Re: st: about residuals and coefficients

["Seed, Paul" <paul.seed@kcl.ac.uk>]

Ronan Conroy asks: 

"Purely pragmatically, I tend to teach the 'other things held constant'
interpretation because it's a good first-pass in understanding
multivariate models, and it's not doing any real-life violence to the
interpretation of the data that I can see (has anyone examples where
it's plain misleading?).

I tend to say 'After adjusting for other things', and point out that we can 
only adjust for the things that have been measured, and only to the extent to 
which they have been measured.  Adjustment is therefore incomplete, 
and a large movement in the estimate following adjustment may be a sign that 
the relationship is spurious.

Here's one artificial example:
Using the auto data set, I find the unadjusted 
regression coefficient for mpg on price is -239 $/mpg (p=0.000).
If I adjust for weight, repair record, and foreign (i.e. non-US), 
this becomes non-significant (+24 $/mpg, p=0.765).

However imagine that weight is for some reason measured very inaccurately, 
(achieved for the example by adding a large random element to weight).
Now I find that the "adjusted" regression coefficient (in fact only partially adjusted)
is -199 $/mpg, p=0.009.

In health research, many things are measured via biochemistry or questionnaires, 
both of which can result in considerable inaccuracy, even before allowing for 
variation over time.  In practice, most "adjusted" estimates are only partially 
adjusted.  The epidemiology literature is probably full of false associations 
that are due to partially adjusted confounding, but identifying them is very 
difficult.  

It should be said that this is only one of a large number of potential pitfalls 
in epidemiology, which taken together explain why Randomised Controlled Trials 
are preferred wherever possible.


One reference for those who are interested: 

Matthias Egger, Martin Schneider, George Davey Smith.
Meta-analysis. Spurious precision? Meta-analysis of observational studies
BMJ VOLUME 316 10 JANUARY 199
http://pubmedcentralcanada.ca/pmcc/articles/PMC2665367/pdf/9462324.pdf


BW

Paul

*******************************
* Begin Stata example *
*******************************
clear 
set seed 984
sysuse auto
qui su weight
gen wt_50 = weight + invnorm(uniform())*r(sd)

regress price mpg
regress price mpg weight
regress  price mpg wt_50
*******************************
* End Stata example *
*******************************



> Date: Sat, 7 Sep 2013 11:16:03 +0000
> From: Ronan Conroy <rconroy@rcsi.ie>
> Subject: Re: st: about residuals and coefficients
> 
> On 2013 MFómh 7, at 01:25, David Hoaglin wrote:
> 
> > I usually suggest the following wording: The coefficient of Xj is the
> > average change in Y per unit increase in Xj after adjusting for
> > simultaneous linear change in the other predictors in the model in
> the
> > data at hand.  It would be nice to have something simpler, but in
> > general nothing simpler will do.  I suggest "per unit increase in Xj"
> > because the coefficient is a sort of slope, and a change of one unit
> > may not be meaningful in the particular set of data.
> 
> Or, if writing for people who think in language: The coefficient of the
> predictor variable is the change we expect associated with a one-unit
> increase in the predicted variable, after we adjusted for the expected
> effect that this change will have on the other predictor variables in
> the model.
> 
> Or can someone suggest a better way of phrasing this?
> 


Paul T Seed, Senior Lecturer in Medical Statistics, 
Division of Women's Health, King's College London
Women's Health Academic Centre, King's Health Partners 
(+44) (0) 20 7188 3642.



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