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From |
Weihua Guan <wguan@stata.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: mistake in weibhet_glfa |

Date |
Thu, 22 Aug 2002 11:30:58 -0500 |

--Alfonso Miranda<alfonsomirand4@yahoo.com> wrote: > While doing an extension of the code weibhet_glfa I > found what I believe is a mistake in the > log-likelihood expression of this code. [...] > I have written down the log-likelihood obtaining (in an intermediate > expression that helps comparison) the expression: > > Logl = ln{1+theta*exp[-x’b*p]*t^(p)}^(-(1/theta + d)) > + ln{exp[-x’b*p]*p*t^(p-1)} > (3) > > Where x is the vector of observed characteristics and > b is its corresponding vector of coefficients. In the > weibhet_glfa code the log-likelihood is written as > > Logl = ln{1+theta*exp[-x’b*p]*t^(p)}^(-(1/theta + d))+ > + ln {exp[-x’b*p]*p*t^(p)} > (4) [...] Alfonso noticed the difference between the equation (3) and (4), and wondered whether -weibhet_glfa- made a mistake. Short answer: ML will get the same estimated parameters from either equation(3) or (4). The program chooses equation (4) because the log-likelihood value will keep constant in this form when the scale of survival time changes. Long answer: Compared the two equations, the difference lies on a term ln(t). Note that ML finds the estimates by the first derivatives (of b and p), and the term ln(t) will not be presented in the first derivatives. The two log-likelihood functions will result in the same estimation results except the log-likelihood value. In fact, the program is written in this way on purpose such that the value of the log-likelihood will be invariant to the scale of survival time. Let's see a simple example with Weibull model: . use http://www.stata-press.com/data/r7/cancer, clear (Patient Survival in Drug Trial) . stset studytime, fail(died) (output omitted) . streg drug age, dist(weibull) nolog failure _d: died analysis time _t: studytime Weibull regression -- log relative-hazard form No. of subjects = 48 Number of obs = 48 No. of failures = 31 Time at risk = 744 LR chi2(2) = 35.92 Log likelihood = -42.662838 Prob > chi2 = 0.0000 [...] Now we rescale the time variable "_t" by 100 and re-estimate the model: . replace _t = _t/100 _t was byte now float (48 real changes made) . streg drug age, dist(weibull) nolog failure _d: died analysis time _t: studytime Weibull regression -- log relative-hazard form No. of subjects = 48 Number of obs = 48 No. of failures = 31 Time at risk = 7.439999977 LR chi2(2) = 35.92 Log likelihood = -42.662838 Prob > chi2 = 0.0000 [...] The value of log-likelihood does not change when the scale of _t changes. This trick can be also found in some other models with -streg-. To get back the original value of log-likelihood, we can simply adjust the calculated log-likelihood by the term ln(t). . gen double lnt = _d*ln(_t) . summarize lnt if e(sample) . di e(ll) - r(sum) Weihua Guan <wguan@stata.com> Stata Corp. * * 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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