The Common Factor Model
PSY544 – Introduction to Factor Analysis
Week 4

Homework!
•Homework assignment 1 will be out this week
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•I’ll send you an email, along with the deadline
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The data model in factor analysis



The data model in factor analysis



The data model in factor analysis



The data model in factor analysis



The data model in factor analysis



The data model in factor analysis
•The data model represents a random observation in the population. It is intended to explain the
structure of the raw data (i.e., the scores on manifest variables)
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•However, it contains a LOT of unknowns
•While we observe the manifest variables x and we can at least estimate the population means μ, the
remaining terms in the equation are unknown to us
•We do not know the latent scores z and u, in fact we cannot know them, since latent variables are
unobservable
•Similarly, we do not know Λ, the matrix of factor loadings – we are unaware of how the
unobservable latent variables affect the (observable) manifest variables

The data model in factor analysis
•Well, that’s kind of a pickle.
•So, do we just, like, go home now?
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•Maybe. Or we can help ourselves with some tricks.
•We have already established that the latent variable scores are unobservable, so we might want to
give up on trying to solve for them in the data model equation
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•Maybe if we turn the problem around, we can get rid of z and u completely and focus on Λ

The data model in factor analysis
•We could use the data model, along with some assumptions, to derive a covariance structure model
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•The data model is accompanied by assumptions about the joint distribution of the elements in z and
u and implies a model for the population covariance matrix. The model for the covariance matrix is
known as the covariance structure and is intended to explain the variances and covariances of the
manifest variables, not the raw data.
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•Before we proceed to derive the covariance structure model, we’ll talk about the important
distributional assumptions and lay down some notational rules.

Assumptions



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Notation



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Deriving the mean and covariance structures



Covariance structure



An example
•Remember the example correlation matrix I have shown earlier?
•(4 performance measures: paragraph comprehension, vocabulary, arithmetic skills, and mathematical
problem solving)
•Don’t forget – correlation matrix is just a special kind of covariance matrix!
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An example



Communality



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure



Correlation structure