rzStructural_Equation_Modeling.ppt ok this is AMOS

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

• Measurement Models

• Structural Models

• Simultaneous Models

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