Re: st: about residuals and coefficients

[Lucas <lucaselastic@gmail.com>]

Dear David,

This is why I do not understand why you prefer the "per unit increase"
phrasing. Many (probably most) analyses use cross-sectional data.
Thus, nothing is increasing or decreasing. The coefficients describe
the relationships, but there is no reason to suspect -- just on the
basis of cross-sectional data -- that change in an X will lead to the
slope's change in Y.

For example, if I regress earnings on yrs of education and age, that
doesn't mean that a 30 year old with 12 years of schooling will be
expected to increase their earnings by the increment of the slope for
years of education by going to college for 1 year.

It seems to me of the two potential disservices we can do to students,
teaching them "per unit increase" is far more misleading than teaching
them "other things constant" because at least the latter is an
accurate representation of what the cross-sectional data can allow.

Think about it like this.  If my model is:

$ = b_0 + b_1 Yrs Ed + b_2 Age + e

then the model summarizes two planes.  The plane for YrsEd has a
constant slope, i.e., the slope of the plane for Yrs Ed does not vary
regardless of where you are on the plane for Age.  And, vice versa. If
for theoretical, prior research, or other reasons I estimate:

$ = b_0 + b_1 Yrs Ed + b_2 Age + b_3 Age^2 + e

then the "plane" for Age has become a curved surface which means its
slope varies for values of Age. Still, the slope for YrsEd  is
constant.  So, the interpretation of the YrsEd slope seems unchanged.
And so on.

Of course, observational data does not usually fix the values of the
independent variables, and experimenters can (up to a point). But
there are other ways of addressing this than changing the
interpretation so that it is either inaccurate or unduly confusing.

Anyway, if we want to be as faithful as possible to what the data can
say, we should avoid "per unit change" in favor of "per unit
difference" because for cross-sectional data -- i.e., what is usually
used -- change is obviously beyond the ability of the data to support.

Other issues (e.g., being on the support vs. extrapolating off the
support) obviously come in as well.

Sam


On Wed, Sep 18, 2013 at 6:38 AM, David Hoaglin <dchoaglin@gmail.com> wrote:
> Richard,
>
> I'm not enthusiastic about "on an all other things being equal basis"
> (or "ceteris paribus") because many people would interpret it as
> "other things being constant."  And "two otherwise identical people"
> seems the same as "other things [predictors in the model] being
> constant."
>
> The key is what the data support.  The data on the other predictors
> may allow a comparison between male and female without extrapolating.
> That is, one may be able to set the other predictors to relevant
> values and compare the predicted average scores for male and female.
> In an extreme situation, however, the male and female data on one or
> more of the other predictors may have no overlap.  The adjustment for
> those predictors would have a substantial impact, and the data would
> not support statements about "two otherwise identical people, one male
> and one female."  It would be important to understand that feature of
> the data and its implications for interpreting the adjusted difference
> between male and female.
>
> David Hoaglin
>
> On Wed, Sep 18, 2013 at 9:53 AM, Richard Williams
> <richardwilliams.ndu@gmail.com> wrote:
>
>
>> I am not sure if David would like this any better, but I often use phrasings
>> along the lines of "On an all other things being equal basis" or "if you had
>> two otherwise identical people, one male and one female, how would you
>> expect their scores to differ?" The "held constant" phrasing seems a bit
>> nonsensical for variables like gender -- gender can't (or at least usually
>> doesn't) change, but it is possible to have people who have similar or
>> identical values on the other independent variables but who differ in their
>> gender.
>>
>> The comparisons should be reasonable -- I probably would not say something
>> like "If you had two otherwise identical people, one a multi-billionaire and
>> the other broke and penniless..."
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