Factor Analysis Ulf H. Olsson Professor of Statistics

Ulf H. Olsson From the Kaplan book Page 40 – 45 (ch.3) Page 24 – 30 (ch.2) Page 48 – 53 (ch.3)

Ulf H. Olsson The (linear) Factor Model

Ulf H. Olsson Nine Psychological Tests(EFA)

Ulf H. Olsson Nine Psychological Tests(CFA)

Ulf H. Olsson The (linear) Factor Model Assumptions/convenient assumptions

Ulf H. Olsson CFA

Ulf H. Olsson Parameter Function

Ulf H. Olsson Rotation

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 CFA The covariance matrices:

Ulf H. Olsson Estimation of the parameters Minimizing a “fit function”

Ulf H. Olsson Introduction to the ML-estimator See page for normal distribution

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 See page based on the normal distribution

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 CFA and ML

Ulf H. Olsson Testing Exact Fit

Ulf H. Olsson ML – chi-square test N=218; # Vars.=9; # free parameters = 21; Df = 24; Likelihood based chi-square =

Ulf H. Olsson CFA and GLS (fit function) 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 ML and GLS are asymptotically equivalent If The models holds in the population The observed variables are multivariate normal

Ulf H. Olsson Large-sample properties

Ulf H. Olsson CFA example NPV-data set Chi-square tests Modification indices T (Z)-values df

Ulf H. Olsson Simple example of 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