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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 The documents accompanying this transmission may contain confidential health or business information. This information is intended for the use of the individual or entity named above. If you have received this information in error, please notify the sender immediately and arrange for the return or destruction of these documents. ________________________________________________________________ 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/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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