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Bob,
=0D
I do not know anything about Brunswick's =
lens model. I would be interested to know how
he determines what is a core construct. I =
suspect that if he uses regression weights, ie
beta weights, in a stepwise manner, then the
weights are calculated in a "mob rule" manner.
This notion of Mob rule has relevance to the
weakness of factor analysis.Your other posting
addressed factor analysis, so I will explain this
notion of mob rule within the context of both.
=0D
In forward stepwise regression the goal is to
find the simplest model for predicting a variable.
Recall that prediction does not mean the same
as causal explanation. Now, the notion of mob =
rule derives from the fact that the stepwise
regression simply attempts to explain the most
variance. But the most may be causally
irrelevant due to an over sampling of parallel
tests, that may or may not be of relevance
to the larger question..
=0D
Consider a model in which we simulate a
factor structure concerning the reddness
of red ribbons handed out randomly to
subjects and the subjects levels of
introversion. Our larger question is "What =
are our subjects personalities?" 'Lets =
say we have six different expert judges
rate each person's ribbon on how red it is,
using seven point scales. Assume the dye
saturation allows for distinctions of redness. =
We therefore have six parallel tests of the
variable redness (R1 thru R6). These should =
differ only randomly, depending on such things
as the judges differing use of rating scales and =
individual differences in the judgement of color
saturation. Some judges may rate a particular
ribbon 5 on redness, others may give it a 7. =
What emerges form all this is a factor =
composed of the six parallel tests of redness. =
=0D
Now, let us also simulate the introversion =
scores of our subjects, using 4 measures of =
introversion. Put the six redness measures
and the four introversion measures together
and we have ten measures.There are more =
redness measures.The first dominant factor =
in this data will be redness. This is for two =
reasons. First is mob rule- there are more =
redness measures than introversion measures. =
Second, the redness measures probably are =
more reliable. There will probably be less =
measurement error in the redness measures =
(and consequently Cronbach's alpha based =
on the redness measures will be higher than
alpha based on the introversion measures). =
=0D
A similar thing would happen if we created =
"personality" total scores by summing up all =
the ten scores on the 6 redness tests =
and 4 introversion tests and predict them in a
stepwise regression. Since most of the variance =
in the "personality" total scores
would be driven by the redness tests, redness
would get the strongest weight in the regression.
More over, the regression solution would pick
the redness measure that correlates highest with
the total score, and dump the other redness tests =
as redundant. This dumping would suggest to the
naive person that the other tests were not
involved in the core generation of the "personality"
total score. This would be a false conclusion.
The other 5 redness tests were involved but
received just slightly more attenuation due to
irrelevant random measurement errors. But =
they are none the less dropped from the naive
causal model. The same simplification would =
occur in the selection of the variable to =
represent the introversion factor. We would end
up with a two variable stepwise equation,
containing the best predictors of the total =
"personality" scores. =
=0D
I hope you were uneasy at the thought of
adding up 6 redness scores with 4 introversion
scores to et "personality" scores.. Since the =
red ribbons were handed out to
our subjects at random, it is very unlikely
that much of interest psychologically would =
emerge from our clumping redness and
introvesion measures into a "personality"
score. True, the more introverted ones might =
prefer a less saturated red, but otherwise, there
is nothing natural about the clumping. In other
words, for the most part, our "personality" scores =
are artificial or syncretistic.They would not
represent a true dialectical synthesis, since
saturation of redness and introversion scores
do not have much "chemistry" between them.
They are logically disparate and occur together =
incidentally. If we did a Slater analysis and plotted
the subjects in the combined space of redness
and introversion, we would not be tapping a "natural"
dialectic. The factors probably shouldn't even be
potted together. (This line of reasoning goes on
but I'll stop here, to see if you follow me so far.)
=0D
Now concerning your averaging correlations
by subject, following Dalgleish's method. =
The question is, 'Which is best, to average =
across all subjects and then correlate versus
correlating by subject and then averaging =
correlations across all subjects
(Dalgleish's method)?' This is off the top of my
head and more thought is needed to be sure
but I think I see a possible explanation for your
regressions being simpler with the Dalgleish
method. Did you square the correlations-
retaining signs- before averaging them? If not,
then larger correlations will be given =
disproportionate weight in the averaging.
Remember that it is the square of the
correlation that reflects the percent of variance
accounted for. By not squaring them, you
essentially shrink the smaller variances
artifactually. Something like this is done in =
certain factor rotation methods, i.e.promax
(It think). But the promax method has subsequent
safe guards built into it so that unexplained
variance is not simply artifactually dumped from
the eigenvalues.
=
The Dalgleish method may be useful, however,
in putting the subjects original ratings on a =
standard scale. Simply averaging over subjects
ratings will handicap the contributions of
subjects who use less extreme ratings. However,
standardizing the ratings by subject first, and
then deriving one set of correlations from the =
concatenated standardized ratings, seems to =
be an easier approach- than the Dalgleish
method.
=0D
Regards,
=0D
Bill
=
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