Re: st: Relative Importance of predictors in regression

[David Muller <davidmull@gmail.com>]

I may be misunderstanding or mischaracterising David Hoaglin's
problems with the term "holding constant" for describing adjustment
for covariates in multiple regression, so forgive me for interjecting
if I am off the mark.

I think the main issue is that the data used to fit the model won't
necessarily support a difference/change in one variable with all other
variables held constant. This is trivially the case when, for
instance, both x and x^2 are used as predictors. When data are sparse
or continuous it is also unlikely that there will be observations that
differ on one variable but are _identical_ on all others.

Personally, I don't think this is a big deal. If one sees regression
coefficients as differences in conditional expectations, then the
"held constant" interpretation is just a model-based interpolation or
extrapolation. It's up to the person fitting and interpreting the
model to justify any such extrapolation.

All the best,
David Muller


On 6 November 2013 01:19, Lucas <lucaselastic@gmail.com> wrote:
> 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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