Structural Equation Modeling: An Overview P. Paxton

What are Structural Equation Models? Also known as: – Covariance structure models – Latent variable models – “LISREL” models – Structural Equations with Latent Variables

What are Structural Equation Models? Special cases: ANOVA Multiple regression Path analysis Confirmatory Factor Analysis Recursive and Nonrecursive systems

What are Structural Equation Models? SEM associated with path diagrams intelligence test 1 test 2test 3test 4test 5 δ1δ1 δ2δ2 δ3δ3 δ4δ4 δ5δ5

What are Structural Equation Models? Latent variables, factors, constructs Observed variables, measures, indicators, manifest variables Direction of influence, relationship from one variable to another Association not explained within the model

What are Structural Equation Models? Depress 1Depress 2Depress 3 Self ratingMD rating# visits to MD Self rated closeness Spousal rating Kids rating Family support depression Physical health δ1δ1 δ2δ2 δ3δ3 ε4ε4 ε 5ε 5 ε 6ε 6 ε1ε1 ε 2ε 2 ε 3ε 3 ζ1ζ1 ζ2ζ2

What are Structural Equation Models? What can you do with these models? – Latent and Observed Variables – Multiple indicators of same concept – Measurement error – Restrictions on model parameters – Tests of model fit

What are Structural Equation Models? What can’t you do? – Prove causation – Prove a model is “correct” All models Models consistent with data Models consistent with reality (Mueller 1997)

Notation ε1ε1 y1y1 ε2ε2 y2y2 ε3ε3 y3y3 ε4ε4 y4y4 ε5ε5 y5y5 ε6ε6 y6y6 ε7ε7 y7y7 ε8ε8 y8y8 δ1δ1 x1x1 δ2δ2 x2x2 δ3δ3 x3x3 η1η1 ξ1ξ1 η2η2 ζ1ζ1 ζ2ζ2 β 21 γ 21 γ 11 λ1λ1 λ2λ2 λ3λ3 λ4λ4 λ5λ5 λ6λ6 λ7λ7 λ8λ8 λ9λ9 λ 10 λ 11 ξ 1 = industrialization η 1 = democracy time 1 η 2 = democracy time 2 x1-x3 = indus. indicators, e.g., energy y1-y4 = democ. indicators time 1 y5-y8 = democ. indicators time 2

Notation η Latent Endogenous Variable ξ Latent Exogenous Variable ζ Unexplained Error in Model x & y Observed Variables δ & ε Measurement Errors λ, β, & γ Coefficients

Notation Two components to a SEM – Latent variable model Relationship between the latent variables Measurement model Relationship between the latent and observed variables

Notation Covariance Matrixes of Interest: – Φ – Ψ – Θ δ – Θ ε

Example: Trust in Individuals Trust in Individuals people are helpful (x1) people can be trusted (x2) people are Fair (x3) 1 ξ1ξ1 δ1δ1 δ2δ2 δ3δ3 λ 11 λ 21

Latent Variables Variables of Interest Not directly measured Common – Intelligence – Trust – Democracy – Diseases – Disturbance variables

Three Types of SEM Classic Econometric Multiple equations One indicator per latent variable No measurement error

Classic Econometric Citations y3 Quality rating y4 Publications y2 Size of dept. y1 Private x1 β 43 β 42 β 41 β 32 β 31 γ 31 γ 41 γ 11

Classic Econometric associations 1980 associations 1990 democracy 1982 trust 1980 democracy 1991 trust 1990 industrialization 1980 Noncore position Ethnic homogeneity

Recursive / Nonrecursive Recursive – Direction of influence one direction No reciprocal causation No feedback loops – Disturbances not correlated Nonrecursive – Either reciprocal causation, feedback loops, or correlated disturbances

Recursive y2x1y3 y2 x3 x1 y1 x2

Nonrecursive x2y1 x1y2 y3 y2 x3 x1 y1 x2

Confirmatory Factor Analysis Latent variables Measurement error No causal relationship between latent variables x = vector of observed indicators Λ x = matrix of factor loadings ξ = vector of latent variables δ = vector of measurement errors

Trust in Individuals people are helpful (x1) people can be trusted (x2) people are Fair (x3) 1 ξ1ξ1 δ1δ1 δ2δ2 δ3δ3 Confirmatory Factor Analysis λ 11 λ 21

General Model Includes latent variable model – Relationship between the latent variables And measurement model – Relationship between latent variables and observed variables

General Model Latent Variable Model η = vector of latent endogenous variables ξ = vector of latent exogenous variables ζ = vector of disturbances Β = coefficient matrix for η on η effects Γ =coefficient matrix for ξ on η effects

General Model Measurement Model x = indicators of ξ Λ x = factor loadings of ξ on x y = indicators of η Λ y = factor loadings of η on y δ = measurement error for x ε = measurement error for y

General SEM ε1ε1 y1y1 ε2ε2 y2y2 ε3ε3 y3y3 ε4ε4 y4y4 ε5ε5 y5y5 ε6ε6 y6y6 ε7ε7 y7y7 ε8ε8 y8y8 δ1δ1 x1x1 δ2δ2 x2x2 δ3δ3 x3x3 η1η1 ξ1ξ1 η2η2 ζ1ζ1 ζ2ζ2 β 21 γ 21 γ 11 λ1λ1 λ2λ2 λ3λ3 λ4λ4 λ5λ5 λ6λ6 λ7λ7 λ8λ8 λ9λ9 λ 10 λ 11 ξ 1 = industrialization η 1 = democracy time 1 η 2 = democracy time 2 x1-x3 = indus. indicator, e.g., energy y1-y4 = democ. indicators time 1 y5-y8 = democ. indicators time 2

Six Steps to Modeling Specification Implied Covariance Matrix Identification Estimation Model Fit Respecification

Specification Theorize your model – What observed variables? How many observed variables? – What latent variables? How many latent variables? – Relationship between latent variables? – Relationship between latent variables and observed variables? – Correlated errors of measurement?

Identification Are there unique values for parameters? Property of model, not data 10 = x + y x = y 2, 8 -1, 11 4, 6

Identification Underidentified Just identified Overidentified

Identification Rules for Identification – By type of model Classic econometric – e.g., recursive rule Confirmatory factor analysis – e.g., three indicator rule General Model – e.g., two-step rule

Identification Identified? Yes, by 3-indicator rule. Trust in Individuals people are helpful (x1) people can be trusted (x2) people are Fair (x3) 1 ξ1ξ1 δ1δ1 δ2δ2 δ3δ3 λ 11 λ 21

Model Fit Component Fit – Use Substantive Experience Are signs correct? Any nonsensical results? R 2 s for individual equations Negative error variances? Standard errors seem reasonable?

Model Fit How well does our model fit the data? The Test Statistic (Χ 2 ) – T=(N-1)F – df=½(p+q)(p+q+1) - # of parameters p = number of y’s q = number of x’s – Σ=Σ(θ) – Statistical power

Model Fit Many goodness-of-fit statistics – T b = chi-square test statistic for baseline model – T m = chi-square test statistic for hypothesized model – df b = degrees of freedom for baseline model – df m = degrees of freedom for hypothesized model

Model Fit Χ 2 = 223, df=5, p=.000 IFI =.87 RMSEA =.25 N=801

Respecification Theory! – Dimensionality? – Correct pattern of loadings? – Correlated errors of measurement? – Other paths? Modification Indexes Residuals:

Respecification Χ 2 = 3.8, df=2, p=.15 IFI = 1.0 RMSEA =.03 N=801

Useful References Book from which this talk is drawn: Bollen, Kenneth A Structural Equations with Latent Variables. New York: Wiley. Ed Rigdon’s website: Archives of SEMNET listserv: bama.ua.edu/archives/semnet.html