Introduction to structural equation modeling using the sem command
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1 Introduction to structural equation modeling using the sem command Gustavo Sanchez Senior Econometrician StataCorp LP Mexico City, Mexico Gustavo Sanchez (StataCorp) November 13, / 33
2 Outline Outline Structural Equation Models (SEM): Applications, concepts and components Mediation Model Measurement Models SEM Model Other Models Gustavo Sanchez (StataCorp) November 13, / 33
3 Outline Outline Structural Equation Models (SEM): Applications, concepts and components Mediation Model Measurement Models SEM Model Other Models Gustavo Sanchez (StataCorp) November 13, / 33
4 Outline Outline Structural Equation Models (SEM): Applications, concepts and components Mediation Model Measurement Models SEM Model Other Models Gustavo Sanchez (StataCorp) November 13, / 33
5 Structural Equation Models: Applications, Concepts and Components Applications SEM: Applications Psychology (e.g. Behavioral analysis, depression) Sociology (e.g. Social network, work environment) Marketing (e.g. Consumer satisfaction, new products development) Academic research (e.g. Analysis of learning abilities) Medicine (e.g. Sleep disorders, population health services) And more Gustavo Sanchez (StataCorp) November 13, / 33
6 Structural Equation Models: Applications, Concepts and Components Applications SEM: Applications Example: Path diagram for a SEM model Gustavo Sanchez (StataCorp) November 13, / 33
7 Structural Equation Models: Applications, Concepts and Components Applications SEM: Applications Models Linear regression ANOVA Multivariate regression Simultaneous equation models Path analysis Simultaneous equation models Mediation analysis Confirmatory factor analysis Reliability estimation Full structural equations models Multiple indicators and multiple causes (MIMIC) Latent growth curve Multiple group models Gustavo Sanchez (StataCorp) November 13, / 33
8 Structural Equation Models: Applications, Concepts and Components Concepts SEM: Concepts: SEM Structural equation modeling was developed by geneticists (Wright 1921) and economists (Haavelmo 1943; Koopmans 1950, 1953) so that qualitative cause-effect information could be combined with statistical data to provide quantitative assessment of cause-effect relationships among variables of interest Pearl (2000). SEM is a class of statistical techniques used for estimating and testing hypotheses on causal relationships among a set of associated variables A significant number of models can be expressed as particular cases of structural equation models. Gustavo Sanchez (StataCorp) November 13, / 33
9 Structural Equation Models: Applications, Concepts and Components Concepts SEM: Concepts: SEM Structural equation modeling was developed by geneticists (Wright 1921) and economists (Haavelmo 1943; Koopmans 1950, 1953) so that qualitative cause-effect information could be combined with statistical data to provide quantitative assessment of cause-effect relationships among variables of interest Pearl (2000). SEM is a class of statistical techniques used for estimating and testing hypotheses on causal relationships among a set of associated variables A significant number of models can be expressed as particular cases of structural equation models. Gustavo Sanchez (StataCorp) November 13, / 33
10 Structural Equation Models: Applications, Concepts and Components Concepts SEM: Concepts: SEM Structural equation modeling was developed by geneticists (Wright 1921) and economists (Haavelmo 1943; Koopmans 1950, 1953) so that qualitative cause-effect information could be combined with statistical data to provide quantitative assessment of cause-effect relationships among variables of interest Pearl (2000). SEM is a class of statistical techniques used for estimating and testing hypotheses on causal relationships among a set of associated variables A significant number of models can be expressed as particular cases of structural equation models. Gustavo Sanchez (StataCorp) November 13, / 33
11 Structural Equation Models: Applications, Concepts and Components Concepts and components SEM: Concepts and components: Types of variables Observed and Latent A variable is observed if it is a variable in your dataset A variable is latent if it is not observed. It is not in your dataset but you wish it were Errors are a special case of latent variables Exogenous and Endogenous A variable, observed or latent, is exogenous (determined outside the system) if paths only originate from it (no path points to it) A variable, observed or latent, is endogenous (determined by the system) if any path points to it. Gustavo Sanchez (StataCorp) November 13, / 33
12 Structural Equation Models: Applications, Concepts and Components Concepts and components SEM: Concepts and components: Types of variables Observed and Latent A variable is observed if it is a variable in your dataset A variable is latent if it is not observed. It is not in your dataset but you wish it were Errors are a special case of latent variables Exogenous and Endogenous A variable, observed or latent, is exogenous (determined outside the system) if paths only originate from it (no path points to it) A variable, observed or latent, is endogenous (determined by the system) if any path points to it. Gustavo Sanchez (StataCorp) November 13, / 33
13 Structural Equation Models: Applications, Concepts and Components Concepts and components SEM: Concepts and components: Types of variables Observed and Latent A variable is observed if it is a variable in your dataset A variable is latent if it is not observed. It is not in your dataset but you wish it were Errors are a special case of latent variables Exogenous and Endogenous A variable, observed or latent, is exogenous (determined outside the system) if paths only originate from it (no path points to it) A variable, observed or latent, is endogenous (determined by the system) if any path points to it. Gustavo Sanchez (StataCorp) November 13, / 33
14 Structural Equation Models: Applications, Concepts and Components Concepts and components SEM: Concepts and components: Path diagrams A Path diagram is a graphical representation of the model Boxes contain observed variables Ovals contain latent variables Circles contain the equation errors Straight arrows represent effects from one variable to another Curved arrows indicate correlation between a pair of variables. sem (income lag_consumption -> consumption) /// > (income -> investment), /// > cov(e.investment*e.consumption) Gustavo Sanchez (StataCorp) November 13, / 33
15 Structural Equation Models: Applications, Concepts and Components Concepts and components SEM: Concepts and components: Path diagrams A Path diagram is a graphical representation of the model Boxes contain observed variables Ovals contain latent variables Circles contain the equation errors Straight arrows represent effects from one variable to another Curved arrows indicate correlation between a pair of variables. sem (income lag_consumption -> consumption) /// > (income -> investment), /// > cov(e.investment*e.consumption) Gustavo Sanchez (StataCorp) November 13, / 33
16 Structural Equation Models: Applications, Concepts and Components Concepts and components. sem (income lag_consumption -> consumption) (income -> investment), /// > cov(e.investment*e.consumption) nolog nodescribe (1 observations with missing values excluded) Structural equation model Number of obs = 91 Estimation method = ml Log likelihood = OIM Coef. Std. Err. z P> z [95% Conf. Interval] Structural consumption <- income lag_consumption _cons investment <- income _cons var(e.consumption) var(e.investment) cov(e.consumption, e.investment) LR test of model vs. saturated: chi2(1) = 1.11, Prob > chi2 = Gustavo Sanchez (StataCorp) November 13, / 33
17 Structural Equation Models: Applications, Concepts and Components Concepts and components. estat gof Fit statistic Value Description Likelihood ratio chi2_ms(1) model vs. saturated p > chi chi2_bs(5) baseline vs. saturated p > chi estat eqgof Equation-level goodness of fit Variance depvars fitted predicted residual R-squared mc mc2 observed consumption investment overall mc = correlation between depvar and its prediction mc2 = mc^2 is the Bentler-Raykov squared multiple correlation coefficient Gustavo Sanchez (StataCorp) November 13, / 33
18 Mediation model Example 1: Mediation models The explanatory variables may have a direct effect on the outcome and also an indirect effect that is transmitted by a mediator variable The traditional mediation analysis was based on a series of linear regressions with no correlated errors (Baron and Kenny (1986)) With SEM we can fit one simultaneous equation model and get estimates for the indirect and total effects The model can be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
19 Mediation model Example 1: Mediation models The explanatory variables may have a direct effect on the outcome and also an indirect effect that is transmitted by a mediator variable The traditional mediation analysis was based on a series of linear regressions with no correlated errors (Baron and Kenny (1986)) With SEM we can fit one simultaneous equation model and get estimates for the indirect and total effects The model can be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
20 Mediation model Example 1: Mediation models The explanatory variables may have a direct effect on the outcome and also an indirect effect that is transmitted by a mediator variable The traditional mediation analysis was based on a series of linear regressions with no correlated errors (Baron and Kenny (1986)) With SEM we can fit one simultaneous equation model and get estimates for the indirect and total effects The model can be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
21 Mediation model Example 1: Mediation models The explanatory variables may have a direct effect on the outcome and also an indirect effect that is transmitted by a mediator variable The traditional mediation analysis was based on a series of linear regressions with no correlated errors (Baron and Kenny (1986)) With SEM we can fit one simultaneous equation model and get estimates for the indirect and total effects The model can be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
22 Mediation model Example 1: Mediation models Example from Alan Acock (2013) The researcher wants to analyze whether children with better span skills at four have advantages in their academic development for math at a later age Variables attention4: span of attention at 4 math7: performance on math at 7 read7: performance on reading at 7 math21: performance on math at 21. sem (math7 <- attention4) (read7 <- attention4) (math21 <- attention4 math7 read7) Gustavo Sanchez (StataCorp) November 13, / 33
23 Mediation model Example 1: Mediation models Example from Alan Acock (2013) The researcher wants to analyze whether children with better span skills at four have advantages in their academic development for math at a later age Variables attention4: span of attention at 4 math7: performance on math at 7 read7: performance on reading at 7 math21: performance on math at 21. sem (math7 <- attention4) (read7 <- attention4) (math21 <- attention4 math7 read7) Gustavo Sanchez (StataCorp) November 13, / 33
24 Mediation model. sem (math7 <- attention4) (read7 <- attention4) (math21 <- attention4 math7 read7), /// > nolog noheader nodescribe (92 observations with missing values excluded) OIM Coef. Std. Err. z P> z [95% Conf. Interval] Structural math7 <- attention _cons read7 <- attention _cons math21 <- math read attention _cons var(e.math7) var(e.read7) var(e.math21) LR test of model vs. saturated: chi2(1) = 23.72, Prob > chi2 = Gustavo Sanchez (StataCorp) November 13, / 33
25 Mediation model. estat teffects,compact nodirect Indirect effects OIM Coef. Std. Err. z P> z [95% Conf. Interval] Structural math21 <- attention Total effects OIM Coef. Std. Err. z P> z [95% Conf. Interval] Structural math7 <- attention read7 <- attention math21 <- math read attention Gustavo Sanchez (StataCorp) November 13, / 33
26 Measurement models - one factor Example 2.1: Measurement model - one factor The researcher is interested in a latent variable (e.g. Consumer satisfaction, verbal abilities, alienation) The model specifies the relation between latent variables and measured indicator variables Modification indices are normally used to refine the model The model can also be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
27 Measurement models - one factor Example 2.1: Measurement model - one factor The researcher is interested in a latent variable (e.g. Consumer satisfaction, verbal abilities, alienation) The model specifies the relation between latent variables and measured indicator variables Modification indices are normally used to refine the model The model can also be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
28 Measurement models - one factor Example 2.1: Measurement model - one factor The researcher is interested in a latent variable (e.g. Consumer satisfaction, verbal abilities, alienation) The model specifies the relation between latent variables and measured indicator variables Modification indices are normally used to refine the model The model can also be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
29 Measurement models - one factor Example 2.1: Measurement model - one factor The researcher is interested in a latent variable (e.g. Consumer satisfaction, verbal abilities, alienation) The model specifies the relation between latent variables and measured indicator variables Modification indices are normally used to refine the model The model can also be incorporated as part of a larger model Gustavo Sanchez (StataCorp) November 13, / 33
30 Measurement models - one factor Example 2: Measurement model - one factor Example with Holzinger and Swineford (1939) Data The researcher wants to analyze the verbal ability based on indices associated to tests on word classification, sentence completion and paragraph comprehension Variables: Verbal: latent variable for verbal ability wordc: scores on word classification test sentence: scores on sentence completion test paragraph: scores on paragraph comprehension test general: scores on general information test Gustavo Sanchez (StataCorp) November 13, / 33
31 Measurement models - one factor. sem (Verbal -> wordc sentence paragraph general) OIM Coef. Std. Err. z P> z [95% Conf. Interval] Measurement wordc <- Verbal 1 (constrained) _cons sentence <- Verbal _cons paragraph <- Verbal _cons general <- Verbal _cons var(e.wordc) var(e.sentence) var(e.paragr ~ h) var(e.general) var(verbal) Gustavo Sanchez (StataCorp) November 13, / 33
32 Measurement models - one factor. sem (Verbal -> wordc sentence paragraph general), standardize OIM Standardized Coef. Std. Err. z P> z [95% Conf. Interval] Measurement wordc <- Verbal _cons sentence <- Verbal _cons paragraph <- Verbal _cons general <- Verbal _cons var(e.wordc) var(e.sentence) var(e.paragr ~ h) var(e.general) var(verbal) 1... Gustavo Sanchez (StataCorp) November 13, / 33
33 Measurement models - one factor. sem (Verbal -> wordc sentence paragraph general), standardize nomeans OIM Standardized Coef. Std. Err. z P> z [95% Conf. Interval] Measurement wordc <- Verbal sentence <- Verbal paragraph <- Verbal general <- Verbal var(e.wordc) var(e.sentence) var(e.paragr ~ h) var(e.general) var(verbal) 1... Gustavo Sanchez (StataCorp) November 13, / 33
34 Measurement models - two factors Example 2.2: Measurement model - two factors Holzinger and Swineford (1939) Data Variables Verbal: latent variable for verbal ability wordc: scores on word classification test sentence: scores on sentence completion test paragraph: scores on paragraph comprehension test general: scores on general information test Memory: latent variable for Memory condition wordr: scores on word recognition test number: scores on number recognition test figurer: scores on figurer recognition test object: scores on object-number test Gustavo Sanchez (StataCorp) November 13, / 33
35 Measurement models - two factors Example 2.2: Measurement model - two factors Holzinger and Swineford (1939) Data Variables Verbal: latent variable for verbal ability wordc: scores on word classification test sentence: scores on sentence completion test paragraph: scores on paragraph comprehension test general: scores on general information test Memory: latent variable for Memory condition wordr: scores on word recognition test number: scores on number recognition test figurer: scores on figurer recognition test object: scores on object-number test Gustavo Sanchez (StataCorp) November 13, / 33
36 Example 2.2: Measurement model - two factors Gustavo Sanchez (StataCorp) November 13, / 33
37 . sem (Verbal -> wordc sentence paragraph general) /// > (Memory -> wordr numberr figurer objectn), standardize /// > cov( Memory*Verbal) nomeans noheader nocnsreport nolog Endogenous variables Measurement: wordc sentence paragraph general wordr numberr figurer objectn Exogenous variables Latent: Verbal Memory OIM Standardized Coef. Std. Err. z P> z [95% Conf. Interval] Measurement wordc <- Verbal sentence <- Verbal paragraph <- Verbal general <- Verbal Gustavo Sanchez (StataCorp) November 13, / 33
38 wordr <- Memory numberr <- Memory figurer <- Memory objectn <- Memory var(e.wordc) var(e.sentence) var(e.paragraph) var(e.general) var(e.wordr) var(e.numberr) var(e.figurer) var(e.objectn) var(verbal) 1... var(memory) 1... cov(verbal,memory) LR test of model vs. saturated: chi2(19) = 41.13, Prob > chi2 = Gustavo Sanchez (StataCorp) November 13, / 33
39 . estat mindices Modification indices Standard MI df P>MI EPC EPC Measurement sentence <- paragraph <- numberr <- figurer <- Memory Memory Verbal Verbal cov(e.wordc,e.paragraph) cov(e.wordc,e.figurer) cov(e.sentence,e.numberr) EPC = expected parameter change Gustavo Sanchez (StataCorp) November 13, / 33
40 Based on estat mindices, let s add figurer and numberr to the equation for verbal, and also cov(e.wordc*e.paragraph): (but we should first check the theoretical framework) sem (Verbal -> wordc sentence paragraph general figurer numberr ) /// (Memory -> wordr numberr figurer objectn), standardize nocnsreport /// nolog nomeans cov( Memory*Verbal) noheader cov(e.wordc*e.paragraph) Compare the chi2 ms before and after adding the suggestions from estat mindices:. estat gof /* Before adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(19) model vs. saturated p > chi estat gof /* After adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(16) model vs. saturated p > chi Gustavo Sanchez (StataCorp) November 13, / 33
41 Based on estat mindices, let s add figurer and numberr to the equation for verbal, and also cov(e.wordc*e.paragraph): (but we should first check the theoretical framework) sem (Verbal -> wordc sentence paragraph general figurer numberr ) /// (Memory -> wordr numberr figurer objectn), standardize nocnsreport /// nolog nomeans cov( Memory*Verbal) noheader cov(e.wordc*e.paragraph) Compare the chi2 ms before and after adding the suggestions from estat mindices:. estat gof /* Before adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(19) model vs. saturated p > chi estat gof /* After adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(16) model vs. saturated p > chi Gustavo Sanchez (StataCorp) November 13, / 33
42 Based on estat mindices, let s add figurer and numberr to the equation for verbal, and also cov(e.wordc*e.paragraph): (but we should first check the theoretical framework) sem (Verbal -> wordc sentence paragraph general figurer numberr ) /// (Memory -> wordr numberr figurer objectn), standardize nocnsreport /// nolog nomeans cov( Memory*Verbal) noheader cov(e.wordc*e.paragraph) Compare the chi2 ms before and after adding the suggestions from estat mindices:. estat gof /* Before adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(19) model vs. saturated p > chi estat gof /* After adding suggestions from estat mindices */ Fit statistic Value Description Likelihood ratio chi2_ms(16) model vs. saturated p > chi Gustavo Sanchez (StataCorp) November 13, / 33
43 Structural equation model Example 3: Structural equation model The model Integrates two components A structural component that specifies the relationship among the latent variables A measurement component that specifies the relationship between the latent variables and the corresponding indicators Modification indices are normally used to refine the model The model can incorporate causal relationships among the latent variables Gustavo Sanchez (StataCorp) November 13, / 33
44 Structural equation model Example 3: Structural equation model The model Integrates two components A structural component that specifies the relationship among the latent variables A measurement component that specifies the relationship between the latent variables and the corresponding indicators Modification indices are normally used to refine the model The model can incorporate causal relationships among the latent variables Gustavo Sanchez (StataCorp) November 13, / 33
45 Structural equation model Example 3: Structural equation model (Fictitious data) Gustavo Sanchez (StataCorp) November 13, / 33
46 Structural equation model Example 3: Structural equation model (Fictitious data) Gustavo Sanchez (StataCorp) November 13, / 33
47 Other models Example 4: Dynamic Factor Unobserved (factor) effect on unemployment in four different regions use dfactor (D.(west south ne midwest) =, ) (Unemp_factor = ) sem (D.(west south ne midwest) <- Unemp_factor),var(Unemp_factor@1) Gustavo Sanchez (StataCorp) November 13, / 33
48 Other models Example 5: VAR model VAR model for consumption and investment with income as an exogenous variable use var dlconsumption dlinvestment,lags(1/2) exog(dlincome) sem (dlconsumption <- l.dlconsumption l2.dlconsumption /// l.dlinvestment l2.dlinvestment dlincome) /// (dlinvestment <- l.dlconsumption l2.dlconsumption /// l.dlinvestment l2.dlinvestment dlincome), /// cov(e.dlinvestment*e.dlconsumption) Gustavo Sanchez (StataCorp) November 13, / 33
49 Summary Summary Structural Equation Models (SEM): Applications, Concepts and components Mediation Model Measurement Model SEM Model Other Models What is next? Generalization (gsem) Extensions for nonlinear models Multilevel models and more Gustavo Sanchez (StataCorp) November 13, / 33
50 Summary Summary Structural Equation Models (SEM): Applications, Concepts and components Mediation Model Measurement Model SEM Model Other Models What is next? Generalization (gsem) Extensions for nonlinear models Multilevel models and more Gustavo Sanchez (StataCorp) November 13, / 33
51 References References Acock, Alan Discovering structural equation modeling using Stata. Stata Press Holzinger, K. J. and F. Swineford, A study in factor analysis: The stability of a bi-factor solution. Supplementary Education Monographs, 48. Chicago, IL: University of Chicago. Haavelmo, T The statistical implications of a system of simultaneous equations. Econometrica 11 (1), Koopmans, T. C When is an Equation System Complete for Statistical Purposes? (in Statistical Inference in Dynamic Economic Models, T. C. Koopmans (ed.), Cowles Commission Monograph 10, Wiley, New York, 1950, pp ) pp Koopmans, T.C Identification Problems in Economic Model Construction (in Studies in Econometric Method, Cowles Commission Monograph 14), New Haven, Yale University Press Pearl J Causality: Models, Reasoning, and Inference. Cambridge University Press. Wright, Sewall S Correlation and causation. Journal of Agricultural Research 20: Gustavo Sanchez (StataCorp) November 13, / 33
52 References Questions - Comments Gustavo Sanchez (StataCorp) November 13, / 33
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