CFAandSEM验证性因子分析与结构方程建模.pdf

structural equation modelingqg 1 correlation and regression covariance structure analysis, structural equation modeling and lisrelequation modeling and lisrel what is sem? how are estimations obtained? the idea of goodness-of-fit model identification testing nested modelstesting nested models kenneth law 同济大学 2010 the idea of cfa 2 confirmatory factoranalysisconfirmatory factor analysis (exploratory factor analysis) efa kenneth law 同济大学 2010 two measurement modelstwo measurement models fairness classical measurement model 3 fairness perceptionx1= 1+ 1 x2= 1+ 2 latent 2 1 2 citd l x1= 111+ 1 item2item1 x1x2 latent co-generic measurement model x2= 211+ 2 latent kenneth law 同济大学 2010 exploratory factor analysis (efa)exploratory factor analysis (efa) 4 factors var f1f2f3h2 60060236 f f1 1f f2 2 x1.60 -.06 .02.36 x2.81.12 -.03.67 x3.17.73.08.60 x4.01.65 -.04.42 4 x5.03.10.87.65 x6.12.22.65.47 x1x2x6x3x4x5 f f3 3 kenneth law 同济大学 2010 confirmatory factor analysis (cfa)confirmatory factor analysis (cfa) 5 factors varf1f2f3 6000x1.6000 x2.8100 x30.730 x40.650 f f3 3f f1 1f f2 2 fit 4 x500.87 x600.65 x1x2x6x3x4x5 fit kenneth law 同济大学 2010 confirmatory factor analysis (cfa)confirmatory factor analysis (cfa) x = + fairness i trust in si 6 x1= 111+ 1 x2= 211+ 2 x3= 322+ 3 perceptionsupervisor x3 322+ 3 x4= 422+ 4 x1x2x3x4 item2 item3item1 item4 1 x2x3x4 1. my supervisor is fair. 2. my supervisor treats us without biases. 3. i believe that my supervisor will protect me. 4. i trust my supervisor. kenneth law 同济大学 2010 confirmatory factor analysis (cfa)yy() x1= 111+ 1 7 x1 111 1 x2= 211+ 2 x3= 322+ 31 1+ 3 x4= 422+ 4 x1x2x3x4 item2 item3item1 item4 1 x2x3x4 no cross loadings 1. my supervisor is fair. 2. my supervisor treats us without biases. no cross loadings (cross loading is common in personality inventories) 3. i believe that my supervisor will protect me. 4. i trust my supervisor. kenneth law 同济大学 2010 confirmatory factor analysis (cfa)confirmatory factor analysis (cfa) x1= 111+ 1 8 11111 x2= 211+ 2 x3= 322+ 3 + x4= 422+ 4 let the errors of x2 2. try to reproduce the observed data using our tit abc a estimates; 3. if we can perfectly reproduce the observation using our estimates, we have an estimated model with 100% fit. the parameters are assumed to be b c p highly believable. kenneth law 同济大学 2010 a b = 1 c theoretical model 25 bc 1 2 c = 2 r b = 1 model abc rab= .61 r= 42 rab 1 rac= 2 rbc = 12 a12 b112 c212 rac= .42 rbc= .35 obi problem 121 2 bb observation abc a b b bb .61 * .42 = .2562 .65 * .45 = .2925 .63 * .53 = .3339 optimal (total 121 2 bb b c 222 121 2 .61.42.35minimumgoodness of fit abbbb p( error is lowest) 1122121 2 ; ; bbbb 121 2 .61.42.35minimumgoodness of fit bbbbb kenneth law 同济大学 2010 a b = 1 c = 2 26 bc 1 2 2 rab= 1 theoretical rab= .61 r= 42 rac= 2 rbc = 12 theoretical model rac= .42 rbc= .35observedest1est2est3 rab= .61 obi rac= .42 rbc= .35 gdf fit093807620952 observation goodness-of-fit.0938.0762.0952 goodness of fit = (rab-rab)2+ (rac-rac)2+ (rbc-rbc)2 kenneth law 同济大学 2010 a = 1 theoretical model 27 bc 1 1 c = 2 r b = 1 b rab= .61 r= 42 rab 1 rac= rbc = abc a12 b13+12 c212 rac= .42 rbc= .35 obi 212 problem observation abc a + problem .61 * .42 = .2562 ; b3 = .0938 b1* b2= b1*b2+ b3 optimal (total b c 0goodness of fit agoodness of fit b p( error is lowest) kenneth law 同济大学 2010 path model r rand r 28 x1 r14 r13 and r34 r24 r23 and r34 transformational leadership leader-membersubordinate exchange (lmx)performance transactional x3 x4 leadership x2 kenneth law 同济大学 2010 structural equation modelingstructural equation modeling transformational 29 1 1 1 x 2 x 1 y y 1 1 2 leadership 1 2 y 2 2 lmxjob performance 3 x 2 3 y 4 y 3 4 3 2 4 x 4 2 4 4 transactional leadershipleadership kenneth law 同济大学 2010 a 30 r b = 1 theoretical model abcbc 1 2 rab 1 rac= 2 rbc = 12 a12 b112 c212abc a a b c problem 61 * 42 = 2562 observation .61 .42 = .2562 .65 * .45 = .2925 .63 * .53 = .3339 b1* b2= b1*b2 optimal (total error is lowest) 222 121 2 .61.42.35minimumgoodness of fit abbbb ) 121 2 .61.42.35minimumgoodness of fit bbbbb kenneth law 同济大学 2010 = 1 a 31 1 c = 2 r b = 1 b bc 1 theoretical model rab 1 rac= rbc= abc a12 b13+12 c212 abc a 212 a b c problem + problem .61 * .42 = .2562 ; b3 = .0938 b1* b2= b1*b2+ b3 optimal (total observation 0goodness of fit agoodness of fit b p( error is lowest) kenneth law 同济大学 2010 an example trial valuescorrelationscriterion an example 1 rabracrbc d2 observed.61.42.23 iterations 1.5.5.50.50.25.018900 (.61-.50)2+(.42-.50)2+(.23-.25)2 32 a 1a.501.5.501.50.2505.018701 1b.5.501.50.501.2505.019081 2.6.5.60.50.30.011400 2a.601.5.601.50.3005.011451 ()()() bc 1 2 2b.6.501.60.501.3006.011645 3.6.4.60.40.24.000600 3a.601.4.601.40.2404.000589 3b.6.401.60.401.2406.000573 bc rab= .61 4.6.41.60.41.246.000456 4a.601.41.601.41.2464.000450 4b.6.411.60.411.2466.000457 5.61.41.61.41.2501.000504 ab rac= .42 rbc= .23 5a.601.41.601.41.2464.0004503 5b.602.41.602.41.2468.0004469 5c.601.411.601.411.2470.0004514 6.602.41.602.41.2468.0004469 6a.603.41.603.41.2472.0004459 6b.602.411.603.411.2474.0004485 7.603.409.603.409.2462.0004480 kenneth law 同济大学 2010 an example trial valuescorrelationscriterion an example a 1 rabracrbc d2 observed.61.42.23 iterations 1.5.5.50.50.25.018900 33 a 1 2 1.5.5.50.50.25.018900 1a.501.5.501.50.2505.018701 1b.5.501.50.501.2505.019081 2.6.5.60.50.30.011400 2a.601.5.601.50.3005.011451 bc 2b.6.501.60.501.3006.011645 3.6.4.60.40.24.000600 3a.601.4.601.40.2404.000589 3b.6.401.60.401.2406.000573 rab= .61 rac= .42 4.6.41.60.41.246.000456 4a.601.41.601.41.2464.000450 4b.6.411.60.411.2466.000457 5.61.41.61.41.2501.000504 rbc= .235a.601.41.601.41.2464.0004503 5b.602.41.602.41.2468.0004469 5c.601.411.601.411.2470.0004514 6.602.41.602.41.2468.0004469 6a.603.41.603.41.2472.0004459 6b.602.411.603.411.2474.0004485 7.603.409.603.409.2462.0004480 minimum value of the fit function kenneth law 同济大学 2010 lisrel estimation an examplelisrel estimation an example 34 b b observed var-cov matrix b ac 1 abc a1.0 b.451.0 * * ab bcabbc bra crbrra c.25.351.0 2 var( )var( ) var( )var()var( ) abab aa br ara 22 12 var( )1 var( )(.45).20 var( )1;.45;.25; ab if our estimates are thatathen a br 22 2 var( )var()var( ) cov( , )cov( ,)cov( , )var( ) cov( , )cov( ,)cov( , )var( ) cov()cov() ab bcab bc ababab ab bcab bcab bc cr r ar ra a ba r ara ara a ca r r ar ra ar ra b cr a r r ar r 2 cov()var( )a ar ra 22 2 22 ( )() var( )var( )(.45) (.25).02 cov( , )var( ).45 cov( , )var( )(.45)(.25).16 ()( ) ab ab bc ab ab bc cr ra a bra a cr ra b 2 ( 45) ( 25)07 cov( , )cov(,) abab bcab b b cr a r r ar rcov( , )var( ) cab bc a ar ra 2 cov( , )var( ) ab bc b cr ra 2 (.45) (.25).07 var(a)var(b)var(c)cov(ab)cov(ac)cov(bc) obs1.01.01.0.45.25.35 kenneth law 同济大学 2010 est1.0.20.02.45.16.07 .00.80.98.00.09.28total 2 2.00.64.95.00.01.081.67 lisrel estimation an examplelisrel estimation an example 35 obd var(a)var(b)var(c)cov(ab)cov(ac)cov(bc) observed var-cov matrix obs1.01.01.0.45.25.35 11.20.02.45.16.071.6735 21 122035017081 642221.1.22.03.50.17.081.6422 31.2.24.03.54.19.09.16364 41.3.26.03.59.20.091.6564拟合指数 51.4.28.03.63.22.101.7013 61.5.30.04.68.24.111.7719 71.6.32.04.72.25.111.8681 minimum estimation error b 81.7.34.04.77.27.121.9897 91.8.36.04.81.28.132.1367 101 938058630132 3093 b ac 1 kenneth law 同济大学 2010 101.9.38.05.86.30.132.3093 estimated var-cov matrix the covariance matrixthe covariance matrix variance covariance matrix ( ) 36 x1 x2 x3 x4 y1 y2 y3 y4 x1 1.5 x2 .551.3 x3 .24 .22 2.1 variance-covariance matrix ( ) 1 x transformational reliability x4 .19 .21 .601.8 y1 .32 .35 .31 .38 1.1 y2 .24 .21 .29 .33 .651.4 y3 .11 .10 .38 .42 .44 .42 1.9 y4 .19 .17 .21 .18 .41 .47 .491.2 1 1 x 2 x 3 x 1 2 1 y 2 y 3 y lmxperformance y 2 3 x 4 x 4 y transactional actual observations theoretical relationship kenneth law 同济大学 2010 the estimation procedurep 37 x1= 11+ 1 x2= 21+ 2 x3= 32+ 3 1 1 x 2 x 1 1 y 2 y 3 y 33 2 3 x4= 42+ 4 y1= 51+ 5 + 2 3 x 4 x 2 4 y estimated structure of y2= 61+ 6 y3= 72+ 7 y4= 82+ 8 structure of covariance matrix var(x)= 2 var( ) + var( ) 1= 11+ 9 2= 21+ 32+ 10 var(x1) = 12var(1) + var(1) cov(x1, x2) = cov(11+ 1 , 32+ 3) = 1 3cov(1, 2) 10 kenneth law 同济大学 2010 the estimation procedurep 1 1 x 2 x 1 y x1 x2 x3 x4 y1 y2 y3 y4 38 x1= 11+ 1 x2= 21+ 2 x = + 2 2 3 x 1 2 2 y 3 y 4 y x1 1.5 x2 .551.3 x3 .24 .22 2.1 x4 .19 .21 .601.8 y1 .32 .35 .31 .38 1.1 y2 .24.21 .29.33.651.4 x3= 32+ 3 x4= 42+ 4 y1= 51+ 5 4 x y2 .24 .21 .29 .33 .651.4 y3 .11 .10 .38 .42 .44 .42 1.9 y4 .19 .17 .21 .18 .41 .47 .491.2 var(x1) = 12var(1) + var(1) cov(x1, x2) = 1 3cov(1, 2) estimated structure of i y2= 61+ 6 y3= 72+ 7 y4= 82+ 8 estimated covariance matrix observed covariance matrix covariance matrix y4 8 2 8 1= 11+ 9 2= 21+ 32+ 10 compare with matrix matrix goodness of fit index = | | kenneth law 同济大学 2010 a simplified example of fita simplified example of fit x = + 1 1 x 1 y x1 x2 x3 x4 y1 y2 y3 y4 x11 5 39 x1= 11+ 1 x2= 21+ 2 x3= 32+ 3 1 2 2 x 3 x 1 2 2 y 3 y 4 y x1 1.5 x2 .24 1.3 x3 .13 .22 2.1 x4 .19 .21 .60 1.8 y1 .32 .35 .31 .38 1.1 y2 .24 .21 .29 .33 .65 1.4 x4= 42+ 4 y1= 51+ 5 y2= 61+ 6 2 4 x observed correlation between and24 y3 .11 .10 .38 .42 .44 .42 1.9 y4 .19 .17 .21 .18 .41 .47 .49 1.2 y3= 72+ 7 y4= 82+ 8 1= 11+ 9 x1and x2= .24 estimated correlation between 1 11+ 9 2= 21+ 32+ 10 x1and x2based on the estimated set of( 1, 2, 3, 1 1. ) = .05 gdf fit i d| | goodness of fit index = | | kenneth law 同济大学 2010 gdof figoodness-of-fit fit value function (f) 40 fit value function (f) = (n-1) f best estimates of parameter set ( 1, 2, 3, 1 1. ) ( 1, 2, 3, 1 1. ) kenneth law 同济大学 2010 goodness of fit indices 41 goodness of fit indices the goodness of fit index (gfi) do not depend on sample size explicitly (note: )dhh bhd l fithe sampling distribution still depends on n) and measure how much better the model fits as compared with no model at all (i.e. all parameters are zero). , ( ) 1 , (0) f f s gfi s the adjusted goodness of fit index (agfi) adjusts for degrees of freedom. (1) (1) 2 1 k k gfi d agfi k is the number of variables; d is the degrees of freedom of the model. goodness-of-fit statistics 42 goodness of fit statistics fn 1 2 ( )1 i d d f pnfi f i (1)/ (1)/1 ii ii nf df d nf d nnfi i f f nfi1 (1);0 (1);0 1 ii max nf d max nfd cfi i 0 /(1);0fmax f dn rmsea dd i (1);0 ii max nfd dd f is the minimum value of the fit function for the estimated model; fiis the minimum value of the fit function for the independencemodel (when all the correlations and covariances are zero); d is the degrees of freedom of the model.g transformational 1 1 1 x 2 x 1 y y 1 1 2 transformational leadership 43 1 2 y 2 2 lmx job performance 3 x 2 3 y 4 y 3 4 3 transactional leadership job performance 2 4 x 4 2 4 4 based on the observed covariance matrix, find a set of best estimates of parameter set ( 1, 2, 3, ) ti iith fitlfti to minimize the fit value function kenneth law 同济大学 2010 44 44 lisrel programlisrel program kenneth law 同济大学 2010 the simplis languagegg 45 kenneth law 同济大学 2010 46 title: test program variable: 46 observed variables: x1 x2 x3 x4 x5 x6 latent variables: a b latent variables: a b correlation matrix: 1.0 .50 1.0 52 43 1 0 .52 .43 1.0 .22 .16 .15 1.0 .12 .31 .20 .69 1.0 .26 .13 .29 .66 .72 1.0 sample size: 139 raw data from file: test.txt your model p relationship: x1 x2 x3 = a x4 x5 x6 = bx4 x5 x6 b admissibilities = off iterations = 1000 p th dipath diagram end of problem kenneth law 同济大学 2010 47 title: test program variable: your model 47 observed variables: x1 x2 x3 x4 x5 x6 latent variables: a b x1= 1a + 1 x2= 2a + 2 latent variables: a b correlation matrix: 1.0 .50 1.0 52 43 1 0 x3= 3a + 3 x4= 4b + 4 .52 .43 1.0 .22 .16 .15 1.0 .12 .31 .20 .69 1.0 .26 .13 .29 .66 .72 1.0 sample size: 139 444 x5= 5b + 5 x6= 6b + 6 p relationship: x1 x2 x3 = a x4 x5 x6 = b a b x4 x5 x6 b admissibilities = off iterations = 1000 p th di xxx path diagram end of problem x1x2x3 x4x5x6 kenneth law 同济大学 2010 48 48 lisrel outputlisrel output kenneth law 同济大学 2010 49 your model 49 your model x1= 1a + 1 x2= 2a + 2x2 2a + 2 x3= 3a + 3 b +x4= 4b + 4 x5= 5b + 5 x6= 6b + 6 a b x1x2x3 x4x5x6 kenneth law 同济大学 2010 50 50 kenneth law 同济大学 2010 51 51 kenneth law 同济大学 2010 52 your model 52 x1= 1a + 1 x2= 2a + 2 x3= 3a + 3 x4= 4b + 4 444 x5= 5b + 5 x6= 6b + 6 a b xxx .76 x1x2x3 x4x5x6 kenneth law 同济大学 2010 53 53 model 2 rmsea (root mean square error of ii)approximation) tucker-lewis index (tli, nnfi) incremental fit index (cfi) kenneth law 同济大学 2010 54 54 a b x1x2x3 x4x5x6 the modification indicesmodification indices suggest to add an error covarianceadd an error covariance between and decrease in chi-squaredecrease in chi-squarenew estimate x5 x115.4 -0.17 x5x220 60 21x5 x2 20.6 0.21 x6 x2 13.1 -0.17 kenneth law 同济大学 2010 scaling in semscaling in sem 55 x1x2x3x4x1x2x3x4 kenneth law 同济大学 2010 scaling in semscaling in sem 56 x1= 111+ 1 + x2= 211+ 2 x3= 322+ 3 x = + x4= 422+ 4 x1x2x3x4 kenneth law 同济大学 2010 scale by the first item of each factorscale by the first item of each factor 57 x1= 111+ 1 + x2= 211+ 2 x3= 322+ 3 x = + x4= 422+ 4 x1x2x3x4 kenneth law 同济大学 2010 scale by standardization 58 scale by standardization x1= 111+ 1 x = + 2 2 1 1 2 1 x2= 211+ 2 x3= 322+ 3 x = + x4= 422+ 4 x1x2x3x4 kenneth law 同济大学 2010 lisrel programming scalinglisrel programmingscaling 59 relationships: obsx1 obsx2 = latent x 1*obsx1 obsx2 = latent x1 obsx1 obsx2 latent x (.5) obsx1 obsx2 = latent x kenneth law 同济大学 2010 other issues in sem 60 60 other issues in sem 1.degrees of freedom 2.identification 3nested model3.nested model 4.missing data in sem 5.single indicatorg 6.parceling of indicators 7.factorial invariance in cross-cultural research kenneth law 同济大学 2010 other issues in sem 61 61 other issues in sem 1.degrees of freedom 2.identification 3nested model3.nested model 4.missing data in sem 5.single indicatorg 6.parceling of indicators 7.factorial invariance in cross-cultural research kenneth law 同济大学 2010 degrees of freedomg 62 x1x2x3 x4x5x6 x7x8x9 number of variance-covariance terms = number of parameters to be estimated = df fd 9(10)/2 = 45 21 degrees of freedom = 45-21 = 24 kenneth law 同济大学 2010 degrees of freedomg 63 x1x2x3 nbfiit3(4)/26number of variance-covariance terms = number of parameters to be estimated = degrees of freedom = 3(4)/2 = 6 7 1 kenneth law 同济大学 2010 other issues in sem 64 64 other issues in sem 1.degrees of freedom 2.identification 3nested model3.nested model 4.missing data in sem 5.single indicatorg 6.parceling of indicators 7.factorial invariance in cross-cultural research kenneth law 同济大学 2010 the issue of identification in sem 65 fairness trust in x fairness perception supervisor x1x2x3x4 x xxxx x11.3.4.6 x2.31.5.7 x3.4.51.3 x1x2x3x4 x3.4.51.3 x4.6.7.31 1. my supervisor is fair. 2. my supervisor treats us without biases. 10 observed statistics 9 estimated parameters kenneth law 同济大学 2010 3. i believe that my supervisor will protect me. 4. i trust my supervisor. identification in sem 66 fairness