Re: st: Relative Importance of predictors in regression

[Lucas <lucaselastic@gmail.com>]

Hi Rich,

You offer an opportunity that perhaps will help David H. clarify what
he meant as well.

Here's the deal--Imagine 4 imaginary people, one male w/ 8 yrs schl,
one male w/ 9 yrs schl, one female w/ 8 yrs schl, one female w/ 9 yrs
schl.  Given the following translation of the original model I
offered:

Y=b1*YrsSchl+b2*Male

and according to the "held constant" interpretation, here are the
following (and correct) expected values:

1)    Male, 8 Yrs Schl => E(Y) = b1*8+b2
2)    Male, 9 Yrs Schl => E(Y) = b1*9+b2
3)Female, 8 Yrs Schl => E(Y) = b1*8
4)Female, 9 Yrs Schl => E(Y) = b1*9

"Hold constanting" sex by comparing males w/ different years of
schooling--subracting case 1 from case 2 yields:

E(Y2)-E(Y1)=(b1*9+b2)-(b1*8+b2)
                   =9b1-8b1
                   =b1

"Holding constant" education by comparing males and females with the
same years of schooling--subtracting case 4 from case 2, yields:

E(Y2)-E(Y4)=(b1*9+b2)-(b1*9)
                  =b2

Thus, the held constant interpretation means that b1 reflects the
difference in Y associated with a 1 year difference in Yrs Schl, once
other variable(s) in the model are "held constant", and b2 reflects
the difference in Y associated with sex, once other variable(s) in the
model are "held constant."

David H.'s claims imply the calculations above are incorrect, for he
claims that we can *never* use the hold constant interpretation.  And
the hold constant interpretation is embedded in the calculations above
because, in fact, we are holding constant all the other variables. It
seems that instead of regarding the model estimation as properly
accounting for any purely empirical (as opposed to logical, e.g., X
and X^2) associations between the X's, we have to come back in after
model estimation and again account for any association between the
X's.  This is obviously necessary in models for categorical variables,
which is why one must interpret the magnitude of coefficients in light
of the location of other variables in the model. But David H. is
saying this is also true of OLS.

David H. may be correct.  I am open to being persuaded--I am not
invested in a particular answer.  But, at this point I remain
unpersuaded. And a citation to his point would really really help.

Thanks a bunch!
Sam

On Wed, Nov 6, 2013 at 7:43 AM, Richard Goldstein
<richgold@ix.netcom.com> wrote:
> Hi,
>
> I have not been paying any particular attention to this thread but the
> most recent contribution caught my eye
>
> Sam writes, "In other cases, however, the held constant interpretation
> seems completely reasonable (e.g., E(Y)=b1*YrsSchl+b2*Sex)"
>
> this confuses me: the effect of sex is the same regardless of whether
> YrsSchl changes or does not change (and also for YrsSchol regardless of
> whether the value of Sex changes) so how can the "held constant
> interpretation" be reasonable?
>
> Maybe you only typed a shorthand of what you meant but, as worded, I do
> not agree with you.
>
> Rich
>
> On 11/6/13, 10:26 AM, Lucas wrote:
>> David M.,
>>
>> Thanks for weighing in.  Maybe your doing so will help out.  Indeed,
>> what you say is how I have interpreted this issue in the past.
>> Clearly, in some cases (e.g., X and X^2) one cannot hold one variable
>> constant and difference the other.  In other cases, however, the held
>> constant interpretation seems completely reasonable (e.g.,
>> E(Y)=b1*YrsSchl+b2*Sex). [Parenthetically, this is structurally the
>> same as saying "change is relevant for some models, impossible to
>> reference for others"--i.e., content matters.]
>>
>> What piqued my interest is David H. indicated he had a mathematical
>> expression that would straightforwardly show that "held constant" is
>> always wrong.  Yet, after asking for it for a couple of days, it still
>> has neither been conveyed nor has a citation been provided (well, two
>> textbooks were cited, but it was unclear which, if either, had the
>> expression or just a differently interpretable derivations).  That's
>> more than a little disappointing.
>>
>> Perhaps someone else has the expression.  If so, it'd be great to
>> either see it or be pointed to where it can be found.
>>
>> Or, perhaps there is no such expression.  No disrespect intended.
>> But, we cannot accept a claim--or expect our students or clients to
>> accept a claim--on the basis of someone saying, "I have the evidence
>> here, I just can't show it to you."
>>
>> Sam
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