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Structural equation modeling (SEM)

What is SEM?

View the complete list of SEM features

SEM stands for structural equation modeling. SEM is a notation for specifying structural equations, a way of thinking about them, and methods for estimating their parameters.

SEM encompasses a broad array of models from linear regression to measurement models to simultaneous equations, including along the way confirmatory factor analysis (CFA), correlated uniqueness models, latent growth models, and multiple indicators and multiple causes (MIMIC).

Stata’s sem fits linear SEMs, and its features are described below. gsem provides extensions to linear SEMs that allow for generalized-linear models and multilevel models.

Features

  • Use GUI or command language to specify model.

  • Standardized and unstandardized results.

  • Direct and indirect effects.

  • Goodness-of-fit statistics.

  • Tests for omitted paths and tests of model simplification including modification indices, score tests, and Wald tests.

  • Predicted values and factor scores.

  • Linear and nonlinear (1) tests of estimated parameters and (2) combinations of estimated parameters with CIs.

  • Estimation across groups is as easy as adding group(sex) to the command. Test for group invariance. Easily add or relax constraints across groups.

  • SEMs may be fitted using raw or summary statistics data.

  • Maximum likelihood (ML) and asymptotic distribution free (ADF) estimation. ADF is also known as generalized method of moments (GMM). Missing at random (MAR) data supported via FIML.

  • Robust estimate of standard errors and standard errors for clustered samples available.

  • Support for survey data including sampling weights, stratification and poststratification, and clustered sampling at one or more levels.

SEM Builder or commands, it’s your choice

Enter your model graphically,

or use the command syntax 

. sem (L1 -> m1 m2)
 (L2 -> m3 m4)
 (L3 <- L1 L2)
 (L3 -> m5 m6 m7)

It’s the same model either way.

Stata’s SEM Builder uses standard path notation.

In command syntax, you type the path diagram. Capitalized names are latent variables. Lowercased names are observed variables. You can type arrows in either direction. The above model could be equally well typed as 

. sem (m1 m2 <- L1)
 (L2 -> m3 m4)
 (L3 <- L1 L2)
 (L3 -> m5 m6 m7)

and order does not matter, and neither does spacing: 

. sem (m1 m2 <- L1) (L2 -> m3 m4) (L3 -> m5 m6 m7) (L3 <- L1 L2)

You can specify paths individually, 

. sem (m1 <- L1) (m2 <- L1) (L2 -> m3) (L2 -> m4) (L3 -> m5) (L3 -> m6) (L3 -> m7) (L3 <- L1) (L3 <- L2)

or combined, 

. sem (m1 m2 <- L1) (L2 -> m3 m4) (L3 -> m5 m6 m7) (L3 <- L1 L2)

Let's see it work

Let’s fit a structural model with a measurement component using data from Wheaton, Muthén, Alwin, and Summers (1977): 

. webuse sem_sm2, clear
(Structural model with measurement component)
. ssd describe
 Summary statistics data from https://www.stata-press.com/data/r17/sem_sm2.dta
 Observations: 932 Structural model with measurem..
 Variables: 13 25 May 2020 11:45
 (_dta has notes)
Variable name Variable label
educ66 Education, 1966 occstat66 Occupational status, 1966 anomia66 Anomia, 1966 pwless66 Powerlessness, 1966 socdist66 Latin American social distance, 1966 occstat67 Occupational status, 1967 anomia67 Anomia, 1967 pwless67 Powerlessness, 1967 socdist67 Latin American social distance, 1967 occstat71 Occupational status, 1971 anomia71 Anomia, 1971 pwless71 Powerlessness, 1971 socdist71 Latin American social distance, 1971
. notes _dta: 1. Summary statistics data from Wheaton, B., Muthen B., Alwin, D., & Summers, G., 1977, "Assessing reliability and stability in panel models", in D. R. Heise (Ed.), _Sociological Methodology 1977_ (pp. 84-136), San Francisco: Jossey-Bass, Inc. 2. Four indicators each measured in 1966, 1967, and 1981, plus another indicator (educ66) measured only in 1966. 3. Intended use: Create structural model relating Alienation in 1971, Alienation in 1967, and SES in 1966.

Below we will demonstrate

Show me, fitting the model

Simplified versions of the model fit by the authors of the referenced paper appear in many SEM software manuals. One simplified model is

You can also readily fit this model using the following command: 

. sem 
 (anomia67 pwless67 <- Alien67) /// measurement piece
 (anomia71 pwless71 <- Alien71) /// measurement piece
 (Alien67 <- SES) /// structural piece
 (Alien71 <- Alien67 SES) /// structural piece
 ( SES -> educ occstat66), nolog /// measurement piece

And the results are 

Endogenous variables
 Measurement: anomia67 pwless67 anomia71 pwless71 educ66 occstat66
 Latent: Alien67 Alien71
Exogenous variables
 Latent: SES
Structural equation model Number of obs = 932
Estimation method: ml
Log likelihood = -15246.469
 ( 1) [anomia67]Alien67 = 1
 ( 2) [anomia71]Alien71 = 1
 ( 3) [educ66]SES = 1

   OIM 
   Coefficient std. err. z P>|z| [95% conf. interval] 
 Structural   
 Alien67   
 SES   -.6140404 .0562407 -10.92 0.000 -.7242701 -.5038107 
 Alien71   
 Alien67   .7046342 .0533512 13.21 0.000 .6000678 .8092007 
 SES   -.1744153 .0542489 -3.22 0.001 -.2807413 -.0680894 
 Measurement   
 anomia67   
 Alien67   1 (constrained) 
 _cons   13.61 .1126205 120.85 0.000 13.38927 13.83073 
 pwless67   
 Alien67   .8884887 .0431565 20.59 0.000 .8039034 .9730739 
 _cons   14.67 .1001798 146.44 0.000 14.47365 14.86635 
 anomia71   
 Alien71   1 (constrained) 
 _cons   14.13 .1158943 121.92 0.000 13.90285 14.35715 
 pwless71   
 Alien71   .8486022 .0415205 20.44 0.000 .7672235 .9299808 
 _cons   14.9 .1034537 144.03 0.000 14.69723 15.10277 
 educ66   
 SES   1 (constrained) 
 _cons   10.9 .1014894 107.40 0.000 10.70108 11.09892 
 occstat66   
 SES   5.331259 .4307503 12.38 0.000 4.487004 6.175514 
 _cons   37.49 .6947112 53.96 0.000 36.12839 38.85161 
 var(e.anomia67)  4.009921 .3582978 3.365724 4.777416 
 var(e.pwless67)  3.187468 .283374 2.677762 3.794197 
 var(e.anomia71)  3.695593 .3911512 3.003245 4.54755 
 var(e.pwless71)  3.621531 .3037908 3.072483 4.268693 
 var(e.educ66)  2.943819 .5002527 2.109908 4.107319 
 var(e.occstat66)  260.63 18.24572 227.2139 298.9605 
 var(e.Alien67)  5.301416 .483144 4.434225 6.338201 
 var(e.Alien71)  3.737286 .3881546 3.048951 4.581019 
 var(SES)  6.65587 .6409484 5.511067 8.038482 
LR test of model vs. saturated: chi2(6) = 71.62 Prob > chi2 = 0.0000

Notes:

  1. Measurement component: In both 1967 and 1971 anomia and powerlessness are used to measure endogenous latent variables representing Alienation for the same two years. Education and occupational status are used to measure the exogenous latent variable SES.

  2. Structural component: SES->Alien67 and SES->Alien71, and Alien67->Alien71.

  3. The model vs. saturated chi-squared test indicates the model is a poor fit.

Show me, modification indices

That the model is a poor fit leads us to looking at the modification indices: 

. estat mindices
Modification indices


   Standard


   MI df P>MI EPC EPC



 Measurement   


 anomia67   


 anomia71   51.977 1 0.00 .3906425 .4019984


 pwless71   32.517 1 0.00 -.2969297 -.2727609


 educ66   5.627 1 0.02 .0935048 .0842631



 pwless67   


 anomia71   41.618 1 0.00 -.3106995 -.3594367


 pwless71   23.622 1 0.00 .2249714 .2323233


 educ66   6.441 1 0.01 -.0889042 -.0900664



 anomia71   


 anomia67   58.768 1 0.00 .429437 .4173061


 pwless67   38.142 1 0.00 -.3873066 -.3347904



 pwless71   


 anomia67   46.188 1 0.00 -.3308484 -.3601641


 pwless67   27.760 1 0.00 .2871709 .2780833



 educ66   


 anomia67   4.415 1 0.04 .1055965 .1171781


 pwless67   6.816 1 0.01 -.1469371 -.1450411



 cov(e.anomia67,e.anomia71)  63.786 1 0.00 1.951578 .5069627


 cov(e.anomia67,e.pwless71)  49.892 1 0.00 -1.506704 -.3953794


 cov(e.anomia67,e.educ66)  6.063 1 0.01 .5527612 .1608845


 cov(e.pwless67,e.anomia71)  49.876 1 0.00 -1.534199 -.4470094


 cov(e.pwless67,e.pwless71)  37.357 1 0.00 1.159123 .341162


 cov(e.pwless67,e.educ66)  7.752 1 0.01 -.5557802 -.1814365


EPC is expected parameter change.

Notes:

  1. There are lots of statistically significant paths we could add to the model.

  2. Some of those statistically significant paths also make theoretical sense.

  3. Two in particular that make sense are the covariances between e.anomia67 and e.anomia71, and between e.pwless67 and e.pwless71.

Show me, refitting the model

Let’s refit the model and include those two previously excluded covariances: 

. sem 
 (anomia67 pwless67 <- Alien67) /// measurement piece
 (anomia71 pwless71 <- Alien71) /// measurement piece
 (Alien67 <- SES) /// structural piece
 (Alien71 <- Alien67 SES) /// structural piece
 ( SES -> educ occstat66) /// measurement piece
 , cov(e.anomia67*e.anomia71) 
 cov(e.pwless67*e.pwless71) nolog

And the results are 

Endogenous variables
 Measurement: anomia67 pwless67 anomia71 pwless71 educ66 occstat66
 Latent: Alien67 Alien71
Exogenous variables
 Latent: SES
Structural equation model Number of obs = 932
Estimation method: ml
Log likelihood = -15213.046
 ( 1) [anomia67]Alien67 = 1
 ( 2) [anomia71]Alien71 = 1
 ( 3) [educ66]SES = 1


   OIM


   Coefficient std. err. z P>|z| [95% conf. interval]



 Structural   


 Alien67   


 SES   -.5752228 .057961 -9.92 0.000 -.6888244 -.4616213



 Alien71   


 Alien67   .606954 .0512305 11.85 0.000 .5065439 .707364


 SES   -.2270301 .0530773 -4.28 0.000 -.3310596 -.1230006



 Measurement   


 anomia67   


 Alien67   1 (constrained)


 _cons   13.61 .1126143 120.85 0.000 13.38928 13.83072



 pwless67   


 Alien67   .9785952 .0619825 15.79 0.000 .8571117 1.100079


 _cons   14.67 .1001814 146.43 0.000 14.47365 14.86635



 anomia71   


 Alien71   1 (constrained)


 _cons   14.13 .1159036 121.91 0.000 13.90283 14.35717



 pwless71   


 Alien71   .9217508 .0597225 15.43 0.000 .8046968 1.038805


 _cons   14.9 .1034517 144.03 0.000 14.69724 15.10276



 educ66   


 SES   1 (constrained)


 _cons   10.9 .1014894 107.40 0.000 10.70108 11.09892



 occstat66   


 SES   5.22132 .425595 12.27 0.000 4.387169 6.055471


 _cons   37.49 .6947112 53.96 0.000 36.12839 38.85161



 var(e.anom~67)  4.728874 .456299 3.914024 5.713365


 var(e.pwle~67)  2.563413 .4060733 1.879225 3.4967


 var(e.anom~71)  4.396081 .5171156 3.490904 5.535966


 var(e.pwle~71)  3.072085 .4360333 2.326049 4.057398


 var(e.educ66)  2.803674 .5115854 1.960691 4.009091


 var(e.occs~66)  264.5311 18.22483 231.1177 302.7751


 var(e.Alien67)  4.842059 .4622537 4.015771 5.838364


 var(e.Alien71)  4.084249 .4038995 3.364613 4.957802


 var(SES)  6.796014 .6524866 5.630283 8.203105



 cov(e.anom~67,  


 e.anomia71)  1.622024 .3154267 5.14 0.000 1.003799 2.240249


 cov(e.pwle~67,  


 e.pwless71)  .3399961 .2627541 1.29 0.196 -.1749925 .8549847


LR test of model vs. saturated: chi2(4) = 4.78 Prob> chi2 = 0.3111

Notes:

  1. We find the covariance between e.anomia67 and e.anomia71 to be significant(Z=5.14).

  2. We find the covariance between e.pwless67 and e.pwless71 to be insignificant at the 5% level (Z=1.29).

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