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Structural Equation Modeling: An Overview P. Paxton

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

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

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

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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

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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

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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

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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)

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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

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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

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

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

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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

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

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

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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

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

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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

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Recursive y2x1y3 y2 x3 x1 y1 x2

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Nonrecursive x2y1 x1y2 y3 y2 x3 x1 y1 x2

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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

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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

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

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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

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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

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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

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

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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?

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

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Identification Underidentified Just identified Overidentified

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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

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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

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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?

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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

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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

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

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

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

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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

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