Chapter 20.2 in “Standard Handbook of Environmental Science, Health, and Technology” J.H. Lehr (Editor), McGraw-Hill (New York); 2000





Bernard L. COHEN


University of Pittsburgh, Pittsburgh, PA 15260, USA



            All estimates of the cancer risk from low level radiation are based on the linear-no threshold theory (LNT) which is based solely on largely discredited concepts of radiation carcinogenesis, with no experimental verification in the low dose region of the most important applications.. These risk estimates are now leading to the expenditure of tens of billions of dollars to protect against dangers whose existence is highly questionable. It is therefore of utmost importance to test the validity of this theory.

            A definitive answer to the validity of LNT in the low dose region must be based on human data, but to obtain statistically indisputable data requires much larger numbers of subjects than can be obtained from occupational, accidental, or medical exposures.  The obvious source is natural radiation.  If one attempts to use natural gamma radiation, which varies somewhat with geography, one is faced with the problem that LNT predicts that only a few percent of cancers are due to natural radiation; whereas there are unexplained differences of tens of percent for different geographic areas.  For example, the percentage of all deaths that are from cancer varies in U.S. from 22% in New England to 17% in the Rocky Mountain States (where radiation levels are highest).  Another problem is that gamma ray backgrounds vary principally with geographic regions, and there are also many potential confounding factors that may vary with geography.  Nevertheless, there have been attempts to study effects of gamma ray background on cancer rates, and in general either no effect or an inverse relationship has been found.  For example, no excess cancer has been found in the high radiation areas of India or Brazil.  But all such effects can easily be explained by potential confounding factors.

            A much more favorable situation is available for radon in homes.  According to LNT, it is responsible for at least 10% of all lung cancers, and a known confounder, cigarette smoking, is responsible for nearly all of the rest.  Another advantage is that levels of radon in homes vary much more widely than natural gamma radiation.

            There have been numerous case-control studies of the relationship between radon in homes and lung cancer but the results from different studies have been inconsistent and this work has given no statistically significant information on the validity of LNT in the low dose region which we define here as below 5 pCi/L which corresponds to 20-50 cSv (whole body equivalent dose) over a lifetime. A different approach, specifically designed for testing LNT, was carried out by the present author and is described in the following sections.


Original 1995 paper

            My group at University of Pittsburgh developed an elaborate study designed specifically to test LNT (1). We briefly review it here. We compiled hundreds of thousands of radon measurements from several sources to give the average radon level, r, in homes for 1729 U.S. counties, well over half of all U.S. counties and comprising about 90% of the total U.S. population.  Plots of age-adjusted lung cancer mortality rates, m, vs these r are shown in Fig. 1 where, rather than showing individual points for each county, we have grouped them into intervals of r (shown on the base-line along with the number of counties in each group) and we plot the mean value of m for each group, its standard deviation indicated by the error bars, and the first and third quartiles of the distribution.  Note that when there is a large number of counties in an interval, the standard deviation of the mean is quite small. We see, in Fig. 1a,c, a clear tendency for m to decrease with increasing r, in sharp contrast to the increase expected from the supposition that radon can cause lung cancer, shown by the line labelled “Theory”.        

            One obvious problem is migration: people do not spend their whole lives and receive all of their radon exposure in their county of residence at time of death where their cause of death is recorded.  However, it is easy to correct the theoretical prediction for this, and the “Theory” lines in Fig. 1 have been so corrected.  As part of this correction, data for Florida, California, and Arizona, where many people move after        retirement, have been deleted, reducing the number of counties to 1601.  (This deletion does not affect the results.)

            A more serious problem is that this is an “ecological study”, relating the average risk of groups of people (county populations) to their average exposure dose. Since most dose-response relationships have a “threshold” below which there is little or no risk, the disease rate depends largely on the fraction of the population that is exposed above this threshold, which is not necessarily closely related to the average dose which may be far below the threshold. Thus, in general, the average dose does not determine the average risk, and to assume otherwise is what epidemiologists call “the ecological fallacy”. However, it is easily shown that the ecological fallacy does not apply in testing a linear-no threshold theory (LNT). This is familiar from the well known fact that, according to LNT,  population dose in person-rem determines the number of deaths; person-rem divided by the population gives the average dose, and number of deaths divided by the population gives the mortality rate which is the average risk. These are the quantities plotted in Fig. 1.  Other problems with ecological studies have been discussed in the epidemiology literature, but these have also been investigated and found not to be applicable to our study. The most important of these problems are discussed below.

            Epidemiologists normally study the mortality risk to individuals, m’, from their exposure dose, r’, so we start from that premise using the BEIR-IV version of LNT (in simplified form; full treatment in Ref.1).

                        m’ = an ( 1 + b r’)                            non-smokers

                        m’ = as ( 1 + b r’ )                               smokers

where an and as are constants determined from national lung cancer rates, and b is a constant determined from studies of miners exposed to high radon levels.

            Summing these over all people in the county and dividing by the population gives

                        m = [ S as + (1 - S) an ] ( 1 + b r )                                      (1)

where m and r have the county average definitions given above in the presentation of Fig. 1, and S is the smoking prevalence--the fraction of the adult population that is smokers. Eq. (1) is the prediction of the LNT theory we are testing here (we also show that  our test applies not only to the BEIR-IV version but to all other LNT theories); note that it is derived by rigorous mathematics from the risk to individuals, with no problem from the ecological fallacy.

            The bracketed term in Eq.(1), which we call m0, contains the information on smoking prevalence, so m/m0 may be thought of as the lung cancer rate corrected for smoking. Fig. 1b,d show m/m0 vs r. We fit the data (I.e. all 1601 points) to

                        m/m0 = A + B r                                                                     (2)

deriving values of B . The theory lines are from Eq. (1) with slight renormalization. It is clear from Fig. 1b,d that there is a huge discrepancy between measurements and theory. The theory predicts B = +7.3% per pCi/L, whereas the data are fit by B = -7.3 (+/- 0.6) and -8.3 (+/- 0.8) % per pCi/L for males and females respectively. We see that there is a discrepancy between theory and observation of about 20 standard deviations; we call this “our discrepancy”.

            All explanations for our discrepancy that we could develop or that have been suggested by others have been tested and found to be grossly inadequate.  We review some of the details of this process here.

            There may be some question about the radon measurements, but three independent sources of radon data, our own measurements, EPA measurements, and measurements sponsored by various states governments, have been used and each gives essentially the same results. These three sets of data correlate well with one another, and by comparing them, we can estimate the uncertainties in each and in our combined data set; these indicate that uncertainties in the radon data are not a problem.

            Another potential problem is in our values of smoking prevalence,S. Three different and independent sources of data on smoking prevalence were used, and all result in essentially the same discrepancy with LNT seen in Fig. 1b,d. Nevertheless, since cigarette smoking is such an important cause of lung cancer, one might think that uncertainties in S-values can frustrate our efforts. Analysis shows that the situation is not nearly so unfavorable. The relative importance of smoking and radon for affecting the variation of lung cancer rates among U.S. Counties may be estimated by use of the BEIR-IV theory. For males, the width of the distribution of S-values, as measured by the standard deviation (SD) for that distribution, is 13.3% of the mean, and according to BEIR-IV a difference of 13.3% in S would cause a difference in lung cancer rates of 11.3%; whereas the SD in the width of the distribution of radon levels for U.S. Counties is 58% of the mean which, according to BEIR-IV, would cause a difference in lung cancer rates of 6.6%. Thus, the importance of smoking for determining variations in lung cancer rates among counties is less than twice (11.3/6.6) that of radon. Smoking is not as dominant a factor as one might intuitively think it is.

            Even more important for our purposes is the fact that smoking prevalence, S, can only influence our results to the extent that it is correlated with the average radon levels in counties. Thus we are facing a straightforward quantitative question: How strong a correlation between S and r, CORR-r, would be necessary to explain our discrepancy. If we use our best estimate of the width of the distribution of S-values for U.S. counties, even a perfect negative correlation between radon and smoking prevalence, CORR-r = -1.0, eliminates only half of the discrepancy. If the width of the S-value distribution is doubled, making it as wide as the distribution of lung cancer rates, which is the largest credible width since other factors surely contribute to lung cancer rates, an essentially perfect negative correlation, CORR-r = -0.90,  would be required to explain the discrepancy and to cut the discrepancy in half requires Corr-r = - 0.62.

            How plausible is such a large |CORR-r|? There is no obvious direct relationship between S and r, so the most reasonable source of a correlation is through confounding by socioeconomic variables (SEV). We studied 54 different SEV to find their correlation with r, including population characteristics, vital statistics, medical care, social characteristics, education, housing, economics, government involvements, etc. The largest |CORR-r| was 0.37, the next largest was 0.30, and for 49 of the 54 SEV, |CORR-r| was less than 0.20. Thus a |CORR-r| for smoking prevalence, S, even approaching 0.90, or even 0.62,  seems completely incredible. We conclude that errors in our S-values can do little to explain our discrepancy.

            In another largely unrelated study (2), we found that the strong correlation between radon exposure and lung cancer mortality (with or without S as a covariate), albeit negative rather than positive, is unique to lung cancer; no remotely comparable correlation was found for any of the other 32 cancer sites.  We conclude that the observed behavior is not something that can easily occur by chance.

            To investigate effects of a potential confounding variable, data are stratified into quintiles on the values of that variable, and a regression analysis is done separately for each stratum.  Since the potential confounder has nearly the same value for all counties in a given stratum, its confounding effect is greatly reduced in these analyses.  An average of the slopes, B,  of the regression lines for the five quintiles then gives a value for B which is largely free of the confounding under investigation.

            This test was carried out for the 54 socioeconomic variables mentioned above, and none was found to be a significant confounder.  In all 540 regression analyses (54 variables x 5 quintiles x 2 sexes), the slopes, B, were negative and the average B value for the five quintiles was always close to the value for the entire data set. Incidently, this means that the negative correlation between lung cancer rates and radon exposure is found if we consider only the very urban counties, or if we consider only the very rural counties; if we consider only the richest counties, or if we consider only the poorest; if we consider only the counties with the best medical care, or if we consider only those with the poorest medical care; and so forth for all 54 socioeconomic variables.  It is also found for all strata in between, as, for example, considering only counties of average urban-rural balance, or considering only counties of average wealth, or considering only counties of average medical care, etc.

            The possibility of confounding by combinations of socioeconomic variables was studied by multiple regression analyses and found not to be an important potential explanation for the discrepancy.

            The stratification method was used to investigate the possibility of confounding by geography, by considering only counties in each separate geographical region, but the results were similar for each region. The stratification method was also used to investigate the possibility of confounding by physical features such as altitude, temperature, precipitation, wind, and cloudiness, but these factors were of no help in explaining the discrepancy. The negative slope and gross discrepancy with LNT theory is found if we consider only the wettest areas, or if we consider only the driest; if we consider only the warmest areas, or if we consider only the coolest; if we consider only the sunniest, or if we consider only the cloudiest; etc

            The effects of the two principal recognized factors that correlate with both radon and smoking were calculated in detail: (1) urban people smoke 20% more but average 25% lower radon exposures than rural people; (2) houses of smokers have 10% lower average radon levels than houses of non-smokers.  These were found to explain only 3% of the discrepancy.  Since they are typical of the largest confounding effects one can plausibly expect, it is extremely difficult to imagine a confounding effect that can explain the discrepancy.  Requirements on such an unrecognized confounder were listed, and they make its existence seem extremely implausible.


Updates on original paper

            Our 1995 paper was based on lung cancer rates for 1970-1979, the latest age adjusted data available at that time. Recently, age adjusted lung cancer rates for 1979-1994 have become available. When these are used, the slopes, B, are changed from -7.3 to -7.7 % per pCi/L for males, and from -8.3 to -8.2 for females. Since there are more lung cancer cases included, the standard deviations of these B-values are reduced, increasing the discrepancy with the predictions of LNT to about 30 standard deviations.

            The 54 socioeconomic variables (SEV) used in the original paper were from the 1980 Census. About 450 new SEV from the 1990 Census have now been introduced and investigated in substantial depth. None of these SEV had |CORR-r| >0.45, and extensive stratification studies led to the conclusion that none of these additional SEV can help to explain our discrepancy.


The Ecological Study issue   

            Most criticisms of our study have been based on generalized criticisms of ecological studies. The most important of these is called “cross level bias” (3). On this basis, in a presentation to NCRP (Feb.17, 1998), Jay Lubin dismissed my work as useless by a mathematical demonstration showing that an ecological study does not do an adequate job in handling a confounding factor. This problem was addressed  in some detail in Ref, 1 and 4 where I describe it as “the ecological fallacy for confounding factors (CF)”. The classical “ecological fallacy” arises from the fact that the average dose does not, in general, determine the average risk, but I avoid this problem by designing my study as a test of the linear-no threshold theory (LNT) -- in LNT, the average dose does determine the average risk. Use of separate and independent risks for smokers and non-smokers avoids this problem for smoking prevalence. However, this problem does arise for other CF -- the average value of a CF does not adequately determine its confounding effects, as demonstrated mathematically by Lubin.

            For example, consider annual income as a CF that might confound the radon vs lung cancer relationship -- maybe very poor people have lower radon levels and for unrelated reasons, have higher lung cancer rates than others. As Lubin’s demonstration shows, average income is not necessarily a measure of what fraction of the population is very poor. A case-control study, in principle, selects cases and controls of matched incomes (although this is not always done, and is still less frequently done well).

            My approach to this problem is to use a large number of CFs. For the example under discussion, I use as CF the fraction of the population in various income brackets, <$5000/y, $5000-$10,000/y, ..........., >$150,000/y (10 intervals in all). In addition, I consider combinations of adjacent brackets, and other related characteristics such as the fraction of the population that is below the poverty line, the percent unemployment, etc.

We have found that smoking prevalence, which is very strongly correlated with lung cancer, must have at least a 35% correlation (Corr-r = -0.6) with radon to have a significant effect, but none of the above CF have a correlation larger than 7%. This is convincing evidence that income is not an important confounder of the lung cancer vs radon relationship.

            It is not difficult to devise a model in which cross level bias could nullify our results, verifying Lubin’s mathematical proof. For example, we might suppose that those with an income that is an integral multiple of $700 have 50 times lower radon and 50 times higher lung cancer rates than average. I have no data to show that this is not the case. But such a model is not acceptable for two reasons:

            (1) It is not plausible

            (2) It would also not be taken care of in case-control studies (they             don’t match incomes with that precision).

What is needed is a model that avoids these two limitations. These limitations are effectively corollaries to Lubin’s mathematical proof.

            Of course annual income is not the only CF that must be considered.. Another example is age distribution. Case-control studies match cases and controls by age, and as Lubin’s mathematical demonstration shows, average age in a county does not handle this problem in our study. Of course I do use age-adjusted mortality rates which take care of the gross aspects of that problem, but there are limitations in the age-adjustment process. My solution is to use as CF the percent of the population in each age bracket, <1y, 1-2y, ......., 80-84Y, >85y, 31 age brackets in all, and to also use combinations of adjacent age brackets. None of these age brackets had correlations with radon above 4% with the exception of the >85y bracket where the correlation was 7.7 %.. This was further investigated by stratification, using five strata of 320 counties each and determining the slopes, B, (cf. Eq. (2) above) of the lung cancer vs radon relationship for each stratum. As we go from the stratum with the lowest to the stratum with the highest percent of population with age >85y, Bvalues for males were -10.1, -6.4, -6.1, -4.7, and -7.2 % per pCi/L, and for females they were -6.3, -2.0, -9.1, -3.5, -10.7 % per pCi/L, whereas LNT predicts B=+7.3% per pCi/L. Since the value of B is negative and grossly discrepant with the LNT prediction for all cases, and there is no consistent trend in its variations, I conclude that the correlation between radon and elderly people cannot explain our discrepancy. I can’t prove this mathematically, but I can’t concoct a not-implausible model in which variations of radon and lung cancer with age helps substantially to explain our discrepancy. As Lubin’s proof shows, it is possible to concoct a model to explain our discrepancy --e.g. we might assume that those born on the first day of a month have 50 times higher radon levels and 50 times lower lung cancer rates than others -- but that does not satisfy our two corollaries to Lubin’s proof. 

            There are few, if any, other bases on which case-control studies match cases and controls, but in my study I gave similar treatments to a host of other potential confounding factors -- educational attainment, urban vs rural differences, ethnicity, occupation, housing, medical care, family structures, etc, etc. I have found nothing that can help substantially to explain our discrepancy.

            Aside from cross-level bias, more generalized and less specific discussions of limitations of ecological studies have appeared in the literature and have been used to criticize our study. But there are many very important differences between  our work and other ecological studies. One such difference is in the quantity of data involved. Most ecological studies involve 10-20 (or less) groups of people, whereas ours involves 100 times that number (1729  counties). Not only does that give a tremendous improvement in statistical accuracy, but it allows much more elaborate and sophisticated analyses to be done, including consideration of large numbers of potential confounding factors and use of stratification techniques.

            A more important difference is that our work avoids the “ecological fallacy”; I know of no other ecological study that contains that feature. That alone makes our paper very different from the others, and should earn it the right to be considered free from the prejudice attached to consideration of other ecological studies.

            Ecological studies are normally viewed as being fast, simple to carry out, and inexpensive, but none of these adjectives applies to our project. It was the focus of my research effort for many years. Our radon measurements extended over six years and involved hundreds of assistants with millions of dollars in salaries, and the completely separate EPA and State-sponsored measurements we used were comparably elaborate. Our data analysis efforts involved dozens of assistants and several years of their efforts and mine. Without the power of modern computers and software packages, which have not been available until quite recently, such analyses would have been completely impractical. I know of no other ecological study to which any of the considerations of this paragraph would apply.

            Since any deep understanding of how radon causes lung cancer must be based on its effects on individuals, it is essential to study the problem in terms of risks to individuals, which seems contrary to the ecological approach. However our treatment is based on risks to individuals (cf derivation of Equation (1) above). That theory is then developed by rigorous mathematics to obtain the prediction, Eqn. (1), we use to compare with observations. This is a time-honored procedure in science; for example, Newton’s famous formula,

                                                 F = m a  

is rarely tested by direct measurements of acceleration,a , but rather, the formula is developed mathematically to determine distance travelled vs time which is much easier to measure, as a test of the theory.


Other published proposals for explaining our discrepancy

            The BEIR-VI Report pointed out that no consideration had been given to variations in intensity of smoking, and proposed a model in which this is expressed as the ratio, k, of one pack per day to two pack per day smokers. To evaluate this proposal, one must recognize that there is surely no direct causal relationship between k and radon levels, r, so any correlation between the two must arise from socioeconomic variables, SEV. How large a correlation between k and r, Corr(k,r), is not completely implausible? For the 500 SEV we have studied, the largest |Corr(SEV,r)| is 0.45. 

            Some indication of intensity of smoking in various states is included in cigarette sales (cs) data, available from tax collection records for each year. From these data, Corr(cs,r) varied between -0.14 and -0.29 between 1960 and 1975. From this and the maximum |Corr(SEV,r)| , it seems reasonable to conclude that |Corr(k,r)| larger than 0.5 would be highly implausible. The effect of Corr(k,r) = -0.5 is to change the slope B in Fig. 1b from -7.3 to -5.0, and even Corr(k,r) = -0.8 gives B =-2.3, still a long way from the LNT prediction B =+7.3 . Of course there is no reason to believe that Corr(k,r) is negative, and a positive Corr(k,r) would change B in the opposite direction.

            Field et al  pointed out that radon gas levels in homes is not the same as exposures to radon progeny which determine the dose, because the latter are affected by time spent in homes, exposures in other places, the ratio of radon progeny to radon gas, etc. and these may be correlated with radon levels. To investigate this, we define a modifying factor, f, by

                        r(effective) = r (1 + f)

and use r(effective) rather than r in determining B. The results depend on two features of f, the width of the distribution of f-values and Corr(f,r). As a rather extreme example, assuming a perfect correlation, Corr(f,r)=1.0

and w=0.7, changes B only from -7.3 to -3.7. Of course there is no reason to believe that the effects would be anywhere near that large, or that they do not change B in the opposite direction.

            Letters-to-the-Editor have proposed that our discrepancy might be explained by confounding by population, or by population density -- an expression of the urban-rural differences. However stratification of our data into 10 deciles, or even finer stratifications on the basis of these variables, showed practically no evidence of a change in B-values. In these fine stratifications, the counties in most strata have essentially the same populations or population densities, but still the analysis for these counties with the same population, or with the same population density, gives a large negative value of B.


Negative slopes and  conflict with data from case-control studies

            It is frequently suggested that the negative slopes in our data for m vs r (I.e. m decreases with increasing r) are incredible and are in conflict with the results of the case-control studies. It should be recognized at the outset that case-control studies investigate the causal relationship between radon exposure and lung cancer, whereas our work has the much more limited objective of testing the linear-no threshold theory; if that theory fails as we have concluded, “the ecological fallacy” becomes relevant and our results cannot be directly interpreted as representing the risks to individuals. We have therefore never claimed that Fig. 1 gives risks to individuals, or that low level exposure to radon is protective against lung cancer. Our only conclusion is that LNT fails very badly, grossly over-estimating the cancer risk of low level radiation.

            However, if one insists on interpreting our data as representing the dose-response relationship to individuals, it should be recognized that the negative slopes in our data are entirely based on radon exposures  in the range r=0-3.5 pCi/L (0- 130 Bq/m3), whereas the case-control studies give essentially no statistically meaningful information on the slope in this region. Detailed analysis shows that there is no discrepancy between our data and the case-control studies in that region.


The smoking-radon interaction

            Our Eq. (1), derived to relate lung cancer rate, m, to r and S is

                        m = [ S as + (1-S) an ] (1 + B r )

where a and B are constants. This has given many the impression that we have assumed some special and simple (linear) relationship between smoking and radon exposure in causing lung cancer. Despite the appearance of the above equation, that is not the case, as we now demonstrate.

            BEIR-IV considers smokers and non-smokers as separate “species”, each with its own lung cancer risks. The relationship between radon and smoking in causing lung cancer in an individual can be infinitely complex. In utilizing the BEIR-IV model to mathematically derive the mortality rate for a county, the fraction of the county population that smokes, S, logically arises and the result is the above formula. Note that S is not the intensity of smoking by an individual, but it is simply the fraction of the population that smokes cigarettes,  the fraction of the population that is in that “species”.

            If counties kept separate statistics on cause of death for smokers and non-smokers, S would not be involved. We could do two completely separate and independent studies for smokers and non-smokers. It is only because counties do not keep separate statistics that we must combine these two studies, and this introduces the relative sizes of the two groups which is represented by S.




            Since no other plausible explanation has been found after years of effort by myself and others, I conclude that the most plausible explanation for our discrepancy is that the linear-no threshold theory fails, grossly over-estimating the cancer risk in the low dose, low dose rate region. There are no other data capable of testing the theory in that region.

            An easy answer to the credibility of this conclusion would be for someone to suggest a potential not implausible explanation based on some selected variables. I (or he) will then calculate what values of those variables are required to explain our discrepancy. We can then make a judgement on the plausibility of that explanation. To show that this procedure is not unreasonable, I offer to provide a not-implausible explanation for any finding of any other published ecological study. This alone demonstrates that our work is very different from any other ecological study, and therefore deserves separate consideration.



Caption for Figure


Fig. 1: This is the same as Fig. 1 of Reference 1 which is posted on this web site







1. Cohen,B.L.Test of the linear-no threshold theory of radiation

carcinogenesis for inhaled radon decay products”, Health Physics 68: 157-174; 1995.


2. Cohen, B.L. Relationship between exposure to radon and various types of cancer. Health Phys. 65:529-531; 1993


3. Greenland,S.; Robins,J. Ecologic studies: biases, misconceptions, and counter examples. Am. J. Epidemiol. 139:747-760; 1994


4. Cohen,B.L. Problems in the radon vs lung cancer test of the linear-no threshold theory and a procedure for resolving them. Health Phys. 72:623-628;1997