Week Eighteen - Exploratory Factor Analysis

What is special and useful about the correlations between factors you get after an obliquely rotated model?

They are free from measurement error (because they were estimated within the model).

What's Principal Component Analysis often used for?

To create indices e.g. for health systems.

How is Principal Component Analysis different from Factor Analysis?

Use PCA when wanting to form new composite variables without really caring about underlying causes (e.g. when you have have lots of correlated predictors, you want to make a weighted indices with) In any other case, use factor analysis

Know guidelines for minimum sample size for epa

n = > 200 is preferred. Rules of thumb: Ratio of 1:10. Ten respondents per variable When expecting weak relationships: Ratio of 1:20. Twenty respondents per variable.

In exploratory factor analysis, x₁ x₂ means what, and what other letter should have a corresponding subscript?

x₁ x₂ means a score on a certain variable, there should be a corresponding error (unique part) denoted as E₁ E₂ etc

In single factor models, why does the fact that the direction of the factor loadings (+/-) is indeterminate not matter?

Because it's just a matter of sign, you can flip the axis to reflect the concept (and sign) that you want e.g. extroversion not introversion, heat not coolness

What is a crossloading? What's the cut off for seriousness?

A crossloading exists when an item loads not just on the factor you expect or want it to, but on another. These can be called non-trivial if they're small but still meaningful. .32 is somewhat of a cut off because at that point a separate factor is explaining more than 10% of variance in the item.

A factorially simple variable... A factorially complex variable...

A factorially simple variable *responds to one factor only* (all but one factor loadings are 0). A factorially complex variable *is influenced by multiple factors* (has several non-zero factor loading on other factors).

What is an extraction method? Which should you never use?

A method that fits your model to the data and helps indicate how many factors you have e.g. ULS, Maximum Likliehood. Never use default SPSS of principal component analysis, it's not even factor analysis.

In EPA, what are independent clusters?

A set of factorially simple indicators that collectively form an independent cluster of items that only respond to one factor. This is desirable in EPA e.g. big five has five independent clusters.

Give an example two psychological phenomena which might impede good factor analysis.

Acquiescence bias e.g. can show up as a factor in itself (even if only expecting one factor) if some group of people tend to just agree with everything. (i.e. measuring a latent tendency to agree with everything). Halo effects, if people just tend to overgeneralise based on one factor themselves e.g. Do I like this person.

How can you identify if a single factor model is sufficient in EPA?

By fitting the model and looking at the residual correlations, if there appear to be lots of correlations way above 0.05 left, you'l want to keep adding factors until correlations are mostly <0.05

DEFINITION: Rotation is...

DEFINITION: Rotation is a transformation of parameters (factor loadings in this case) to approximate an independent clusters structure (usually). That is, to have as many factorially simple variables as possible. e.g. a situation in which the first factor loads on all items, and the following factors only on their specific items.

Why do factor loadings have two subscripts in EFA?

Each factor will have a corresponding factor loading, and that's where 1, 2, 3 k etc come from on the factor loadings. Each will also also be unique to a particular item (1 2 3 i etc). Each variable will also have it's own factor loading on each factor hence lambda

What's the purpose of exploratory factor analysis?

Exploratory Factor Analysis (EFA) is the process of finding a factor model that best explains the pattern of covariances between observed variables

What are the steps of principal component analysis? What are eigenvalues? Why might it be useful for regression?

For regression, you can potentially transform loads of correlated predictors into separate components explaining lots of variance but uncorrelated!

When examining a scree plot how do you determine how many factors?

From wherever the rubble definitely seems to begin, minus one. *Unsure on this one.

What two things should you look at to check if factor analysis is worthwhile and; What three things would you look at to check how many factors you've got?

Good for FacAnal: Correlations Thurstone's measure of sampling adequacy How many factors? Scree plot Goodness of fit chi-squared Residuals Definitely never use default SPSS option of all eigenvalues above 1.

What's the rule of parsimony in EPA?

If you have two models that explain variance identically, always choose the one with less factors (i.e. simpler).

If you rotated __________ you need the pattern matrix to interpret your results if you rotated orthogonally you need the _____________ matrix

If you rotated obliquely you need the *pattern* matrix to interpret your results if you rotated orthogonally you need the *rotated factor* matrix

Single factor models are identified so long as you have.... If you only have ______ variables there's no point as this situation is described by a ___________, there'd also be no point running a _____________ with this amount of variables because...

Single factor models are identified so long as you have *three or more variables.* If you only have *two* variables there's no point as this situation is described by a *correlation*, there'd also be no point running a *regression* with this amount of variables because *you can always draw a straight line between two points.*

What is the basis of the solution to the rotation problem in multi-factor models?

The ability to move the axes. While the factor loadings have a relationship with each other in multidimensional space, they're not locked in relation to the axis.

The aim of principal components analysis (PCA) is ...

The aim of principal components analysis (PCA) is *to present the original variables through a (smaller) set of linear combinations*

What does it mean for a model to be identified?

The parameters (i.e. coefficients: Factor loadings, error variance) of the model can be uniquely determined from the data. To be identified, a model has to have no more parameters to estimate than the known pieces of information (i.e. it can't have more things to estimate than it has bits of information; this would not be a simplification of the data).

Why does the the indeterminacy of the direction of the factor loadings (+/-) become a problem in exploratory multiple-factor models ?

The problem is that it becomes very difficult to interpret. The factor loadings are no longer on a single axis, but locked against each other in multi-dimensional space. Consider like this: In the Factor Matrix output it's clear what one factor is, but how can we interpret these factors 2 and 3 which seem to have a mix of positive and negative. There's not any obvious pattern of three factors that we were expecting.

How is multicolinearity related to model identification?

With multicolinearity you can't tell how much total variance is truly explained by each variable, some variance might belong other correlating variables, or all of it or none. That is, you can't uniquely identify the co-efficient/parameter from the data.

Can factors correlate?

Yes they can

In EFA, how does rotation work and what are two main types. Which is preferred if you have to choose one and why?

You turn the axis so that each factor is close to zero on one dimension, and clearly directional in another. The two types are orthogonal and oblique. Orthogonal will be used when factors are independent from each other (i.e. orthogonal!) and oblique when you allow them to correlate. Oblique is preferred because you only gain information, don't lose e.g. if no correlations between factors it will just show no correlations.

What is data reduction? What's one example of it?

You're trying to logically find a linear combination of the original variables that will explain *most* of the variation while making data simpler. A favorable tradeoff between simplicity and information. e.g. instead of two variables explaining two similar things, you have one which explains most of what both together do. Principal components analysis.