Submitted for publication in J. Radiol. Prot, July 2002
Response to “The potential for bias in Cohen’s ecological analysis of lung cancer and residential radon”
I am writing in response to the critique by Lubin  of my test of the linear-no threshold theory (hereafter LNT). That paper has been cited as providing an explanation for the discrepancy between my observational findings and the predictions of LNT, thus negating my test. Therefore, to stay within the length restrictions on Letters-to-the-Editor, I limit my comments to addressing that issue.
In the simplified form adopted by Lubin, my procedure  starts with the lung cancer mortality risk for an individual male, m’, expressed in the BEIR-IV version  of LNT as
ms’ = as (1 + b r’) smokers
mn’ = an (1 + b r’) non-smokers
where r’ is his radon exposure, and a and b are constants given in BEIR-IV.
Mathematically summing the experiences of all males in a county, leads to
m = [S as + (1 – S) an ] (1 + b r)
where m is the county lung cancer mortality rate, r is the average radon exposure in the county, and S is the fraction of adult males in the county that smoke cigarettes. Defining the important symbol R as
R = as / an
and applying minor corrections, this may be expressed as
M = m / an [R S + (1 – S)] = A + B r (1)
where M is defined by the equation on the left, and A and B are constants, with A close to 1.0 and, according to BEIR-IV,
B = +7 LNT – theory (2a)
In units of % per 37 Bq/m3 (% per pCi/L).
Data on M and r are available for 1601 U.S. counties and I use them to test the LNT theory by fitting them to Eqn. (1) to determine an observational value of B. The result is
B = -7.3 (+/-0.56) Observation (2b)
which is discrepant with the LNT-theory prediction, (2a), by 26 standard deviations.
The Scientific Method requires that theories be tested, where possible, with observational data, and that any discrepancies between observation and predictions of the theory must be given at least a possible explanation that is not implausible; failing that, the theory is invalid. In a series of papers, I have attempted to find such a possible plausible explanation for the discrepancy between (2a) and (2b), including evaluations of suggestions offered by others, but none of these explanations has been successful.
Lubin’s recent suggestion  is that the discrepancy can be explained by systematic differences in average radon exposure to smokers, rs and non-smokers, rn. within each county. I previously treated this general suggestion  by defining the effective average radon exposure, re , as
re = [R S rs + (1-S) rn ] / [R S + (1 – S)]
where the two terms in the numerator are the weighted radon exposures to smokers and non-smokers, and the denominator is the sum of the weightings. This differs from the measured county average radon exposure, r
r = S rs + (1 – S) rn
If rs and rn are not equal, re instead of r should be used to fit the data with Eqn. (1), which would change the observational value of B, and thus hopefully eliminate the discrepancy between (2a) and (2b). Up to this point, these are the same as the procedures used by Lubin .
I defined 
x = rs / rn
and treated as variable parameters the national average value of x, the width of the distribution of x-values among U.S. counties, and the correlation between x and r. I explored the effects of varying these parameters within and well beyond their plausible ranges, but the maximal effect was to change the observational value of B from B = -7.3 to B = -4.3, still very discrepant with the LNT theory value, B = +7. I concluded that possible systematic differences between radon exposures to smokers and non-smokers could not explain the discrepancy between (2a) and (2b).
Lubin’s paper  that I am responding to here demonstrates that the discrepancy between (2a) and (2b) can be explained if both R and x vary in a systematic way with r. In particular, this requires that R decreases in a systematic way with increasing r to 65% of its value at low r, and that x varies with r in a very complex way, some samples of which are in the following Table derived from Fig. 2 of Reference 1:
r 30 50 75 90 110 150 200
x 7 2.7 1.0 2.2 1.0 0.36 0.27
The behavior for r < 30, which includes about 20% of our data, is even more complex, but I could not read the plot with sufficient accuracy to list values here.
Lubin’s paper  does not address the issue of plausibility. But it is extremely important to recognize that “The Scientific Method” requires that an explanation be found for the discrepancy between predictions of the theory and observation that is not implausible. We therefore now consider the plausibility of Lubin’s proposed explanation.
The quantity R is the ratio of lung cancer risks for smokers and non-smokers, irrespective of their exposure to radon. It is applicable to individuals. According to Lubin’s proposal, the ratio of these risks is lower if they live in a county with high average radon levels than if they live in a county with low average radon levels, even if they have the same personal exposure to radon, or even if they have no personal exposure to radon. Surely an individual’s risk depends on his personal exposure to radon, not on the average exposure to people in his county of residence. The variation of R with r proposed in Reference 1 is therefore completely implausible.
Average radon exposure in a county, r, depends on geological factors, house characteristics, and ventilation practices (e.g. window opening). The only way I can imagine these being systematically different for smokers and non-smokers, thus affecting the value of x, is in ventilation practices. But it seems inconceivable that these can vary systematically and very sharply over a very wide range with average radon exposure in the county, especially since these average radon levels were not known to the people controlling the ventilation. The systematic variation of x with r illustrated in the above Table is surely completely implausible.
Importance of plausibility
Quite aside from its importance in applying The Scientific Method, plausibility is of utmost importance in all epidemiological studies. For example, the choice of confounding factors to be considered in a study is based completely on plausibility considerations; a plausible confounding factor that is not considered can easily invalidate any such study. It is therefore difficult to understand how such wild violations of plausibility could have escaped consideration in Reference 1.
 Lubin, J H 2002 The potential for bias in Cohen’s ecological analysis of lung cancer and residential radon J. Radiol. Prot. 22:141-148
 Cohen, B L 1995 Test of the linear-no threshold theory of radiation carcinogenesis for inhaled radon decay products Health Phys. 68:157-174
 National Academy of Sciences Committee on Biological Effects of Ionizing Radiation 1988 Health risks of radon and other internally deposited alpha emitters (BEIR-IV) Washington, DC; National Academy Press
 Cohen, B L 1998 Response to Lubin’s proposed explanations of our discrepancy Health Phys. 75:18-22