Factor analysis (PCA)
in action
Thought for the day:
“Does one learn better by understanding the
abstract definition or by actually doing the
activity?”
My answer: Both together works best
Demonstration of the process
• I’d like to show you how to do an EFA on a
dataset.
• Further, I’d like to show you how messy it
can be. A bit like finger painting.
• This dataset has to do with individualism
and collectivism.
I and C
• Harry Triandis’s book, Individualism and
collectivism (1995), is a classic in that it
defines I and C and summarizes several
decades of research concerning these
constructs.
• He says that a “subjective culture” becomes
organized around a central theme and this is
the basis for I and C. It’s a way of thinking
about the self in relation to others.
Definition of Collectivism
• Collectivism is: 1) emphasis on the views,
needs, and goals of the ingroup rather than
the self; 2) emphasis on behaviour
determined by social norms and duties
rather than by pleasure or personal
advantage; 3) common beliefs that are
shared by the ingroup; and 4) willingness to
cooperate with the ingroup.
Definition of Individualism
• Individualism is a social pattern that consists of: 1)
loosely linked individuals who view themselves as
independent of collectives; 2) are primarily
motivated by their own preferences, needs, rights,
and contracts; 3) give priority to their personal
goals over the goals of others; and 4) emphasize
rational analyses of the advantages and
disadvantages of associating with others.
A huge amount of research
• I and C were initially laid down by Hofstede in his
cross-cultural study of business people around the
world in 1980.
• Still, the I/C distinction has proven useful in crosscultural research.
• Psychometrically, what do we know?
– Few good measures exist.
– There is an assumption now that I and C are orthogonal.
– Triandis wanted to combine I and C with another
dimension (see the next page).
Vertical vs. horizontal
• Triandis conceptualized I and C as crossed with V and H.
He borrowed the categorization from Markus and
Kitayama (1991).
• M & K described a two-by-two categorical system of the
self:
– Independent vs. interdependent; and
– Same vs. different.
• In the same way, Triandis thinks of V & I this way:
– In collectivist societies, horizontal refers to a sense of social
cohesion; and vertical refers to serving the ingroup.
– In individualist societies, horizontal refers to treating others as
individuals; and vertical refers to competition being healthy.
• In short, verticality refers to inequality and the importance
of rank; and horizontality refers to basic equality.
Triandis’s taxonomy
verticality
I sacrifice my self-interest for the
sake of the group.
Winning is everything.
individualism
collectivism
I prefer to be direct and forthright
with people.
Group harmony is important.
horizontality
Key things to notice
• First point: no item is pure I, C, V, or H. All items
are combinations of either I/C and V/H. Not the
usual way to do it.
• It looks as though I and C are opposites, i.e., not
orthogonal. Is that true?
• Is looks as though V and H are opposites too. Is
that true?
• Will we find in an EFA the pattern described
above? Good question.
• Note that I’m actually doing something “wrong”
here, i.e., doing an EFA on a previously identified
factor structure. I should do a CFA (later).
Let’s try the first run
• It identifies 9 eigenvalues greater than 1.0.
And that is completely useless. Notice that
it did not converge—bad sign.
• An examination of the scree plot suggests
either 3 or 4 factors.
• Let’s begin with 4 factors.
4-factor solution
• Notice that it actually converged (in 10 iterations-which is so-so). Accounts for 38% of the variance.
• A scan of the obtained factor loadings shows the
following:
–
–
–
–
1st factor: 7 HC; 2 HI; 1 VC
2nd factor: 7 VI
3rd factor: 5 VC; 1 HC 1 HI (rev)
4th factor: 4 HI; 1 HC
• Some order here, but it’s still sort of messy.
3-factor solution
• Accounted for 33% and converged in 12
iterations. Percent variance is a bit low.
• Let’s look at the factors:
– 1st factor: 6 HC; 6 VC
– 2nd factor: 8 VI
– 3rd factor: 5 HI
• Seems like we have a single collectivism factor
whereas the two individualisms are distinguished
from each other. Not a bad result for a Western
sample.
2-factor solution
• Accounts for 24.5% of the variance and
converged in 3 iterations.
– 1st factor: 7 HC; 5 HI; 1 VC
– 2nd factor: 8 VI; 3 VC
• Umm, what does this mean? Seems like we
have a horizontal factor and a vertical
factor. It doesn’t split by I and C.
Okay, so what’s the best solution?
Remember that I check the Cronbach’s alphas
and correlations.
• 4-factor solution: as = .75; .79; .66; .49
• 3-factor solution: as = .73; .79; .70
• 2-factor solution: as = .78; .78
• 1-factor solution: a = .75
So which is best? I like the 3-factor solution.
Let’s check the correlations.
Are there any high correlations?
Factor 1
Factor 2
Factor 2
.15*
Factor 3
.12
.03
• So, the 3-factor solution looks pretty good.
• 2-factor: r = .06.
Nothing is perfect
• Several things to notice:
– The FA programme does not tell you what the
optimal factor solution is, one has to discern it
from the various signs.
– Involves a number of FA runs. Then you follow
up with Cronbach’s alphas and correlations.
– The final solution is debatable. Other
researchers might choose different factor
structures.