Re: st: Relative Importance of predictors in regression

[Lucas <lucaselastic@gmail.com>]

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

On Mon, Nov 4, 2013 at 2:28 PM, David Hoaglin <dchoaglin@gmail.com> wrote:
> Hi, Sam.
>
> I'm not sure what you mean by "the mathematical expression for 'held
> constant,'" other than setting each of the other predictors to some
> particular value.
>
> The general interpretation of the coefficient of a predictor in a
> multiple regression is that it tells how the dependent variable
> changes per unit increase in that predictor, adjusting for
> simultaneous linear change in the other predictors in the data at
> hand.  If the model has n observations on Y and p predictors
> (including the constant, if present), and the data on the predictors
> form the columns of the (full-rank) matrix X, the mathematics of
> ordinary least squares involves projecting Y on the subspace of
> n-space spanned by the columns of X.  Nothing in that process of
> projection holds the other predictors constant at particular values.
>
> Regards,
>
> David Hoaglin
>
> On Mon, Nov 4, 2013 at 3:39 PM, Lucas <lucaselastic@gmail.com> wrote:
>> What would be the mathematical expression for "held constant"? And
>> what is the mathematical expression to which you are comparing it that
>> leads you to reject "held constant"?
>>
>> Thanks a bunch!
>>
>> Sam
> *
> *   For searches and help try:
> *   http://www.stata.com/help.cgi?search
> *   http://www.stata.com/support/faqs/resources/statalist-faq/
> *   http://www.ats.ucla.edu/stat/stata/
*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/faqs/resources/statalist-faq/
*   http://www.ats.ucla.edu/stat/stata/