1、結(jié)構(gòu)方程模型第二講課件結(jié)構(gòu)方程模型第二講課件1、模型與假設(shè)測(cè)量模型 結(jié)構(gòu)模型 2STRUCTURAL EQUATION MODELING1、模型與假設(shè) 結(jié)構(gòu)模型 4STRUCTURAL EQUA假設(shè)e 與h；d與x；z與x不相關(guān)； e，d，z兩兩不相關(guān)且均值為0；協(xié)方差矩陣:cov(x)=F; cov(z)=Y; cov(e)= Qe; cov(d)= Qd3STRUCTURAL EQUATION MODELING假設(shè)e 與h；d與x；z與x不相關(guān)； e，d，z兩兩不相關(guān)且2、模型的估計(jì)LISREL 基于協(xié)方差矩陣PLS (Partial Least Square)基于主成份SEM (LISR

2、EL; PLS): 第二代數(shù)據(jù)分析技術(shù)Bagozzi and Fornell, 19824STRUCTURAL EQUATION MODELING2、模型的估計(jì)LISREL 基于協(xié)方差矩陣6STRUCTU2.1 協(xié)方差結(jié)構(gòu)觀測(cè)變量(y, x)的協(xié)方差矩陣為一般結(jié)構(gòu)S(q)，所以也叫協(xié)方差結(jié)構(gòu)分析。5STRUCTURAL EQUATION MODELING2.1 協(xié)方差結(jié)構(gòu)觀測(cè)變量(y, x)的協(xié)方差矩陣為一般結(jié)構(gòu)2.2 模型的識(shí)別定義：如果S(q1)=S(q2) 必有 q1 = q2 ，則稱結(jié)構(gòu)方程模型為可識(shí)別的(identified).考慮方程S(q)S，如果方程數(shù)小于參數(shù)個(gè)數(shù)，則必有參數(shù)不

3、能由已知量表示出來，此時(shí)模型為不可識(shí)別的(under identified).(S是p維矩陣，方程個(gè)數(shù)有多少個(gè)？)6STRUCTURAL EQUATION MODELING2.2 模型的識(shí)別定義：如果S(q1)=S(q2) 必有 如果一個(gè)參數(shù)不能由已知量表示出來，則稱該參數(shù)是不可識(shí)別的(under identified);如果一個(gè)參數(shù)能且只能由已知量的一個(gè)表達(dá)式表示，則稱該參數(shù)是恰好識(shí)別的(just identified);如果一個(gè)參數(shù)可以由已知量的兩個(gè)以上表達(dá)式表示，則稱該參數(shù)是超識(shí)別的(over identified);如果至少有一個(gè)參數(shù)是超識(shí)別的，則模型是超識(shí)別的；如果至少有一個(gè)參數(shù)是不

4、可識(shí)別的，則模型是不可識(shí)別的。7STRUCTURAL EQUATION MODELING如果一個(gè)參數(shù)不能由已知量表示出來，則稱該參數(shù)是不可識(shí)別的(u因子模型識(shí)別法則8STRUCTURAL EQUATION MODELING因子模型識(shí)別法則10STRUCTURAL EQUATION 2.3、參數(shù)估計(jì)目的：總體協(xié)方差矩陣S(q)與樣本協(xié)方差矩陣S盡可能接近。擬合函數(shù)(fit function)F(S,S(q) 1) 非負(fù)； 2）連續(xù)； 3）F(S,S(q)0當(dāng)且僅當(dāng)S(q)=S。估計(jì)：找出 使得擬合函數(shù)取得最小值。9STRUCTURAL EQUATION MODELING2.3、參數(shù)估計(jì)目的：總體

5、協(xié)方差矩陣S(q)與樣本協(xié)方差矩陣1、極大似然估計(jì)(Maximum Likelihood, ML)假設(shè)觀測(cè)誤差是多元正態(tài)分布，則擬合函數(shù)為基本性質(zhì)：1) ML估計(jì)是漸近無偏的(asymptotically unbiased)2) ML估計(jì)是一致估計(jì)(consistent);3) ML估計(jì)是漸近有效的(asymptotically efficient);4) ML估計(jì)是漸近正態(tài)分布：10STRUCTURAL EQUATION MODELING1、極大似然估計(jì)(Maximum Likelihood, M5) ML估計(jì)的最優(yōu)擬合值漸近卡方分布，即其中p*=p(p+1)/2，t為自由參數(shù)的個(gè)數(shù)。這個(gè)結(jié)

6、果可以用于整個(gè)模型的檢驗(yàn)。H0: S = S(q)。證明可參看(Bollen, 1989)11STRUCTURAL EQUATION MODELING5) ML估計(jì)的最優(yōu)擬合值漸近卡方分布，即13STRUCTU2、未加權(quán)最小二乘估計(jì)(unweighted Least Squares, ULS)(意義：殘差矩陣S-S(q)全部元素的平方和)ULS估計(jì)是一致估計(jì)，但是不是漸近有效的；沒有尺度不變性； ULS估計(jì)的最優(yōu)擬合值不是漸近卡方分布。12STRUCTURAL EQUATION MODELING2、未加權(quán)最小二乘估計(jì)(unweighted Least S3、廣義最小二乘估計(jì)(Generaliz

7、ed Least Squares, GLS)(意義：殘差矩陣S-S(q)全部元素的加權(quán)平方和，其權(quán)重為樣本協(xié)方差矩陣的逆矩陣)可以證明：當(dāng)誤差假設(shè)為正態(tài)分布時(shí)，GLS估計(jì)與ML估計(jì)是漸近等價(jià)的。因此， GLS估計(jì)具有ML估計(jì)一樣的漸近性質(zhì)。13STRUCTURAL EQUATION MODELING3、廣義最小二乘估計(jì)(Generalized Least S4、廣義加權(quán)最小二乘估計(jì)(Generally weighted Least Squares, WLS)其中：s為由樣本協(xié)方差矩陣S的所有下半對(duì)角元素組成的向量，稱為“拉直”向量，記為s =Vecs(S) = (s11,s21,s22,s31

8、,s32,s33,spp) s =Vecs(S) = (s11,s21,s22,s31,s32,s33,spp)W為p*維正定矩陣， p*=p(p+1)/2。14STRUCTURAL EQUATION MODELING4、廣義加權(quán)最小二乘估計(jì)(Generally weighte特別，1）取 W = SS， : Kronecker 乘積，則WLS 估計(jì)化為GLS；2）取 W = S(qML)S(qML)，則WLS 估計(jì)化為ML；一般，Browne(1982,1984)建議W取wgh,ij = mghij sghsij其中wghij是4階樣本中心矩。這是一種漸近與分布無關(guān)的估計(jì)(asymptoti

9、cally distribution-free, ADF)，具有許多與ML估計(jì)相同的漸近性質(zhì)。15STRUCTURAL EQUATION MODELING特別，17STRUCTURAL EQUATION MODEL2.4、模型評(píng)價(jià)目的：評(píng)價(jià)模型擬合的好壞。方法：擬合指數(shù)，對(duì)模型進(jìn)行整體評(píng)價(jià)； 測(cè)定系數(shù)，評(píng)價(jià)模型對(duì)數(shù)據(jù)的解釋能力；參數(shù)檢驗(yàn)，評(píng)價(jià)參數(shù)的顯著性。16STRUCTURAL EQUATION MODELING2.4、模型評(píng)價(jià)目的：評(píng)價(jià)模型擬合的好壞。18STRUCTU2.4.1 擬合指數(shù)擬合指數(shù)，也叫擬合優(yōu)度統(tǒng)計(jì)量(Goodness-of-fit statistics)，反映模型擬合好壞

10、。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 (RMR)17STRUCTURAL EQUATION MODELING2.4.1 擬合指數(shù)擬合指數(shù)，也叫擬合優(yōu)度統(tǒng)計(jì)量19STRU1、Chi-Square (c2) c2(n-1)F( )。已經(jīng)證明，對(duì)ML, GLS和 WLS估計(jì)，在一定條件下，

11、漸近趨向于c2分布，自由度為(p*-t)， p*=p(p+1)/2，t為自由參數(shù)的個(gè)數(shù)。判斷：c2越小，說明擬合越好。當(dāng)卡方檢驗(yàn)顯著時(shí)(p-值0.1)，模型擬合不好；如果不顯著，模型可以接受。注1：對(duì)ML和GLS，數(shù)據(jù)輸入為樣本協(xié)方差時(shí)，c2正確；如果是相關(guān)系數(shù)矩陣，則只有模型具有度量不變性(scale-invariant)才能給出正確的c2值。對(duì)WLS，還需給出正確的權(quán)陣W才可以。18STRUCTURAL EQUATION MODELING1、Chi-Square (c2)20STRUCTURAL 2、Goodness-of-fit indices (GFI) Adjusted Goodne

12、ss-of-fit index (AGFI) 擬合優(yōu)度指數(shù) Joreskog & Sorbom (1981)給出：其中p*=p(p+1)/2，d為模型的自由度。一般認(rèn)為GFI大于0.9時(shí)，擬合良好。19STRUCTURAL EQUATION MODELING2、Goodness-of-fit indices (GFI3、Root mean square error of approximation (RMSEA) 近似誤差均方根 (Steiger & Lind, 1980)其中df 是卡方的自由度。c2-df 稱為離中參數(shù)(Noncentrality parameter,NCP; Steige

13、r, 1980)?？傮w差距函數(shù)(Population Discrepancy Function, PDF):20STRUCTURAL EQUATION MODELING3、Root mean square error of ap4、Standardized Root mean square residual (SRMR) 標(biāo)準(zhǔn)化殘差均方根注1：其他擬合指數(shù)參看侯杰泰（2019）注2：以上指數(shù)僅反映整個(gè)模型的擬合程度。整個(gè)模型擬合很好，不表示每個(gè)關(guān)系符合得也很好。21STRUCTURAL EQUATION MODELING4、Standardized Root mean squar2.4.2 測(cè)

14、定系數(shù)類似于回歸分析中的R2(Coefficient of Determinant)1）第i個(gè)方程的測(cè)定系數(shù):其中 是第i個(gè)方程的殘差的方差的估計(jì)值， 是第i個(gè)變量的樣本方差。方程的測(cè)定系數(shù)用于評(píng)價(jià)第i個(gè)方程對(duì)數(shù)據(jù)的解釋能力。22STRUCTURAL EQUATION MODELING2.4.2 測(cè)定系數(shù)類似于回歸分析中的R2(Coeffic2）整個(gè)模型的測(cè)定系數(shù)其中|Y|是Y的行列式，|S|是S的行列式。計(jì)算時(shí)Y一般用的估計(jì)值， S用擬合的協(xié)方差矩陣或者樣本協(xié)方差矩陣代替。測(cè)定系數(shù)在01之間，越大越好。測(cè)定系數(shù)與方程個(gè)數(shù)有關(guān)，因此，建議用于評(píng)價(jià)方程，評(píng)價(jià)總體模型還是擬合指數(shù)為優(yōu)。23STRU

15、CTURAL EQUATION MODELING2）整個(gè)模型的測(cè)定系數(shù)25STRUCTURAL EQUATI模型修正指數(shù)(Modification Index, MI): 通過對(duì)自由參數(shù)增加、減少、變動(dòng)，引起的卡方的改變量。在LISREL中通過MI命令，每個(gè)固定參數(shù)都會(huì)給出一個(gè)修正指數(shù)，它等于當(dāng)該參數(shù)設(shè)為自由參數(shù)時(shí)所減少的卡方值。24STRUCTURAL EQUATION MODELING模型修正指數(shù)(Modification Index, MI)由于每個(gè)參數(shù)都會(huì)給出標(biāo)準(zhǔn)誤(standard error)，因此可以對(duì)參數(shù)進(jìn)行顯著性檢驗(yàn)。也就是檢驗(yàn)參數(shù)是否為零。比如，檢驗(yàn)結(jié)果兩個(gè)潛在變量之間的系

16、數(shù)不顯著，就應(yīng)該固定該參數(shù)為零，然后修正模型并重新估計(jì)。2.4.3 參數(shù)檢驗(yàn)25STRUCTURAL EQUATION MODELING由于每個(gè)參數(shù)都會(huì)給出標(biāo)準(zhǔn)誤(standard error)，3、模型的另一種估計(jì)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 r

17、egression. Consequently, PLS is more suited for predictive applications and theory building, in contrast to covariance-based SEM. 26STRUCTURAL EQUATION MODELING3、模型的另一種估計(jì)方法：PLS (Partial LeasConditions when you mightconsider using PLSDo you work with theoretical models that involve latent constructs?

18、Do you have multicollinearity problems with variables that tap into the same issues?Do you want to account for measurement error?Do you have non-normal data?27STRUCTURAL EQUATION MODELINGConditions when you mightconsDo you have a small sample set?Do you wish to determine whether the measures you dev

19、eloped are valid and reliable within the context of the theory you are working in?Do you have formative as well as reflective measures?Conditions when you mightconsider using PLS (Cont)28STRUCTURAL EQUATION MODELINGDo you have a small sample set29STRUCTURAL EQUATION MODELING31STRUCTURAL EQUATION MOD

20、ELINGThe basic PLS algorithm forLatent variable path analysis Stage 1: Iterative estimation of weights and LV scores starting at step #4,repeating steps #1 to #4 until convergence is obtained.Stage 2: Estimation of paths and loading coefficients. Stage 3: Estimation of location parameters.30STRUCTUR

21、AL EQUATION MODELINGThe basic PLS algorithm forLa31STRUCTURAL EQUATION MODELING33STRUCTURAL EQUATION MODELINGComputer SoftwaresLVPLSPLS-GUIPLS- Graph (Wynne W. Chin )32STRUCTURAL EQUATION MODELINGComputer SoftwaresLVPLS34STRUCConsiderations when choosingbetween PLS and LISREL ObjectivesTheoretical c

22、onstructs - indeterminate vs. defined Epistemic relationshipsTheory requirementsEmpirical factorsComputational issues - identification & speed33STRUCTURAL EQUATION MODELINGConsiderations when choosingbObjectives Prediction versus explanation34STRUCTURAL EQUATION MODELINGObjectives Prediction versus

23、Theoretical constructs -Indeterminate versus defined For PLS - the latent variables are estimatedas linear aggregates or components. Thelatent variable scores are estimated directly.If raw data is used, scoring coefficients areestimated. For LISREL - Indeterminacy35STRUCTURAL EQUATION MODELINGTheore

24、tical constructs -IndetEpistemic 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 covaria

25、nce matrix to a prediction/reconstruction forecast of the raw data matrix36STRUCTURAL EQUATION MODELINGEpistemic relationships LatentTheory requirements LISREL expects strong theory(confirmation mode) PLS is flexible37STRUCTURAL EQUATION MODELINGTheory requirements LISREL exEmpirical factors Distrib

26、utional assumptions PLS estimation is a “rigid” technique thatrequires only “soft” assumptions about the distributional characteristics of the raw data. LISREL requires more stringent conditions.38STRUCTURAL EQUATION MODELINGEmpirical factors DistributioEmpirical factors (continued) Sample Size depe

27、nds on power analysis, butmuch smaller for PLS PLS heuristic of ten times the greater of thefollowing 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.39STRUCTURAL EQUATION MODELINGEmpirical factors (continued) Empirical factors (continued)Types of measuresPLS can use categorical through ratio measuresLISREL generally expects interval level, otherwise need PRELIS preprocessing.