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Re: st: Elasticity


From   Richard Goldstein <richgold@ix.netcom.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Elasticity
Date   Wed, 20 Apr 2005 15:15:29 -0400

The use of ratios in very problematic here.  There was a discussion
on this list several years ago -- below is my final statement at that
time, one that I continue to believe is correct:

Recently there was an incomplete discussion of the use of ratios in
regression.  I submit the following as a form of completion (and in part
because I feel guilty about not completing and then criticizing,
privately, someone who had submitted something incomplete to the list).

Ratios are often used in regression to "adjust" or "standardize" for
some factor such as size.  One can divide the ways this is used into two
classes, one of which is acceptable and the other of which is
(generally) not acceptable.

1. Acceptable:  If every variable in the regression is divided by the
   same factor there is no problem.  This is done for example, when
   turning everything into a "per capita" measurement; another example
   is weighted regression.  One needs, however, to be clear regarding
   what is meant by "every variable".  Say your regression has two
   predictors (X and Z) and you want to control for population size
   (POP); the basic regressions looks like (suppressing the subscript
   for individual observations):

          Y = b0 + b1X + b2Z + e

   When adjusted for population size, the regression should look like:

          Y/POP = b0/POP + b1(X/POP) + b2(Z/POP) + e/POP

   Leaving out any of these terms will cause problems.  See Stata's
   write-up on weighted regression for more on this.  (Note that
   inclusion of a constant in this last model is called for in the case
   where the first model includes b3POP.)

2. Unacceptable:  Sometimes it makes no sense to divide all variables by
   the denominator of the ratio; for example, in many health studies
   there is a desire to control for the size of the individual by using
   BMI (body mass index:  wt/(ht^2)) as a predictor; another example
   occurs in the study of strength where the desire is to adjust
   strength by the size of the muscle (or muscle fiber); note in the
   latter case that the ratio will now be the response variable.  If the
   set of predictors include any demographic variables (e.g., sex, age),
   then clearly one will not want to divide the demographic predictor by
   the denominator of the ratio.  The issue here is mostly easily, I
   think, seen by observing that the ratio is an interaction term, but
   that the regression does not (usually) include the accompanying main
   effect terms; this is, among other things, a violation of the
   "marginality" principle (fn. 1).  In general, one does not want to
   automatically include an interaction term without its component
   parts.  Further, the inclusion of an interaction term has
   implications about the form of the adjustment:  use of BMI without
   either height or weight has implications for the way that size is
   adjusted and these implications may be wrong.  The answer is to
   multiply out the ratio; e.g., if the ratio is in the response
   variable, multiply everything on the right by the denominator; if the
   ratio is in a predictor, add the component main effects to the model
   and see if the interaction (ratio) adds anything.  A good discussion
   of this case, with explicit advice, can be found in Kronmal, R.A.
   (1993), "Spurious correlation and the fallacy of the ratio standard
   revisited," _Journal of the Royal Statistical Society, series A_,
   156:  379-392.

-----------------------------

1. For example, including one main effect but not the other implies that
   the intercept but not the slope is independent of the other main
   effect.  For more, see Nelder, J.A.  (1998), "The selection of terms
   in response-surface models -- How strong is the weak-heredity
   principle?", _The American Statistician_, 52:  315-8.

Rich Goldstein

Rita Luk wrote:

Hi Statalist,

Please excuse me that this is not exactly a question about Stata, but I
think professionals here can help.

For a regression equation:  X/Y = a + bZ + cK + error where X, Y, Z and K
are variables. Given estimates of the coefficients b and c, what is the
formula for the marginal effect of Z on X ie. dX/dZ, and the formula for the
elasticity of X with respect to Z.

I know how to get the elasticity when the Dependent is a variable in level,
but do not know how when the dependent is a ratio.

Your information on the formula or reference materials are much appreciated.


Thank you very much for your precious time.

Rita Luk
Research Officer
Ontario Tobacco Research Unit
Toronto, Ontario

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