A Brief Look at Confirmatory Factor Analysis

1. A Brief Look at Confirmatory Factor Analysis • Structural Equation Models • For the economists, this usually means multi-equation systems of regression models • For the social scientist, this usually means the analysis of covariance – in a multi-equation structure

2. LISREL • Means Linear Structural Relations • Also called • Structural Equation Modeling • Confirmatory Factor Analysis • Latent Variable Analysis • Much different than Exploratory Factor Analysis

3. Latent Variable Analysis • SEM is best understood as latent Variable Analysis • Frequently we wish to describe or explain things that cannot be observed directly • Justice • Belief systems • Democracy • Sometimes, we have partial measures for larger constructs that might be more adequately described with a measurement model.

4. Measures • The variables that we use to describe this underlying, or latent, variable are called • Manifest, or • Observed variables

5. Exploratory Factor Analysis • In the old days (the 60’s and 70’s) these measured variables were ‘dumped’ into a factor analysis program to see how the variables would ‘load’ • Each dimension, or factor, would capture the covariation of the variables most closely related, and Factor scores would be developed. • This was a data exploration unguided by theory. • Whatever the factors produced was accepted as the appropriate factor analysis of the data.

6. Confirmatory Factor Analysis • In the modern sense, the researcher starts with a set of beliefs (assumptions) about the interrelationships of the variables, • And then tests, or confirms, the theoretical expression of the model. • The resulting model has two pieces: • The measurement model expressing the relationships between the observed variables and the underlying unobserved or latent variables • The structural relationships between the latent variables

7. Causal Determination in SEM • As with regression we are often concerned with causation • Exogenous variables are caused outside the model – independent variables. • Endogenous variables are causally determined within the model – dependent variables.

8. The SEM Model • The SEM mathematical model • Comprised of three pieces: • The Measurement Model for the Exogenous , or X, variables. • The Measurement Model for the Endogenous , or Y, variables. • The Structural Equation Model – the relationships between X and Y.

9. It’s all Greek to me - 1 • The Measurement Model for the Exogenous, or X, variables. • Where: • x is a qx1 vector of observed exogenous variables, •  (xi) is an nx1 vector of latent exogenous variables, • x (Lambda-x) is a qxn matrix that related n exogenous factors to each of the q observed variables that measure them, and •  (delta) is qx1 vector of measurement errors in x.

10. It’s all Greek to me - 2 • The Measurement Model for the Endogenous, or Y, variables • Where: • y is a px1 vector of observed endogenous variables, •  (eta) is an mx1 vector of latent endogenous variables, • Y(Lambda-y) is a pxm matrix that related m endogenous factors to each of the p observed variables that measure them, and •  (epsilon) is px1 vector of measurement errors in y.

11. It’s all Greek to me - 3 • The Structural Equation Model – the relationships between X and Y. • Where •  (gamma) is an mxn matrix of coefficients that relates the n exogenous factors to the m endogenous factors, •  (beta) is an mxm matrix of coefficients that relates the endogenous variables to each other, and •  (zeta) is an mx1 vector of residuals representing errors in the equations relating  and .

12. For Instance

15. Models Covariance • The covariation between Y and X is:

16. Diagnostics