Alternate explanations

[David Scherer <scherer@uiuc.edu>]

Dr. Cohen has asked for a plausible, quantifiable explanation for his
results.  If he allows the use of internal, confounding variables there
would be numerous ways to do this.  I will illustrate the methodology with
migration as such a confounder.

People who die in County A may have lived in any other county.  Thus for
County A, the average radon dose is a linear combination of all counties in
the data set.  Thus, (D a) = Sum-over-i (W ai * C i) where (D a) is the
radon Dose for people who die in County A,  (W ai) are migration weighting
factors, and (C i) are the radon concentration in County I.  
Each (W ij) is the lifetime fraction of time spent in County J by people
who die in Count I. This produces a large system of simultaneous linear
equations.  I am confident that some set of Wai will result in a linear
relationship between (Dose i) and  (M i), the smoking-corrected lung cancer
mortality in County I.

Procedurally, you start with the (M i), and derive a set of doses (D i) =
(risk) * (M i).  Here (risk) is the risk coefficient you want to use.  You
plug these (D i) into the matrix equation above and sovle for the matrix
elements (M ij).  However, you need to modify the smoking corrrection by
the same set of (W ij).  Perhaps this could be solved simultaneously of
iteratively.  The explanation is "plausible" if the diagonal elements are
fairly large and the other elements are generally smaller as you move from
further the main diagonal.

You could follow a similar procedure with other confounding variables
(socioeconomic status, employment, family history, smoking, et cetera ad
infinitum).  But they remain conjecture - these are not real, empirical
data.  (The absence of these data is one weakness of the experimental
design.)  It is obvious that this procedure will not produce any
information about radon risk; it's a numbers game.  But, it does show that
internal variables could offer other explanations for Dr. Cohen's results.

I have no interest in performing the calculation, and I waive any claim to
the $5K bounty.

Regards,
Dave Scherer
scherer@uiuc.edu