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_,
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.
Rita Luk wrote:
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.
Ontario Tobacco Research Unit
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