Additional Topics
PSY544 – Introduction to Factor Analysis
Week 13

Additional Topics
•Today’s lesson will be a bit of an amorphic cross-over
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•We’ll talk about some topics that exceed the basics of FA that we have learned during the semester
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•If we had more time, the topics presented today would be presented in a more thorough way over the
course of multiple days, but…

Model comparison
•In many cases, you will have multiple models that are all plausible candidates. Your goal might be
to select one of them – the one which is superior to the rest.
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•You should compare interpretability
•You should compare model-data fit
•Ideally, you should compare both

Direct model comparison
•In order to compare models directly, the compared models
must be nested.
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•Model A is nested within Model B if Model A is a special case of Model B, or Model B is a general
case of Model A.
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•More specifically, Model A is nested within Model B if Model A can be obtained by imposing
additional restrictions on Model B.
•The free parameters of Model A are a subset of those of Model B.

Direct model comparison
•Sounds arcane?
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•Some examples of nested models:
•Any (usual) restricted model with m factors is nested within an unrestricted model with m factors
•An orthogonal model with m factor is nested within an oblique model with m factors (if restricted,
they must have the same loading structure)
•A model where two parameters are constrained to be equal is nested within the model without this
restriction.

Direct model comparison
•Nested models:
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•Will have more degrees of freedom
•Will have fewer free parameters (that’s the same thing)
•Will have equal or greater value of the same discrepancy function (the model have the same or
greater discrepancy from data)

Direct model comparison



Direct model comparison



Indirect model comparison



Information criteria
•Remember the log-likelihood from way back when we talked about maximum likelihood estimation?
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•The log-likelihood is usually a relatively large, negative number. The smaller it gets (the more
negative it gets), the smaller the likelihood. In case data is the same, worse models will result
in smaller likelihood.
•Sometimes, a deviance is calculated: -2*log-likelihood (so, a relatively large, positive number).
The larger the deviance, the smaller the likelihood (the more the model deviates from data)
•Deviance is used to calculate the so-called information criteria.
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Information criteria



Information criteria



Information criteria
•You can only compare models on their information criteria if the models were fit to the same data
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•Moreover, even if the information criteria values differ for two models, we don’t know how much is
too much – there is no “effect size” for information criteria.
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•So treat the AIC and BIC as sources of information, but keep the above in mind.
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Bi-factor model
•What is the bi-factor model?
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•1) All items load on a single “general” factor
•2) All items also load on one, and only one, additional “specific” factor
•3) All factors are uncorrelated
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•So, the Λ matrix has m columns, where one of these columns is full of free parameters and the
remaining m-1 columns contain free parameters each for a set of MVs, these sets do not overlap. The
Φ matrix is diagonal.
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Bi-factor model
•Why can the bi-factor model be useful?
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•It’s a “multidimensional unidimensional model” J
•It might have interesting interpretations
•It usually fits better than a 1-factor model
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