Structural Equation Modeling

What is SEM Swiss Army Knife of Statistics Can replicate virtually any model from “canned” stats packages (some limitations of categorical variables) Can test process (B mediates the relationship between A and C) Can test competing models Simultaneous estimation of multiple equations Error Free estimates (if used correctly)

Regression Logic Estimated using Ordinary Least Squares (OLS) Assumes normally distributed, independent error Why? Not for estimating B’s, but for estimating the Standard Error for B’s!

SEM Logic Y X1X1 X2X2 β1β1 β2β2 β0β0 r e 1 This will result in the same parameter estimates

Symbols One headed arrows: AB “A causes B” AB Two headed arrows: “A is correlated with B” also referred to as “unanalyzed relationship”

More Symbols Important Note: No arrows between variables fixes the relationship to zero. It is an argument that there is no relationship, not just one that you are not analyzing. AB Is the same as AB 0

More Symbols X An observed variable (i.e. in your dataset) F An un-observed variable (i.e. not in your dataset) This is called a latent trait. If used correctly, these guys can get you error free estimates of parameters

Latent Trait In a confirmatory factor analysis: Latent trait is the shared variance Remember, error doesn’t correlate with anything! F X1X2X3

You Already Know About Latent Traits! You just might not know that you know MANOVA F Y1 Y2 Y3 X1 e F would be the first characteristic root

Lets Practice Write out this model as a regression equation SES Obesity Height e 1 r β1β1 β2β2 Days Walked to School β3β3 r r

Estimation Maximum Likelihood: “Plug In” numbers until you reach a point where the error is minimized (which is the same as where the likelihood is maximized) Have to make one very strong assumption: multivariate normality But you can violate this to some degree without causing major problems (and most models will).

Problem When Estimating SEM

Model Fit Remember, we are just drawing the representation of a bunch of equations You can ask the question, at the end, how well these equations recreate the correlation matrix The test of this is the Chi-Squared test for Goodness of Fit (surprise, surprise!) This stat only works when the N size is between 50 and 200

Other Fit Indices RMSEA: Root Mean Square Error. This works well for larger sample sizes. It is a parsimony adjusted measure (favors simpler models, punishes unneeded parameters) Values lower that.10 considered adequate, lower than.05 considered good CFI- Comparative Fit Index (values closer to 1 are better)

Assumptions Now in Maximum Likelihood World Iteratative method Multivariate Normality Much stronger assumption than regression Estimating Population Coefficients: not a good small sample method (some newer methods of estimation may solve this though) Same problems with multicollinearity (results in a mathematical error)

On To More Exciting Models! Remember our good friend ANCOVA? Analysis of Covariance: Linear Dependent Variable, Categorical Independent Variable with a linear covariate Common form: Did treatment condition predict post-test after controlling for pre-test? More powerful than repeated measures t-test (assumes some measurement error). Just another incarnation of our even better friend, the general linear model! (Like many popular statistics!) This guy is 1 for treatment, 0 otherwise

Let’s Draw an ANCOVA using Path Analysis PrePost Treatment (1,0) e 1 r β1β1 β2β2 β0β0 Major Problem: Error! How can we judge effects when 20+ percent of our measures are error variance? Attenuation AND Inflation

Solution? What if I had all of the answers to all of the items making up the scales of my pre and post test? Or multiple sub scales making up a larger scale? ANCOVA becomes something more exciting: Latent Pre I1I2I3 Latent Post I1I2I3 Treatment (0,1)

MIMIC Models These are called MIMIC models, or Multiple Indicator, Multiple Cause Models Our treatment effect is (theoretically) not attenuated by error Latent Pre I1I2I3 Latent Post I1I2I3 Treatment (0,1)

More MIMIC These can get more interesting than an ANCOVA Latent Pre I1I2I3 Latent Post I1I2I3 Treatment (0,1) 2 nd Latent Pre 2 nd Latent Post I1I2I3I1I2I3 (This model is arguing that the treatment only changes the first latent variable while the second latent variable is just influenced by the first)

Latent Mediation B I1I2I3 C I1I2I3 A I1I2I3

Latent Mediation B I1I2I3 C I1I2I3 A I1I2I3

One More Interesting Model Latent Mediation Models