Structural Equation Modeling

Structural Equation Modeling

• Diagramming P. 374, Box 11.1—

Rules for Causal Diagrams• Exogenous vs. Endogenous Variables

• Direct and Indirect Effects

• Residual Variables

Observed Latent Residual, uncorrelated

Structural Equation Modeling

• Measurement Models

Structural Equation Modeling

• Structural Models

Structural Equation Modeling

• Simultaneous Models

Structural Equation Modeling

Path Analysis

• Path Analysis estimates effects of variables in a causal system.

• It starts with structural Equation—a mathematical equation representing the structure of variables’ relationships to each other.

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Structural Equation ModelingPath Analysis• Regress endogenous variables on their predictors• Variables are stated in terms of Z-scores• Standardized coefficients are the path coefficients• Path coefficient from the residual to a variable is √1 – R2

(the unexplained variation)• Squaring the path from a residual to a variable gives 1 –

R2 % variance not explained by the model

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Structural Equation ModelingPath Analysis• Path coefficients also account for your correlation matrix.• Decomposition gives you parts of the correlation between variables• Correlation between two variables is the additive value of the paths

from one to the other. Along the way, paths are multiplied if a variable(s) intervenes along the way. Intuitively, this makes sense. If X Y Z, and the two coefficients are .5 each, the correlation of X and Z is .25. This makes sense because if X goes up one standard deviation, Y goes up .5. Z will go up .5 for every one unit increase in Y, meaning that it will go up .25 for a .5 unit increase in Y, or a one unit increase in X. (Like gears turning in sequence).

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Structural Equation ModelingPath Analysis• P. 384 for mathematical way to decompose• But really, all you need to do is trace paths. P.

386 for rules (let’s go through these)• If you specify a potential path as zero, you can

test to see if the correlations between variables as hypothesized in that model match the real correlations

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Structural Equation ModelingConfirmatory Factor Analysis• Same thing we proposed last week w/EFA and Cronbach’s Alpha• But now let’s assume the factor is standardized with a variance of 1,

where an indicator’s loading equals 1—this is actually a way to create a scale for your factor.

• Variance of X will equal correlation with the factor plus error.• The correlation of a pair of observed variables loading on a factor is

the product of their standardized factor loadings.

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Structural Equation ModelingConfirmatory Factor Analysis• MLE replicates the covariance matrix by generating “parameters”

(really statistics) that come as close as possible to the observed covariance matrix (see Vogt).

• Weights are generated. They have unstandardized coefficient interpretations and do not represent the correlations with the factor. The biggest is not necessarily the best.

• We’ll discuss

loadings later.

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Structural Equation ModelingConfirmatory Factor Analysis• Next, we should test the fit of the model.• This test compares the generated predicted covariance matrix with

the real covariance matrix in the sample data.• The “fit function” multiplied by N-1 is distributed as a 2 with degrees

of freedom = [k(k+1)/2] – t, where t is the number of independent parameters.

• Some do not use 2 as a test statistic--think of 2 as equal to (S – E), where S is the sample covariance matrix and E is the expected from MLE.

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Structural Equation ModelingConfirmatory Factor Analysis• Some do not use 2 as a test statistic--think of 2 as equal to (S – E),

where S is the sample covariance matrix and E is the expected from MLE.

• Rejecting 2 has a “badness of fit” interpretation: Null: S = E

Alternative: S E

p > .05 means fail to reject the null, the model has a good fit.

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Structural Equation ModelingConfirmatory Factor Analysis• Larger samples produce greater chance of rejecting the

null, saying the fit is bad.• Therefore, some use the ratio of 2 /df and try to get this

ratio under 2. 2 = 0 would imply perfect fit.

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Structural Equation ModelingConfirmatory Factor Analysis• Other measures of fit that don’t depend on sample size:

– GFI: Analogous to R2, Fit of this model versus fit of no model (where all parameters equal zero). You try for .95 or higher.

GFI = [1 – (unexplained variability/total variability)]– AGFI: GFI adjusted for model complexity. More complex

models reduce GFI more.

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Structural Equation ModelingConfirmatory Factor Analysis• Parameters

– When model fit is good, you can begin focusing on “parameters.” Each has a standard error associated with it.

– The AMOS output refers to critical ratios. A critical ratio is the parameter divided by its standard error. (a lot like a z test) If your critical ratio is 1.96 or larger, your parameter is significant.

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Structural Equation ModelingConfirmatory Factor Analysis• Loadings--Check to see if your loadings are equal.

– Completely standardizing the model gives correlations and standardized coefficients, allowing you to compare them with each other on strength of association with the factor.

– The square of the loading plus the square of the effect of the error term equals 1. All variation is coming from two sources.

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Structural Equation ModelingConfirmatory Factor Analysis

Remember the concept of scaling with variations in item weights? SEM allows for “automatic” weighting.

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Structural Equation ModelingIn SEM, interpret path coefficients the way you would with OLS

regression (or path analysis), which would give the same results.

Testing whether one model is better than another is simply a 2 test with degrees of freedom of the difference in df between the two models.

Null: Models are the same

Alternative: Models are different

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Structural Equation ModelingTesting whether one model is better than another is simply a 2 test

with degrees of freedom of the difference in df between the two models.

Null: Models are the sameAlternative: Models are different If 2 is not significant, the models are equivalent. Therefore, use the

more restricted model with higher degrees of freedom. If 2 is significant, the parameters cannot be the same between models.

No constraint should be added.

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Structural Equation ModelingIf 2 is not significant, the models are equivalent. Therefore, use the

more restricted model with higher degrees of freedom. If 2 is significant, the parameters cannot be the same between models.

No constraint should be added.This indicates a way to test whether two parameters are equal—just

constrain them to be equal and compare the fit.Ways to test whether parameters are zero—just constrain them to be

zero and compare the fit.

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