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RE: st: RRR with CI from logit model


From   "FEIVESON, ALAN H. (AL) (JSC-SK) (NASA)" <alan.h.feiveson@nasa.gov>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   RE: st: RRR with CI from logit model
Date   Tue, 2 Nov 2004 15:12:22 -0600

Following up on what Constantine Daskalakis wrote - 

The delta method is based on a linear approximation to the function of the
random variable about it's mean. When you try to use it (the delta method)
on a nonlinear function of a random variable which can be very far from it's
mean, you get nonsense. For example, suppose X ~ N(mu, sig^2) (normal with
mean 4 and variance 1). Then the (true)variance of exp(t*X) can be shown to
be

var_true = exp(2*mu*t + 2*t*t*sig^2) - exp(2*mu*t + t*t*sig^2)

This follows from the moment-generating function E(exp(tX)) which for a
N(mu, sig^2) random variable is exp(mu*t + 0.5*t*t*sig^2)



On the other hand using the delta method, we get the variance of g(X) is
approximated by Var(X)*[g'(mu)]^2

For the above example, we get g(X) = exp(t*X) and g'(X) = t*exp(t*X) so
g'(mu)=t*exp(t*mu). Therefore,

var_approx = [t*t*sig^2]*exp(2*mu*t)

Now, just for fun, I compared these two variances for t = 0(0.1)19 with mu=4
and sig^2=1. Here's what I got:


    +---------------------------+
     |   t  var_true  var_approx |
     |---------------------------|
  1. |   0          0          0 |
  2. |  .1   .0225919   .0222554 |
  3. |  .2   .2103865   .1981213 |
  4. |  .3   1.135862   .9920861 |
  5. |  .4   4.995238   3.925205 |
     |---------------------------|
  6. |  .5   19.91172   13.64954 |
  7. |  .6    75.4706   43.74376 |
  8. |  .7   279.1179   132.5089 |
  9. |  .8   1023.232   385.1809 |
 10. |  .9   3757.346   1084.939 |
     |---------------------------|
 11. |   1   13923.38   2980.958 |
 12. | 1.1   52359.96   8027.437 |
 13. | 1.2   200706.3   21261.29 |
 14. | 1.3   787029.9   55532.74 |
 15. | 1.4    3166629   143335.6 |
     |---------------------------|
 16. | 1.5   1.31e+07   366198.3 |
 17. | 1.6   5.59e+07   927276.9 |
 18. | 1.7   2.46e+08    2329716 |
 19. | 1.8   1.12e+09    5812800 |
 20. | 1.9   5.31e+09   1.44e+07 |
     +----------------------

Note that once t exceeds .25 or so, the approximation goes bad. For large
values of t, the approximation is complete garbage.

Al Feiveson



-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Constantine
Daskalakis
Sent: Tuesday, November 02, 2004 12:09 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: RRR with CI from logit model


At 12:42 PM 11/2/2004, Michael Ingre wrote:
>On 2004-11-02, at 18.19, Constantine Daskalakis wrote:
>
>>Look at the estimates and estimated standard errors for the different 
>>situations in your example. You'll probably find that the estimated RRR 
>>increases but its estimated standard error goes to hell (increases much 
>>more). This is a problem with Wald-type tests that has been pointed out 
>>before (eg, see Hauck & Donner, JASA 1977, w/ corrrection in 1980). The 
>>delta method (especially for anti-log functions of coefficients) seems to 
>>exacerbate that.
>
>Thank you, I think you got it. But what is the solution to the 
>problem?  Am I stuck with just OR? Is there a way to calculate RRRs with 
>CIs directly from predicted probabilities with CIs instead?
>
>Michael

The trouble is that you are computing a quantity (RRR) that is 
"non-standard" (ie, non-linear) from the logistic regression model.
Wald-type tests/CIs via the delta method often perform poorly.

If you insist on logistic regression, try bootstrap perhaps? You may not do 
better, but I doubt you can do worse.

Or, as others have suggested, try another model where your quantity is a 
"natural" result.

CD





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________________________________________________________________
Constantine Daskalakis, ScD
Assistant Professor,
Biostatistics Section, Thomas Jefferson University,
211 S. 9th St. #602, Philadelphia, PA 19107
    Tel: 215-955-5695
    Fax: 215-503-3804
    Email: c_daskalakis@mail.jci.tju.edu
    Webpage: http://www.jefferson.edu/medicine/pharmacology/bio/  

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