Re: st: Relative Importance of predictors in regression

[Lucas <lucaselastic@gmail.com>]

Dear David,

I am confused.  You first write the following (emphasis capitalization added):

"I would add a note of caution, however.  Nathans et al. (and many
others) interpret a beta weight (or a regression coefficient more
generally) in a way that involves holding all the other predictor
variables constant.  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."

In response I 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!"

It seemed to me both pieces of information would be necessary for
someone to rule that one is appropriate and the other wrong (or, at
least, it should be demonstrable that the wrong one has no formal
expression).  To this David replied:

"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."

This latter reply suggests David and I agree that a mathematical
expression will be an equation--not a derivation.  I responded,
writing:

"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."

In that message I introduced a critique of David's use of change when
difference is generally correct--the aim of doing so was to suggest
that maybe we all can cut each other some slack.  I had expected David
to just say, "Sure, yeah, that's right, my bad" but David resists this
obvious fact. Okay, fine--it's a general discussion, but he prefers to
use the specific language. Anyway, David does address the request for
a mathematical expression by responding that:

"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."

I have not offered a representation because I have not maintained one
is right and the other wrong, so it seems I would not be required to
distinguish two things I am not sure can be distinguished. In an
effort to understand David's point, every response I have written
since has been asking for one simple thing: Where can I find this
point made in n-dimensional geometry?

Other matters are not directly relevant--David won't accept that if
you have 2 terms, one general, and one specific, the general applying
everywhere, the specific applying in a smaller subset, one should use
the general language.  Pedagogically and scientifically this seems
obvious.  Okay.  This just means this is not the ideal speech
community one might have hoped.  Still, I ask--which of the two
textbooks David mentioned have the n-dimensional expression David
intimated existed? Do either of them have it?  Both?  Neither? If
neither, is there another citation to which I (we?) could turn? Just
answering this question with the relevant citation(s) would be
immensely helpful. Of course, it is not your job to be helpful.  But
you've made this point several times on statalist, which led me to
think you might want people to get the point.  I'm asking for help in
getting the point.  Rather than more analogies and your plain text
derivations (which you indicate are intrinsically sub-optimal), a
citation I (and perhaps others) can peruse would be incredibly
helpful.

Again, thanks a bunch!

Sam

On Tue, Nov 5, 2013 at 9:26 AM, David Hoaglin <dchoaglin@gmail.com> wrote:
> Dear Sam,
>
> It would help communication if you explained, as specifically as
> possible, what sort of "mathematical expression" you are looking for.
>
> The material in my previous message that you reject as a "mathematical
> manipulation" needs only one further step, involving straightforward
> algebra: In the result of regressing the Y-residuals on the
> X2-residuals, multiply out the right-hand side, rearrange the equation
> to leave only Y on the left-hand side, and compare the result term by
> term against the original model.  Since the adjustments for the
> contributions of the other predictors are shown explicitly, the
> interpretation of b2 is clear.  Please explain how you would interpret
> the demonstration differently.
>
> The fact that regression coefficients are a type of slope does not
> provide any basis for the "held constant" interpretation.  I do not
> see the connection between a regression model and your analogy of the
> position of two people on a hill.  Please explain further.
>
> When you said that I "retain one mis-interpretation of the regression
> model that is extremely elementary and easily corrected," I assume you
> are referring to the distinction that you make between "change" and
> "difference."  I explained earlier that I would use words appropriate
> to the particular context and application, so I am not making any
> mis-interpretation.
>
> I remind you that you have not offered any mathematical expression for
> the "held constant" interpretation.
>
> Regards,
>
> David Hoaglin
>
> On Tue, Nov 5, 2013 at 9:37 AM, Lucas <lucaselastic@gmail.com> wrote:
>> Hi David,
>>
>> I am looking for the mathematical expression you indicated would make
>> it clear which interpretation is correct. The mathematical
>> manipulation isn't very helpful, because someone who interprets the
>> issue differently than you do before can interpret this demonstration
>> differently than you do. So, do either of those books have the
>> mathematical expression you mentioned? If so, I'll check it out.
>>
>> On change vs. difference, discrete things change or do not, and
>> non-discrete things change or do not.  The distinction between "change
>> and difference" is orthogonal to the distinction between "discrete and
>> non-discrete."
>>
>> Indeed, the analogy you deploy to support the change interpretation,
>> using slopes and hills, is one reason people say "held constant."  The
>> difference (slope) between my height on the hill and Joe's height on
>> the hill is distinct from (and independently estimable given) our
>> horizontal placement on the hill. Horizontal placement, thus, is "held
>> constant." If this is incorrect, it shows why analogies are less
>> helpful than mathematical expressions. Thus, my request for the
>> mathematical expression you indicated was available.
>>
>> I do not understand why you retain one mis-interpretation of the
>> regression model that is extremely elementary and easily corrected,
>> but are adamant that everyone else is wrong if they use (what you
>> call) another mis-interpretation of the model, a mis-interpretation
>> that 1)can be shown with straightforward mathematical expressions but
>> then 2)seems so complex that it cannot be written in plain text.
>>
>> Anyway, please let me know which of those textbooks have the
>> mathematical expression you referenced earlier.  I'll pull it from the
>> library and take a look
>>
>> Thanks!
>>
>> Sam
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