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st: Re: ratios first or last?
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.
----- Original Message -----
From: "David Airey" <firstname.lastname@example.org>
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.
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