Re: st: Relative Importance of predictors in regression

[William Buchanan <william@williambuchanan.net>]

Hi Sam,

By its very definition a constant does not vary.  So unless you've sampled only males with a varying number of years of schooling, you have not held gender constant; you may have adjusted your estimates for the influence of being a male, but this is not the same.  I would argue that in the best of cases, your illustration below indicates that you've included a sex fixed-effect.

Billy


On Nov 6, 2013, at 1:22 PM, Lucas <lucaselastic@gmail.com> wrote:

> Hi Rich,
> 
> Depends on which of us you ask.  I'd say if you compare a male w/ 9
> YrsSchl and a male w/ 8YrsSchl you've held sex constant and b1 is the
> difference in Y associated with that one year difference in schooling.
> I think David H. would say that you've held nothing constant.  Is
> that a correct interpretation of your claim, David H.?
> 
> Sam
> 
> On Wed, Nov 6, 2013 at 9:08 AM, Richard Goldstein
> <richgold@ix.netcom.com> wrote:
>> Hi Sam,
>> 
>> using your example, the effect of comparing a male with 9 years of
>> schooling to a female with 8 years of schooling is b1, correct? So what
>> is held constant?
>> 
>> Rich
>> 
>> On 11/6/13, 11:42 AM, Lucas wrote:
>>> 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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