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Introduction to Factor Analysis
& Exploratory Factor
Analysis
- Introduction to Factor Analysis
- An Aside About Principal Component Analysis
- Exploratory Factor Analysis
Introduction to Factor Analysis
Introduction
- “Measure one thing, and measure it well.”
- Internal consistency of an instrument:
- Testing the unitary nature of an instrument
- But what if it doesn’t measure one thing well?
- What if an instrument measures more than one thing?
Factors
- Often, instruments measure more than one thing
- E.g., a test of content knowledge may measure more than one content
area:
- Health promotion & maintenance
- Psychosocial integrity
- Physiological integrity
- Safe & effective care environment
Factors (cont.)
- A respondent may know one content area better than others
- E.g., know “psychosocial integrity” better than “health promotion
& maintenance”
- So items related to “psychosocial integrity” should correlate
together well
- And correlate less well with items about “health promotion &
maintenance”
Factors (cont.)
- Items that intercorrelate are assumed to measure the same,
underlying trait
- I.e., sample the same domain
- A domain, of course, is a theoretical,
non-ostensible (endogenous) construct
- The collection of measured items, though, creates a score
- The linear combination of related items is a factor
score
Factors (cont.)
- One can therefore measure if all items on an instrument indeed
measure the same factor
- And whether, instead, an instrument measures more than one
factor
- By testing how well items correlated with—“load”
onto—a given factor
- Factor loadings are thus the correlations / standardized regression
coefficients between items and the factors they load onto
Factors (cont.)
- A factor, then, is that which related items measure
- Factor vs. domain:
- Both are not directly measured
- A factor is nonetheless indirectly quantifiable through the
items
- A domain is only sampled, and rarely fully measurable
- The nature of that domain is investigated through content (and
construct) analysis
Factors (cont.)
- Factor as linear combination
Whole Instrument=x1+x2+x3+x4
Whole
Instrument=1x1+1x2+1x3+1x4
Factor1=1x1+1x2+0x3+0x4
Factor2=0x1+0x2+1x3+1x4
Factors (end)
- Factor as linear combination (cont.)
Whole Instrument=x1+x2+x3+x4
Whole
Instrument=1x1+1x2+1x3+1x4
Factor1=.7x1+.5x2+.1x3+.2x4
Factor2=.1x1+.1x2+.6x3+.8x4
Types of Factor Analysis
- Factor analysis is, in fact, a very broad field
- Generally (and here), though, we will focus on two types:
- Exploratory factor analysis
- Letting the data suggest a possible factor structure
- Confirmatory factor analysis
- Using hypotheses to test a factor
structure
EFA & PCA
- Factor analysis is an example of “data reduction”
- I.e., attempting to find a more parsimonious explanation for a
series of responses & their concomitant data
- Reduce the number of explanatory variables while loosing the least
important information
EFA & PCA (cont.)
- Factor analysis is similar to principal
component anlaysis (PCA)
- Which differs
from factor analysis in not attempting to find an underlying,
theoretical structure
- PCA also assumes there is no unique variance to items
- That all variance is shared variance
- Factor analysis assumes there is shared variance
& unique variance
- The “unique” being those underlying factors
EFA & PCA (end)
Explore and test the presence &
structure of possible latent (non-ostensible) constructs |
Reduce list of items (or variables) to a
linear combination of a smaller set of components |
I.e., make inferences about what the items
measure |
I.e., explain data more succinctly |
Exploratory Factor Analysis
Main Steps to
Exploratory Factor
Analysis
- Extraction
- Rotation
- Interpretation
EFA: Rotation
- Factors may be naturally orthogonal:
Factor1=1x1+1x2+0x3+0x4
Factor2=0x1+0x2+1x3+1x4
- But more often, they are not entirely so:
Factor1=.7x1+.5x2+.1x3+.2x4
Factor2=.1x1+.1x2+.6x3+.8x4
EFA: Rotation (cont.)
- Orthogonal factors are at right angles to each
other
- If the extracted factors are intercorrelated,
- We can “rotate”
them to be uncorrelated
![](efa_files/figure-revealjs/unnamed-chunk-1-1.png)
EFA: Rotation (cont.)
- Non-orthogonal factors:
- The factors intercorrelate
- The level of intercorrelation can be chosen
- This is called an oblique solution
![](efa_files/figure-revealjs/unnamed-chunk-2-1.png)
EFA: Rotation (end)
- There are many
methods one can use to rotate factors
- And thus the items that are loaded onto each factor
- Researchers can try out several
rotations,
- But usually trying out a couple suffices
- E.g., one or two each of oblique & orthogonal
- Exploring rotations can help interpret factors
Common Orthogonal
Rotations
- Varimax
- Attempts to simplify interpretations of factors
- By making the factor loadings for each item as close to 0 or as
close to 1 as possible
- Quartimax
- Attempts to simplify interpretation of items
- By maximizing the variance of the loadings between factors
- Quartimax accentutates differenecs between factors
- Changing proportion of variance factors explains
Common Oblique Rotations
- Promax
- Tends to give rather easily-interpreted loadings onto the
factors
- Direct Oblimin
- Attempts to find the best axis for each factor
- Promax tends to produce
more orthogonal factors than direct oblimin
EFA: Interpretation
- EFA does not directly test what domains are being represented
- It can only suggest which items are most related to the proffered
factors
- It is up to us to decide what constructs underlie the factors
- In large part by evaluating items loadings
- E.g., from different extractions & rotations
Final Thoughts on EFAs
- EFA is highly data driven
- And based on subjectivity
- It is therefore best for initial analyses
- EFA is not to be discounted
- But it is, frankly, often abused
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Introduction to Factor Analysis& Exploratory Factor
Analysis