Re: st: Relative Importance of predictors in regression

[David Hoaglin <dchoaglin@gmail.com>]

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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