Re: st: Relative Importance of predictors in regression

[David Hoaglin <dchoaglin@gmail.com>]

Hi, Sam.

Thank you for the additional discussion.

Your comments on "change" vs. "difference" make clear the challenge of
choosing terminology that applies to a wide range of applications.  It
may be impossible to satisfy everyone.

The use of the word "change" in the interpretation is, of course, not
wrong.  One must apply any interpretation appropriately in the
particular context.  In cross-sectional data it is clear that one
cannot change the characteristics of any individual and that "change"
must (as you pointed out) mean "difference": Change in the value of a
particular variable must mean that one shifts to another individual.
The difference between the two individuals' values on that variable
may be 1 unit, or it may necessarily be some other amount, either
because 1 unit is not a meaningful change in the context of the data
or because no individuals in the data have such a value.  And, if
other individuals do have values of the particular variable that are 1
unit greater, their values of some of the other variables may differ
from those of the initial individual.

I do have all the necessary mathematical expressions for the proper
general interpretation.  A plain-text message, however, is not
suitable for displaying them.  I am not aware of a mathematical
representation of the "held constant" interpretation in the
n-dimensional geometry in which ordinary least squares operates.  It
is easy to represent the "held constant" interpretation in the
p-dimensional geometry, but that is not the relevant geometry.  The
absence of a representation for the "held constant" interpretation in
the n-dimensional geometry is evidence for its lack of validity.  If
you have a suitable representation in mind, I would be interested in
seeing it.

Regards,

David Hoaglin

On Mon, Nov 4, 2013 at 6:05 PM, Lucas <lucaselastic@gmail.com> wrote:
> I asked my question because you wrote:
>
>> The "held constant" part of that interpretation
>> is not correct.  Straightforward mathematics shows that it does
>> not reflect the way that multiple regression actually works
>
> I presumed you had a mathematical representation of the two
> interpretations and could then show that the former is wrong because
> the actual regression model is accurately represented by the latter.
> However, instead of a formula, you provided more text, which is
> necessarily somewhat imprecise.
>
> For example, you keep talking about change.  I could pound on that,
> because in cross-sectional data--the dominant form of data people use
> with regression modeling--nothing is changing.  The values *differ*
> across cases; they do not change.  So, your interpretation of the
> coefficient as representing change in Y associated with change in X
> is, it would seem, wrong--the coefficient represents the *difference*
> in Y associated with a *difference* in X.  These observations are not
> trivial.  If I regress cross-sectional son's height on father's
> height, that does not mean stretching the father will raise the son's
> height.  However, if *change* were truly implicated by the coefficient
> it would. But, instead of writing in every time you say this I just
> presume you really understand there is no *change* going on and you
> are simplifying (or maybe slipping) during a discussion amongst
> knowledgeable users of the method.
>
> Which leads me back to my question. Setting the issue of change vs.
> difference aside, I still wonder: what is the mathematical
> representation that makes it clear that your interpretation is right
> and "held constant" is absolutely wrong?
>
> Thanks a bunch!
>
> Sam
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