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Factor Analysis

Factor Analysis. Ulf H. Olsson Professor of Statistics. Home work. Down load LISREL 8.8. Adr.: http://www.ssicentral.com/ Read: David Kaplan: Ch.3 (Factor Analysis) Read: Lecture Notes. J-te order Moments. Skewness Kurtosis. Kurtosis. Factor Analysis. Exploratory Factor Analysis (EFA)

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Factor Analysis

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  1. Factor Analysis Ulf H. Olsson Professor of Statistics

  2. Home work • Down load LISREL 8.8. Adr.: http://www.ssicentral.com/ • Read: David Kaplan: Ch.3 (Factor Analysis) • Read: Lecture Notes Ulf H. Olsson

  3. J-te order Moments • Skewness • Kurtosis Ulf H. Olsson

  4. Kurtosis Ulf H. Olsson

  5. Factor Analysis • Exploratory Factor Analysis (EFA) • One wants to explore the empirical data to discover and detect characteristic features and interesting relationships without imposing any definite model on the data • Confirmatory Factor Analysis (CFA) • One builds a model assumed to describe, explain, or account for the empirical data in terms of relatively few parameters. The model is based on a priori information about the data structure in form of a specified theory or hypothesis Ulf H. Olsson

  6. The EFA model Ulf H. Olsson

  7. EFA • Eigenvalue of factor j • The total contribution of factor j to the total variance of the entire set of variables • Comunality of variable i • The common variance of a variable. The portion of a variable’s total variance that is accounted for by the common factors Ulf H. Olsson

  8. EFA-How many factors to retain • Based on theory • Eigenvalues 1 • Checking the rows in the pattern matrix Ulf H. Olsson

  9. Factor Solutions • Principal Factor Method • Extracts factors such that each factor accounts for the maximum possible amount of the variance contained in the set of variables being factored • No distributional assumptions • Maximum Likelihood • Will be treated in detail later • Multivariate normality Ulf H. Olsson

  10. Rotation of Factors • The objective is • To achieve a simpler factor structure • To achieve a meaningful structure • Orthogonal rotation • Oblique Rotation Ulf H. Olsson

  11. Rotation • Varimax • Major objective is to have a factor structure in which each variable loads highly on one and only one factor. • Quartimax • All the variables have a fairly high loading on one factor • Each variable should have a high loading on one other factor and near zero loadings on the remaining factors Ulf H. Olsson

  12. Rotation The rationale for rotation is very much akin to sharpening the focus of a microscope in order to see the details more clearly Ulf H. Olsson

  13. The CFA model • In a confirmatory factor analysis, the investigator has such a knowledge about the factorial nature of the variables that he/she is able to specify that each xi depends only on a few of the factors. If xi does not depend on faktor j, the factor loading lambdaij is zero Ulf H. Olsson

  14. CFA • If does not depend on then • In many applications, the latent factor represents a theoretical construct and the observed measures are designed to be indicators of this construct. In this case there is only (?) one non-zero loading in each equation Ulf H. Olsson

  15. CFA Ulf H. Olsson

  16. CFA Ulf H. Olsson

  17. CFA • The covariance matrices: Ulf H. Olsson

  18. Nine Psychological Tests(EFA) Ulf H. Olsson

  19. Nine Psychological Tests(CFA) Ulf H. Olsson

  20. Introduction to the ML-estimator Ulf H. Olsson

  21. Introduction to the ML-estimator • The value of the parameters that maximizes this function are the maximum likelihood estimates • Since the logarithm is a monotonic function, the values that maximizes L are the same as those that minimizes ln L Ulf H. Olsson

  22. Introduction to the ML-estimator • In sampling from a normal (univariate) distribution with mean  and variance 2 it is easy to verify that: MLs are consistent but not necessarily unbiased Ulf H. Olsson

  23. Two asymptotically Equivalent Tests Likelihood ratio test Wald test

  24. The Likelihood Ratio Test Ulf H. Olsson

  25. The Wald Test Ulf H. Olsson

  26. Example of the Wald test • Consider a simple regression model Ulf H. Olsson

  27. Likelihood- and Wald. Example from Simultaneous Equations Systems • N=218; # Vars.=9; # free parameters = 21; • Df = 24; • Likelihood based chi-square = 164.48 • Wald Based chi-square = 157.96 Ulf H. Olsson

  28. CFA and ML k is the number of manifest variables. If the observed variables comes from a multivariate normal distribution, and the model holds in the population, then Ulf H. Olsson

  29. Testing Exact Fit Ulf H. Olsson

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