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In This Chapter We Will Cover

In This Chapter We Will Cover. Models with multiple dependent variables, where the independent variables are not observed. This is called Factor Analysis. We cover The factor analysis model A factor analysis example Measurement properties of the unobserved variables

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In This Chapter We Will Cover

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  1. In This Chapter We Will Cover • Models with multiple dependent variables, where the independent variables • are not observed. This is called Factor Analysis. We cover • The factor analysis model • A factor analysis example • Measurement properties of the unobserved variables • Maximum Likelihood estimation of the model • Some interesting special cases When statistical reasoning is applied to factor analysis, as it will be in this chapter, we often call this Confirmatory Factor Analysis.

  2. Regression with Multiple Dependent Variables These matrices have only one column in univariate regression analysis Y = XB +

  3. Comparing Regression with Factor Analysis Looking at a typical row corresponding to the data from subject i:

  4. We Transpose It and Drop the Subscript i From the previous slide we have Transpose both sides to get Then dropping the subscript i altogether gets us to y = Bx + e

  5. The Factor Analysis Model Observed variables Unique Factors Factor Loadings Common Factors

  6. Assumptions of the Model Random inputs of the model:  ~ N(0, )  ~ N(0, ) Cov(, ) = 0

  7. Now We Can Deduce the V(y) Named  Named  Assumed 0 We end up with only components 1 and 4 from the above equation V(y) =  + 

  8. Variables Description y1 Measurement 1 of B y2 Measurement 2 of B y3 Measurement 3 of B y4 Measurement 1 of C y5 Measurement 2 of C y6 Measurement 3 of C A Simple Example to Get Us Going

  9. The Pretend Example in Matrices

  10. 21 y1 y4 42 11 21 52 y2 y5 1 2 31 62 y3 y6 Graphical Conventions of Factor Analysis • Note use of • boxes • circles • single-headed arrows • double-headed arrows • unlabeled arrows

  11. Two Alternative Models Assume I have a model with just one y and one . My model is then y =  +  Now assume you have a model y = ** +  where * = a∙ and * = /a Whose model is right?

  12. Ambiguity in the Model My Model y =  +  V(y) = 2 +  Your Model y = ** +  where * = a∙ and * = /a but

  13. Resolving the Ambiguity by Setting the Metric Plan A Plan B

  14. Degrees of Freedom The General Alternative The Model H0:  =  +  HA:  = S 4 ’s 3 ’s 6 ’s 13 parameters

  15. ML Estimation of the Factor Analysis Model The likelihood of observation i is The likelihood of the sample is Because eaeb = ea+b

  16. The Log of the Likelihood

  17. The Log of the Likelihood Under HA LA = constant -

  18. The Likelihood Ratio From LA From L0 S, 0 n , 

  19. The Single Factor Model if V() = 11 = 1  =  +  The latent variable  is called a true score The model is called congeneric tests

  20. Even More Restrictive Models with More Degrees of Freedom -equivalent tests Parallel tests

  21. Multi-Trait Multi-Method Models 1 2 2 y11 y21 y31 y12 y22 y32 y13 y32 y33 4 5 6

  22. The MTMM Model in Equations

  23. Goodness of Fit According to Bentler and Bonett (1980) Define HA:  = S H0:  =  +  HS:  =  (with  diagonal) Perfect Fit (1) No Fit (0) (for off-diagonal) Then we could have

  24. Goodness of Fit

  25. Modification Indices

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