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1、结构方程模型及其应用 第二讲 模型的估计,Wang Shu Jia Business School University of Shenzhen 2019年7月19日星期五,1、模型与假设 测量模型,结构模型,PAGE3,STRUCTURAL EQUATION MODELING,假设,e 与h；d与x；z与x不相关； e，d，z两两不相关且均值为0； 协方差矩阵: cov(x)=F; cov(z)=Y; cov(e)= Qe; cov(d)= Qd,PAGE4,STRUCTURAL EQUATION MODELING,2、模型的估计,LISREL 基于协方差矩阵 PLS (Partial Le
2、ast Square)基于主成份 SEM (LISREL; PLS): 第二代数据分析技术Bagozzi and Fornell, 1982,PAGE5,STRUCTURAL EQUATION MODELING,2.1 协方差结构,观测变量(y, x)的协方差矩阵为,一般结构S(q)，所以也叫协方差结构分析。,PAGE6,STRUCTURAL EQUATION MODELING,2.2 模型的识别,定义：如果S(q1)=S(q2) 必有 q1 = q2 ，则称结构方程模型为可识别的(identified). 考虑方程S(q)S，如果方程数小于参数个数，则必有参数不能由已知量表示出来，此时模型为
3、不可识别的(under identified). (S是p维矩阵，方程个数有多少个？),如果一个参数不能由已知量表示出来，则称该参数是不可识别的(under identified); 如果一个参数能且只能由已知量的一个表达式表示，则称该参数是恰好识别的(just identified); 如果一个参数可以由已知量的两个以上表达式表示，则称该参数是超识别的(over identified); 如果至少有一个参数是超识别的，则模型是超识别的；如果至少有一个参数是不可识别的，则模型是不可识别的。,PAGE8,STRUCTURAL EQUATION MODELING,因子模型识别法则,PAGE9,ST
4、RUCTURAL EQUATION MODELING,2.3、参数估计,目的：总体协方差矩阵S(q)与样本协方差矩阵S尽可能接近。 拟合函数(fit function)F(S,S(q) 1) 非负； 2）连续； 3）F(S,S(q)0当且仅当S(q)=S。 估计：找出 使得拟合函数取得最小值。,1、极大似然估计(Maximum Likelihood, ML) 假设观测误差是多元正态分布，则拟合函数为 基本性质： 1) ML估计是渐近无偏的(asymptotically unbiased) 2) ML估计是一致估计(consistent); 3) ML估计是渐近有效的(asymptoticall
5、y efficient); 4) ML估计是渐近正态分布：,5) ML估计的最优拟合值渐近卡方分布，即 其中p*=p(p+1)/2，t为自由参数的个数。 这个结果可以用于整个模型的检验。 H0: S = S(q)。 证明可参看(Bollen, 1989),2、未加权最小二乘估计(unweighted Least Squares, ULS) (意义：残差矩阵S-S(q)全部元素的平方和) ULS估计是一致估计，但是不是渐近有效的；没有尺度不变性； ULS估计的最优拟合值不是渐近卡方分布。,3、广义最小二乘估计(Generalized Least Squares, GLS) (意义：残差矩阵S-S
6、(q)全部元素的加权平方和，其权重为样本协方差矩阵的逆矩阵) 可以证明：当误差假设为正态分布时，GLS估计与ML估计是渐近等价的。 因此， GLS估计具有ML估计一样的渐近性质。,4、广义加权最小二乘估计(Generally weighted Least Squares, WLS) 其中：s为由样本协方差矩阵S的所有下半对角元素组成的向量，称为“拉直”向量，记为 s =Vecs(S) = (s11,s21,s22,s31,s32,s33,spp) s =Vecs(S) = (s11,s21,s22,s31,s32,s33,spp) W为p*维正定矩阵， p*=p(p+1)/2。,特别， 1）取
7、 W = SS， : Kronecker 乘积，则WLS 估计化为GLS； 2）取 W = S(qML)S(qML)，则WLS 估计化为ML； 一般，Browne(1982,1984)建议W取 wgh,ij = mghij sghsij 其中wghij是4阶样本中心矩。这是一种渐近与分布无关的估计(asymptotically distribution-free, ADF)，具有许多与ML估计相同的渐近性质。,PAGE16,STRUCTURAL EQUATION MODELING,2.4、模型评价,目的：评价模型拟合的好坏。 方法： 拟合指数，对模型进行整体评价； 测定系数，评价模型对数据的解
8、释能力； 参数检验，评价参数的显著性。,PAGE17,STRUCTURAL EQUATION MODELING,2.4.1 拟合指数,拟合指数，也叫拟合优度统计量 (Goodness-of-fit statistics)，反映模型拟合好坏。 1、Chi-Square (c2) 2、Goodness-of-fit index (GFI) 3、Adjusted Goodness-of-fit index (AGFI) 4、Root mean square error of approximation (RMSEA) 5、Standardized Root mean square residual
9、(RMR),1、Chi-Square (c2) c2(n-1)F( )。已经证明，对ML, GLS和 WLS估计，在一定条件下，渐近趋向于c2分布，自由度为(p*-t)， p*=p(p+1)/2，t为自由参数的个数。 判断：c2越小，说明拟合越好。当卡方检验显著时(p-值0.1)，模型拟合不好；如果不显著，模型可以接受。 注1：对ML和GLS，数据输入为样本协方差时，c2正确；如果是相关系数矩阵，则只有模型具有度量不变性(scale-invariant)才能给出正确的c2值。 对WLS，还需给出正确的权阵W才可以。,2、Goodness-of-fit indices (GFI) Adjuste
10、d Goodness-of-fit index (AGFI) 拟合优度指数 Joreskog & Sorbom (1981)给出： 其中p*=p(p+1)/2，d为模型的自由度。 一般认为GFI大于0.9时，拟合良好。,3、Root mean square error of approximation (RMSEA) 近似误差均方根 (Steiger Steiger, 1980)。 总体差距函数(Population Discrepancy Function, PDF):,4、Standardized Root mean square residual (SRMR) 标准化残差均方根 注1：其
11、他拟合指数参看侯杰泰（2004） 注2：以上指数仅反映整个模型的拟合程度。整个模型拟合很好，不表示每个关系符合得也很好。,PAGE22,STRUCTURAL EQUATION MODELING,2.4.2 测定系数,类似于回归分析中的R2(Coefficient of Determinant) 1）第i个方程的测定系数: 其中 是第i个方程的残差的方差的估计值， 是第i个变量的样本方差。 方程的测定系数用于评价第i个方程对数据的解释能力。,2）整个模型的测定系数 其中|Y|是Y的行列式，|S|是S的行列式。计算时Y一般用的估计值， S用拟合的协方差矩阵或者样本协方差矩阵代替。 测定系数在01之
12、间，越大越好。 测定系数与方程个数有关，因此，建议用于评价方程，评价总体模型还是拟合指数为优。,模型修正指数(Modification Index, MI): 通过对自由参数增加、减少、变动，引起的卡方的改变量。 在LISREL中通过MI命令，每个固定参数都会给出一个修正指数，它等于当该参数设为自由参数时所减少的卡方值。,PAGE25,STRUCTURAL EQUATION MODELING,由于每个参数都会给出标准误(standard error)，因此可以对参数进行显著性检验。也就是检验参数是否为零。 比如，检验结果两个潜在变量之间的系数不显著，就应该固定该参数为零，然后修正模型并重新估计
13、。,2.4.3 参数检验,PAGE26,STRUCTURAL EQUATION MODELING,3、模型的另一种估计方法：PLS (Partial Least Square),PLS, the second major SEM technique, is designed to explain variance, i.e., to examine the significance of the relationships and their resulting R2, as in linear regression. Consequently, PLS is more suited for
14、predictive applications and theory building, in contrast to covariance-based SEM.,PAGE27,STRUCTURAL EQUATION MODELING,Conditions when you might consider using PLS,Do you work with theoretical models that involve latent constructs? Do you have multicollinearity problems with variables that tap into t
15、he same issues? Do you want to account for measurement error? Do you have non-normal data?,PAGE28,STRUCTURAL EQUATION MODELING,Do you have a small sample set? Do you wish to determine whether the measures you developed are valid and reliable within the context of the theory you are working in? Do yo
16、u have formative as well as reflective measures?,Conditions when you might consider using PLS (Cont),PAGE30,STRUCTURAL EQUATION MODELING,The basic PLS algorithm for Latent variable path analysis,Stage 1: Iterative estimation of weights and LV scores starting at step #4,repeating steps #1 to #4 until
17、 convergence is obtained. Stage 2: Estimation of paths and loading coefficients. Stage 3: Estimation of location parameters.,PAGE32,STRUCTURAL EQUATION MODELING,Computer Softwares,LVPLS PLS-GUI PLS- Graph (Wynne W. Chin ),PAGE33,STRUCTURAL EQUATION MODELING,Considerations when choosing between PLS a
18、nd LISREL,Objectives Theoretical constructs - indeterminate vs. defined Epistemic relationships Theory requirements Empirical factors Computational issues - identification & speed,PAGE34,STRUCTURAL EQUATION MODELING,Objectives, Prediction versus explanation,PAGE35,STRUCTURAL EQUATION MODELING,Theore
19、tical constructs - Indeterminate versus defined,For PLS - the latent variables are estimated as linear aggregates or components. The latent variable scores are estimated directly. If raw data is used, scoring coefficients are estimated. For LISREL - Indeterminacy,PAGE36,STRUCTURAL EQUATION MODELING,
20、Epistemic relationships,Latent constructs with reflective indicators - LISREL & PLS Emergent constructs with formative indicators - PLS By choosing different weighting “modes” the model builder shifts the emphasis of the model from a structural causal explanation of the covariance matrix to a predic
21、tion/reconstruction forecast of the raw data matrix,PAGE37,STRUCTURAL EQUATION MODELING,Theory requirements, LISREL expects strong theory (confirmation mode) PLS is flexible,PAGE38,STRUCTURAL EQUATION MODELING,Empirical factors, Distributional assumptions PLS estimation is a “rigid” technique that r
22、equires only “soft” assumptions about the distributional characteristics of the raw data. LISREL requires more stringent conditions.,PAGE39,STRUCTURAL EQUATION MODELING,Empirical factors (continued),Sample Size depends on power analysis, but much smaller for PLS PLS heuristic of ten times the greate
23、r of the following two (ideally use power analysis) -construct with the greatest number of formative indicators -construct with the greatest number of structural paths going into it LISREL heuristic - at least 200 cases or 10 times the number of parameters estimated.,PAGE40,STRUCTURAL EQUATION MODEL
24、ING,Empirical factors (continued),Types of measures PLS can use categorical through ratio measures LISREL generally expects interval level, otherwise need PRELIS preprocessing.,PAGE41,STRUCTURAL EQUATION MODELING,Computational issues - Identification,Are estimates unique? Under recursive models - PL
25、S is always identified LISREL - depends on the model. Ideally need 4 or more indicators per construct to be over determined, 3 to be just identified. Algebraic proof for identification.,PAGE42,STRUCTURAL EQUATION MODELING,Computational issues - Speed,PLS estimation is fast and avoids the problem of negative variance estimates (i.e., Heywood cases) PLS needs less computing time and memory. The PLS-Graph program can handle up to 400 indicators. Models with 50 to 100 are estimated in a matter of seconds.,PAGE43,STRUCTURAL EQUATION MODELING,PAGE44,STRUCTURAL EQUATION MODELING,Any Questions?,