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1 1 Stata SEM LightStone 1 5 SEM Stata Alan C. Acock, Discovering Structural Equation Modeling Using Stata, Revised Edition, Stata Press. Introduction to confirmatory factor analysis 9 Stata14

2 2 1 SEM p,. y ij = z i1 b 1j + z i2 b 2j + + z iq b qj + e ij y ij j i z ik k (common factor) i b kj (factor loadings) e ij (unique factor) y ij p q Tarlov et al. (1989) bg2.dta 6, 5. 1, 5. use bg2,clear. des 1 Alan C. Acock, Discovering Structural Equation Modeling Using Stata, Revised Edition, Stata Press Stata.

3 3 1 Contains data from obs: 568 Physician-cost data vars: 7 11 Feb :54 size: 14,768 (_dta has notes) storage display value variable name type format label variable label clinid int %9.0g Physician identifier bg2cost1 float %9.0g Best health care is expensive bg2cost2 float %9.0g Cost is a major consideration bg2cost3 float %9.0g Determine cost of tests first bg2cost4 float %9.0g Monitor likely complications only bg2cost5 float %9.0g Use all means regardless of cost bg2cost6 float %9.0g Prefer unnecessary tests to missing tests Sorted by: clinid bg2cost1 bg2cost2 bg2cost3 bg2cost4 bg2cost5 bg2cost6,. factor bg2cost1-bg2cost6 (obs=568) Factor analysis/correlation Number of obs = 568 Method: principal factors Retained factors = 3 Rotation: (unrotated) Number of params = 15 Factor Eigenvalue Difference Proportion Cumulative Factor Factor Factor Factor Factor Factor LR test: independent vs. saturated: chi2(15) = Prob>chi2 = Factor loadings (pattern matrix) and unique variances Variable Factor1 Factor2 Factor3 Uniqueness bg2cost bg2cost bg2cost bg2cost bg2cost bg2cost

4 (Eigenvalue) 1 1: 2:. y ij = z i1 b 1j + z i2 b 2j + z i3 b 3j + e ij z ik ( ), b kj ( ) (Uniqueness) z ik,. 1- =. saturated independent 1.2 (principal component factor analysis) ( ), 2.. z = a 1 x 1 + a 2 x 2, x :nlsy97cfa.dta 20.?

5 4 1 x1: x2: x3: x4: x5: x6: x7: x8: x9: x10: (Conservative). use "nlsy97cfa.dta", clear. codebook x1-x10, compact ( ) Conservative Conservative x1-x10. factor x1-x10, pcf (obs=1,617) Factor analysis/correlation Number of obs = 1,617 Method: principal-component factors Retained factors = 2 Rotation: (unrotated) Number of params = 19 Factor Eigenvalue Difference Proportion Cumulative Factor Factor Factor Factor Factor Factor Factor Factor Factor Factor LR test: independent vs. saturated: chi2(45) = Prob>chi2 =

6 Factor loadings (pattern matrix) and unique variances Variable Factor1 Factor2 Uniqueness x x x x x x x x x x (Eigenvalue) 1 2 (loadings) 3.91 = 10 1 factor 2 i 0.4 (0.3 ), ( ). 10,. factor x1-x9, pcf (obs=1,625) Factor analysis/correlation Number of obs = 1,625 Method: principal-component factors Retained factors = 1 Rotation: (unrotated) Number of params = 9 Factor Eigenvalue Difference Proportion Cumulative Factor Factor Factor Factor Factor Factor Factor Factor Factor

7 6 1 LR test: independent vs. saturated: chi2(36) = Prob>chi2 = Factor loadings (pattern matrix) and unique variances Variable Factor1 Uniqueness x x x x x x x x x Uniqueness, ( ) 61%. x1, 61% Uniqueness, 9 ( ) Conservative? α α = k r 1 + (k 1) r

8 alpha x1-x9,item label Test scale = mean(unstandardized items) item-test item-rest interitem Item Obs Sign corr. corr. cov. alpha Label x GOVT RESPONSIBILITY - PROVIDE JOBS 2006 x GOVT RESPNSBLTY - KEEP PRICES UND CTRL 2006 x GOVT RESPNSBLTY - HLTH CARE FOR SICK 2006 x GOVT RESPNSBLTY -PROV ELD LIV STAND 2006 x GOVT RESPNSBLTY -PROV IND HELP 2006 x GOVT RESPNSBLTY -PROV UNEMP LIV STAND 2006 x GOVT RESPNSBLTY -REDUCE INC DIFF 2006 x GOVT RESPNSBLTY -PROV COLL FIN AID 2006 x GOVT RESPNSBLTY -PROV DECENT HOUSING 2006 Test scale mean(unstandardized items) Test scale α = 0.81( 0.70 ), x1 α ( ),. r (= 0.17) 40 α 0.8. egen conserve = rowmean(x1-x9) ( ) conserve 7,097 ( 8,985 ) conserve

9 8 1. summarize conserve, detail conserve Percentiles Smallest 1% 1 1 5% % Obs 1,888 25% Sum of Wgt. 1,888 50% Mean Largest Std. Dev % % Variance % Skewness % Kurtosis , 0.51, 1 1,888. histogram conserve, norm freq ( ) Conservative.,

10 conservf1. factor x1-x9, pcf ( ). predict conservf1 (regression scoring assumed) Scoring coefficients (method = regression) Variable Factor1 x x x x x x x x x ,, conserve conservf1 do hist01.do (hist01.do ) histogram conserve, norm freq name(a, replace) /// xtitle(mean Conservatism Score) ylabel(0(25)175) histogram conservf1, norm freq name(b, replace) /// xtitle(factor Score on Conservatism) ylabel(0(25)175) graph combine A B

11 10 1 ( ),,,., ( ) 1.3 (CFA:Conirmatory Factor Analysis) ( 1), CFA Conservative Conservative 9 (x 1 x 9 ) : ϵ 1 ϵ 9, ( ) /SEM/, SEM SEM sembuilder

12 Conservative, x1-x9, SEM / (ml) mlmv:, adf:

13 12 1. sem (Conservative->x1-x9) (7360 observations with missing values excluded) Endogenous variables Measurement: Exogenous variables Latent: x1 x2 x3 x4 x5 x6 x7 x8 x9 Conservative Fitting target model: ( ) Structural equation model Number of obs = 1,625 Estimation method = ml Log likelihood = ( 1) [x1]conservative = 1 OIM Coef. Std. Err. z P> z [95% Conf. Interval] Measurement x1 <- Conservative 1 (constrained) _cons x2 <- Conservative _cons x3 <- Conservative _cons x4 <- Conservative _cons x5 <- Conservative _cons x6 <- Conservative _cons x7 <- Conservative _cons x8 <- Conservative _cons x9 <- Conservative _cons var(e.x1) var(e.x2) var(e.x3) var(e.x4) var(e.x5) var(e.x6) var(e.x7) var(e.x8) var(e.x9) var(conservative)

14 LR test of model vs. saturated: chi2(27) = , Prob > chi2 = x1 = α 1 + β 1 Conservative +ϵ 1 x1 x9 9 ϵ Conservative, Conservative, 7,360 x1,... x9 x1 β 1 x1., β, 7 ( ). sem (Conservative -> x7 x1-x6 x8 x9 ) β( 1 )

15 14 1. sem, standardized Structural equation model Number of obs = 1,625 Estimation method = ml Log likelihood = ( 1) [x1]conservative = 1 OIM Standardized Coef. Std. Err. z P> z [95% Conf. Interval] Measurement x1 <- Conservative _cons x2 <- Conservative _cons x3 <- Conservative _cons x4 <- Conservative _cons x5 <- Conservative _cons x6 <- Conservative _cons x7 <- Conservative _cons x8 <- Conservative _cons x9 <- Conservative _cons var(e.x1) var(e.x2) var(e.x3) var(e.x4) var(e.x5) var(e.x6) var(e.x7) var(e.x8) var(e.x9) var(conservative) 1... LR test of model vs. saturated: chi2(27) = , Prob > chi2 = saturated model

16 ,, Conservative 1, x1 0.55, SEM. estat gof,stats(all) Fit statistic Value Description Likelihood ratio chi2_ms(27) model vs. saturated p > chi chi2_bs(36) baseline vs. saturated p > chi Population error RMSEA Root mean squared error of approximation 90% CI, lower bound upper bound pclose Probability RMSEA <= 0.05 Information criteria AIC Akaike s information criterion BIC Bayesian information criterion Baseline comparison CFI Comparative fit index TLI Tucker-Lewis index Size of residuals SRMR Standardized root mean squared residual CD Coefficient of determination

17 16 1 chi2 ms(27) SEM saturated model( ) : CFA. SEM saturated model,., chi2 bs(36) baseline model saturated model : baseline model saturated model( ) ( ) baseline model., RMSEA Root Mean Squared Error of Approximation 0.05, 0.08 RMSEA = T (N 1) df T = max (model chi-squared df, 0), N., CFA,? CFI(comparative fit index) CFI:89.8%. ( ), 89.8%

18 Standardized Root Mean Squared Residual(SRMR) SRMA = , CD(Coefficient of Determination). 1. SEM Conservative, / /. sem x1-x9 ( ).estat framework,fitted Fitted covariances of observed variables observed Sigma x1 x2 x3 x4 x5 observed x x x x x x x x x observed Sigma x6 x7 x8 x9 observed x x x x Fitted means of observed variables observed mu x1 x2 x3 x4 x5 mu observed mu x6 x7 x8 x9 mu

19 18 1 9, 45 ( 35+ 9) 8 ( 1 )+ 9 +Conservative =18, = 27,, saturated model ( ), ( )?. sem (Conservative -> x1-x9) ( ). estat mindices Modification indices Standard MI df P>MI EPC EPC cov(e.x1,e.x2) cov(e.x1,e.x4) cov(e.x1,e.x6) cov(e.x1,e.x7) cov(e.x1,e.x8) cov(e.x2,e.x5) cov(e.x2,e.x6) cov(e.x2,e.x9) cov(e.x3,e.x4) cov(e.x3,e.x5) cov(e.x3,e.x6) cov(e.x3,e.x7) cov(e.x3,e.x9) cov(e.x4,e.x6) cov(e.x4,e.x7) cov(e.x4,e.x8) cov(e.x4,e.x9) cov(e.x6,e.x7) cov(e.x6,e.x8) cov(e.x6,e.x9) cov(e.x7,e.x9) cov(e.x8,e.x9) EPC = expected parameter change

20 MI ( ) P>MI MI (x3 x4 ) 2, Conservartive codebook x3 x4,compact. codebook x3 x4,compact Variable Obs Unique Mean Min Max Label x GOVT RESPNSBLTY - HLTH CARE FOR SICK 2006 x GOVT RESPNSBLTY -PROV ELD LIV STAND ,,, Conservartive 1. sem (Conservative -> x1 x3-x7 x9),cov(e.x3*e.x4) variance(conservative@1)

21 20 1 (7354 observations with missing values excluded) Endogenous variables Measurement: Exogenous variables Latent: x1 x3 x4 x5 x6 x7 x9 Conservative Fitting target model: ( ) Structural equation model Number of obs = 1,631 Estimation method = ml Log likelihood = ( 1) [var(conservative)]_cons = 1 OIM Coef. Std. Err. z P> z [95% Conf. Interval] Measurement x1 <- Conservative _cons x3 <- Conservative _cons x4 <- Conservative _cons x5 <- Conservative _cons x6 <- Conservative _cons x7 <- Conservative _cons x9 <- Conservative _cons var(e.x1) var(e.x3) var(e.x4) var(e.x5) var(e.x6) var(e.x7) var(e.x9) var(conservative) 1 (constrained) cov(e.x3,e.x4) LR test of model vs. saturated: chi2(13) = 56.02, Prob > chi2 =

22 model vs.saturated? estat gof CFI 95%? RMSEA 0.05? SRMR 0.08? ? AIC

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