Re: st: Dealing with exposure differences using logarithmic offsets versus rescaling

[Phil Schumm <pschumm@uchicago.edu>]

On Aug 14, 2013, at 6:07 PM, Dimitriy V. Masterov <dvmaster@gmail.com> wrote:
> I am trying to test my intuition that logarithmic offsets should yield the same estimates as dividing the outcome by exposure and using exposure to weight.

<snip>

> gen lnt1 = ln(t1);
> lab var lnt1 "Treatment Offset (ln of Days)";
> 
> gen bipd1 = bi1/t1;
> gen gmbpd1 = gmb1/t1;
> 
> /* Simple Poisson Model for BI */
> poisson bi1 i.treat, offset(lnt1) robust;
> poisson bipm1 i.treat [iw=t1], robust;


This is a function of the fact that for the poisson family, the variance is proportional to the mean, and in the case of no overdispersion, equal to the mean.  So, in your case, the mean and variance of Y are equal to

    e^[XB + log(exposure)]

which is exactly the model you are fitting when using log(exposure).  If you divide Y by exposure, then the mean and variance of Y become

    mean = e^[XB]
    var  = e^[XB] x 1/exposure

which is the model you are fitting without the offset and using exposure as the weight.


> /* GMB using GLM */
> glm gmb1 i.treat, offset(lnt1) link(log) family(gamma) robust;
> glm gmbpm1 i.treat [iw=t1], link(log) family(gamma) robust;


In contrast, for the gamma family, the variance is proportional to the *square* of the mean.  Thus, unlike with the poisson, in this case you are fitting two different models to the data.  Which is better is an empirical question, which you should determine by examining the mean and variance of Y relative to the linear predictor and its components (initially through plots, and then possibly via more formal methods).


-- Phil


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