Re: st: Is it valid to use the individual ratios (i.e. Xi/Yi) in the dependent or independent part of a regression model?

[David Hoaglin <dchoaglin@gmail.com>]

Dear Jinn-Yuh,

Your analysis is not actually concerned with the ratio distribution as such.

Your first message, at the start of this thread, assumed that X and Y
were normally distributed.  Real data, however, are never normal, so
results for a ratio of normal variables are mainly of theoretical
interest.

In particular, from Phil Clayton's comments, it seems that urinary
albumin and urinary creatinine are not close to having normal
distributions.  You could investigate the distribution shape of
urinary albumin, urinary creatinine, and ACR empirically in your
patients, though they are probably not a random sample from any
well-defined population.

In your analyses, if ACR is the dependent variable, a more important
consideration is the distribution of the variation of the part of ACR
that remains after the contributions of the predictor variables have
been removed.  In other words, the residual variation (or the
so-called "error term").  You can investigate that also, by using the
residuals from the fitted regression model.

If ACR is an explanatory variable (I avoid the term "independent
variable" because "independent variables" are seldom "independent" in
any reasonable sense), the underlying theoretical distribution is much
less important than the way ACR is distributed in your patients.

David Hoaglin

On Sun, May 27, 2012 at 9:23 AM,  <guhjy@kmu.edu.tw> wrote:
> Dear David:
> Thank you for your detailed explanations.
> It helps me a lot.
> P.S.: The motivation for my question was the fact that the ratio
> distribution is very complicated
> (http://en.wikipedia.org/wiki/Ratio_distribution) with no means and
> variances (http://www.mathpages.com/home/kmath042/kmath042.htm).
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