Epidemiologic studies of underground miners exposed to radon have convincingly established that radon-decay products are carcinogenic and that exposure to these products at levels previously found in mines increases lung-cancer risk (NCRP 1984a,b; NRC 1988; Samet 1989; Lubin and others 1994a). Lubin and others (1994a) conducted a pooled analysis of data from 11 major studies of underground miners (identified as the studies from Colorado, Czechoslovakia, China, Ontario, Newfoundland, Sweden, New Mexico, Beaverlodge, Port Radium, Radium Hill, and France). That analysis included over 2,700 lung-cancer cases among 68,000 miners representing nearly 1.2 million person-years of observation. Lubin and others (1994) found that the relationship between the relative risk (RR) of lung-cancer and cumulative exposure to radon progeny was generally consistent with linearity within each cohort. However, estimates of the excess relative risk (ERR) due to exposure to radon progeny varied substantially among the cohorts. For example, the ERR following exposure to 100 working level months (WLM) varied from 0.16 in China to 5.06 in Radium Hill. The precision of these estimates was highly variable from cohort to cohort.

The combined effect of radon exposure and tobacco smoke on lung-cancer risk has been discussed in the literature (Lundin and others 1971; Whittemore and McMillan 1983; Moolgavkar and others 1993; Lubin 1994) (See also appendix C.) Although the Colorado Plateau uranium miners study has revealed a synergistic effect between exposure to radon and cigarette smoking, this interaction is not well characterized (NRC 1988; Lubin 1994). Data from two large studies, the

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Appendix A
Risk Modeling and Uncertainty Analysis
INTRODUCTION
Epidemiologic studies of underground miners exposed to radon have convincingly established that radon-decay products are carcinogenic and that exposure to these products at levels previously found in mines increases lung-cancer risk (NCRP 1984a,b; NRC 1988; Samet 1989; Lubin and others 1994a). Lubin and others (1994a) conducted a pooled analysis of data from 11 major studies of underground miners (identified as the studies from Colorado, Czechoslovakia, China, Ontario, Newfoundland, Sweden, New Mexico, Beaverlodge, Port Radium, Radium Hill, and France). That analysis included over 2,700 lung-cancer cases among 68,000 miners representing nearly 1.2 million person-years of observation. Lubin and others (1994) found that the relationship between the relative risk (RR) of lung-cancer and cumulative exposure to radon progeny was generally consistent with linearity within each cohort. However, estimates of the excess relative risk (ERR) due to exposure to radon progeny varied substantially among the cohorts. For example, the ERR following exposure to 100 working level months (WLM) varied from 0.16 in China to 5.06 in Radium Hill. The precision of these estimates was highly variable from cohort to cohort.
The combined effect of radon exposure and tobacco smoke on lung-cancer risk has been discussed in the literature (Lundin and others 1971; Whittemore and McMillan 1983; Moolgavkar and others 1993; Lubin 1994) (See also appendix C.) Although the Colorado Plateau uranium miners study has revealed a synergistic effect between exposure to radon and cigarette smoking, this interaction is not well characterized (NRC 1988; Lubin 1994). Data from two large studies, the

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China tin miners study and the Colorado Plateau uranium miners study, indicated that the lung-cancer risk associated with the combined exposure is greater than the sum of the risks associated with each factor individually, evidence of a synergistic effect between radon and tobacco smoke in the induction of lung-cancer. Data on tobacco use, available for 6 of 11 cohorts, are summarized in Table A-1. In the Colorado, Newfoundland and New Mexico studies, detailed data on tobacco use including duration, intensity, and cessation are available, whereas studies in China and Radium Hill identify individuals only as ever-smokers or never-smokers.
Since publication of the report by Lubin and others (1994a), studies of the Chinese tin miners, and of the Czech, Colorado, and French uranium miners have been updated or modified (Lubin and others 1997). Modifications of these four data sets are described in Table A-2. In addition, there has also been a reassessment of exposure for a nested case-control series within the Beaverlodge cohort of uranium miners, including all lung-cancer cases and matched control subjects (Howe and Stager 1996). For the Beaverlodge miners, exposure estimates were about 60% higher than the original values. Because of the computational and conceptual difficulties of merging case-control data with cohort data, only the data from the Beaverlodge cohort study with the original exposure estimates were used in the BEIR VI analysis.
In the first part of this appendix, we consider models for describing the relationship between exposure to radon and lung-cancer risk. We begin with a review of risk models developed by other investigators. In order to lay the foundation for the committee's risk model, we then discuss methods for combining data from different sources, including random-effects and two-stage methods. Those methods are then used in a combined reanalysis of the updated data from the 11 miner cohorts considered previously by Lubin and others (1994a).
By using a random-effects model, the overall effect of radon on lung-cancer risk can be described by fixed regression coefficients, and variation across cohorts characterized by random regression coefficients (Wang and others 1995). Two-stage regression analysis represents an alternative to random-effects methods, which benefits from an element of numerical simplicity. Although both methods are considered, the emphasis in the report is on the computationally simpler two-stage method.
In the second part of the appendix we focus on uncertainties in predictions of risk. There are many sources of uncertainty in health-risk assessments. Epidemiologic data on exposed human populations can be subject to considerable uncertainty. Retrospective exposure profiles are difficult to construct, particularly with chronic diseases such as cancer for which exposure data many years prior to disease ascertainment are needed. For example, radon measurements in homes taken today may not reflect past exposures because people change residences, make building renovations, or change their lifestyle such as sleeping with the bedroom window open or closed, and because of inherent variability in radon

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TABLE A-1 Characteristics of 11 underground miner studiesa
Location
Type of mine
Number of miners
Period of follow-up
Data available on smoking
China
Tin
17,143
1976–87
Smoker: yes/no [missing on 24% of subjects, 25 (out of 907) non-smoking lung-cancer cases.]
Czechoslovakia
Uranium
4,284
1952–90
Not available
Colorado, USA
Uranium
3,347
1950–90
Cigarette use: duration, rate, cessation [unavailable after 1969, 25 (out of 294) non-smoking cases.]
Ontario, Canada
Uranium
21,346
1955–86
Not available
Newfoundland, Canada
Fluorspar
2,088
1950–84
Type of product, and duration, cessation (available for 48% of subjects, including 25 cases.)
Sweden
Iron
1,294
1951–91
Type of product, and amount, cessation (from 35% sample of active miners in 1972, supplemented by subsequent surveys)
New Mexico, USA
Uranium
3,469
1943–85
Cigarette use: duration, rate, cessation (available through time of last physical examination)
Beaverlodge, Canada
Uranium
8,486
1950–80
Not available
Port Radium, Canada
Uranium
2,103
1950–80
Not available
Radium Hill, Australia
Uranium
2,516
1948–87
Smoke status: ever, never, unknown (available for about half subjects, 1 non-smoking case.)
France
Uranium
1,785
1948–86
Not available
a Lubin and others (1994a).

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TABLE A-2 Summary of new information on 5 miner cohorts
Cohort
Updated information
Related reference
Chinese tin miners
New information indicated miners worked 313 days/yr before 1953, 285 days/yr from 1953–84, and 259 days/yr from 1985.
Unpublished information
Czech uranium miners
Exposure histories re-evaluated and follow-up improved. There were 705 lung-cancer cases, compared to 661 in the previous analysis. Cohort was enlarged from 4,284 to 4,320 miners, including all miners who entered 1948–59.
Tomásek and others 1994
Colorado uranium miners
Follow-up extended from December 31, 1987 to December 31, 1990. In updated data, there were 336 lung-cancer deaths < 3,200 WLM used in the pooled analysis (and 377 total cases), compared to 294 lung-cancer deaths < 3,200 WLM (and 329 total cases).
Hornung and others 1995
Beaverlodge uranium minersa
Re-calculation of WLM exposures, but limited to a nested case-control sample.
Howe and Stager 1996
French uranium miners
Corrections of some (non-lung-cancer) outcomes and exposure data. Changes were not extensive.
Unpublished information
a The most recent update of the Beaverlodge data was not included in the BEIR VI analysis.

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measurements. Exposures in prospective studies may also be uncertain. In addition to errors in exposure ascertainment, errors in disease diagnosis are also possible. In epidemiologic studies using computerized record linkage to link exposure data from one database with health status in another database, even vital status can be in error (Bartlett and others 1993).
In addition to identifying sources of uncertainty, the committee attempted a quantitative analysis of uncertainty in radon risk estimates. This analysis is conducted within the general framework developed by Rai and others (1996) for quantitative uncertainty analysis in health risk assessment. Since not all sources of uncertainty and variability could be fully characterized, the committee acknowledges that this analysis is necessarily incomplete. Nonetheless, the committee felt that this analysis is informative, and provides a basis for further research in this area.
PREVIOUS RISK MODELS
A number of different exposure-response models may be used to describe the relationship between lung-cancer risk and exposure to radon (Krewski and others 1992). Analyses of miner cohorts have been largely based on empirical models that describe risk as a linear or linear-quadratic function of exposure. Those analyses have formed the basis for estimates of risks of exposure to radon prepared by the National Research Council (NRC), the National Commission on Radiological Protection and Measurements (NCRP), and the International Commission on Radiological Protection (ICRP). Previous estimates of risk prepared by those organizations are reviewed here.
Empirical Models
Numerous studies of lung-cancer in radon-exposed underground miners have been published, although until recently the number of distinct populations had been small and the total follow-up time and numbers of lung-cancer cases limited (NRC 1988). Those analyses are described in detail in appendix D. The relatively small numbers of lung-cancer cases in individual studies have hindered the evaluation of temporal patterns and other determinants of risk. Early efforts at risk modeling were limited by the lack of data and, as a result, investigators relied on summaries of the miner studies. A complete review of the earlier risk estimates was provided in the BEIR IV Report (NRC 1988). In the current report, we review the most relevant efforts.
National Research Council (1980)
The NRC BEIR III committee based their modeling efforts on results of miner studies (NRC 1980). The model assumed a linear relationship between

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exposure and the absolute excess risk of lung-cancer. The absolute or excess risk (ER) model represents lung-cancer mortality as r(x,z,w) = ro(x) + g(z,w), where ro(x) is the background lung-cancer rate, and g(z,w) is the effect of exposure. Here, w denotes cumulative exposure, x is a vector of covariates which affect the background lung-cancer rate, and z is a vector of covariates that may modify the exposure-response relationship. The excess risk varied by categories of attained age, <35, 35–49, 50–65, >65 yrs, with 0, 10, 20, 30 excess cases per 106 person-years per unit of exposure in WLM. In addition, the model specified a minimum latent period of 15–20 yr for those exposed at ages 15–34 or 10 yr for those exposed above age 34. The derivation of this model and the method of combining the available miner data were not described. The model did not directly account for the effects of smoking.
National Council on Radiation Protection and Measurements (1984)
The National Council on Radiation Protection and Measurements (NCRP) committee's Report 77 (NCRP 1984a) and its Report 78 (NCRP 1984b) adopted the excess-risk model of Harley and Pasternak (1981). That model was based on the following assumptions.
The latent interval is 5 yr for persons first exposed at ages 35 yr and older, and (40-u) yr for persons exposed under age 35 yr, where u is the age at first exposure.
Following a latent interval, disease rate declines exponentially with time since exposure.
lung-cancer is rare before age 40 yrs.
The median age at lung-cancer occurrence for miners is age 60 yr for never-smokers and
50 yr for ever-smokers.
The minimum time from initial cell transformation to clinical detection is 5 yr.
For an annual exposure at age u, the excess lung-cancer risk at age t > u (and t > 40 years of age) is taken to be
A(t,u) = Re-m(t-u) S(t) / S(u), (1)
where R is the excess-risk coefficient per WLM, S(u) is the probability of survival to a specified age, and m is the rate of removal of transformed stem cells due to repair or cell death. The NCRP committee fixed m = ln(2)/20 yr-1, corresponding to a 20-yr half-life for exposure effects. The exponential term reduces the exposure effect with time after exposure, while the survival probability adjusts for competing causes of mortality. Lifetime risk for exposure at age u is obtained by integrating over age (t) from age 40 yr to some specified life-span. For chronic

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exposures in years u1, ... ,un, lifetime risks are obtained by summing risks for each of the annual exposures.
Parameters for the NCRP model were values assumed to be ''reasonable" from published data, but not based on a direct evaluation of the miner data. Although the choices of R and m are critical in applying the model, the NCRP provides little guidance on their selection. The 20-yr half-life was selected as being "representative for extrapolation." S was taken from 1978 World Health Organization tables for the U.S. population. NCRP 78 had the first model to incorporate a reduction of risk with time-since-exposure.
In the NCRP model, the joint effects of radon-progeny exposure and smoking were considered additive. That is, radon exposure had the same effect on the excess risk, regardless of smoking status. The increased absolute excess risk with radon-progeny exposure was added to the background lung-cancer risk in ever-smokers or in never-smokers.
International Commission on Radiological Protection (1987)
Report 50 of the International Commission on Radiological Protection presented a risk model for indoor radon exposure (ICRP 1987). The ICRP model was based on a simple constant excess relative risk model of the form
RR(w) = 1 + ßw, (2)
where w is total cumulative exposure allowing for a lag interval of 10 years. No accommodation for variation in the exposure-response parameter ß with other factors was included. The value ß was taken as 0.7% per WLM, a representative value for the excess relative risk of the available miner studies after adjustment for exposure-dose differences between mines and homes. Based on the study of atomic-bomb survivors and on dosimetric considerations, the model assumed a greater effect for radon-progeny exposure at young ages; ß was set at 2.1% per VVLM for exposures occurring under the age of 20 years.
In the ICRP model, the joint effects of radon-progeny exposure and smoking were assumed multiplicative. Thus, for smoking status-specific risk estimation, the radon relative-risk model was applied separately to ever-smokers and to never-smokers. Similarly, the same model was applied to the background rates for males and for females.
Thomas and Others (1985)
In recent years, the number of data sets on radon-exposed miners has increased and follow-up has lengthened for the cohorts developed initially. Thus, direct synthesis of multiple miner studies has become increasingly important and informative for defining the form of the risk model and estimating the values of its parameters. The first joint analysis of results of epidemiologic studies using

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modern statistical methodology was carried out by Thomas and others (1985). (See also Thomas and McNeill 1982, an earlier and more detailed report on which the more recent work was based.) They carried out a meta-analysis of 5 miner studies. They fit relative risk and excess (or attributable) risk models, including various "cell killing" and "non-linear" models to summary data from cohort studies from Czechoslovakia (now the Czech Republic), Colorado, Ontario, Newfoundland, and Sweden. Original data from these cohorts with more extensive follow-up are included in the analysis conducted by Lubin and others (1994a) and by this BEIR VI committee. Thomas and others found no significant deviation from a linear exposure-response relationship, although inferences on curvilinearity were somewhat dependent on the choice of referent population. Fitting a linear excess relative-risk (ERR) model, Thomas and others (1985) estimated the ERR to increase by 2.28% per WLM, implying a doubling of risk at 44 WLM. The ERR per WLM was found to vary with attained age. The risk with combined exposure to smoking and to radon progeny, while consistent with a multiplicative model, was most consistent with a relationship intermediate between additive and multiplicative. The analyses were necessarily limited by the extent of the data and not having access to original data. However, many of these results presaged subsequent work.
National Research Council (1988)
A comprehensive assessment of risk from underground exposure to radon progeny was carried out by the National Research Council's Biological Effects of Ionizing Radiation IV committee (BEIR IV). That committee conducted a pooled analysis of data from four cohort studies of underground miners including the studies of Colorado, Ontario, Beaverlodge, and Sweden (NRC 1988). The Beaverlodge and Swedish data sets were the same as in the current analysis, while less extensive data were available from the other 2 studies. The BEIR IV committee used regression methods similar to those used in this report. The committee found that excess risk did not increase in a simple fashion with exposure, either in direct proportion to background (a constant ERR model) or at a constant level above background (that is, a constant attributable or excess risk [ER] model). Rather, the risk varied with two time-dependent factors: time since exposure and attained age. The analysis showed that lung-cancer risk increased linearly with cumulative exposure to radon progeny, and that the exposure-response trend declined with attained age and with time since exposure. The committee evaluated other potentially important covariates, such as age at first exposure and exposure duration, but risk patterns were not consistent across studies. Although the constant ERR model was not compatible with the data, the ERR per WLM was estimated to be 1.34% (corresponding to a doubling of risk at 75 WLM) under this model.
Analyses of the combined effects of radon-progeny exposure and smoking in

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the Colorado and New Mexico (a case-control subset of data included in the current analysis) miners were also presented in the BEIR IV Report. Results indicated that while a multiplicative risk relationship between the two factors could not be excluded, an intermediate relationship between additive and multiplicative was most consistent with the data.
International Commission on Radiological Protection (1993)
In a 1993 report on risks from radon-progeny exposure in homes and at work (ICRP 1993), ICRP did not provide its own risk model, but used the so-called "GSF model" developed by Jacobi and others (1992). That model was related to a "smoothed" version of the BEIR IV model (NRC 1988). Compared to the BEIR IV model, the GSF model provided a monotonic variation of the excess relative risk with age. The GSF model had the same general structure as the BEIR IV model, with the effect of exposure adjusted by time since the exposure occurred, but with the exposure-response relationship determined by age at exposure, as opposed to attained age. For attained age a, exposure we, occurring at age ae, and time since exposure f, the ERR at age a was defined as:
Exposures within 4 years f = a - ae = 4) were assumed to have no impact on the RR of lung-cancer. The function s was not explicitly defined in the ICRP Report, but was described as a decreasing function of age at exposure, taking values of 0.036 per WLM for age at exposure of 20 years and 0.017 per WLM for age at exposure of 60 years. The effect of time since exposure on risk was modeled through the function defined as
According to the ICRP report, risk projections based on the GSF model were similar to those of the BEIR IV model (ICRP 1993).
The ICRP approach was notable in two respects. First, in contrast to the earlier ICRP risk model, which was based on a constant relative risk in cumulative exposure (ICRP 1987), no modification to the exposure-response relationship was included for exposures received at ages 20 years and under. The previous ICRP model postulated a 3-fold greater exposure-response for young ages. Second, the ICRP model assumed an equal (absolute) excess risk in males and females. Thus, the model assumed that the radon-related excess risk in males should be directly added to the background lung-cancer rate in females. This additive feature of the model results in a markedly greater relative risk in females than in males.

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National Cancer Institute (1994)
The analysis and risk model published by Lubin and others (1994a, 1995b), which served as the starting point for the current report, utilized methods similar to those of the BEIR IV committee (NRC 1988). This approach assumes that the time to death from lung-cancer is distributed in a fashion so that follow-up time to a key event was piece-wise exponential, that is, death rates are constant within fixed time intervals and exposure categories. Note that death times can be censored due to loss to follow-up or study termination. This assumption was considered appropriate for two reasons: 1) variability in lung-cancer mortality rates within each time interval and exposure category was small relative to the variability between intervals; and 2) disease rates within time intervals and exposure categories were well-characterized by the average rate. The method allowed for use of external referent rates (although they were not used in Lubin and others 1994a, 1995b or in the current analysis) and for the modeling of excess disease rates. A full discussion of these models, including regression of the standardized mortality ratio, is given in Breslow and others (1983) and Breslow and Day (1987).
Relative-risk regression procedures were applied to data summarized in a multi-way table, consisting of events, person-years, and summary variables for each cell of the cross-tabulation. Analyses were conducted using the EPICURE package of computer programs (Preston and others 1991). Data were cross-classified by various factors, depending on the cohort and on the variables being analyzed. For a typical cohort, data were cross-classified by attained age (<40, 40–44, ..., 65–69, = 75 years), calendar period (< 1950, 1950–54, ..., 1980–84, = 1985), estimated exposure (0, 1–49, 50–99, 100–199, 200–399, 400–799, = 800 WLM), duration of radon-progeny exposure (<5, 5–9, 10–14, =15 years), age at first radon progeny exposure (< 10, 10–19, 20–29, = 30 years) and other mining experience (no, yes). For each cell of the table, the number of observed lung-cancer deaths, the number of person-years, and the mean (weighted by person-years) for the cross-classification variables, such as cumulative exposure, exposure duration, attained age, and age at first exposure, were computed. For pooling purposes, data were further cross-classified by cohort. A 5-year lag period was assumed.
Whenever possible, similar categories for variables that specified dimensions of the person-year tables were established across the cohorts; however, because of intrinsic differences among the cohorts, this was not always possible. For example, some exposures in the Newfoundland cohort were as high as 21 Jhm-3 (6,000 WLM), while all exposures in the Radium Hill cohort were under 0.35 Jhm-3 (100 WLM).
The risk model was developed with the following approach. Suppose the lung-cancer mortality rate is given by r(x,z,w), and depends on cumulative exposure, w, a vector of covariates, x, which described the background lung-cancer

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rate, and a vector of covariates, z, which may modify the exposure-response relationship. The relative risk, r, was expressed as a product of the background disease rate among nonexposed, denoted ro(x), and an exposure-response function, RR(z,w). The background rate ro depends on x while the exposure-response function RR depends on z, which may include one or more components of x, as well as w. This general relative risk model can be written as
r(x,z,w) = r0(x)RR(z,w). (5)
The background lung-cancer rate was modeled as r0(x) = exp(ax), with x being the vector of controlling variables and a the corresponding parameter vector. Components of x typically included indicator variables for age group, calendar period, and cohort, as well as variables describing other mine exposures. Main effects and all higher order interactions were included.
Specific models were fit for RR. A linear RR model in w was fitted, namely,
RR = 1 + βw. (6)
Here, β is a parameter which describes increase in ERR per unit increase in w (ERR/exposure). More generally, a model for the assessment of a broad range of exposure-response relationships was defined as
RR = (1 + βwk)eθw. (7)
Again, β reflects the overall ERR/exposure, the parameter, θ measures the exponential deviation from linearity (sometimes referred to in the radiation effects literature as a "cell killing" parameter), and k is a parameter to describe departure from linearity. This general model includes the linear ERR model (k = 1, θ = 0), the linear-exponential model (k = 1) and the "non-linear" model considered by Thomas and others (1995) (θ 0). Tests for improvement in model fit in relation to θ and k were carried out using likelihood ratio procedures.
An important goal of the analysis was to examine variations of the exposure-response trend with other variables, that is, to test whether β varied within categories of other factors, such as attained age, age at first exposure, duration and rate of exposure, and time since last exposure. In epidemiologic terms, they evaluated the components of the covariates vector z as an effect modifier. Suppose a particular covariate z had J categories with values z1, ... , z J. Variation in the exposure-response relationship within levels of z was assessed by fitting model (6) and comparing its deviance with model (8) below which included J exposure-response parameters, namely,
RR = 1 + βjw, (8)
where βj was the ERR/exposure within category zj. Under the null hypothesis of no effect modification, the difference in the model deviances was approximately

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categories including all solid tumors, digestive cancers, and cancers of the breast and thyroid. In most cases, the strongest evidence for a dependence of risk on age at exposure was based on a comparison of risks in those exposed as children (under age 20 years) and those exposed later in life.
Data from the miner cohorts on exposures in childhood are limited primarily to the China cohort. Of 813 lung-cancer deaths occurring in miners initially exposed under age 20 years, 735 occurred in this cohort, and all 54 of the lung-cancers occurring in miners exposed under age 10 yrs were from the China cohort. In fact, a large percentage of miners in the China cohort were first exposed at very young ages, with only 25% (245) of the lung-cancers occurring in those with first exposure at age 20 yrs and older. Analyses by Lubin and others (1994a) show statistically significant variations in ERR per unit exposure with age at first exposure in the China cohort, although the pattern was not consistent. Clearly the uncertainty in the lung-cancer risks in adults resulting from exposure in childhood is much greater than for risks resulting from exposure in adulthood Not only is there the possibility that the ERR per unit exposure for childhood exposures might differ from that predicted by the committee's models, it is also possible that the pattern of decline in risks differs from that observed for adult exposures. It is noted, for example, that the committee's preferred models are based on the assumption that risks persist for a lifetime, but there are no data to validate this assumption for exposure in childhood.
Dependence of Risks on Smoking Status
Limitations in the available data on smoking make it difficult to evaluate the modifying effect of smoking on radon risks. Because of the need to extrapolate from miners with a high proportion of ever-smokers to the general population with a lower proportion of ever-smokers, uncertainties in adjustments for smoking may even affect estimates of average population risks. Such uncertainties have a particularly strong effect on estimates that specifically address risks in ever-smokers and never-smokers. Because of limitations in available data, it was not possible to develop a model that took account of the amount smoked, degree of inhaling, and changes in smoking habits over time.
Uncertainties in the Model for Estimating Differences in Radon-Progeny Dosimetry in the Mines and in Homes
Several factors that affect the lung dosimetry of radon progeny differ between mines and homes. These differences must be accounted for in using risk estimates based on underground miners to estimate risks for persons exposed in homes. The parameter summarizing these differences is often referred to as the K-factor. This factor can be expected to vary among individuals, and its average value may also be subject to uncertainty. Uncertainty in the K-factor is discussed

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in more detail in appendix B (dosimetry). Variability in the K-factor was included in the committee's Monte Carlo simulations.
Uncertainty in the K-factor values arises from several sources. These include the measurement error in the size-dependent concentrations of airborne radioactivity collected in the sampling devices, the error in deconvoluting a size distribution for the measured activity fractions, and the uncertainty in the relative fractions of time assigned to the various locations in the mine. The error in determining the collected activity concentration depends on the statistics of the counts used to estimate the amounts of the 3 decay products, and the variation in the pump flow. Thus, there is variation in the in-home measurements since the 222Rn concentrations ranged from about 30 Bqm-3 up to 800 Bqm-3, depending on the home being studied. For typical airborne activity concentrations, the uncertainties in the individual concentrations are of the order of 5 to 10% and for PAEC, the errors are 3 to 5%.
These errors then propagate in a non-linear manner since they are input values to the algorithms used to estimate the activity-weighted size distributions. In general, this inversion process is ill-posed since it is an undetermined problem for which a unique solution is not possible. It is known from simulation exercises (Ramamurthi and others 1990) that these algorithms can find acceptable solutions although not necessarily the "true" solution. Thus, it is not possible to definitively determine the overall uncertainties in the size distributions. Similarly, it is also extremely difficult to precisely determine the uncertainty in the times for various mining activities. Although an exact uncertainty cannot be assigned to each K-factor value, it is estimated that the values in the central portion of the distribution should not have errors in excess of 25%. Thus, the variability in K is larger than its uncertainty.
Uncertainties Relating to Background Exposures
The risk models developed by the committee based on its analysis of data from the 11 miner cohorts are based on occupational radon exposures. However, miners were also exposed to radon in their homes and outdoors. Although these additional exposures contribute to their lung-cancer risk, the residential and ambient exposures experienced by miners are much lower than their occupational exposures. Consequently, the impact of these non-occupational exposures on the coefficients in the committee's risk model is expected to be negligible.
Ambient exposures also need to be considered when evaluating residential radon risks, since people are exposed to radon in outdoor air as well as in their homes. Again, since ambient exposures are not widely available, it is not possible to adjust for the effects of outdoor exposures on residential radon risks. However, since ambient exposures are much lower than typical indoor exposures, any adjustment for outdoor exposures when evaluating residential radon risks is likely to be small.

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Uncertainties in the Demographic Data Used to Calculate Lifetime Risks
An additional source of uncertainty in lifetime risks involves the application of the committee's risk model to obtain estimates of risks for the U.S. population. These calculations require assumptions about the age distribution of the population, life expectancy, and baseline age and sex-specific lung-cancer mortality rates. As described in chapter 3, current U.S. life table and vital statistics data were used for this purpose. It was assumed that these data were appropriate both now and in the future. Rather than evaluate uncertainty from this source, it seems preferable simply to state that the lifetime risk estimates presented in this report are appropriate only for a population with these demographic characteristics. If changes occur in the future, or risk projection for other populations are desired, these estimates will need to be recalculated to reflect these modifications.
QUANTITATIVE UNCERTAINTY ANALYSIS
Currently, there is a trend in risk assessment towards a more complete characterization of risk using quantitative techniques for uncertainty analysis (Bartlett and others 1996; Morgan and others 1990; NRC 1994b). The results of these analyses can be summarized in the form of a distribution of possible risks within the exposed population, taking into account as many sources of uncertainty and variability as possible. This distribution gives an indication of maximal and minimal risks that might be experienced by different individuals, and the relative likelihood of intermediate risks between these two extremes.
Finley and Paustenbach (1994) discuss the benefits and disadvantages of probabilistic exposure assessment compared to using point estimates. Point estimates are simple to interpret, but provide no indication of level of confidence. Probability distributions provide risk analysts with a more complete picture of the possible range of exposure, but are more complex to determine and to use in decision making. Edelmann and Burmaster (1996) show that distributions with the same 95th percentiles could have dramatically different shapes, and that risk-management decisions based on these distributions may be different. Recently, considerable effort has been extended in estimating distributions of risk for use in health-risk assessment (Finley and others 1994; Ruffle and others 1994).
Uncertainty exists in all stages of the risk-assessment process (Small 1994; Dakins and others 1994). An integrated environmental health-risk assessment model includes source characterization, fate and transport of the substance, exposure media, biological modeling, and estimation of risk using dose-response modeling. Each component of the integrated risk assessment is subject to uncertainty. Often what is known about input parameters may be admissible ranges, shape of the distribution, or the type of data. To estimate uncertainty in the output, uncertainty distributions are associated with the input parameters. Distributions characterizing the uncertainty associated with each adjustment factor are developed

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using scientific knowledge to the extent possible These distributions can then be sampled using Monte Carlo methods to provide estimates of the output distributions. Monte Carlo methods have been used for estimating the impact of uncertainty in adjustment factors on estimation of human population thresholds for non-carcinogens (Baird and others 1996). Though convenient and easy to use with modern computing technology, Monte Carlo methods should be carefully monitored to ensure the integrity of the results (Burmaster and Anderson 1994).
Distinguishing between uncertainty and variability is necessary but sometimes difficult (Hoffman and Hammonds 1994; Bogen 1995). One approach for describing uncertainty and variability in lognormal random variables is to plot a range of probability distributions in two dimensions, representing uncertainty and variability, thereby permitting the impact of uncertainty to be visualized (Burmaster and Korsan 1996). Uncertainty can sometimes be reduced by collecting more information, but variability cannot. Information on metabolic activation, detoxification, and DNA repair was recently considered in evaluating interindividual variability (Hattis and Barlow 1996; Hattis and Silver 1994). It was shown that empirical studies of biological parameters are useful in establishing uncertainty factors for heterogeneity in individual risk. It was also noted that the interindividual variability tends to be overstated due to the presence of measurement error; adjustments can be made using empirical Bayes shrinkage estimators (Goddard and others 1994) and other statistical techniques.
Quantitative Analysis of Uncertainty and Variability
Rai and others (1996) have developed a general framework for the analysis of uncertainty and variability in risk. In the most general case, the risk R is defined as function
R = H(X1, X2, ... , Xp) (31)
of p risk factors X1, ... , Xp. Each risk factor X1 may vary within the population of interest according to some distribution with probability density function fi(Xi ¦ i), conditional upon the parameteri. Uncertainty in Xi is characterized by a distribution for i, where is the true value of the parameter. If i is a vector valued, gi is a multivariate distribution. Here, it is assumed that the forms of the distributions f and g are known. The case in which the form of f or g is unknown introduces another level of complexity which remains to be addressed.
If i is a known constant and the distribution fi is not concentrated at a single point, Xi exhibits variability only. On the other hand, Xi is subject to both uncertainty and variability if both i and fi are stochastic. When fi is concentrated at a single point i, andi is stochastic, Xi is subject to uncertainty but not variability. Consequently, the variables X1, ... Xp can be partitioned into three groups: variables subject to uncertainty only, variables subject to variability only, and variables subject to both uncertainty and variability.

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Rai and Krewski (1998) consider the special case of a multiplicative risk model
R = X1 X2 ... Xp. (32)
The multiplicative model is applicable in many situations encountered in practice, and affords a number of simplifications in the analysis. In particular, the multiplicative risk model in (32) is simplified by applying the logarithmic transformation. Xi = log Xi (i = 1, ...p). After transformation, the multiplicative model can be re-expressed as an additive model
where R* = log R. Assume that each Xi has mean mi and variance Var(Xi), where
and
The expected risk on logarithmic scale in , with variance
UNCERTAINTY IN POPULATION ATTRIBUTABLE RISK
In the present application, uncertainty in estimates of the population attributable risk of lung-cancer due to residential radon exposure is of primary interest. The general approach to uncertainty analysis proposed by Rai and others (1996) is used for this purpose. As discussed previously, the attributable risk (AR) is not subject to variability, since it is a measure of population rather than individual risk.
Following Levin (1953) and Lubin and Boice (1989), the attributable risk of lung-cancer due to radon exposure is defined as the proportion of lung-cancer deaths attributable to radon progeny. For continuous risk factors, the AR can be written as

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Here, fw is the marginal probability density function of the exposure distribution, reflecting variability in residential radon concentrations. R(w) is the lifetime risk of lung-cancer for a lifetime exposure to radon progeny at a yearly level w in the presence of competing risks, and
is the lifetime relative risk. Note that lifetime excess relative risk, ERR(w) = RR(w)-1, appears in the integrand (36).
The K-factor also influences the lifetime risk of lung-cancer. To accommodate the effect of the K-factor, equation (36) can be modified as
Here the joint marginal distribution of w and K, fw,k(w,k), is the product of the distributions of w and K, since we assume that these two variables are statistically independent.
Let hi and be the lung-cancer and overall mortality rates for age group i, respectively, in a referent population. Furthermore, let ei be the excess relative risk due to exposure w to radon progeny for age group i. Here, we consider two types of models for ei:
and
where
The factors w5–14,w15–24 and w25+ represent cumulative radon-progeny exposures received 5–14, 15–24 and more than 25 years prior to disease diagnosis, respectively. Note both (39) and (40) are multiplicative models of the type (32).
The remaining risk factors in (39) and (40) are redefined as

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and
Models (39) and (40) express the age-specific excess relative risk using a risk factor γwl for exposure concentration and an alternative risk factor γdur for exposure duration, respectively. These correspond to the committee's exposure-age-concentration and exposure-age-duration models.
The lifetime risk of lung-cancer is given by the sum of the risk of lung-cancer death each year:
Here
and
is the probability of surviving year i for an individual with exposure w given that the individual survived up to year i-1.
The model for the AR in (38) depends on the uncertainty and variability distributions of w and K. We assume that w has lognormal distribution with geometric mean 24.8 Bqm-3 and geometric standard deviation 3.11, and has uncertainty only in the geometric standard deviation. We also assume that K has a lognormal distribution with geometric mean 1.0 and geometric standard deviation 1.5, and has no uncertainty. Since there is no closed form expression for the integrand in model (38), we approximate the integral by summing over the ranges of values for w and K.

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This approximate version of the attributable risk model depends on the risk factors in either (39) or (40). The uncertainty in these factors is characterized by lognormal or log-uniform distributions as summarized in Table A-8. These uncertainty distributions were based on the committee's judgment as to the likely range of values for each of these factors. By postulating an uncertainty distribution for each of the factors in the model, the committee acknowledges that all factors are subject to some degree of uncertainty. Although statistical uncertainty in the estimates of the parameters in the committee's performed models is included, distributions also reflect other sources of uncertainty as discussed previously in this appendix.
Uncertainty distributions of the AR were obtained with Monte Carlo sampling (Rai and others 1996). Although computationally intensive, the analysis is straightforward. First, a set of model parameter values was obtained by sampling from a multivariate normal distribution with mean equal to the estimated parameter values and the values in the covariance matrix given in Table A-9 and based on the standard statistical software package S-Plus. (The Monte Carlo simulation was done on a log scale, with each risk factor Xi in (39) and (40) replaced by exp[Ef(X*i)].) Second, the attributable risk was calculated as described previously in this appendix. Repeating this procedure 10,000 times in case I and 1,000
TABLE A-8a Uncertainty and variability distributions for risk factors in the exposure-age-concentration model for excess relative risk
Risk factor
Variability
Uncertainty
Model parameters
Constant
a ~ N (µ, S)a
Exposure to radon w
LNb (gmc = 24.8, gsdd = 3.11)
K-factor
LN (gm = 1, gsd = 1.5)
gm = 1.00, gsd~LUe (1.2, 2.2)
TABLE A-8b Uncertainty and variability distributions for risk factors in the exposure-age-duration model for excess relative risk
Risk Factor
Variability
Uncertainty
Model parameters
Constant
a ~ N (µ, S)f
Exposure to radon w
LN (gm = 24.8, gsd = 3.11)
K-factor
LN (gm = 1, gsd = 1.5)
gm = 1. 00; gsd ~ LU(1.2, 2.2)
a Multivariate normal distribution with µ and S specified in Table A-9a.
b LN: Log-Normal.
c gm: geometric mean.
d gsd:geometric standard deviation = defined as exps, where s denotes the standard deviation of logeX.
e LU(a,b): Log-uniform distribution, with, with logeX uniformly distributed between logea and logeb.
f Multivariate normal distribution with µ and S specified in Table A-9b.

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TABLE A-9a Parameters for uncertainty distributions for risk factors in exposure-age-concentration modela
I. Estimated Values of Parametersb
Parameter
β
θ15–24
θ25+
55–64
65–74
75+
γ0.5–1.0
γ1.0–3.0
γ3.0–5.0
γ5.0–15
γ15.0+
Value
-2.57
0.77
0.51
-0.56
-1.23
-2.38
-0.72
-0.98
-1.13
-1.80
-2.21
II. Covariance Matrixc
β
θ15–24
θ25+
55–64
65–74
75+
γ0.5–1.0
γ1.0–3.0
γ3.0–5.0
γ5.0–15.
γ15.0+
β
9.47
θ15–24
-0.36
0.77
θ25+
-0.04
0.24
0.42
55–64
-2.87
-0.10
-0.15
5.71
65–74
-3.18
-0.17
-0.33
2.85
10.87
75+
-3.44
-0.19
-0.54
2.90
3.20
87.65
γ0.5–1.0
-5.57
-0.10
-0.02
0.14
0.42
0.83
8.24
γ1.0–3.0
-6.36
-0.12
-0.11
0.15
0.53
0.97
5.88
6.93
γ3.0–5.0
-6.58
-0.16
-0.10
0.18
0.59
1.08
5.83
6.69
7.30
γ5.0–15.0
-6.90
-0.05
-0.09
0.26
0.61
0.81
5.69
6.51
6.67
7.84
γ15.0+
-7.04
-0.02
-0.08
0.27
0.54
0.50
5.63
6.44
6.64
7.33
8.59
a Interindividual variability in both the level of exposure to radon and the K-factor is also characterized by lognormal distributions. The parameters of the two distributions were determined from national data on the distribution of radon in U.S. homes and from data on a sample of homes used to estimate the K-factor.
b Except for θ15–24 and θ25+ values are on log scale.
c Except for θ15–24 and θ25+ values are on log scale. All values were multiplied by 100.

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TABLE A-9b Parameters for uncertainty distributions for risk factors in exposure-age-duration model
I. Estimated Values of Parametersa
Parameter
β
θ15–24
θ25+
55–64
65–74
75+
γ5–14
γ15–24
γ25–34
γ35+
Value
-5.20
0.72
0.44
-0.65
-1.29
-2.07
1.02
1.49
1.89
2.32
II. Covariance Matrixb
β
θ15–24
θ25+
55–64
65–74
75+
γ5–14
γ15–24
γ25–34
γ35+
β
7.98
θ15–24
-0.30
0.98
θ25+
-0.01
0.25
0.44
55–64
-2.07
-0.11
-0.21
4.32
65–74
-2.16
-0.20
-0.39
2.10
9.60
75+
-2.43
-0.24
-0.59
2.15
2.43
95.37
γ5–14
-5.06
-0.21
-0.14
0.31
0.37
0.54
4.60
γ15–24
-5.56
-0.39
-0.23
0.31
0.54
0.93
4.67
5.94
γ25–34
-5.66
-0.40
-0.15
0.20
0.45
0.83
4.73
5.60
6.75
γ35+
-5.65
-0.37
-0.12
-0.15
0.18
0.65
4.76
5.61
6.03
7.26
a Except for θ15–24 and θ25+, values are on log scale.
b Except for θ15–24 and θ25+, values are on log scale. All values were multiplied by 100.

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times in cases II and III (which require considerably more computational effort) produced uncertainty distributions for the AR.
The median of the uncertainty distribution for the AR is shown in Table A-10 along with 95% uncertainty limits covering the central mass of distribution. These limits range from 9.1–28.2% for the exposure-age-concentration model and from 6.8–21.0% for the exposure-age-duration model.
TABLE A-10a Impact of uncertainty and variability on uncertainty intervals for population attributable risk for males
Exposure-age-concentration model
Exposure-age-duration model
Source
AR
95%U.I.
AR
95%U.I.
Uncertainty when K = 1
0.148
(0.091, 0.238)
0.103
(0.068, 0.158)
Uncertainty incorporating variability in K
0.150
(0.097, 0.224)
0.106
(0.077, 0.178)
Uncertainty incorporating variability and uncertainty in K
0.159
(0.095, 0.259)
0.111
(0.081, 0.194)
TABLE A-10b Impact of uncertainty and variability on uncertainty intervals for population attributable risk for females
Exposure-age-concentration model
Exposure-age-duration model
Source
ARa
95%U.I.
ARa
95%U.I.
Uncertainty when K = 1
0.160
(0.099, 0.256)
0.111
(0.079, 0.179)
Uncertainty incorporating variability in Kb
0.169
(0.104, 0.278)
0.119
(0.084, 0.192)
Uncertainty incorporating variability and uncertainty in Kc
0.173
(0.104, 0.282)
0.125
(0.088, 0.210)
a Median of uncertainty distribution
b K ~ LN(gm = 1, gsd = 1.5)
c K ~ LN(gm = 1, gsd ~ LU(1.2, 2.2))