Factor analysis (PCA) in action

1
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

2
Demonstration of the process

• Id like to show you how to do an EFA on a
  dataset.
• Further, Id like to show you how messy it can
  be. A bit like finger painting.
• This dataset has to do with individualism and
  collectivism.

3
I and C

• Harry Triandiss 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. Its a way of thinking about
  the self in relation to others.

4
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.

5
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.

6
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
  cross-cultural 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).

7
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.

8
Triandiss 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
9
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 Im actually doing something wrong
  here, i.e., doing an EFA on a previously
  identified factor structure. I should do a CFA
  (later).

10
Lets try the first run

• It identifies 9 eigenvalues greater than 1.0. And
  that is completely useless. Notice that it did
  not convergebad sign.
• An examination of the scree plot suggests either
  3 or 4 factors.
• Lets begin with 4 factors.

11
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 its still sort of messy.

12
3-factor solution

• Accounted for 33 and converged in 12 iterations.
  Percent variance is a bit low.
• Lets 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.

13
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
  doesnt split by I and C.

14
Okay, so whats the best solution?

• Remember that I check the Cronbachs 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.
  Lets check the correlations.

15
Are there any high correlations?

• Factor 2 Factor 3
• Factor 1 .15 .12
• Factor 2 .03
• So, the 3-factor solution looks pretty good.
• 2-factor r .06.

16
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 Cronbachs alphas and correlations.
• The final solution is debatable. Other
  researchers might choose different factor
  structures.