It's not that the denominator is skewed right, it's that the ratio itself
varies between smaller and larger denominators. Quoting myself:
"If you find that the ratio of averages is giving larger estimates, then
the individual ratios tend to be larger with larger values of x."
Michael Blasnik
[email protected]
----- Original Message -----
From: "David Airey" <[email protected]>
To: <[email protected]>
Sent: Sunday, November 23, 2003 3:11 PM
Subject: re: st: ratios first or last?
> This is helpful, Michael. So, if I understand this correctly, for
> denominators that are skewed right, we should expect the ratio of
> averages to give a greater animal estimate than the average of ratios?
> I'll test this.
>
> Thank you for these thoughts.
>
> -Dave
>
> > The ratio of averages gives greater weight to observations with larger
> > values in the denominator than the average of the ratios. In fact,
> > you can
> > get both estimates from weighted regressions:
> >
> > ratio of averages:
> > reg y x [aw=1/x], nocons
> >
> > average of ratios:
> > reg y x [aw=1/x^2], nocons
> >
> > which allows you to see the implicit weighting of each approach. Each
> > estimator can be considered the "best" (BLUE) estimator for a given
> > assumption about the variance. The ratio of averages approach is
> > commonly
> > referred to as a "ratio estimator" in survey sampling literature. In
> > most
> > applications, the average of ratios approach is thought to be less
> > representative of the variance structure.
> >
> > If you find that the ratio of averages is giving larger estimates,
> > then the
> > indivdual ratios tend to be larger with larger values of x.
> >
> > Michael Blasnik
> > [email protected]
> >
> > ----- Original Message -----
> > From: "David Airey" <[email protected]>
> > To: <[email protected]>
> > Sent: Saturday, November 22, 2003 8:27 PM
> > Subject: st: ratios first or last?
> >
> >
> > > Recently I was surprised to find a difference between two methods of
> > > calculating a ratio during an experiment. Each animal has two
> > measures
> > > taken repeatedly over time. The ratio is of the two measures. I could
> > > take the ratio at each time point, and then average the ratios to get
> > > my animal ratio. Alternatively, I could average each of the two
> > > measures and then form a ratio of the two averages, again getting my
> > > animal ratio. The second method consistently gets a higher ratio than
> > > the first method. Why would this occur? The second method is standard
> > > in my literature base.
> > >
> > > -Dave
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