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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 Equationa mathematical

equation representing the structure of variables

relationships to each other.

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Structural Equation Modeling

- Path 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 v1 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 Modeling

- Path 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 Modeling

- Path Analysis
- P. 384 for mathematical way to decompose
- But really, all you need to do is trace paths.

P. 386 for rules (lets 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 Modeling

- Confirmatory Factor Analysis
- Same thing we proposed last week w/EFA and

Cronbachs Alpha - But now lets assume the factor is standardized

with a variance of 1, where an indicators

loading equals 1this 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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. - Well discuss
- loadings later.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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(k1)/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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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 gt .05 means fail to reject the null, the model

has a good fit.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory Factor Analysis
- Other measures of fit that dont 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- Confirmatory Factor Analysis
- Remember the concept of scaling with variations

in item weights? SEM allows for automatic

weighting.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- In 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

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- 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
- 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.

Factor

1

X1

X3

X2

e1

e2

e3

Structural Equation Modeling

- 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. - This indicates a way to test whether two

parameters are equaljust constrain them to be

equal and compare the fit. - Ways to test whether parameters are zerojust

constrain them to be zero and compare the fit.

Factor

1

X1

X3

X2

e1

e2

e3

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