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From |
"FEIVESON, ALAN H. (AL) (JSC-SK) (NASA)" <[email protected]> |

To |
"'[email protected]'" <[email protected]> |

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: [email protected]
[mailto:[email protected]]On Behalf Of Constantine
Daskalakis
Sent: Tuesday, November 02, 2004 12:09 PM
To: [email protected]
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: [email protected]
Webpage: http://www.jefferson.edu/medicine/pharmacology/bio/
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