Structural Equation Modeling

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Structural Equation Modeling
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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

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Structural Equation Modeling

• Measurement Models

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Structural Equation Modeling

• Structural Models

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Structural Equation Modeling

• Simultaneous Models

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Structural Equation Modeling

• Confirmatory Factor Analysis
• Remember the concept of scaling with variations
  in item weights? SEM allows for automatic
  weighting.

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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

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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.

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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.

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