Factor Analysis Ulf H. Olsson Professor of Statistics

Ulf H. Olsson Home work Down load LISREL 8.8. Adr.: Read: David Kaplan: Ch.3 (Factor Analysis) Read: Lecture Notes

Ulf H. Olsson J-te order Moments Skewness Kurtosis

Ulf H. Olsson Kurtosis

Ulf H. Olsson 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 The EFA model

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

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

Ulf H. Olsson 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 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 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 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 CFA

Ulf H. Olsson CFA

Ulf H. Olsson CFA The covariance matrices:

Ulf H. Olsson Nine Psychological Tests(EFA)

Ulf H. Olsson Nine Psychological Tests(CFA)

Ulf H. Olsson Introduction to the ML-estimator

Ulf H. Olsson 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 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

Two asymptotically Equivalent Tests Likelihood ratio test Wald test

Ulf H. Olsson The Likelihood Ratio Test

Ulf H. Olsson The Wald Test

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

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

Ulf H. Olsson 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 Testing Exact Fit