Determining the

1
Determining the Of PCs

• Remembering the process
• Some cautionary comments
• Statistical approaches
• Mathematical approaches
• Nontrivial factors approaches
• Help thats coming later

2
How the process really works

• Heres the series of steps we talked about
  earlier.
• factors decision
• Rotate the factors
• interpreting the factors
• factor scores

These steps arent made independently and done
in this order!
Considering the interpretations of the factors
can aid the factors decision!
Considering how the factor scores (representing
the factors) relate to each other and to
variables external to the factoring can aid both
the factors decision and interpretation.
3
Some cautionary comments

• Remember that the factors decision ...
• is influenced by the particular variables in the
  analysis
• so, unless you are working with a closed set of
  variables, there probably isnt a real
  factors
• the whole story includes how the factors
  changes with what variable additions and
  deletions
• how do these change your interpretation of
  factors and variables
• isnt independent of interpretability
• a factor is only real if its meaningful
• be cautious of both making up and missing
  meaning

4
Some cautionary comments, cont.

• agreement across decision rules is helpful
• well talk about several decision rules, each of
  which is flawed in known ways
• replication is convincing
• split-half and hold-out sampling can help
• separate-sample replication is more convincing
• convergence research is more convincing
• not just replicating, but correctly anticipating
  what will be the results of adding, deleting
  variables across samplings
• Remember this is Exploratory factoring
• explore consider alternative factor solutions
• Want to be really convincing? Use confirmatory
  factoring!!

5
Statistical Procedures

• PC analyses are extracted from a correlation
  matrix
• PCs should only be extracted if there is
  systematic covariation in the correlation
  matrix
• This is know as the sphericity question
• Note the test asks if there the next PC should
  be extracted
• There are two different sphericity tests
• Whether there is any systematic covariation in
  the original R
• Whether there is any systematic covariation left
  in the partial R, after a given number of factors
  has been extracted
• Both tests are called Bartletts Sphericity Test

6
Statistical Procedures, cont.

• Applying Bartletts Sphericity Tests
• Retaining H0 means dont extract another
  factor
• Rejecting H0 means extract the next factor
• Significance tests provide a p-value, and so a
  known probability that the next factor is 1 too
  many (a type I error)
• Like all significance tests, these are influenced
  by N
• larger N more power more likely to reject H0
  more likely to keep the next factor ( make a
  Type I error)
• Quandary?!?
• Samples large enough to have a stable R are
  likely to have excessive power and lead to
  over factoring
• Be sure to consider variance, replication
  interpretability

7
Mathematical Procedures

• The most commonly applied decision rule (and the
  default in most stats packages -- chicken egg
  ?) is the ? gt 1.00 rule heres the logic
• Part 1
• Imagine a spherical R (of k variables)
• each variable is independent and carries unique
  information
• so, each variable has 1/kth of the information in
  R
• For a normal R (of k variables)
• each variable, on average, has 1/kth of the
  information in R

8
Mathematical Procedure, cont.

• Part 2
• The trace of a matrix is the sum of its
  diagonal
• So, the trace of R (with 1s in the diag) k (
  vars)
• ? tells the amount of variance in R accounted for
  by each extracted PC
• for a full PC solution ? ? k (accounts for all
  variance)
• Part 3
• PC is about data reduction and parsimony
• trading fewer more-complex things (PCs - linear
  combinations of variables) for fewer more-simple
  things (original variables)

9
Mathematical Procedure, cont.

• Putting it all together (hold on tight !)
• Any PC with ? gt 1.00 accounts for more variance
  than the average variable in that R
• That PC has parsimony -- the more complex
  composite has more information than the average
  variable
• Any PC with ? lt 1.00 accounts for less variance
  than the average variable in that R
• That PC doesnt have parsimony -- the more
  complex composite has more no information than
  the average variable

10
Mathematical Procedure, cont.

• There have been examinations the accuracy of this
  criterion
• The usual procedure is to generate a set of
  variables from a known number of factors (vk
  b1kPC1 bfkPCf, etc.) --- while varying N,
  factors, PCs communalities
• Then factor those variables and see if ? gt 1.00
  leads to the correct number of factors
• Results -- the rule works pretty well on the
  average, which really means that it gets the
  factors right some times, underestimates
  sometimes and overestimates sometimes
• No one has generated an accurate rule for
  assessing when which of these occurs
• But the rule is most accurate with k lt 40, f
  between k/5 and k/3 and N gt 300

11
Nontrivial Factors Procedures

• These common sense approaches became increasing
  common as
• the limitations of statistical and mathematical
  procedures became better known
• the distinction between exploratory and
  confirmatory factoring developed and the crucial
  role of successful exploring became better
  known
• These procedures are more like judgement calls
  and require greater application of content
  knowledge and persuasion, but are often the
  basis of good factorings !!

12
Nontrivial factors Procedures, cont.

• Scree -- the junk that piles up at the foot of
  an glacier
• a diminishing returns approach
• plot the ? for each factor and look for the
  elbow
• Old rule -- factors elbow (1966 3 below)
• New rule -- factors elbow - 1 (1967 2
  below)

• Sometimes there isnt a clear elbow -- try
  another rule
• This approach seems to work best when combined
  with attention to interpretability !!

? 4 2 0
PC 1 2 3 4 5 6
13
An Example
A buddy in graduate school wanted to build a
measure of contemporary morality. He started
with the 10 Commandments and the 7 Deadly
Sins and created a 56-item scale with 8
subscales. His scree plot looked like How many
factors?
?
1? big elbow at 2, so 67 rule suggests a
single factor, which clearly accounts
for the biggest portion of variance 7? smaller
elbow at 8, so 67 rule suggests 7 8? smaller
elbow at 8, 66 rule gives the 8 he was looking
for also 8th has ? gt 1.0 and 9th had ? lt 1.0
0 1 10 20
1 8 20
40 56

• Remember that these are subscales of a central
  construct, so..
• items will have substantial correlations both
  within and between subscales
• to maximize the variance accounted for, the
  first factor is likely to pull in all these
  inter-correlated variables, leading to a large ?
  for the first (general) factor and much smaller
  ?s for subsequent factors
• This is a common scree configuration when
  factoring items from a multi-subscale scale!

14
Nontrivial factors Procedures, cont.

• of variance accounted for
• keep the factors necessary to account for
  enough variance -- 75 to 90 are common goals
• Interpretability -- meaningfulness of resulting
  PCs
• Depends greatly upon content knowledge
• Beware factoring illusions
• Were good at finding patterns, even when
  theyre not really there
• Rotational Survival -- akin to meaningfulness
• Consider different factors with different types
  of rotation -- see which factors keep showing
  up
• Replicability -- split, holdout, or independent
  samples
• What PCs appear consistently across factorings?
• Jack-knifing
• Re-sampling from a single dataset looking for
  consistency of factors

15
Help thats coming later

• If you have a reasonably clear factor structure
  all the different ways of deciding the factors
  are likely to give the same result (except maybe
  statistical likely to over-factor with N)
• Remember that what the factors are can be very
  important in deciding how many factors there
  are
• Consider the different interpretations of the
  factors from the different -of-factors solutions
• we can also look at the correlations between the
  factors to help with these decisions
• Remember that what the factors do can be very
  important in deciding how many factors there
  are
• you can look at how factors from the different
  -of-factor solutions correlate with other
  variables that are not in the factor analysis