Structural Equation Modeling Natasha Hudek and Stephanie Rattelade

Other Names for SEM Causal modeling Causal analysis Simultaneous equation modeling Analysis of covariance structures Path analysis Confirmatory factor analysis

Structural Equation Modeling Confirmatory Approach Combines Factor Analysis and multiple regression Data = Model + Residual Path Diagram – a visual representation of the variables and their relationships

Key Terms Latent Variable – cannot be observed and is not measured directly Observed Variable - measurable variables Exogenous – causes changes in other variables Endogenous – dependent and mediating variables Parameters – the relationships between each variable to be estimated

Relationships between Variables Association – a relationship between two variables that is nondirectional Direct Effect – a directional relationship between two variables Indirect Effect – effect of an exogenous variable on a endogenous variable through a mediating variable Total effect – the sum of direct and indirect effects on a variable

Path Diagram 1 E1 E2 1 1 SAT School Success Income Education 1 E3

Path Diagram Legend Circles/Elipses: Latent variables/factors (not directly measured) and error terms Rectangles/Squares: Observed variables (measured) Lines indicate relationships between variables

Path Diagram Legend Single arrow: directional relationship between an IV and DV Double arrow: unanalysed relationship/covariance between 2 variables Also represents variable’s variance Stars indicate free parameters, numbers represet fixed parameters

Model Components Measurement model – shows the relationships between observed and latent variables Structural model – the relationships between latent variables Recursive model – unidirectional links between variables Nonrecursive model – reciprocal links between variables

Models First-Order Factor Model Second-Order Factor Model

When to Use SEM 1 + IVs and 1 + DVs IVs and DVs can be continuous or discrete, latent or observed Combines factor analysis and multiple regression Must have a hypothesis, SEM is confirmatory, not exploratory

When to Use SEM Testing a model: How well does the model fit the data? Testing a theory: Which model/theory fits the data better? Variances and reliabilities: Amount of variance in the DVs accounted for by the IVs Reliability of observed/measured variables Parameter estimates: What is the coefficient predicting one DV from one IV

When to Use SEM Mediating/moderating relationships Group differences in model fit: Does the model fit 2 or more groups? Differences across time: Longitudinal differences within and between people Latent Growth Curve Modeling Nested models: IVs predict DVs at various levels of multilevel models

Assumptions Large sample size No missing data Multivariate normality No outliers Linearity (among observed variables) No multicollinearity and singularity Residuals should be small and their covariances should be symmetrical

Note on Covariance Algebra Covariance algebra can be used to solve SEM models Each DV has it’s own equation Several steps are required in the calculation of each covariance For complex models, these calculations become time consuming

Step 1: Specification 1 E2 E1 1 1 SAT (V2) School Success (F1) Income (V1) Education (V3) 1 E3

Step 1: Specification Relationships defined in the model are converted into equations/matrices to be estimated Bentler-Weeks method uses every variable in the model as an IV or DV, and estimates: Regression coefficients, and Variances and covariances of IVs

Step 1: Specification Regression coefficients: h = bh + gx Where, for example, h = a 3x1 vector, b = a 3x3 matrix, g = a 3x4 matrix, and x = a 4x1 vector, so h = b h + g x = + Ex: v2 = 0v2 + 0v3 + *f1 + 0v1 + 0e1 + 1e2 + 0e3, so v2 =*f1 + e2

Step 1: Specification Variances and covariances of the IVs: f = a 4x4 matrix f =

Step 2: Estimation Population parameters are estimated to produce a covariance matrix for the model Start values are used to make initial guesses at the coefficients and variances to be estimated For example: = , = , =

Step 2: Estimation Observed/measured variables are extracted from the full parameter matrices using a selection matrix (G): For observed DVs: Y = Gy*h = For observed IVs: X = Gx*x = v1 Rewriting the regression equation to express the DVs as a linear combination of IVs, we get: h = (I – b)-1gx

Step 2: Estimation Using the formula We get the covariance matrix between the DVs For example: =

Step 2: Estimation Using the formula We get the covariance matrix between the IVs and DVs For example: =

Step 2: Estimation Using the formula We get the covariance matrix between the IVs and For example: =

Step 2: Estimation We then combine , , and to get an initial covariance matrix = This is what we get after one iteration Iterations continue until the function converges, and a solution is reached Our example took 12 iterations

Step 2: Estimation The final estimated parameters for the example are: = , = , = And the final estimated population covariance matrix is: = The final residual matrix, where S is the sample covariance matrix, is: S - =

Step 2: Estimation 1 391.84 E1 E2 1 1 SAT (V2) School Success (F1) 20.49 2.56 Income (V1) .35 Education (V3) .14 1 E3

Step 3: Evaluation Model “fit” is evaluated using c2 In the example, convergence occurred at c2 = 0.47 Can also compute c2 from, c2 = i(n-1), where I is the minimum value obtained by convergence, and n is sample size Sample size affects c2 value

Step 3: Evaluation To get degrees of freedom, So, df = 6 – 5 = 1 df = # data points - # parameters to be estimated # data points = v(v+1)/2 = 3(3+1)/2 = 6, where v is # of measured variables So, df = 6 – 5 = 1 c2 (1) = .47, p = .49

Step 3: Evaluation Since our model fist the data well, we can look at individual relationships Converting our 3 estimated regression parameters to z-scores: Income  School Success = .347/.150 = 2.31, p = .02 School Success  SAT = 20.938/5.093 = 4.11, p = .000 School Success  Education = .143/.023 = 6.22, p = .000

Step 4: Modification If you are happy with the evaluation results, you can leave your model as is. If your c2 test proves significant, you may want to make revisions to your model and compare the old and new models This is also done when testing a new theory versus a previous theory

SEM Programs EQS SAS CALIS LISREL

AMOS

Creating a Path Diagram

Adding Parameters

Running the Output

Testing Models Default Model – model to be tested Independence model – goodness-of-fit tested at 0 Saturated model – goodness-of-fit tested at 1

Unstandardized Outputs

Standardized Outputs

Outputs

Outputs

Variance-Covariance Matrix of Sample

Model Fit Summary

Interpretation We can conclude that our hypothesized model fits the data Next Steps: Identify areas of misfit Modify model relationships and retest