[From Bruce Abbott (971219.1155 EST]
Richard Kennaway (971219.15:00) --
Bruce Abbott (971218.1505 EST)
If I discover that 50% of the variation in GPA can be accounted for by
variation in IQ then I can cut the average error in my prediction of GPA in
half by knowing the IQ.
Don't you mean "average squared error"?
Yes. R-squared represents the proportion of variance accounted for;
variance is the average squared error.
Bruce Nevin (971218.1745) --
The 50% correlation is perfectly fine as long as it refers the particular
population for which the correlation was found. The problem begins when you
apply it to other populations, and becomes acute when you apply it to
populations of one, to individuals, whether they are in the original
population or not. Then you might as well flip a coin -- once for each
individual. And such population measures are routinely applied to
individuals, aren't they? In my experience, they are routinely presented as
significant generalizations about "people", where the term "people" (or the
like) is ambiguous between population and individual, ambiguous to the
point of prevarication IMO.
If the person in question is a member of the population for which the
correlation applies (e.g., a high-school student applying for admission to
college), then for the person who has to make a decision about this
individual (e.g., admit to college or not) the information may be _much_
better than a flip of the coin, in the sense that it leads to the correct
decision being made more often than would be the case absent this
information. Thus, if students who have a high-school GPA above a certain
value are much more likely to succeed in college than those whose
high-school GPA below that value, and this is all the information one has
about the person, and there is a limited number of classroom seats available
(i.e., some must be turned down), then the decision-maker can improve the
rate of student success in college by applying this criterion and admitting
only those whose high-school GPA falls above it.
Such a strategy will still admit those who, for various reasons, will not
succeed in college and it will deny admission so some who, for various
reasons, would have succeeded, but that is the price to pay for having only
limited information about the individual. The decision-maker still must
"flip a coin," but by gathering data on student variables that correlate
with degree of success in college, he or she can bias the coin heavily in
favor of making the right decision. By combining information from a number
of sources (e.g., high school GPA, SAT score, letters of recommendation,
etc.), one can usually do better than one could by relying on a single
source (Multiple R higher than any single r).
I'm not a psychologist, so maybe I just don't understand these deep
matters. Help me out. (I may even be able to answer sometime.)
Bruce, what _is_ your field? Linguistics? Judging from the quality of
thought expressed in your writing, I very much doubt that you have any
trouble understanding these "deep matters," psychologist or not.
Regards,
Bruce