One can "just include it in the regression", but the potential problems
for interpretation are surely greater than those indicated.  Inclusion  
of
X1 = T1+E1 may cause X2 to appear significant when in fact it is having
no effect at all.  Or the true effect can be reversed in sign.  This  
happens
because X1 and X2 are correlated.  Maybe this is implicit in what Jon
is saying.

See Carroll, Ruppert and Stefanski:
Measurement Error in Nonlinear Models (2004, pp.52-55).  The error in E1
may need to be fairly large relative to SD(T1) for this to be an  
issue.  My notes
at http://www.maths.anu.edu.au/%7Ejohnm/r-book/2edn/xtras/xtras.pdf
have brief comments, and code that can be used to illustrate the point.

I support Stephen Kolassa's suggestions re using simulation for
sensitivity analysis, though I think this can also be done analytically.

John Maindonald             email: john.maindonald@anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 08/03/2009, at 10:00 PM, r-help-request@r-project.org wrote:

> From: Jonathan Baron <baron@psych.upenn.edu>
> Date: 8 March 2009 5:21:55 AM
> To: Juliet Hannah <juliet.hannah@gmail.com>
> Cc: r-help@r-project.org
> Subject: Re: [R] using a noisy variable in regression (not an R  
> question)
>
>
> If you form categories, you add even more error, specifically, the
> variation in the distance between each number and the category
> boundary.
>
> What's wrong with just including it in the regression?
>
> Yes, the measure X1 will account for less variance than the underlying
> variable of real interest (T1, each individual's mean, perhaps), but
> X1 could still be useful in two ways.  One, it might be a significant
> predictor of the dependent variable Y despite the error.  Two, it
> might increase the sensitivity of the model to other predictors (X2,
> X3...) by accounting for what would otherwise be error.
>
> What you cannot conclude in this case (when you measure a predictor
> with error) is that the effect of (say) X2 is not accounted for by its
> correlation with T1.  Some people try to conclude this when X2 remains
> a significant predictor of Y when X1 is included in the model.  The
> trouble is that X1 is an error-prone measure of T1, so the full effect
> of T1 is not removed by inclusion of X1.
>
> Jon
>
> On 03/07/09 12:49, Juliet Hannah wrote:
>> Hi, This is not an R question, but I've seen opinions given on non R
>> topics, so I wanted
>> to give it a try. :)
>>
>> How would one treat a variable that was measured once, but is known  
>> to
>> fluctuate a lot?
>> For example, I want to include a hormone in my regression as an
>> explanatory variable. However, this
>> hormone varies in its levels throughout a day. Nevertheless, its  
>> levels differ
>> substantially between individuals so that there is information  
>> there to use.
>>
>> One simple thing to try would be to form categories, but I assume
>> there are better ways to handle this. Has anyone worked with such  
>> data, or could
>> anyone suggest some keywords that may be helpful in searching for  
>> this
>> topic. Thanks
>> for your input.
>>
>> Regards,
>>
>> Juliet
>>
>> ______________________________________________
>> R-help@r-project.org mailing list
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>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> -- 
> Jonathan Baron, Professor of Psychology, University of Pennsylvania
> Home page: http://www.sas.upenn.edu/~baron
> Editor: Judgment and Decision Making (http://journal.sjdm.org)


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