Why does Weibull with entry and exit times produce different results from Weibull with duration?

A subscriber to Statalist asked about fitting a Weibull model using weibull and stweib. He got different results and attributed that to use of the two different commands. In fact, the difference was because Weibull has an absolute concept of 0 and because he was estimating different Weibull models.

The subscriber had two measures of time-to-failure:

• duration (variable dura), and
• entry and exit times (variables entry and exit).

What the subscriber observed was that

        . weibull dura x1 x2, dead(censor)                          (1)

and

        . weibull exit x1 x2, dead(censor) t0(entry)                (2)

produced different results. He would have obtained the same individual answers and the same difference had he

        . stset dura, failure(censor)                                        (1')
        . stweib x1 x2

and

        . stset exit, failure(censor) t0(entry)                              (2')
        . stweib x1 x2

What he in fact ran was (1) and (2') and that is what confused him. The difference has nothing to do with weibull vs. stweib; it has to do with the model itself.

The Weibull hazard function is not constant with time. That means that, for a given set of Weibull coefficients

        p_1 = Pr(surviving from     0  to  dura)        *IS NOT EQUAL* 
        p_2 = Pr(surviving from entry  to  entry+dura)

The hazard might be increasing (in which case p_2<p_1) or decreasing (in which case p_2>p_1).