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# Re: st: coefficient interpretation in OLS

 From David Hoaglin To statalist@hsphsun2.harvard.edu Subject Re: st: coefficient interpretation in OLS Date Sun, 19 Aug 2012 08:13:49 -0400

```Clive,

Since the definition of a coefficient in a multiple regression
involves the set of other predictors in the model, Lynn should report
those other variables, whose contributions are being adjusted for.

No "utter waffle" is involved; the proof is straightforward
mathematics.  It would be nice if multiple regression were simpler,
but it is not.  The distortion comes in using the oversimplified
interpretation "with the other variables held constant."  I have no
reluctance to give an audience the longer interpretation, because that
is what multiple regression actually does.  Better that than to
deceive.  One can often dispense with "in the data at hand"; and
instead of "allowing for simultaneous linear change in", one can say
"adjusting for the contributions of" (as I did in my reply to Lynn).

David Hoaglin

On Sun, Aug 19, 2012 at 2:36 AM, Clive Nicholas
> David Hoaglin replied to Lynn Lee:
>
>> In interpreting the coefficients in a multiple regression, two facts
>> are important.
>>
>> 1. The definition of each coefficient includes the set of other
>> predictors in the model.
>>
>> 2. The coefficient of a predictor, say X1, tells how Y changes in
>> response to change in X1 after adjusting for the contributions of the
>> other predictors in the model (in the data at hand).  A coefficient is
>> a slope, so it gives change in Y per unit increase in X1, not
>> necessarily the change in Y when X1 is increased by 1 unit (unless the
>> values of X1 are only 0 and 1).  Some textbooks, unfortunately,
>> interpret the coefficient of X1 as telling how Y changes with an
>> increase of 1 unit in X1 when the other predictors are held fixed, but
>> that is simply not how OLS works; that interpretation is
>> oversimplified and often incorrect.
>>
>> Terry Speed's column in the current issue of the IMS Bulletin
>> discusses both of these points.
>
> I've had time to read Speed's article. The key quotation, on page 13,
> appears to be:
>
> "A lengthy but basically correct interpretation goes like this:
> {b}_{12.3} tells us how X1 responds, on average, to change in X2,
> after allowing for simultaneous linear change in X3 in the data at
> hand."
>
> This may well be right, but it reads and sounds like utter waffle.
> Would you seriously explain an effect, as explained by its
> coefficient, of a continuous variable on another continuous variable
> in front of an audience in this way? I wouldn't, and nor would anyone
> else I know. There has to be a simpler way of saying this that is also
> correct.
>
> If it has to be like this, I'd think I'd advise Lynn simply to say the
> effect of this or that continuous variable is positive or negative and
> leave it at that.
>
> --
> Clive Nicholas
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