1. Slide 9.1 Confirmatory Factor Analysis MathematicalMarketing 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. Slide 9.2 Confirmatory Factor Analysis MathematicalMarketing Regression with Multiple Dependent Variables Y = XB +  These matrices have only one column in univariate regression analysis

3. Slide 9.3 Confirmatory Factor Analysis MathematicalMarketing Comparing Regression with Factor Analysis Looking at a typical row corresponding to the data from subject i:

4. Slide 9.4 Confirmatory Factor Analysis MathematicalMarketing We Transpose It and Drop the Subscript i y = Bx + e Then dropping the subscript i altogether gets us to From the previous slide we have Transpose both sides to get

5. Slide 9.5 Confirmatory Factor Analysis MathematicalMarketing The Factor Analysis Model Observed variables Factor Loadings Common Factors Unique Factors

6. Slide 9.6 Confirmatory Factor Analysis MathematicalMarketing Assumptions of the Model Random inputs of the model:  ~ N(0,  )  ~ N(0,  ) Cov( ,  ) = 0

7. Slide 9.7 Confirmatory Factor Analysis MathematicalMarketing Now We Can Deduce the V(y) Named  Assumed 0 Named  We end up with only components 1 and 4 from the above equation V(y) =  + 

8. Slide 9.8 Confirmatory Factor Analysis MathematicalMarketing A Simple Example to Get Us Going VariablesDescription y1y1 Measurement 1 of B y2y2 Measurement 2 of B y3y3 Measurement 3 of B y4y4 Measurement 1 of C y5y5 Measurement 2 of C y6y6 Measurement 3 of C

9. Slide 9.9 Confirmatory Factor Analysis MathematicalMarketing The Pretend Example in Matrices

10. Slide 9.10 Confirmatory Factor Analysis MathematicalMarketing Graphical Conventions of Factor Analysis y2y2 y3y3 y4y4 y5y5 y6y6 y1y1 11 22 11 21 31 42 52 62  21 Note use of  boxes  circles  single-headed arrows  double-headed arrows  unlabeled arrows

11. Slide 9.11 Confirmatory Factor Analysis MathematicalMarketing 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? Two Alternative Models

12. Slide 9.12 Confirmatory Factor Analysis MathematicalMarketing Ambiguity in the Model My Model y =  +  V(y) = 2  +  Your Model y = *  * +  where  * = a∙  and * = /a but

13. Slide 9.13 Confirmatory Factor Analysis MathematicalMarketing Resolving the Ambiguity by Setting the Metric Plan A Plan B

14. Slide 9.14 Confirmatory Factor Analysis MathematicalMarketing Degrees of Freedom 4 ’s 3  ’s 6  ’s 13 parameters The General Alternative The Model H 0 :  =  +  H A :  = S

15. Slide 9.15 Confirmatory Factor Analysis MathematicalMarketing ML Estimation of the Factor Analysis Model The likelihood of observation i is The likelihood of the sample is Because e a e b = e a+b

16. Slide 9.16 Confirmatory Factor Analysis MathematicalMarketing The Log of the Likelihood

17. Slide 9.17 Confirmatory Factor Analysis MathematicalMarketing The Log of the Likelihood Under H A L A = constant -

18. Slide 9.18 Confirmatory Factor Analysis MathematicalMarketing The Likelihood Ratio From L 0 From L A   S, 0 n  , 

19. Slide 9.19 Confirmatory Factor Analysis MathematicalMarketing The Single Factor Model  = +  if V(  ) =  11 = 1 The latent variable  is called a true score The model is called congeneric tests

20. Slide 9.20 Confirmatory Factor Analysis MathematicalMarketing Even More Restrictive Models with More Degrees of Freedom  -equivalent tests Parallel tests

21. Slide 9.21 Confirmatory Factor Analysis MathematicalMarketing Multi-Trait Multi-Method Models y 21 y 31 y 11 y 22 y 32 y 12 y 32 y 33 y 13 11 22 22 44 55 66

22. Slide 9.22 Confirmatory Factor Analysis MathematicalMarketing The MTMM Model in Equations

23. Slide 9.23 Confirmatory Factor Analysis MathematicalMarketing Goodness of Fit According to Bentler and Bonett (1980) Define H A :  = S H 0 :  =  +  H S :  =  (with  diagonal) Then we could have Perfect Fit (1) No Fit (0) (for off-diagonal)

24. Slide 9.24 Confirmatory Factor Analysis MathematicalMarketing Goodness of Fit

25. Slide 9.25 Confirmatory Factor Analysis MathematicalMarketing Modification Indices