0.64 - - - - VAR 12 - - - - - - 0.62 - - VAR 13 - - - - - - 0.66 - - VAR 14 - - - - - - 0.54 - - VAR 15 - - - - - - - - 0.65 VAR 16 - - - - - - - - 0.72 VAR 17 - - - - - - - - 0.55,KT HAU SEM p. 62,PHI KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 - - - - - KSI