Transcript Determining the # Of PCs - University of Nebraska–Lincoln

Determining the # Of PCs
• Remembering the process
• Some cautionary comments
• Statistical approaches
• Mathematical approaches
• “Nontrivial factors” approaches
• Help that’s coming later
How the process really works…
Here’s the series of steps we talked about earlier.
•
# factors decision
•
Rotate the factors
•
interpreting the factors
•
factor scores
These “steps” aren’t 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.
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 isn’t 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
• isn’t independent of interpretability
• a factor is only “real” if its meaningful
• be cautious of both “making up” and “missing” meaning
Some cautionary comments, cont.
• agreement across decision rules is helpful
• we’ll 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!!
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 “Bartlett’s Sphericity Test”
Statistical Procedures, cont.
• Applying Bartlett’s Sphericity Tests
• Retaining H0: means “don’t 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
Mathematical Procedures
• The most commonly applied decision rule (and the
default in most stats packages -- chicken & egg ?) is
the  > 1.00 rule … here’s 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
Mathematical Procedure, cont.
Part 2
• The “trace” of a matrix is the sum of its diagonal
• So, the trace of R (with 1’s 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)
Mathematical Procedure, cont.
Putting it all together (hold on tight !)
• Any PC with  > 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  < 1.00 accounts for less variance
than the average variable in that R
• That PC “doesn’t have parsimony” -- the more
complex composite has more no information than
the average variable
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 = b1k*PC1 + … +bfk*PCf, etc.) --while varying N, # factors, # PCs & communalities
• Then factor those variables and see if  > 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 < 40, f between k/5 and
k/3 and N > 300
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 !!
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)
 4
2
0
# PC 1 2 3 4 5 6
• Sometimes there isn’t a clear
elbow -- try another “rule”
• This approach seems to
work best when combined
with attention to
interpretability !!
An Example…
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 λ > 1.0 and 9th had λ < 1.0
01
10
λ
20
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
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 λ for the first (general) factor
and much smaller λs for subsequent factors
This is a common scree configuration when factoring items from a multisubscale scale!
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”
• We’re good at “finding patterns”, even when they’re 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
Help that’s 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