Test of the Linear-No Threshold Theory:
Rationale for Procedures

Bernard L. Cohen*

*Address: 100 Allen Hall,

e-mail: blc@pitt.edu

Telephone: (412)624-9245

Fax: (412)624-9163

Running head:

Procedures
in Test of the Linear-No Threshold Theory

A tightly reasoned justification is
presented for the procedures used in our test of the linear-no threshold theory
of radiation carcinogenesis by comparing lung cancer rates with average radon
exposure in

value of a
single CF. The problem of confounding factors on the level of individuals is
addressed. The requirements on a CF for affecting the results are quantified in
terms of its correlations with lung cancer rates and radon levels and it is
shown that the existence of an unknown confounder satisfying these requirements
is highly implausible. Effects of combinations of confounding factors are
treated and shown not to be important. Consideration of plausibility of correlations
is used in several other applications, including treatments of uncertainty in
smoking prevalence, within county differences in radon exposure between smokers
and non-smokers, variations in intensity of smoking, differences between
measured radon levels and actual exposures, etc. Examples are presented for all
applications. The differences between our study and case-control studies, and
the advantages of each for testing the linear-no threshold theory, are
discussed. It is shown that the methods used here cannot be used to determine a
dose-response relationship.

Key words: linear-no threshold, radiation carcinogenesis,
confounding, stratification,
plausibility of correlation, dose-response

1.1. Background

In
a 1995 paper (Cohen 1995), data on lung cancer mortality rate, m, in 1601

These
data are used to test the validity of the linear-no threshold theory
(hereafter, LNT) in the low dose region. This test is very different from other
tests of LNT utilizing case-control studies (Lubin and Boice 1997), which are
really designed to determine a risk vs dose relationship for individual
persons. That obviously requires data on individuals, whereas we have only
average data on groups of individuals, the populations of counties. Such data
on groups are called “ecological data”.

As
an example of the difficulty this represents, consider a situation where the
risk has a sharp threshold at 50 units of dose. The average risk in the county
then depends on the fraction of the population exposed to more than 50 units,
which is not necessarily related to the average dose which might be about 5 units.
Clearly, the average dose does __not __determine the average risk, and is
therefore not useful for determining the risk vs dose relationship. To assume
otherwise is called “the ecological fallacy”. However it is readily
demonstrated mathematically that this particular problem does not arise if the
risk is linearly related to the dose. In that special case, the average dose __does__
determine the average risk. This is familiar to Health Physicists from the
widely used paradigm from LNT that person-sieverts (man-rem) determines the
number of deaths; person-sieverts divided by population is the average dose,
and number of deaths divided by population is the average risk.

The procedure for
testing LNT involves two basic steps. The first step is, assuming LNT to be
valid, to transform the risk vs dose relationship for individuals
mathematically into a relationship between ecological variables, and the second
step is to test that relationship against observation. The first step starts
with the BEIR-IV formula (NAS 1988) for risk to an individual, based on LNT,
and develops it mathematically, summing over all persons in the county. This
and subsequent analyses were done separately for males and females, always
leading to similar results, but for brevity here, we confine our attention
(with a single exception) to males. The result of the mathematical development
(Cohen 1995) for males (with m in units of deaths per year/100,000 population,
and r in units of 37 Bq/m^{3} [pCi/L) is

M
= m / [9 + 99 S] = A + B r Eqn.
(1)

where S is the fraction of adult males that smoke cigarettes, A is
close to 1.0, B = +7.3 (in percent increase per 37 Bq/m^{3} [per pCi/L],
and M is defined by the equation on the left and may be thought of as lung
cancer rate corrected for smoking. The data thus corrected for smoking are
shown in Fig. 1b.

Eqn.(1) is a
relationship between ecological variables – m, r, and S – and hence it
accomplishes our first step. Since it is derived mathematically from the LNT
relationship between variables for individuals, if the latter is valid, Eqn.(1)
must be valid and can be used as a test for the validity of LNT. This use of a
mathematically derived formula to verify the theory from which it is derived is
a time honored procedure in science. Newton’s famous theory relating force
acting on an object, its mass, and its acceleration, **F **=** m a**, was not
directly tested for centuries since acceleration could not be directly
measured; it was rather used to mathematically derive the distance traveled by
the object vs time, which was measured to test the theory.

The fact that Eqn.(1),
a relationship between ecological variables – m, r, and S – is being used to
test LNT represents a radical departure from previous tests. This has far
reaching consequences. The principal previous tests have used case-control
studies for individuals, which require extensive information on these
individuals. But in our approach, no such information is required unless it can
be shown that it might affect the relationship between m, r, and S. The difference
this makes will be illustrated through the rest of this paper.

It is apparent from Fig. 1b that there is a huge discrepancy
between the LNT prediction, B = +7.3, and the fit of Eqn.(1) to the observed
data which gives B = -7.3±0.56, a discrepancy of 26 standard deviations. We
refer to this as “our discrepancy”. The Scientific Method requires that, if a
theory makes predictions that are discrepant with observations and if no
plausible explanation can be found for that discrepancy, the theory is invalid.
If LNT is to survive the test, we must therefore find a plausible explanation
for our discrepancy. The principal purpose of this paper is to describe the
search for such an explanation.

Before proceeding,
it is important to understand that Fig. 1 should not be interpreted to be a
dose-response relationship between radon exposure and lung cancer. As explained
above, to do so would be falling into the trap of “the ecological fallacy”.
There are only two logical alternatives to consider: (1) LNT is valid in which
case a plausible explanation must be found for our discrepancy, or (2) LNT is
not valid, in which case we cannot use these data to determine a dose-response
relationship.

1.2. Confounding Factors (CF)

It is not
unexpected that factors other than smoking and radon exposure can affect the
risk for lung cancer. In principle, these should be included in the BEIR-IV
formula for risk to individuals that we start with, carried through the
mathematical development, and end up represented in Eqn. (1). This would be a
completely unmanageable process, but failure to carry it out does not mean that
the problem is unmanageable. Analogous situations arise universally throughout
science. Few, if any, formulas used by scientists are absolutely exact and complete,
not even **F **=** m a**, but it is conventional to develop
them mathematically to yield other formulas that are useful. In an elementary
Physics course and in several Engineering courses,

In seeking an
explanation for our discrepancy, we must investigate the effects of variable
factors that might, in principle, be included in a complete treatment of the
lung cancer vs radon relationship. If one of these variables does indeed
contribute to the lung cancer risk, and if, for some unrelated reason, it is
correlated with radon levels, it would affect the relationship evident in Fig.
1. That variable would then be said to confound the relationship between M and
r, and would be called a confounding factor (CF). As an illustrative
hypothetical example, suppose that ozone levels in the atmosphere irritate the
lungs and thereby cause lung cancer, and suppose that through some unknown
process ozone scavenges radon out of the air, reducing radon levels. Then
counties with high ozone levels would tend to have high lung cancer rates and low
radon levels, and vice versa for counties with low ozone levels. The variations
in ozone levels among

A lengthy list of
potential CFs could be drawn up, and each of these should be investigated
before a judgment can be made on the validity of LNT. It is this process that
we now describe. We begin by considering smoking prevalence to be known so that
M and r have known values for each county, and later, in Section 4, we come
back to consider potential confounding by smoking variables.

2.
STRATIFICATION

2.1 The stratification method

A straightforward
way to check on whether a particular factor, the value of which is known for
each county, is a CF is to consider only counties for which that factor has the
same value, leaving no possibility for it to confound. The practical
manifestation of this procedure is to stratify the complete data file into many
sub-files on the basis of the factor being investigated. As an example for
which direct data are available, we consider population density (PD) which
might affect lung cancer rates through behavioral patterns and medical
services, and might affect radon levels through house construction
characteristics. In Table 1, the results are shown for stratifying our
entire1601 county data file into 10 deciles (sub-files) of 160 counties each on
the basis of PD; the data in each decile are fitted to Eqn. (1) to obtain a
completely independent value of the slope B of the M vs r regression. Table 1
includes data for females as an example of the general similarity of results
for the two sexes, but data for females will be omitted for brevity in all
further discussions. .From the second column of Table 1, we see that in each
stratum (except the last) the values of PD are very similar, much more similar
than in the data for all U.S. counties, so any confounding by PD is greatly
reduced. This is reiterated in Table 1 by including results of a multivariate
regression of M on r and PD, noting that B-values (the coefficients of r in the
regression) from single and multivariate regression are essentially the same.
Note that the values of B are all negative and generally of the same magnitude
as the value for the entire 1601 county data set. The differences between their
average and the values for the entire data set, B = -7.3 for males and B = -8.3
for females, are well within the standard deviation of the averaging process.
More importantly, there is no evident trend for B to increase or decrease
monotonically with increasing PD; there is little difference in B-values if we
consider only counties with the largest PD, or if we consider only counties
with the smallest PD, or if we consider only counties with average PD. These
facts clearly indicate that confounding by PD is of little help in explaining
our discrepancy.

2.2.
Comparison of stratification with multivariate regression

Since stratification is a somewhat
laborious process, one might ask why not simply do a multivariate regression of
M on r and the CF, and accept the coefficient of r in this regression as the
value of B corrected for confounding? One obvious weakness of multivariate regression
is that it assumes the relationship of M to the CF is a linear one, which may
not be true. But here we offer a treatment which demonstrates another weakness.

We begin by recognizing the fact that
the only way a confounding factor, X, can affect the value of B derived from
fitting data with Eqn.(1) is by systematically causing counties with low M to
have high (or low) r, and vice versa. This would be evidenced by the rankings
of counties in our data file by M, R(M), which has a value ranging from 1 to
1601 for each county, and (for our case) the inverse rankings of these counties
by r, R(r), both being highly correlated, for unrelated reasons, with the
rankings of counties by X, R(X). We refer to these correlations by ranking
(also known as Spearman’s ρ) as CoRR(X,M)
and CoRR(X,r) respectively; that is, in a notation where Corr(a,b)
denotes the coefficient of correlation (Pearson product moment) between a and
b,

CoRR(X,M) = Corr[R(X), R(M)]

and
similarly for CoRR(X,r). __Both__ CoRR(X,M) and CoRR(X,r) must be large for
unrelated reasons if X is to be an important confounding factor.

Using this
background, we now address the issue raised at the beginning of this section,
on the use of multivariate regression. Let us assume that there is a confounding
factor, X, that is causally related to M but has __no__ causal or other
direct relationship to r. It thus cannot confound the relationship between M
and r, and therefore should have no effect on the value of B. But it will
necessarily have a correlation with r because of the correlation of M with r
evident in Figure 1.

To study this effect quantitatively, we take

R(X)
= s R(M) + (1-s) R-random

where R-random is a random
rearrangement of rankings and s can be varied between 0 and 1.0 to get various
CoRR(X,M). Some results for CoRR(X,r) and B, defined here as the coefficient of
r in the multivariate regression of M on r and R(X), follow:

CoRR(X,M)
0
0.30 0.44 0.59 0.73

CoRR(X,r) 0
-0.14 -0.18 -0.23 -0.28

B -7.3 -6.7
-6.0 -4.9 -3.7

We
see that, even though X is not a confounding factor in the relationship between
M and r because it has no causal or other direct relationship with r, its use
in multivariate regression still has an appreciable effect on the resulting
value of B. This demonstrates why the method of stratification is preferable to
multivariate regression in assessing the effects of confounding factors; since with stratification, the value of
the CF is essentially the same for all data in each regression, it cannot
affect that regression.

The
demonstration here does not mean that multivariate regression is a useless
tool. But it indicates that it should be used with some caution.

2.3. Problem: Average value of a CF may not
represent its confounding effects

The
stratification procedure may not eliminate effects of a confounding
relationship because the average value of a CF does not necessarily represent
its confounding effects. For example, average annual income may not represent
the confounding effects of monetary income because its confounding effects may
depend on the fraction of the population that is very poor, or very rich. To
cover this problem, we consider separately as CF the fraction of the population
with income <$5000, $5000 to $10,000, ….., >$150,000 (10 brackets in
all), plus various combinations of adjacent brackets (Cohen 2000a). Since none
of these has a confounding effect, we may conclude that any aspect of annual
income, not just __average__ annual income, can be excluded as an important
CF.

As
a related but different type example, a person’s age is an explicit factor in
the BEIR-IV formula for risk vs dose to an individual, and it is carried
through the mathematical process of deriving Eqn.(1) by showing that variations in age distributions do not
have appreciable effects on that equation (Appendix A of Cohen 1995). But as an
extended treatment of that issue, we consider as CF the fraction of the
population in age groups <1 year, 1-2 years, …..,>85 years (31 groups in
all) plus various combinations of adjacent brackets. Since none of these has an
important confounding effect, we may conclude that age distribution can be
excluded as a CF explaining our discrepancy.

Perhaps
our conclusions here, that annual income and age distribution are not plausible
confounders, is not justified. To consider this possibility, we need a
suggestion for specific plausible dependencies on these for individuals that
would not be reflected in the ecological CF specified above. Despite frequent efforts
to devise or solicit such a suggestion, none has materialized.

2.4. Stratification on
Geography and Environmental Factors

The very wide differences in average radon levels in
counties, evident in Fig. 1, indicates that geography is an important factor in
determining radon levels. Since different geographic regions have many other
different characteristics that may affect lung cancer rates - climate or
ethnicity of the population are plausible examples – geography is a potentially
important CF. This can be investigated with a method akin to stratification by
dividing the nation into sections and doing a separate analysis for each
section.

This was done using Bureau of Census Regions and Divisions,
and using individual states (Cohen 2000a). If we take the average value of B to
represent a corrected true value, the results are as follows: for the 4
national regions, B = -5.2; for the 8 national divisions, B = - 4.1; for the 33
individual states plus 4 combinations of contiguous states (combined so as to
get at least 20 counties in each data file), B = -5.0. This may be interpreted
as indicating that confounding by geography changes the value of B from –7.3 to
about -5.0, but this is still a long way from explaining our discrepancy with
the LNT value, B = +7.3.

Stratification on environmental factors like altitude,
temperature, precipitation, etc was treated in Section K of (Cohen 1995) and
found not to affect the results. The behavior shown in Fig. 1 is found if we
consider only the warmest areas or if we consider only the coolest areas, if we
consider only the wettest or only the driest, etc.

2.5. Screening candidates for
stratification studies

Well over 100
potential CF have been treated by the
stratification method with no progress in resolving our discrepancy. But
stratification is a tedious process and the number of potential confounding
factors is very large. A more rapid screening procedure is desirable. Moreover,
there are numerous potential CF for which no data are available – ozone levels
discussed in Section 1.2 above is an example -
but they cannot simply be ignored. The solution to these problems is
treated in Section 3 below.

3.
OTHER CONFOUNDING FACTOR
ISSUES

3.1. Plausibility requirements on
confounding factors

For subsequent analyses,
it is important to understand quantitatively what is required for a confounding
factor to influence our results. As pointed out in Section 2.2, the only
situation in which a confounding factor X can affect the value of B derived
from fitting data with Eqn (1) is where both CoRR(X,M) and CoRR(X,r) are large
for definite but unrelated reasons. The most effective R(X), was found to be

R_{0}(X)
= Ranking of {0.5 R(M) + 0.5 R(r)}

which, applied to our data file, gives CoRR(X,r) = CoRR(X,M) =
0.82. Lesser correlations can be generated by taking

R(X) =
Ranking of {p R_{0}(X) + (1 - p)
R-random}

where R-random is a random
rearrangement of the rankings, that is of integers between 1 and 1601, and p is
varied between 0 and 1.0 to obtain varying CoRR(X,M)
and CoRR(X,r). For each value of p we utilize stratification on R(X) into
quintiles, fitting the data in each stratum to Eqn (1) to obtain a B-value for
that stratum, and averaging the B-values from the five strata to obtain a
B-value for the entire data set. The results for three different sets of
R-random (generated by the MINITAB statistical package) are listed in Table 2. The
B-values in parentheses there are the coefficients of r from multivariate
regression of M on r and R(X). Note that the B-values obtained in these two
very different ways are in good agreement. In view of our definition of R_{0}(X), CoRR(X,M) and CoRR(X,r) should be nearly the same, but since
these depend somewhat on R-random, both are listed in Table 2.

We see from Table 2 that the different sets of R-random give
consistent results and indicate that for a CF, X, to shift the value of B from
its original value, B = -7.3, to the LNT prediction B = +7.3, requires
CoRR(X,r) __and__ CoRR(X,M) (or, according to further calculations, their
average) to be about 0.75, and even to change the sign of B from - to + ,
accounting for half of our discrepancy, requires these correlations to be about
0.6.

The bottom rows of
Table 2 show that a confounding factor can indeed drastically change the
results of the study, as Lubin (1998) has demonstrated mathematically. But
there is an unstated corollary to Lubin’s mathematical demonstration – the
required values of the CF must be plausible. The issue of plausibility must be
addressed.

How plausible are
the values of a CF leading to the correlations required here? The factors
affecting radon exposure, r, are geology and house construction details, while
the factors affecting M, lung cancer rates corrected for smoking, are human
behavioral and genetic characteristics, so it is very difficult to imagine a
CF, other than geography which was treated in Sec. 2.4, that has a causal
relationship with both of these very different type characteristics. It would
seem that the most likely source of confounding is through socioeconomic
variables, SEV. Our data base includes 530 SEV (Cohen 2000a). For these, the
maximum (in absolute value) CoRR(SEV,r) is 0.486; for only 13 of these 530 SEV
is it >0.4, and for only 49 of them is it
>0.3. The maximum CoRR(SEV,M). is 0.39 and for only 13 of the 530 SEV
is it >0.3. Calculations indicate that the relevant quantity is the average
of the absolute values of CoRR(SEV,r) and CoRR(SEV,M), which we call
“Aver-CoRR” (in all cases, the signs of the two are opposite). The maximum
Aver-CoRR is 0.43, only one other is >0.4, only 11 are >0.35, and only 25
of the 530 SEV have Aver-CoRR >0.3. It thus seems implausible for any CF to
have Aver-CoRR larger than 0.5, and it is surely very highly implausible for
Aver-CoRR to approach 0.75, which is required to explain our discrepancy with
the LNT prediction, or even 0.6 which is required to substantially reduce that
discrepancy.

It should be noted
in passing that all of the strong correlations cited above can be explained as
arising from the urban-rural effect -- urban people smoke more and have lower
radon exposures than rural people. The effects of this on B-values were studied
by stratification on county population, percent urban, etc in Section H of
(Cohen 1995) and no effects on B-values were found. The urban-rural effect
within counties was treated in Section L of (Cohen 1995) and found not to
affect the value of B.

The concept of
plausibility of correlation introduced above is a very powerful one, available
in this study because there are data on such a large number of potential CF,
enough to draw meaningful conclusions about the distribution of their
correlations. For example, it covers ozone as a CF in the hypothetical
situation introduced in Sec 1.2. Ozone level in the atmosphere is related to
urban vs rural factors, importance of manufacturing, prevalence of motor
vehicles and highways, and other variables for which data are available and
included in our above analysis. We may thus conclude that ozone level is not an
important CF even though data are not available on ozone levels in each county.
Similarly, we may conclude that any factor which is related to socioeconomics
may be excluded as a CF that might explain our discrepancy.

3.2. Confounding factors on the level if individuals

There are potential CF on the level of
individuals that might seem not to be represented by ecological variables, as
required in our procedures. As a “far-out” example, one might think this
applies if a man’s lung cancer risk depends on Y = [the product of his annual
income squared, and the number of siblings that he has, raised to the fourth
power]. But, in principle, counties
could keep statistics on the values of Y in its population, and report
averages, Y_{av. . }These
would then be an ecological socioeconomic variable, and it would be reasonable
to expect CoRR(Y_{av},r)
to be in the same range as other CoRR(SEV,r), which would not affect our
results.

There is a substantial literature
pointing out that CF on the level of individuals cannot be adequately
represented in a case-control study by ecological variables (Greenland and Robins 1994, Morgenstern 1995, Stidley and
Samet 1994, Lubin 1998). But before this can be interpreted as invalidating our
study, it must be shown that such a CF can affect the relationship between m,
r, and S, which is the basis for our test of LNT through Eqn. (1). In trying to
go through this process, I have found it to be inevitable that the effect under
consideration can be represented by ecological variables; the effects
considered below in Sec. 4.3, 4.4, and 4.5 are examples, as is the treatment of
Y in the previous paragraph. I would be anxious to address any suggestions for
effects in which this treatment fails.

3.3. Effects of Combinations of
Confounding Factors

Up
to this point we have been considering CFs one at a time. Since any one may
cause small changes in the value of B, is it possible that these small changes
can accumulate and thereby explain our discrepancy? We address that issue here.

The only way in which confounding
factors can affect the results is if the rankings of counties by these factors
is highly correlated for unrelated reasons with R(M) and R(r). But from the
treatment of CF by stratification, it is clear that only one set of such
rankings can enter into the determination of B through its correlations with
R(M) and R(r). This could be an
equivalent set, R(E), based on all relevant CFs. It
is important to recall here that CoRR(X,r) refers to the __inverse__ ranking
of X vs r; the correlations considered in Table 2 require that X be correlated
with M and r in opposite directions. There can just as easily be confounding
factors X that are correlated in the same direction with M and r in which case
the effect is to make the B-value more negative, increasing the discrepancy
with the LNT prediction. For example, with CoRR(X,r) = CoRR(X,M) = 0.75 with correlations
in opposite directions, analysis gives B = +7.7, but for these correlations in
the same direction, the result is B=-16.2 which is a much larger discrepancy
with LNT than is found without confounding, B=-7.3. Thus R(E) is just as likely
to increase our discrepancy as to reduce it.

But improbable as it is, let us
consider the worst case, where all effective CF have correlations with r and M
in opposite directions. If we define R(1), R(2), R(3), …... as the rankings of
the most important CF, the second most important CF, the third most important
CF, ….., a first approximation to R(E) is R(1). Any improvement to R(E) by
making changes to include R(2), R(3), etc will decrease the effectiveness of
R(1), and hence will tend not to be a major improvement. Thus the effect of
combinations of confounding factors would not be much greater than the effect
of the single most important confounding factor.

As
an illustration of this conclusion, the confounding effects of the 530 SEV in
our data base were investigated by multivariate regression of M on r and groups
of SEV. If we use multivariate regression of M on r and the single SEV with the
largest Aver-CoRR, the slope B is changed from its original value, B = -7.3, to B=-4.8. If we expand the
multivariate regression to include 12 variables, r plus the 11 SEV with the
largest Aver-CoRR, the result is B=-4.4, only a slight change from the effect of
the single most important CF.

Since
the combination of all CFs cannot have much more effect than that of the single
most important CF, and we have shown that it is very highly implausible for a
single CF to explain our discrepancy, we conclude that it is also very highly
implausible for a combination of CF to affect the results.

Accepting the result that confounding may change B from –7.3 to –4.4 reduces our discrepancy with the LNT prediction, B=+7.3, by only 20%, not an appreciable improvement. Moreover, it was shown in Section I of (Cohen 1995), using an approach similar to that in Section 2.2, that use of multivariate regression substantially over-estimates the effects of confounding.

4. CONFOUNDING BY SMOKING-RELATED
VARIABLES

4.1. Relative importance of smoking and radon exposure

One might think that smoking is such
a dominant cause of lung cancer that its effects can easily mask the effects of
radon. To address this, we estimate the relative importance of these two
factors in determining lung cancer rates by use of BEIR-IV. The width of the
distribution of S-values for

But 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 radon levels, r. Thus we are facing a straightforward quantitative question: How strong an S-r correlation is needed to affect our results? That question is addressed in the next section.

4.2 Uncertainties in smoking prevalence, S

Smoking prevalence,
S, has a very special place in our analysis due to its explicit inclusion in
Eqn. (1). Since S is involved in the
equation to be fitted, the distribution of S-values, not just its ranking for
various counties, affects results for B. Three very different sources of data
were used to determine S-values for counties – (1) a Bureau of Census survey
for States with an adjustment for urban vs rural differences among the counties
in each state, (2) state cigarette sales tax collections with a similar
adjustment, and (3) lung cancer rates for counties with similar radon levels.
Each of these gives essentially the same results. Nevertheless, the uncertainty
in S-values was still a matter of some concern which was addressed by studying
the correlations between S and r required to explain our discrepancy.

As an initial
approach, values of the best estimated S-values are maintained but these
S-values are reassigned to counties so as to give CoRR(S.r) = -1.0; that is,
the county the lowest r was assigned the highest S, the county with the next
lowest r was assigned the next highest S, and so forth thru our 1601 counties,
ending with the county with the highest r assigned the lowest S. Even with this
perfect inverse correlation, which completely violates any considerations of
plausibility, B is only reduced from its original value, -7.3, to zero, still leaving half of our
discrepancy with the LNT prediction, B = +7.3, unexplained.

Going still
further, the effects are increased if the distribution of S-values is wider.
The maximum not implausible width for the distribution of S-values is the width
of the lung cancer mortality rate (m) distribution, since other factors
influence m in ways that, statistically, would increase that width. With this
increased S-distribution width, centered on the well established national
average for S, S-values are reassigned to each county to give CoRR(S.r) = -1.0;
we call this S-perfect. At the other extreme, these same S-values are randomly
assigned to each county to obtain S-random. Calculations are then done with

S = q S-perfect + (1-q) S-random (2)

where q is various numbers between 0 and 1.0 chosen to obtain
various CoRR(S,r) and coefficients of correlation with r (__not__
correlations by rank), Corr(S,r). The results for three different sets of
S-random are shown in Table 3. We see there that the Corr(S,r) required to
change B to the LNT prediction, B = +7.3, is about 0.9, and just to reduce B
down to zero, eliminating half of the discrepancy, is about 0.62, even with
this substantially increased width of the S-distribution.

How
plausible are these required Corr(S,r)? The most probable source of a
correlation between S and r is through socioeconomic variables, SEV. It
therefore seems reasonable to assume that Corr(S,r) should be in the same range
as Corr(SEV,r) for other SEVs. In our data base of 530 potential confounding
SEV, the largest Corr(SEV,r) is 0.45, only 7 of the 530 are >0.4, and only
15 are >0.35. It thus seems reasonable to assume that values of CoRR(S,r)
larger than 0.5 are implausible. It is surely highly implausible for Corr(S,r)
to approach the values, 0.62 - 0.90, required to help explain our discrepancy,
even if we accept the almost implausibly
increased width of the distribution of S-values which ignores our three
independent sources of data.

4.3. Different r for Smokers and Non-smokers

Another type
problem arises if there is a systematic difference in average radon exposures
for smokers, r_{s,} and non-smokers, r_{n} (Cohen 1998a). Since
smokers are 12 times more at relative risk from radon than non-smokers (NAS
1988), the effective radon level, r_{e}, for the county as a whole for
causing lung cancer is

r_{e }=
[12 S r_{s} + (1 - S) r_{n} ] / [12 S + (1 - S)]

where the two terms in the numerator are the weightings for radon
exposure to smokers and non-smokers, and the denominator is the sum of these
weightings. This differs from the measured average radon level, r,

r = S r_{s}
+ (1-S) r_{n}

If we define x = r_{s }/ r_{n}, the relationship
between the effective and measured radon levels is converted by algebra to

r_{e}
= r (12 S x + 1 - S) / [(x S + 1 - S) (11 S + 1)]

We then use r_{e} instead of r in fitting the data to
determine values of B. In doing this, the parameters that may be varied are the
average value of x (x-average), the width of the distribution of x-values, and
Corr(x,r).

It has been found
(Cohen 1991) that the national average for x is 0.9, but we give some results
for other values of x-average. For the 52 of the 54 SEV considered in (Cohen
1995) that are not proportional to the county population, the average width of
distributions is 26 % of their mean, and for only one of the 52 is it above 50%
-- 55% for “percent of income from government” which is an understandable
special case. On this basis, we consider distributions of x-values to have
width 57% of the mean, which severely stretches the limits of plausibility, and
28% of the mean which is in the region of reasonable plausibility.

Some results are listed in Table 4. The first
five entries explore the effect of x-average using assumptions about the other
factors most favorable for explaining the discrepancy. The remaining entries
use the known value of x-average and explore the effects of the width of the
distribution and of Corr(x,r). The results in Table 4 indicate that it is
highly implausible for systematic differences between radon exposures to
smokers and non-smokers to change B from –7.3 to less than about -5.5, still a
very long way from the LNT prediction, B = +7.3.

4.4. Variations in Intensity of Smoking

The BEIR-IV formula
for cancer risk to an individual, which was the starting point for our test of
LNT, considers only the distinction between smokers and non-smokers, with no
consideration of intensity of smoking. Therefore, that factor is not represented
in Eqn. (1) which is derived from the BEIR-IV formula. But the BEIR-VI Report^{
}(NAS 1999) suggested that Eqn. (1) is deficient in that it ignores
intensity of smoking, and proposes that this be treated by dividing smokers
into two categories, 2 pack/day and 1 pack/day. To study this (Cohen 2000b), we
define

k = ratio
of 2 pack/day to 1 pack/day smokers in a county

f = ratio
of lung cancer risk for 2 pack/day to 1 pack/day

Analysis of available data indicates the plausible values most
favorable for the BEIR-VI suggestion are f = 2.0 and national average for k =
0.4. Using these converts Eqn. (1) to

M = m / [9 - 9S +
84 S {(1 + 2 k)/((1 + k)} ] = ( A + B r
) (3).

Different distributions of k-values were tried but the most
promising was a level distribution between 0 and 0.8, to be consistent with the
national average of 0.4. We assign k-values to counties so as to define k-perfect as assignments for which CoRR(k.r)
= 1.0, and k-random as one where k-values are assigned randomly. We then
generate k-values to be used in fitting Eq. (3) as

k = g k-perfect + (1 - g) k-random

where g is given various
values between 0 and 1.0 to obtain different Corr(k,r). The results are

Corr(k,r) 0 -0.37 -0.52 -0.78 -0.91 -0.93

B -7.6 -5.0 -4.2 -2.5 -0.7 +1.4

In view of our previous discussion on plausibility of
correlations, it seems reasonable to assume that an absolute value of Corr(k,r)
> 0.5 is highly implausible. It is therefore clear that including intensity
of smoking as a confounder can do little to reduce our discrepancy with the LNT
value, B=+7.3. Of course, there is no reason why CoRR(k,r) should not be
negative rather than positive, in
which case the discrepancy with LNT would be increased, so the above table
gives an unbalanced view, emphasizing things that may reduce the discrepancy
with LNT.

(Cohen 2000b) also
considers possible correlations with r for both S and k, using the above
method. For cases where Corr(k,r) = Corr(S,r), as these vary from zero to -0.8,
B increases roughly linearly from -10.0 to +1.3; for example, for Corr(k,r) =
Corr(S,r) =0.4, B=-4.3. Again it is apparent that plausible values of these
correlations can do little to bring B close to the LNT prediction, B =
+7.3.

4.5. Combinations of confounding by smoking and other factors

To go further, we
consider the effects of uncertainties in our S-values combined with an unknown
confounding factor, X. Starting with our best estimates of S-values in a pool,
we reassign S-values from this pool utilizing Eqn (2) to generate sets of
S-values with various Corr(S,r). For each of these sets of S-values, we
determine corresponding sets of M-values, utilizing the left side of Eqn (1).
For each of these sets of M-values, we go through the analysis described at the
beginning of Sec. 3.1, to obtain sets of R(X) with various CoRR(X,R); we then
determine values of B as the coefficient of r in multivariate regression of M
on r and R(X). This gives tables of B-values for various combinations of
Corr(S,r) and CoRR(X,r). By interpolating from these tables, we derive Table 5
which shows values of B obtained for various combinations of C0RR(S,r) and
CoRR(X,r).

Applying our
plausibility limit of 0.5 to both CoRR(S,r) and CoRR(X,r) simultaneously, which
is far less plausible than applying it to only one of the two, and
interpolating in Table 5, we obtain B = -0.2. This is still strongly discrepant
with the LNT prediction, B = +7.3.

5.1 Urban vs Rural Differences

In extensive
studies^{ }(Cohen 1991) of how radon levels vary with socioeconomic
factors, house characteristics, geography, etc), it was found that rural houses
average about 25% higher radon levels than urban houses, whereas urban males
smoke about 25% more frequently than rural males. This problem was treated in Section
L of (Cohen 1995) using a model with the above percentages as parameters, by modifying
the derivation of Eqn.(1) to consider not just the two categories , smokers and
non-smokers, but four categories, urban and rural smokers and urban and rural
non-smokers, each category having its own percentage of the population, lung
cancer rate, and average radon level. These are related by the percent of the
population that lives in urban areas, a known quantity for each county, and m,
r, and S for the county. It was found that the changes in B caused by various
plausible values of the parameters was only a few percent.

5.2. Differences between Radon Exposure and Measured Radon Levels

There have been
suggestions that effective radon exposure, r-effective, may not be the same as
the measured radon level in the home, r-measured; for example, time spent in
the home may vary, or exposures outside the home may be important. We represent
this (Cohen 1998b) as

r-effective
= (1+f) r-measured

The properties of f that can affect our results are the width of
the distribution of f-values among the counties, and the correlation between f
and r, Corr(f,r). We take the distribution to be uniform between –w and +w. To
test the maximal effects of correlations between f and r, values of f can be
assigned to counties such that CoRR(f,r) = 1.0, which is equivalent to
Corr(f,r)=1.0. To check on the effects of no correlation, values of f can be
assigned randomly.

The values of the slope, B, from regression of
M on r-effective, are shown in Table 6, along with standard deviations in the
B-values derived from the regression analysis. We see from Table 6 that no
values of w or Corr(f,r) can help substantially in explaining our discrepancy
with LNT predictions. In fact, when these factors reduce the negative value of
B, they also reduce the standard deviation, so the number of standard
deviations by which B differs from the LNT prediction, B=+7.3, is not reduced

6. Comparison with
case-control studies

Since case-control
study practitioners usually deal with risk to individuals, their studies
require data on CFs for individual persons, obtained by questioning each
involved individual or a close relative or acquaintance. This information is of
key importance in their studies. For example, if annual income is an important
element, the annual income of each individual must be connected to his cancer
or lack of cancer. They may sometimes use ecological data for crude estimates;
in the above example, they might assume the annual income of each individual to
be the average income in his section of the city. But they clearly recognize
this to be an inferior procedure, and label studies that depend on such
procedures as “qualitative”, useful only for suggesting “analytical” studies
that avoid them. Non-epidemiologists have frequently used ecological data
unjustifiably to imply risk vs dose relationships for individuals. It is easy
to show how any paper depending on ecological data for such a purpose can give
false results. It is certainly easy to understand that epidemiologists are
instinctively “turned off” by use of ecological data.

However, our test
of LNT is based on Eqn.(1) which is a relationship between ecological
variables. Analyses are therefore straightforward for ecological CFs.
Fortunately, every potential confounding relationship that seems plausible to
me, or that has been suggested as being plausible by others, can be represented
by ecological variables.

As an example of
the difference between our approach and that of a case-control study, consider
a hypothetical situation in which people of a certain ethnicity, call it
Ethnicity-A, may have a high risk for lung cancer. In our approach, the relevant
variable for determining the county lung cancer rate is the fraction of the
population of Ethnicity-A, an ecological variable; there is no need to know which
individuals in the county are of that ethnicity. If we were trying to find out whether
people of ethnicity-A have an excess cancer risk, it would not be sufficient to
find high lung cancer rates in counties with large fractions of their citizenry
of ethnicity-A. We would have to know whether it is the people of ethnicity-A
in those counties who had excess lung cancer. But in our study, we are simply
testing the consequences of the hypothesis that people of ethnicity-A might
have an excess cancer risk, which might cause counties with large populations
of ethnicity-A to have high lung cancer rates, which could affect our results
(if people of ethnicity-A have systematically low radon exposure). The fact
that we find our results unaffected by fraction of the citizenry of ethnicity-A
does not disprove the hypothesis that people of ethnicity-A may have high
cancer risk, but that is irrelevant to our purpose which is to find CFs that __do__
affect our results.

At least two papers
(Lagarde and Pershagen 1999, Darby, Deo, and Doll 2001) have pointed out cases
where the relationship between lung cancer and radon exposure derived from a
study of individuals gives results different from an ecological study based on
the same data. But these ecological studies involved __no__ treatment of
confounding factors, and the difference between results from these and from the
individual level studies is easily explained by recognizable confounding
factors. That is certainly not the case in our study which involves very
extensive consideration of possible confounding factors.

It is frequently
implied that our study is inferior to case-control studies for testing LNT.
Aside from the fact that case-control studies do not have the statistical power
to test LNT in the low dose region, this ignores the inherent weaknesses in
treatments of CF in case-control studies. An individual’s risk of lung cancer depends on a multitude
of factors on a molecular, cellular, intercellular, hormonal, etc. level that
are not understood, not readily

measurable, and therefore not considered in these studies. There
are also a large number of potential CFs that could be, but are not included
because of time, cost, or other practical limitations. In practice,
case-control studies treat only a very few CFs, often using multivariate
regression which is a process of limited validity, and frequently depending on
marginal statistics.

Our study has many
important advantages over these case-control studies. It treats a far wider
diversity of CF, and even includes a strong argument that an unidentified CF
cannot be important – no such arguments are available in case-control studies
which can easily be rendered invalid by an unrecognized CF. Our study largely
avoids use of multivariate regression with its inherent weaknesses pointed out
in Sec. 2.2 above and elsewhere (Kleinbaum, Kupper, and Muller 1988). It
includes a method for treating cases where no data are available on a required
variable, by use of “plausibility of correlation”. It includes a wide variety of geographic
areas and population characteristics, whereas case-control studies are normally
confined to a single, or at most a few local areas. Statistical uncertainties,
one of the greatest limitations in many case-control studies, are virtually
eliminated.

It should be
understood that the success in treating confounding factors reported here is
due to a combination of fortunate circumstances not present in the great
majority of studies. The very large number of data points, 1601 counties, with
good quality data for each on hundreds of different variables, is highly
unusual. But perhaps more important is the fact that radon levels in homes are
very weakly correlated with most other variables like climate, socioeconomics,
ethnicity, etc that might affect lung cancer rates.

One might question
the fact that our study leans heavily on plausibility arguments, especially
plausibility of correlation. But case-control studies choose the few CFs they
investigate and the control groups they adopt based solely on subjective judgments
of plausibility. The CF analyzed in our study include all of the many hundreds
that are available.

With all these
advantages, the problem in accepting our study is difficult to understand
unless someone can suggest a plausible specific CF that could possibly explain
our discrepancy. It must be specific in order to address the issue of
plausibility, but it is not necessary to show that it does explain our
discrepancy, only that it might possibly explain it. If such a suggestion is
forthcoming, I would be eager to address it. I have tried very hard to solicit
such a suggestion but have had marginal success. Therefore, until such a CF is
suggested, it seems reasonable to conclude that LNT fails our experimental test
and must therefore be invalid in the low dose region covered by Fig. 1.

__REFERENCES__

Cohen, B. L. Variation of radon
levels in __ Health Phys__.60: 631-642;1991

Cohen, B. L.Test of the
linear-no threshold theory of radiation carcinogenesis for inhaled radon decay
products, __Health Phys__. 68:157-174;1995

Cohen, B. L. Response to
Lubin’s proposed explanations of our discrepancy. __Health Phys__.75:
18-22;1998a

Cohen, B.L. Response to criticisms of Smith et al. Health Phys.
75:23-28;1998b

Cohen, B. L. Updates and extensions to tests of the linear-no
threshold theory, __Technology__ 7:657-672;2000a

Cohen, B. L. Testing a
BEIR-VI suggestion for explaining the lung cancer vs radon relationship for
U.S. Counties.__ Health Phys.__ 78:522-527;2000b

Darby,
S., Deo, H., and Doll, R. A parallel
analysis of individual and ecological data on residential radon and lung cancer
in south-west __J. R. Statist. Soc.__ A164, Part 1, 193-203;2001

__Am.
J. Epidemiol.__139:747-760;1994

Kleinbaum, D.G., Kupper,L.L., and Muller,K.E. “Applied Regression
Analysis and Other Multivariable Methods”. PWS-Kent Publishing Co. (

Lagarde, F and Pershagen, G. Parallel analyses of individual and
ecologic data on residential radon, cofactors, and lung cancer in __Am J Epidemiol __149:268-274;1999

Lubin, J. and Boice, J. Lung cancer risk from residential radon;
meta-analysis of eight epidemiologic studies. J Nat’l Cancer Inst 89:49-57;1997

Lubin, J. H. On the discrepancy between epidemiologic studies in
individuals of lung cancer and residential radon and Cohen’s ecologic
regression. __Health Phys__. 75:4-10;1998

Morgenstern, H. Ecologic studies in epidemiology: concepts,
principles, and methods. __Annual Rev. Public Health__ 16:61-81;1995

NAS (__ Health risks of radon and other
internally deposited alpha emitters (BEIR-IV)__.

NAS (__Health effects of exposure
to radon (BEIR-VI), __

Stidley, C. A. and Samet, J. M. Assessment of ecologic regression
in the study of lung cancer and indoor radon.
__Am. J. Radiol.__ 139:312-322;1994

Table 1: Treatment of “County Population Density” (PD) as a
confounding factor by the stratification method. Results are for B using single
regression of M on r as in Eq. (!), and
multivariate (double) regression of M on r and PD, fitting the data to

M = A
+ B r + E PD

where E is a fitting parameter. Bottom line gives the averages of
the columns above and the standard deviation of that average.

County Rank PD range Single Regression Double Regression

__ by PD
__ __(x100/sq.mi)__ __B-male__
__B-female__ __B-male__ __ B-female__

1 - 160 0.003-0.094 -3.7 -6.6 -3.7 -6.4

161- 320 0.095-0.22 -8.0 -7.8 -8.0 -7.9

321- 480 0.22-0.35 -7.0 -8.5 -7.0 -8.5

481- 640 0.35-0.50 -6.4 -9.7 -6.4 -9.8

641- 800 0.50-0.67 -8.9 -8.7 -8.9 -8.7

801- 960 0.67-0.92 -4.3 -4.4 -4.3 -4.4

961-1120 0.93-1.29 -9.2 -6.0 -9.3 -6.0

1121-1280
1.30-2.05 -5.9 -8.1 -5.9 -8.1

1281-1440 2.05-4.11 -0.5 -2.7 -0.5 -2.8

1441-1601
4.12-671.8 -4.5 -7.4 -3.9 -6.2

_____________________ _____ _____
_____ _____

Average ± Std.Dev. -5.8±2.7
-7.0±2.1 -5.8±2.7
-6.9±2.1

Table 2: B values obtained if a confounding factor, X, has various
correlations by ranking with M and r, CoRR(X,M) and CoRR(X,r). The three sets
of results are for three different R-random. The first B-value is from stratification into
quintiles, and value in parenthesis is the coefficient of r in a multivariate
regression of M on r and R(X).

CoRR CoRR CoRR CoRR CoRR CoRR

__ (X.r)__ __(X,M)__ __ B
__ __(X,r)__ __(X,M) __ ** **__ B
__ __(X,r__) (__X,M)__ __ B
__

0.09 0.09
-7.2(-7.2) 0.07 0.12
-7.2(-7.2) 0.08 0.08
-7.2(-7.2)

0.18 0.18
-7.0(-6.9) 0.16 0.21
-6.9(-6.8) 0.16 0.17
-7.0(-6.9)

0.34 0.34
-5.2(-5.5) 0.32 0.37
-5.3(-5.4) 0.33 0.33
-5.1(-5.6)

0.53 0.53
-2.1(-2.2) 0.51 0.55
-2.3(-2.0) 0.52 0.52
-2.1(-2.3)

0.69 0.69
+3.2(4.0) 0.68 0.70
+2.9(+4.0) 0.69 0.69
+3.3(+3.9)

0.79 0.79
10.5(11.3) 0.78 0.79
10.4(11.1) 0.79 0.79
10.8(11.2)

0 81 0.81 13.7(14.4) 0.81 0.82 13.7(14.2) 0.81 0.81 13.8(14.3)

Table 3: B-values obtained if smoking prevalence, S, has various
Corr(S,r), assuming the maximum plausible width for the S-distribution. The
three sets of results are for three different S-random.

__Corr(S,r)__
__CoRR(S,r)__ __ B __ __Corr(S,r)__ __CoRR(S,r)__ __ B __ __Corr(S,r)__ __CoRR(S.r)__ __B __

-0.17 -0.17 -7.1 -0.24 -0.23 -5.5 -0.23 -0.22 -6.0

-0.33 -0.32 -4.7 -0.39 -0.37 -3.2 -0.38 -0.36 -3.7

-0.41 -0.39 -3.5 -0.47 -0.45 -2.0 -0.45 -0.43 -2.6

-0.49 -0.47 -2.3 -0.54 -0.52 -0.9 -0.53 -0.51 -1.4

-0.57 -0.55 -1.1 -0.62 -0.60 +0.3 -0.60 -0.58 -0.2

-0.65 -0.63 +0.1 -0.68 -0.66 +1.4 -0.68 -0.66
+0.9

-0.78 -0.75 +2.7 -0.81 -0.79 +3.8 -0.80 -0.78 +3.3

-0.88 -0.85 +5.5 -0.89 -0.86 +6.4 -0.88 -0.85 +6.0

-0.93 -0.90
+8.6 -0.93 -0.90 +9.3 -0.93 -0.90 +8.9

Table 4: Effects of difference in radon exposure for smokers and
non-smokers, with x = smoker/non-smoker exposures in each county. Table gives
value of B for various choices of the distribution of x-values and Corr(x,r).

__x-average__ __SD(mean) of x__ __Corr(x,r)__ __ B __

0.8 0.57 1.0 -4.9

0.9 0.57 1.0 -4.8

1.0 0.57 1.0 -4.7

1.2 0.57 1.0 -4.5

1.5 0.57 1.0 -4.3

0.9 0.57 0 -6.5

0.9 0.57 0.4 -5.9

0.9 0.57 0.7 -5.5

0.9 0.57 1.0 -4.8

0.9 0.28 0 -7.3

0.9 0.28 0.4 -6.7

0.9 0.28 1.0 -5.6

Table 5: B-values from combined effects of
various CoRR(S,r) and CoRR(X,r)

CoRR(X,r)
__________________CoRR(S,r__)___________________________

____________ __-0.69__ __-0.53__ __-0.37__
__-0.23__ __0.00__

-0.65 6.5 5.5 4.4 3.3 1.5

-0.60 4.4 3.3
2.0 1.0
-1.2

-0.55 2.4 1.1 0.2 -1.0 -2.9

-0.50
1.0 0.0
-1.0 -1.7 -4.5

-0.45
-0.4 -1.6
-2.5 -3.6 -5.5

-0.40
-1.4 -2.4
-3.5 -4.5 -6.8

-0.35 -2.2 -3.2 -4.3
-5.4 -7.6

-0.30
-3.0 -4.0
-5.1 -6.1 -8.3

-0.25
-3.5 -4.5
-5.6 -6.6 -8.7

-0.20 -4.0 -5.1 -6.2 -7.0 -9.2

0.0 -4.8 -5.9 -6.9 -8.0 -10.0

Table 6: Slopes B from regression of M on r-effective for various
values of w and Corr(f,r). The last column is the standard deviation in the
determination of B.

__ w __ __Corr(f,r)__ __ B __ __ SD(B)
__

0 -7.3 0.56

0.2 0 -7.0 0.53

0.5 0 -5.5 0.47

0.8 0 -4.0 0.41

__ 1.0 0 -3.2 0.36__

0.1 +1.0 -6.4 0.48

0.3 +1.0 -5.2 0.39

0.5 +1.0 -4.3 0.33

__ 0.7 +1.0 -3.7 0.28__

0.1 -1.0 -8.5 0.62

0.3 -1.0 -12 0.9

0.5 -1.0 -21 1.4

__CAPTION FOR FIGURE__

Fig.
1: Lung cancer mortality rates before (Fig. 1a) and after (Fig. 1b) correction
for smoking prevalence vs average radon levels in homes, for 1601