1999, Vol. 14, No. 3, 305{337 Analysis of Local Decisions Using …gelman/research/published/lin.pdf · 2005-12-13 · Statistical Science 1999, Vol. 14, No. 3, 305{337 Analysis of

Statistical Science1999, Vol. 14, No. 3, 305–337

Analysis of Local Decisions UsingHierarchical Modeling, Applied to HomeRadon Measurement and RemediationChia-yu Lin, Andrew Gelman, Phillip N. Price and David H. Krantz

Abstract. This paper examines the decision problems associated withmeasurement and remediation of environmental hazards, using the ex-ample of indoor radon (a carcinogen) as a case study. Innovative methodsdeveloped here include (1) the use of results from a previous hierarchicalstatistical analysis to obtain probability distributions with local variationin both predictions and uncertainties, (2) graphical methods to displaythe aggregate consequences of decisions by individuals and (3) alterna-tive parameterizations for individual variation in the dollar value of agiven reduction in risk. We perform cost-benefit analyses for a variety ofdecision strategies, as a function of home types and geography, so thatmeasurement and remediation can be recommended where it is most ef-fective. We also briefly discuss the sensitivity of policy recommendationsand outcomes to uncertainty in inputs. For the home radon example,we estimate that if the recommended decision rule were applied to allhouses in the United States, it would be possible to save the same num-ber of lives as with the current official recommendations for about 40%less cost.

Key words and phrases: Bayesian decision analysis, hierarchical mod-els, small area decision problems, value of information.

1. INTRODUCTION

1.1 Decision-making for Environmental Hazards

Associated with many environmental hazards isa decision problem: whether to (1) perform an ex-pensive remediation to reduce the risk, (2) do noth-

Chia-yu Lin is a graduate student, Department ofBiostatistics, Columbia University, New York, NewYork 10032. Andrew Gelman is Associate Professor,Department of Statistics, Columbia University, NewYork, New York 10027. Phillip N. Price is scientist,Lawrence Berkeley National Laboratory, Berkeley,California 94720. David H. Krantz is Professor, De-partment of Psychology, Columbia University, NewYork, New York 10027.Address all questions to Andrew Gelman, [email protected]. See also http://www.stat.colum-bia.edu/radon for more information.

ing, or (3) take a relatively inexpensive measure-ment of the risk and use this information to decidewhether to (a) remediate or (b) do nothing. This de-cision can often be made at the individual, house-hold, or community level. Performing this decisionanalysis requires estimates for the risks. In partic-ular, the more precise are the local risk estimates,the more feasible it is to construct localized decisionrecommendations that allow attention and effort tobe focused on the individuals, households and com-munities at most risk.

In this paper, we present an analysis of theremediation–measurement decision problem in thecontext of a hierarchical model for estimating riskas a function of location and various covariates.We develop and illustrate our method for the prob-lem of home radon, a recognized cancer risk, forwhich appropriate measurement and remediationstrategies have been and continue to be the subject

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306 LIN, GELMAN, PRICE AND KRANTZ

of debate. In addition to its own importance, theradon problem shares several features with otherenvironmental hazards: (a) the risks are geographi-cally dispersed but have strong spatial patterns; (b)information exists to identify risky areas, but onecannot easily identify individual households at highrisk; (c) it is possible to perform local measurementsto identify the risks of individual households, but itwould be expensive to measure every household.

By “hierarchical model,” we mean a statisticalmodel in which a separate statistical parameteris assigned to each predicted unit (each county, inthe present case). To clarify, consider a standardnonhierarchical approach to predicting mean radonconcentrations by county: measured radon concen-trations are regressed on, say, measured surficialuranium concentration, and two regression coeffi-cients (a slope and an intercept) are determined.These coefficients are then used to predict meanradon concentrations over an area such as theUnited States, with uncertainties estimated by thestandard error of the regression. Although such anapproach yields a different predicted mean radonlevel for each county, this is not a hierarchical modelbecause the underlying statistical parameters—theregression coefficients—do not vary by county.

In contrast, a hierarchical model includes the re-gression coefficients and also, for each county, anadditional coefficient allowing the predicted meanfor that county to differ from the regression predic-tion. To see why this makes sense, consider a countywith many radon measurements, whose observedmean concentration differs substantially from theregression prediction. Using the regression predic-tion in such a county is not reasonable, since (givenenough samples) the prediction is known to be erro-neous. In the hierarchical Bayesian approach, eachcounty’s estimate is a compromise between the re-gression prediction and the measured value, withthe relative weighting determined by the amountof data in the county and the overall accuracy ofthe regression predictions. Moreover, a hierarchicalmodel allows a separate uncertainty estimate foreach county.

1.2 Hierarchical Decision Analysis

Our approach to hierarchical decision analysishas four steps. First, a hierarchical model is fit toavailable data, resulting in a posterior distribu-tion for exposure to the environmental hazard forany given household, as a function of locality andother household information. Second, the problemof “decision-making under certainty” is formulated:for example, for the radon remediation problem, thetradeoff between dollars and lives implies a willing-

ness to remediate at an action level Raction, so that ifthe true exposure were known, one would remediateif and only if it exceeds Raction; depending on varia-tions in risks and risk preferences, Raction can varyamong households. Third, the “decision-making un-der uncertainty” problem is solved: for the radonproblem, the measurement–remediation decisionfor any household is a function of its Raction and itsposterior distribution of exposure level. If additionalinformation is available at the household level—forexample, a previous radon measurement—this canbe incorporated into the posterior distribution. Thefourth step of our analysis is to evaluate the effectof various decision recommendations, in terms ofexpected lives saved and expected costs, if appliedwithin a larger geographic area (for example, theentire United States). Results can also be expressedin terms of expected marginal and aggregate costper life saved. As always in decision analysis, sensi-tivity analysis is then done to see how the estimatedcosts and lives saved vary when assumptions areperturbed.

With the exception of the hierarchical modeling,the above steps follow the standard paradigm ofexpected-value or “Bayesian” decision analysis (see,e.g., Dakins, Toll, Small and Brand, 1996, and En-glehardt and Peng, 1996, for a recent review andexamples). The use of a hierarchical model for thespatially varying hazard allows us to incorporatemodern Bayesian inference into a formal decisionanalysis. The standard regression-modeling ap-proach does allow recommendations to vary as afunction of the predictive variables and thus tovary with location, but hierarchical modeling haslower predictive error than standard regression ap-proaches, and, more importantly, allows additionalvariation in recommendations, related to spatiallyvarying uncertainties.

Our cost-benefit analysis of radon decisionsmakes geographically variable recommendations.The recommended localized actions are more cost-effective than the current single nationwide recom-mendation (see, in particular, Evans, Hawkins andGraham, 1988, who recognize that precise modelingof radon levels should allow targeted recommen-dations), but care is required in summarizing thedecision analysis, which we do using maps and aseries of graphs indicating costs for various decisionoptions.

A characteristic of the hierarchical decision analy-sis is that aggregate outcomes of decision strategiescan no longer be trivially derived from individualrecommendations. In statistical terms, aggregatingrequires averaging over the predictive distributionof the thousands of model parameters that indicate

LOCAL DECISIONS USING HIERARCHICAL MODELING 307

radon risk in the counties. We compute expectedcosts and lives saved by simulation from our pos-terior distribution.

1.3 Outline of This Paper

We develop our hierarchical approach to decisionanalysis in the context of measurement and reme-diation of home radon. Sections 2 and 3 of thispaper provide background on the indoor radon prob-lem and our previous work using hierarchical mod-eling of radon survey data to identify the housesthat are likely to have high radon levels given in-formation at the geographical and house level. Sec-tion 4 addresses the measurement and remediationdecision for individual homeowners, and Section 5presents the estimated aggregate consequences offollowing the recommended strategy and various al-ternatives if applied throughout the United States.For both the individual decisions and aggregate con-sequences, we develop a series of graphical displaysthat are potentially useful for hierarchical decisionproblems in general. In Section 6 we explore thesensitivity of our results to assumptions, and weconclude in Section 7 with a discussion of the spe-cific relevance of our methods to the indoor radonproblem and the general applicability to hierarchi-cal decision problems.

2. THE RADON PROBLEM ANDDECISION OPTIONS

2.1 Health Effects

Radon is a carcinogen, a naturally occurring ra-dioactive gas whose decay products are also radioac-tive, known to cause lung cancer in high concentra-tion and estimated to cause several thousand lungcancer deaths per year in the U.S. (see Nazaroff andNero, 1988, for an overview of the radon problem;Cole, 1993, for a discussion of the governmental re-sponse to it; National Research Council, 1998, foran influential official report).

The well-documented dose-dependent excess oflung cancer among underground miners exposedto radon has convincingly demonstrated that expo-sure to very high concentrations of radon causeslung cancer. Levels in homes are usually lower thanthose in mines, miners are also exposed to othercarcinogens, miners are overwhelmingly smokersand working miners generally breathe both harderand more deeply than people at home, so sev-eral assumptions and extrapolations are neededto estimate cancer risk at typical home levels (seeNational Research Council, 1991 and NationalResearch Council (BEIR VI), 1998, Tables 3–6).These extrapolations (including an assumed linear

dose-response function) suggests that about 15,400additional lung cancer deaths occur annually in theUnited States due to radon, mostly among smokers,though this number is based on an unrealistic com-parison to the number of deaths that would occur ifnobody were exposed to any radon at all.

The miner studies demonstrate statistically sig-nificant elevated cancer risk at doses equivalent tolifetime residence in a home at about 20 picoCuriesper liter (pCi/L). (An alternative notation is the in-ternational standard unit of Bequerels per cubic me-ter; 1 pCi/L = 37 Bq/m3.) Estimates based on theminer studies, on experiments on animals and onbiological and biophysical models suggest that, atleast at high levels, lifetime exposure to each ad-ditional pCi/L of indoor radon adds a lifetime riskof about 0.0134, 0.0026, 0.0088 and 0.0018 of lungcancer for male ever-smokers, male never-smokers,female ever-smokers, and female never-smokers, re-spectively (National Research Council, 1998). SeeTable 1 for the parameters that we use in this pa-per for each sex–smoking category.

The dose-response at low concentrations is dif-ficult to estimate, because all case-control studieshave been fairly small and because lifetime radonexposures are poorly estimated. Although a lineardose-response relation is plausible and is consistentwith case-control data, current data are also consis-tent with a threshold model or even a small benefi-cial effect at low doses (Cohen, 1995; Bogen, 1997;Lagarde et al., 1997; Lubin and Boice, 1997).

Partly from necessity and partly for historical rea-sons, radon researchers use a fairly large and con-fusing assortment of units. For instance, the radia-tion absorbed by the body (the “dose”) depends notjust on the concentration of radon in the air, butalso on the breathing rate and of course on the du-ration of exposure, so there is no simple conversionbetween radon concentration in indoor air and doseabsorbed by a human body. Moreover, radon’s decayproducts, rather than radon itself, deliver most ofthe radiation dose associated with radon, and thedifferential removal of decay products and radonitself can lead to variation in the relative concen-trations of each. For clarity and convenience, wewrite “the radon dose” when we mean “the dose fromradon and its decay products,” where standard pa-rameter estimates have been used to make all ofthe necessary adjustments. See Nazaroff and Nero(1988) for an overview of many of these issues.

We present cumulative exposures in terms ofpCi/L-years, rather than the historical unit (alsonon-SI) which is “working level months (WLM).”The direct conversion, for breathing 1 pCi/L air fora year, is 1 pCi/L-year = 0.26 WLM, but it is stan-


Table 1Estimated absolute and relative risks of lung cancer death for lifetime indoor exposure to radon*

Male Female

Ever-smoker Never-smoker Ever-smoker Never-smoker

Baseline absolute risk at 0 pCi/L 0.07409 0.00579 0.04349 0.00377Excess relative risk for additional pCi/L 0.1149 0.2827 0.1280 0.2998Excess absolute risk for additional pCi/L 0.0134 0.0026 0.0088 0.0018Average numbers in U.S. households 0.30 1.07 0.27 1.16

*Derived from BEIR VI (National Research Council, 1998), which describes the absolute increment in lung cancer risk resulting fromexposure to indoor radon beyond that from exposure to outdoor-background concentration of radon, and under the assumption that aperson spends 70% of his or her time indoors. Model is absolute risk = baseline risk×�1+excess relative risk�, with excess relative riskproportional to radon exposure. Average household populations by sex and smoking status are derived from the Statistical Abstract ofthe United States and CDC (1994), combining populations of children and adults. An ever-smoker is defined as a person who has smokedat least 100 cigarettes or the equivalent in his or her lifetime.

dard to assume that an individual is only at homeabout 70% of the time, so that a home concentra-tion of 1 pCi/L for a year leads to an exposure of0.18 WLM.

2.2 U.S. Residential Radon Concentrationsand Measurements

Residential radon measurements are commonlymade following a variety of protocols. The mostfrequently used protocol in the United States hasbeen the “screening” measurement: a short-term(2–7 day) charcoal-canister measurement madeon the lowest level of the home (often an unoc-cupied basement), at a cost of about $15 to $20.(Under a more recent protocol, measurements aretaken on the lowest living area level of the home.)Short-term measurements made at different timesduring the same season have an approximately log-normal distribution (i.e., the log measurements arenormally distributed) with a geometric standarddeviation (GSD, the exponential of the standarddeviation of the log measurements) of roughly 1.6,primarily due to temporal variation in indoor radonconcentrations.

In addition, because short-term measurementsare usually made on the lowest level of the homeand during the season of highest indoor radonexposure, they are upwardly biased measures ofannual living area average radon level. The magni-tude of this bias varies by season and by region ofthe country and depends on whether the basement(if any) is used as living space (White, Clayton,Alexander and Clifford, 1990; Klotz et al., 1993;Price and Nero, 1996); our estimated correction fac-tors for winter-season, lowest-level measurements,known as “screening” measurements, appear in Ta-ble 2. (If the short-term measurement is not madein winter, then an additional seasonal correction

factor is needed.) Due to the large temporal vari-ability and other sources of variation, a short-termmeasurement can predict the long-term living areaconcentration only to within a factor of 1.8 or so,even after correcting for systematic biases.

A radon measure that is far less common than thescreening measurement, but is believed to be muchbetter for evaluating radon risk, is a twelve-monthintegrated measurement of the radon concentration.By monitoring on every living level of the home(that is, on every floor in which people spend morethan a small amount of time each day), one canmeasure the “annual living area average radon con-centration,” or ALAA. For a typical home with twostories used as living space, such monitoring costsabout $50. These long-term living area measure-ments are not subject to the biases and effects ofday-to-day and seasonal variation that affect screen-ing measurements. A national sample of ALAA mea-surements was collected in the National ResidentialRadon Survey (NRRS) (see Marcinowski, Lucas andYeager, 1994).

The exact relationship between the ALAA concen-trations and the occupant exposures is not known;people spend different amounts of time in differ-ent areas of the home, long-term measurements arestill subject to some error, even on the same floordifferent rooms can have slightly different radonlevels and so on. For the purposes of this paper,we assume that an ALAA measurement (i.e., thearithmetic mean of long-term measurements madeon each occupied level of the home) estimates eachresident’s exposure to within a multiplicative errorwith a geometric mean (GM) of unity and a GSDof 1.2.

The distribution of annual-average living areahome radon concentrations in U.S. houses, as mea-sured in the NRRS, is approximately lognormal


Table 2Correction factors by which one must divide a short-term winter radon measurement to estimate annual-average living-area level*

No Basement is a Basement is notRegion basement living area a living area

New England 2.2 (1.3) 1.7 (1.1) 3.4 (1.3)New York/New Jersey 1.6 (1.3) 1.6 (1.3) 3.0 (1.3)Mid-Atlantic 1.6 (1.1) 1.6 (1.1) 2.8 (1.1)Southeast 1.3 (1.1) 1.9 (1.1) 2.3 (1.1)Midwest 1.2 (1.1) 1.6 (1.1) 2.2 (1.1)South 1.3 (1.1) 1.8 (1.2) 1.7 (1.1)Central Plains 1.5 (1.1) 1.7 (1.1) 3.1 (1.1)Big Sky and Plains 1.2 (1.1) 2.1 (1.1) 3.1 (1.1)Southwest 1.3 (1.1) 1.8 (1.2) 2.6 (1.1)Northwest 1.2 (1.1) 1.9 (1.1) 4.0 (1.1)

*Geometric standard errors of estimation for the correction factors are in parentheses; even if the correction factors were known perfectly,the annual average living area concentration would still be subject to large uncertainty due to temporal variability in the short-termmeasurements. From Price and Nero (1996).

with geometric mean (GM) 0.67 pCi/L and geo-metric standard deviation (GSD) 3.1 (Marcinowski,Lucas and Yaeger, 1994). (Throughout, we use theterm “house” to refer to owner-occupied ground-contact homes.) These data suggest that between50,000 and 100,000 homes have radon concentra-tions in primary living space in excess of 20 pCi/L.This level causes an annual radiation exposureroughly equal to the occupational exposure limit foruranium miners. Thirty years’ occupancy of such ahouse would yield an added estimated risk of lungcancer of about 2.4% among never-smokers and12.1% among ever-smokers. The lung cancer risksfrom radon are very high compared with the risksestimated for other kinds of environmental expo-sures regulated by the EPA (for comparison, seeU.S. Congress, Office of Technology, 1993).

2.3 Radon Remediation

Several radon control techniques have been devel-oped, tested and implemented (Henschel and Scott,1987; Prill, Fisk and Turk, 1990), and long-termperformances of these systems were reported (Turk,Harrison and Sextro, 1991). The currently preferredremediation method for most homes, “sub-slab de-pressurization,” costs about $1000–$1500 to installand requires constant use of a small electric fan;the net present value of such a system is about$2000, including the heating and cooling costs as-sociated with increased ventilation. Although long-term experience with these systems is lacking, forpurposes of our analysis we will assume that sucha system remains effective for 30 years. We arenot aware of any large-scale randomized studies onthe effect of remediation on radon levels, but manysmall nonrandomized studies have been conducted

and are summarized in an EPA report (Henschel,1993). These studies suggest that almost all homescan be remediated to below 4 pCi/L, while reduc-tions under 1 pCi/L are rarely attained with conven-tional methods, for homes with a very wide range ofpreremediation levels. For simplicity, we make theassumption that remediation will reduce radon con-centration to 2 pCi/L. For obvious reasons, little isknown about effects of remediation on houses thatalready have low radon levels; we will assume thatif the initial annual living area average level is lessthan 2 pCi/L, then remediation has no effect.

Recommendations for radon remediation vary bycountry, with Sweden setting a recommended actionlevel for the annual living area average (ALAA) in-door radon concentration of 10 pCi/L and Canadarecommending action at 20 pCi/L, compared to theU.S. level of 4 pCi/L. The current U.S. recommenda-tions, if fully implemented, would cost on the orderof $10–$20 billion in measurement and remediationcosts (see Nero, Gadgil, Nazaroff and Revzan, 1990).In Section 5, we discuss the efficiency of variouspolicies in terms of estimated dollars per life saved.

2.4 Individual and Public Decision Options

One can imagine an ideal world in which home-owners make monitoring and remediation decisionsbased on full knowledge of the current understand-ing of radon risk and remediation costs and effec-tiveness and taking into account their own risk tol-erance and financial state. In the real world, though,there is a substantial cost (in time and hassle) as-sociated with reaching that level of expertise, and itis reasonable for people to follow more general rec-ommendations on whether to take action and whatsort of action to take.


An important policy decision, well outside thescope of this paper, is just how general such recom-mendations should be in practice: should differentrecommendations be made for smokers and non-smokers, for old and young people, for large andsmall families, and so on. Given the estimated risksfrom radon, and the cost and effectiveness of reme-diation, an individual homeowner (or home selleror buyer) can make decisions about measurementand remediation. In addition, national, state andlocal governments can make recommendations forindividual decisions or, for stronger action, can re-quire measurement or remediation of new housesor existing houses, either in the entire country orin targeted areas.

From a long-term public health perspective, anapproach that does not depend on household compo-sition makes some sense; children are born, peopletake up smoking, houses are sold to other ownersand so on, so that basing recommended actions onaverage households is not totally unreasonable. Fol-lowing the procedures of the present paper, one canmake household-specific recommendations based onsuch factors, but we have not done so, choosing in-stead to focus only on geographic variation and afew house construction parameters. Realistically, wethink that if the EPA were to change its radon pol-icy, the most likely change would be to make ge-ographically specific recommendations rather thanto make recommendations that vary at the level ofindividual households.

3. GEOGRAPHIC MODELING OF INDOORRADON LEVELS

3.1 Data Sources and HierarchicalRegression Model

Although radon is thought to cause a large num-ber of deaths compared to other environmentalhazards, the vast majority of houses in the UnitedStates do not have elevated radon levels that wouldbe substantially reduced by remediation. Based onthe NRRS data, about 84% of homes have ALAAconcentrations under 2 pCi/L, and about 90% arebelow 3 pCi/L. A goal of some researchers hasbeen to identify locations and predictive variablesassociated with high-radon homes so that moni-toring and remediation programs can be focusedefficiently. One such effort at the Lawrence Berke-ley National Laboratory used Bayesian hierarchicalmodeling to analyze indoor radon measurements.These models include monitoring data, county indi-cators, a measure of surficial radium concentration,a climatological variable, and house constructioninformation and were fit separately in 10 regions of

the United States (Price, Nero and Gelman, 1996;Price, 1997; Revzan et al., 1998). These modelswere used to fit data from short-term measure-ments, which were calibrated to long-term livingarea averages as described by Price and Nero(1996). Combining short- and long-term measure-ments allowed us to estimate the distribution ofradon levels in nearly every county in the UnitedStates, albeit with widely varying uncertaintiesdepending primarily on the amount of monitoringdata within the county.

Unfortunately (from the standpoint of radon mit-igation programs), indoor radon concentrations arehighly variable even within small areas. Giventhe predictive variables mentioned in the previ-ous paragraph, the radon level of an individualhouse in a specified county can be predicted only towithin a factor of at best about 1.9, with a factorof 2.3 being more typical (Price, Nero and Gelman,1996; Price 1996), a disappointingly large predic-tive uncertainty considering the factor of 3.1 thatwould hold given no information on the home otherthan that it is in the United States. On the otherhand, this seemingly modest reduction in uncer-tainty is still enough to identify some areas wherehigh-radon homes are very rare or very common.For instance, in the mid-Atlantic states, more thanhalf the houses in some counties have long-termliving area concentrations over the EPA’s recom-mended action level of 4 pCi/L, whereas in othercounties fewer than 0.5 percent exceed that level(Price, 1996).

Various monitoring efforts demonstrate that thedistribution of indoor radon concentrations for anarea or region of almost any scale is reasonably wellrepresented by a lognormal distribution, or some-times the sum of two such distributions (Nero, Gad-jil, Nazaroff and Revzan, 1990). Further, a largearea’s distribution is effectively a mixture of the in-dividual distributions of the composite subareas, allof which are reasonably well represented by individ-ual lognormal distributions, with geometric means(GM’s) that vary from one subarea to another (seeNero, Schwehr, Nazaroff and Revzan, 1986; Price,Nero and Gelman, 1996, for example).

In each region of the country, a hierarchical linearregression model at the level of individual countieswas previously fit to the logarithms of home radonmeasurements (see Price, Nero and Gelman, 1996;Price, 1997). We shall apply these models to performinferences and decision analyses for previously un-measured houses i, using the following notation:

Ri = ALAA radon concentration in house i;θi = log�Ri�;


Xi = Vector of explanatory variables (includingcounty-level variables, house-level variables,and county indicators) for house i;

β = Vector of regression coefficients;τ2 = Variance component in the model correspond-

ing to variability between houses conditionalon the predictors;

σ2 = Variance component in the model correspond-ing to measurement variability within ahouse.

Then the unknown θi has the predictive distribu-tion,

θi�X;β ∼N�Xiβ; τ2�:(1)

There is some uncertainty in the coefficients β (par-ticularly for the indicators corresponding to coun-ties with few observations) and a small amount ofposterior uncertainty in the variance components ofthe model. For the purposes of this paper, we needonly know the predictive distribution for any givenθi, averaging over all these uncertainties; it will beapproximately normal (because the variance compo-nents are so well estimated), and we label it as

θi ∼N�Mi; S2i �:(2)

We write Mi = �Xβ̂�i, where β̂ is the posteriormean from the analysis in the appropriate regionof the country. The variance S2

i includes the poste-rior uncertainty in the coefficients β and also thewithin-county variance τ2. The GSD of the unex-plained within-county variation, eτ, is estimated tobe in the range 1.9–2.3 (depending on the region ofthe country) which puts a lower limit on eS. To beprecise, the GSD’s eS of the predictive distributionsfor home radon levels vary from 2.1 to 3.0, and theyare in the range �2:1;2:5� for most U.S. houses (thehouses with eS > 2:5 lie in small-population coun-ties for which little information was available in theradon surveys, resulting in relatively high predic-tive uncertainty within these counties). The GM’sof the house posterior predictive distributions, eM,vary from 0.1 to 14.6 pCi/L, with 95% in the range�0:3;3:7� and 50% in the range �0:6;1:6�. The houseswith the highest prior GM’s are houses with base-ment living areas in high-radon counties; the houseswith lowest prior GM’s have no basements and liein low-radon counties. See Price and Nero (1996)for more details on the characteristics of high- andlow-radon houses.

3.2 Using the Model as Input to Decision Analysis

This paper focuses on decisions, not modeling.For the rest of the paper we work at the individ-ual house level and use the posterior inference for

house i from the model discussed above as our priordistribution for the subsequent analysis. Since weare considering decisions for houses individually,we suppress the subscript i for the rest of the paper.

The distribution (2) summarizes the state ofknowledge about the radon level in a house givenits county and basement information. Now supposea measurement y ∼N�θ; σ2� is taken in the house.(We are assuming an unbiased measurement. Ifa short-term measurement is being used, it willhave to be corrected for the bias shown in Table2, and for an addition seasonal correction factor,if the measurement was not made in winter (e.g.,see Mose and Mushrush, 1997; Pinel, Fearn, Darbyand Miles, 1995). In our notation, y and θ are thelogarithms of the measurement and the true ALAAradon level, respectively. The posterior distributionfor θ is

θ�M;y ∼N�3;V�;(3)

where

3 = M/S2 + y/σ2

1/S2 + 1/σ2; V = 1

1/S2 + 1/σ2(4)

(see, e.g., Gelman, Carlin, Stern and Rubin, 1995).We base our decision analysis of when to measureand when to remediate on the distributions (2)and (3).

Before moving to the decision analysis, we brieflydiscuss the relevance of the hierarchical aspect ofour radon model. In a classical regression model, theestimated distributions of home radon levels varyacross counties because of the geographic variationin the regression predictors (for our model, theseare listed in the first paragraph of Section 3). In thehierarchical regression model, the county estimatesare allowed to vary from the regression prediction,by an amount dependent on the observational datain the county. As a result, the recommended deci-sions within any county depend on the availabledata for that county as well as on the estimatedregression coefficients.

4. INDIVIDUAL DECISIONS ON WHETHER TOMONITOR OR REMEDIATE

The suggestion that every home should monitor ishighly conservative (we might also say highly “pro-tective”), based on the knowledge that homes withelevated radon concentrations have been found inevery state, so the only way to be sure that a homedoes not have an elevated concentration is to test.However, if the risk is low enough [i.e., if the pre-dicted radon level Mi = exp�Xiβ� is low for housei], then even the small cost of monitoring may notbe worthwhile.


We now work out the optimal decisions of mea-surement and remediation conditional on the pre-dicted radon level in a home, the additional risk oflung cancer death from radon, the effects of remedi-ation and individual attitude toward risk. We followa standard approach in decision analysis (see, e.g.,Watson and Buede, 1987) by proceeding in twosteps: first, decision-making under certainty—atwhat level would you remediate if you knew R,your home radon level?—and, second, averagingover the uncertainty in R.

4.1 Decision-making under Certainty

We shall express decisions under certainty inthree ways, equivalent under a linear no-thresholddose-response relationship.

1. The dollar value Dd associated with a reductionof 10−6 in probability of death from lung cancer(the value of a microlife). If one applies a 5% peryear discounting of the value of a life and anexpected twenty-year lag to lung cancer death,then Dd corresponds to a net present value of amicrolife of 1:0520Dd = 2:7Dd.

2. The dollar value Dr associated with a reductionof 1 pCi/L in home radon level for a thirty-yearperiod (the equivalent dollar cost per unit ofradon exposure).

3. The home radon level Raction above which youshould remediate if your radon level is known.

We need to work with all three of these concepts be-cause, depending on the context, either Dd, Dr orRaction will be most relevant for individual decision-making. In any case, the essence of the radon deci-sion is a tradeoff between dollars and lives.

Initially, we make the following assumptions.

• The increase of probability of lung cancer deathis a linear function of radon exposure (consis-tent with current concepts of dose effects inhigh linear-energy-transfer radiation; see Upfal,Divine and Siemiatychi, 1995). The added riskdiffers for smokers and nonsmokers and for malesand females; we use estimates γg; s (g = male orfemale, and s = ever-smoking or never-smoking)for the additional lifetime risk per additionalpCi/L exposure as derived from the Committeeon Health Risks of Exposure to Radon (BEIR VI,National Research Council, 1998); see Table 1.• Remediation takes a house’s annual-average

living-area radon level down to a level Rremed if itwas above that, but leaves it unchanged if it wasbelow that. We shall assume that Rremed has thevalue 2 pCi/L.

• Mitigation costs $2000, including the net presentvalue of future energy cost to run the mitigationsystem.• Decisions will be made based on the consequences

over the next 30 years.• If a measurement is taken, it is a long-term mea-

surement that is an unbiased measure of annual-average living-area exposure with a measurementGSD of 1.2, and it costs $50.

We can now determine the equivalent cost Dr perpCi/L of home radon exposure and the action levelRaction for remediation given the following individ-ual information.

• The numbers of male and female ever-smokersand never-smokers in the house, ng; s; see Table 1.• The dollars Dd that would be paid to reduce the

probability of lung cancer death by one-millionth.From the risk assessment literature, typical val-ues for medical interventions are in the range of$0.1 to $0.5 (see, e.g., Eddy, 1989, 1990; Owens,Harris, Scott and Nease, 1996). Higher values areoften found in other contexts, for example, juryawards for deaths due to negligence, values usedin legislating industrial risks and risk tradeoffsbetween worker wages and fatality risks (see Vis-cusi, 1992, for an excellent survey of values ofrisks to life and health). However, we feel thatthe lower values are reasonable in this case since,like medical intervention, expenditure on radonremediation is voluntary and is aimed at reduc-ing future risk rather than compensating for jobfatality.

For any given household, the equivalent cost perpCi/L, Dr, can be computed as a function of therisk assumed above and the individual parametersand Dd:

Dr =3070

(∑g; s

ng; sγg; s

)106Dd;(5)

where the fraction 30/70 is the ratio of the thirty-year decision period to a seventy-year life ex-pectancy per occupant. For U.S. homes, the averagevalue of

∑g; s ng; sγg; s is 0.0113 (see Table 1). We

can also compute the remediation concentrationRaction, given the equivalent cost and the aboveassumptions of cost and effects of remediation:

Raction =$2000Dr

+Rremed:(6)

4.2 Individual Choice of a RecommendedRemediation Level under Certainty

The U.S., English, Swedish and Canadian recom-mended remediation levels are Raction = 4, 5, 10 and


20 pCi/L, which, with Rremed = 2 pCi/L, correspondto equivalent costs per pCi/L of Dr = $1000, $670,$250 and $111, respectively. Setting the valuesof ng; s to the average numbers of male and fe-male ever-smokers and never-smokers in a U.S.household implies dollar values per microlife ofDd = $0:21, $0.14, $0.05 and $0.02, respectively.This suggests that, to the extent that we believe thestandard estimates of radon risk and remediationeffects, the U.S. and English recommendationsare on the low end for acceptable risk reductionexpenditures, and the Canadian and Swedish rec-ommendations are too cavalier about the radonrisk. However, this calculation obscures the dra-matic difference between smokers and nonsmokers,which arises entirely from the difference in risk perdose associated with the two groups. For example, ahousehold of one male never-smoker and one femalenever-smoker that is willing to spend $0.21 per per-son to reduce the probability of lung cancer by 10−6

should spend $370 per pCi/L of radon reduction,implying an action level of Raction = 7:4 pCi/L. Incontrast, if the male and female are both smokers,they should be willing to spend the much highervalue of $1900 per pCi/L, because of their higherrisk per pCi/L and thus should have an action levelof Raction = 3:1 pCi/L.

Other sources of variation in Raction, in additionto varying risk preferences, are (a) variation in thenumber of smokers and nonsmokers in households,(b) variation in individual beliefs about the risksof radon and the effects of remediation, and (c)variation in the perceived dollar value associatedwith a given risk reduction. From a public policystandpoint, one might wish to ignore the varia-tion attributable to (a), since over the thirty-yearperiod of assumed remediation effectiveness thehousehold composition is likely to change, and in-deed the house is likely to be sold to several sets ofnew owners with possibly different smoking habits.However, as a practical matter, the homeownersare likely to perform remediation only if they fore-see major risk reductions for themselves, or if theyare planning to sell their house and fear that anelevated radon concentration will reduce its value.As illustrated above, a male-female never-smokingcouple might choose an action level of 7.4 pCi/L orhigher, depending on their willingness and ability topay for risk reduction, whereas most smokers maybe more willing to risk lung cancer than are non-smokers and thus might be unwilling to remediateat levels as low as 3.1 pCi/L.

Through the rest of the paper, we use 4 pCi/L asan exemplary value, but rational informed individ-uals might plausibly choose quite different values

of Raction, depending on smoking habits, risk toler-ance, financial resources and the number of peoplein the household.

4.3 Decision-making under Uncertainty

Given an action level under certainty, Raction, wenow address the question of whether to pay for ahome radon measurement and whether to remedi-ate. The decision of whether to measure dependson the prior distribution, (2) of radon level for yourhouse, given your predictors X. The decision ofwhether to remediate depends on the posterior dis-tribution, (3) if a measurement has been taken orthe prior distribution, (2) otherwise. In our compu-tations, we shall make use of the following resultsfrom the normal distribution: if z ∼ N�µ; s2�, thenE�ez� = exp�µ+ �1/2�s2� and E�ez�z > a�Pr�z >a� = exp�µ+ �1/2�s2��1−8��µ+ s2 − a�/s��, where8 is the standard normal cumulative distributionfunction.

The decision tree is set up as three branches. Ineach branch, we evaluate the expected loss in dollarterms, converting radon exposure to dollars usingDr = $2000/�Raction−Rremed� as the equivalent costper pCi/L for additional home radon exposure.

1. Remediate without monitoring. Expected lossis remediation cost + equivalent dollar cost ofradon exposure after remediation,

L1 = $2000+DrE�min�R;Rremed��= $2000+Dr

[RremedPr�R ≥ Rremed�

+ E�R�R < Rremed�Pr�R < Rremed�]

= $2000+Dr

[Rremed8

(M− log�Rremed�

S

)

+ exp(M+ 1

2S2)

·(

1−8(M+S2 − log�Rremed�

S

))]:

(7)

2. Do not monitor or remediate. Expected loss is theequivalent dollar cost of radon exposure,

L2 = DrE�exp�θ�� = Dr exp(M+ 1

2S2):(8)

3. Take a measurement y (measured in log pCi/L).The immediate loss is measurement cost (as-sumed to be $50) and, in addition, the radonexposure during the year that you are takingthe measurement [which is 1/30 of the thirty-year exposure (8)]. The inner decision has twobranches.(a) Remediate. Expected loss is computed as fordecision 1, but using the posterior rather thanthe prior distribution,


L3a = $50+Dr

130

exp(M+ 1

2S2)+ $2000

+Dr

[Rremed8

(3− log�Rremed�√

V

)

+ exp(3+ 1

2V

)

·(

1−8(3+V− log�Rremed�√

V

))];

(9)

where 3 and V are the posterior mean and vari-ance, from equation (4).(b) Do not remediate. Expected loss is

L3b = $50+Dr

130

exp(M+ 1

2S2)

+Dr exp(3+ 1

2V

):

(10)

4.3.1 Decision of whether to remediate given ameasurement. To evaluate the decision tree, wemust first consider the inner decision between 3(a)and 3(b), conditional on the measurement y. Let y0be the point (in log space) at which you will chooseto remediate if y > y0, or do nothing if y < y0. (Be-cause of measurement error, y 6= θ, so ey0 6= Raction.)We shall solve for y0 in terms of the prior mean M,the prior standard deviation S, and the measure-ment standard deviation σ , by solving the implicitequation

L3a = L3b at y = y0:(11)

The expected losses L3a and L3b depend on y0 onlythrough 3 = �M/S2 + y/σ2�/�1/S2 + 1/σ2�, and sowe can solve for y0 by first solving for 30 in (11),then setting

y0 =(

1+ σ2

S2

)30 −

σ2

S2M:(12)

Thus the relation between y0 and M is linear,with the slope depending only on the variance ratioσ2/S2.

Given σ2/S2 and Dr, we solve for 30 numeri-cally, using the bisection method to converge on thevalue of 3 that satisfies (11). Figure 1 shows themeasurement action level ey0 as a function of theperfect-information action level Raction, evaluated atvalues of the prior GM radon level eM ranging from

Fig. 1. Measurement action levels ey0 as a function of the perfect-information action level Raction, evaluated at values of the priorGM radon level eM ranging from 0:5 pCi/L to 4 pCi/L. For agiven Raction and prior GM, find the “measurement threshold”;if the measured value exceeds this threshold, then remediationis recommended. The threshold differs substantially from Ractiononly for low prior GM and high Raction.

0.5 to 4.0. For this example, we have assumed thatσ = log�1:2�, and that S = log�2:3� for all counties.

4.3.2 Deciding whether to measure. We deter-mine the expected loss for branch 3 of the decisiontree by averaging over the prior uncertainty in themeasurement y,

L3 = E�min�L3a;L3b��:(13)

Given �M;S;σ;Dr�, we evaluate this expression asfollows.

1. Simulate 5000 draws of y ∼N�M;S2 + σ2�.2. For each draw of y, compute min�L3a;L3b� from

(9) and (10).3. Estimate L3 as the average of these 5000 values.

Of course, this expected loss is valid only if we as-sume that you will make the recommended optimaldecision once the measurement is taken.

We can now compare the expected losses L1, L2,L3, and choose among the three decisions. Figure2 displays the expected losses as a function of theperfect-information action level Raction for severalvalues of eM. As with Figure 1, we illustrate withσ = log�1:2� and S = log�2:3�. For any value ofM and Raction, the recommended decision is the onewith the lowest expected loss.

For any Raction, we can summarize the deci-sion recommendations as the cut-off levels Mlow


Fig. 2. Expected losses in dollars (including the dollar value of the expected reductions in radon levels) of the three decisions: �1� re-mediate, �2� do nothing, �3� monitor (take a measurement), as a function of the perfect-information action level Raction. The four plotscorrespond to four different values of the prior geometric mean radon level eM.

Fig. 3. Recommended decisions as a function of the perfect-information action level Raction and the prior geometric mean radon level eM;under the simplifying assumption that eS = 2:3. You can read off your recommended decision from this graph and, if the recommendationis “take a measurement,” you can do so and then use Figure 1 (with interpolation or extrapolation if necessary) to tell you whether toremediate. The horizontal axis of this figure begins at 2 pCi/L because remediation is assumed to reduce ALAA radon level to 2 pCi/L,so it makes no sense for Raction to be lower than that value. Wiggles in the lines are due to simulation variability.

and Mhigh for which decision 1 is preferred ifM > Mhigh, decision 2 is preferred if M < Mlow,and decision 3 is preferred if M ∈ �Mlow;Mhigh�.Figure 3 displays these cut-offs as a function ofRaction, and thus displays the recommended deci-

sion as a function of �Raction; eM�, once again under

the simplifying assumption that σ = log�1:2� andS = log�2:3� for all counties. For example, settingRaction = 4 pCi/L leads to the following recommen-dation based on eM, the prior GM of your home


radon based on your county and house type:

• If eM is less than 1.0 pCi/L (which corresponds to68% of U.S. houses), do nothing.• If eM is between 1.0 and 3.5 pCi/L (27% of U.S.

houses), perform a long-term measurement (andthen decide whether to remediate).• If eM is greater than 3.5 pCi/L (5% of U.S. houses),

remediate immediately without measuring. Actu-ally, in this circumstance, short-term monitoringturns out to be (barely) cost-efficient: the reasonfor the recommendation of immediate remedia-tion is that the excess risk associated with oc-cupying the home for a year while a long-termmeasurement is made is not worth bearing, giventhe high likelihood that the home will eventuallybe remediated anyway. However, if a short-termmeasurement is made and is sufficiently low, thenthe home is unlikely to have such an exceptionallyhigh level that one additional year of exposurecarries a large risk. In this case, long-term moni-toring can be performed to determine whether re-mediation is really indicated. We will ignore thisadditional complexity to the decision tree, sinceit occurs rarely and has very little impact on theoverall cost-benefit analysis.

4.4 Decision-Making If a Short-term MeasurementHas Been Taken

We do not in general recommend taking short-term measurements, because long-term measure-ments are much superior in terms of both bias andvariance. However, short-term measurements arequite popular (partly because these are often takenas a condition of sale of a house), and so it is worthconsidering the decision problem in this situation.

In fact, the above decision framework is immedi-ately adaptable to a homeowner who has alreadytaken a short-term measurement. The only changethat needs to be made is that the prior distribu-tion (2) needs to be updated given the informationfrom the short-term measurement. We thus replaceM and S2 in the above formulas by

Mnew =M/S2 + �yst − log b�/σ2

st

1/S2 + 1/σ2st

;

S2new =

11/S2 + 1/σ2

st;

(14)

where yst is the logarithm of the short-term mea-surement, b is the correction factor derived fromTable 2 and σst = log�1:8�. If the short-term mea-surement was not made in winter, then a seasonalcorrection factor will also apply; see, for example,Mose and Mushrush (1997) and Pinel et al. (1995).

At this point, we can return to the procedure de-scribed in the previous sections.

4.5 Summary of the Individual Decision Process

Ideally, an individual homeowner in the UnitedStates can now make a remediation decision usingthe following process:

1. Determine the radon level Raction above whichyou would remediate, if you knew your homeradon level exactly. This value can be chosen inits own right or by choosing a value of Dr basedon the perceived gains from lowering radon levelor by assigning a dollar value Dd to a millionthof a life and computing based on the numberof ever-smokers S and never-smokers N in thehouse. As discussed in Section 4.2, current un-derstanding of the risks of radon and the effectsof remediation suggest that the EPA’s recommen-dation of 4 pCi/L is a reasonable catch-all value,with 8 pCi/L being a more reasonable value fornonsmokers.

2. Look up eM and eS, the GM and GSD of the poste-rior predictive distribution for your home’s radonlevel, as estimated from the hierarchical modeldescribed in Section 3.

3. If a short-term measurement has been taken, up-date the prior distribution using (14) and the biascorrection from Table 2 (and possibly an addi-tional seasonal correction).

4. Calculate the expected losses of decisions (1), (2)and (3) from the formulas in Section 4.3 and, ifdecision (3) is chosen, the recommended mea-surement action level ey0 . The recommendeddecision—that with the lowest expected loss, cor-responds to that indicated in Figure 4.3.2 (withslight alterations depending on the exact valueof S).

5. If decision (3) is chosen, perform a long-termmeasurement. In one year, the measurement ey

is available. Remediate if ey > ey0 .

We are in the process of setting up a website athttp://www.stat.columbia.edu/radon to automatethe steps listed above and supply other informationabout decision-making for radon hazards.

5. AGGREGATE CONSEQUENCE OFDECISION STRATEGY

Now that we have made idealized recommenda-tions, we consider their aggregate effects if followedby all homeowners in the United States. In par-ticular, how much better are the consequencescompared to other policies such as the current one,implicitly endorsed by the EPA, of taking a short-


Fig. 4. Map (a) showing fraction of houses in each county for which measurement is recommended, given the perfect-informationaction level of Raction = 4 pCi/L; (b) expected fraction of houses in each county for which remediation will be recommended, once themeasurement y has been taken. For the present radon model, within any county the recommendations on whether to measure and whetherto remediate depend only on the house type: whether the house has a basement and whether the basement is used as living space. Apparentdiscontinuities across the boundaries of Utah and South Carolina arise from irregularities in the radon measurements from the radonsurveys conducted by those states, an issue we ignore in the present paper.

Fig. 5. Map (a) showing fraction of houses in each county for which measurement is recommended, given the perfect-information actionlevel ofRaction = 8 pCi/L; (b) expected fraction of houses in each county for which remediation will be recommended, once the measurementy has been taken. As with the previous figure, the decision recommendations depend only on county and house type.

term measurement as a condition of a home saleand performing remediation if the measurement ishigher than 4 pCi/L?

5.1 Estimated Consequences of Applying theRecommended Decision Strategy to theEntire United States

Figures 4 and 5 display the geographic pattern ofrecommended measurements (and, after one year,recommended remediations), based on action levelsRaction of 4 and 8 pCi/L, respectively. These recom-mendations incorporate the effects of parameteruncertainties in the models that predict radon

distributions within counties, so these maps wouldbe expected to change somewhat as better pre-dictions become available. Note that these mapsare not based on a single estimated parametersuch as “the probability that a home’s concentra-tion exceeds 4 pCi/L.” Although a discrete actionlevel does play a role in the decision process (af-ter all, each home must either monitor or not, andremediate or not) the benefit of remediation is acontinuous function of the initial radon concen-tration, and that concentration is assumed to bedrawn from a continuous distribution. It is the con-fluence of these continuous distributions and the


discrete willingness-to-remediate point that giverise to the fairly complex expressions for expectedloss in Section 4.3.

From a policy standpoint, perhaps the most sig-nificant feature of the maps is that even if the EPA’srecommended action level of 4 pCi/L is assumed tobe correct—and, as we have discussed, it does leadto a reasonable value of Dd, under standard dose-response assumptions—monitoring is still not rec-ommended in most U.S. homes. Indeed, only 28%of U.S. homes would perform radon monitoring. Ahigher action level of 8 pCi/L, a reasonable valuefor nonsmokers under the standard assumptions,would lead to even more restricted monitoring andremediation: only about 5% of homes would performmonitoring.

5.2 Decision Strategies Considered andEvaluation Criteria

In this section, we shall consider various decisionstrategies.

1. Follow the recommended strategy from Section4.3 (that is, monitor homes with prior mean es-timates above a given level, and remediate thosewith high measurements).

2. Perform long-term measurements on all housesand then remediate those for which the measure-ment exceeds a specified level: ey > Raction.

3. Perform short-term measurements on all housesand then remediate those for which the bias-corrected measurement exceeds a specified level:eyst/b > Raction (with b defined as described inSection 4.4).

4. Perform short-term measurements on all housesand then remediate those for which the uncor-rected measurement exceeds a specified level:eyst > Raction.

We evaluate each of the above strategies in termsof aggregate lives saved and dollars cost, withthese outcomes parameterized by the radon actionlevel Raction. Both lives saved and costs are con-sidered for a thirty-year period. For each strategy,we assume that the level Raction is the same forall houses (this would correspond to a uniform na-tional recommendation) and that 0.30 male and0.27 female ever-smokers and 1.07 male and 1.16female never-smokers live in each house (or, rather,that these are the averages over the thirty-yearperiod).

We also evaluate strategies based on the esti-mated cost per life saved. This aggregate cost perlife is different from the marginal cost per life usedto set the action level Raction in Section 4.2. For ex-ample, as discussed previously, an action level ofRaction = 4 pCi/L approximately corresponds to anet present value of $0.21 per microlife, which corre-sponds to a marginal cost of $210,000 per life saved.However, if the optimal recommendation is followedfor the entire country, the estimated aggregate costper life saved is only $87,000: the aggregate costaverages over the whole population, ranging frommitigations that are barely cost-effective throughmitigations that are highly efficient in terms of riskreduction for a given cost. See also Figure 10 for acomparison of aggregate and marginal costs per lifesaved.

5.3 Modeling the Variation in the Population ofU.S. Homes

Because we use inferences from a hierarchicalmodel, we are able to give different recommen-dations for different houses in the population ascharacterized by location as well as continuouscovariates.

Thus, aggregate effects are determined by addingup the individual decisions over all the ground-contact homes in the country. Considering 3078counties with three house types within each, wehave 3078 × 3 pairs of �M;S� obtained from thehierarchical model fit to the national and stateradon survey data as described in Section 3. Given�M;S;Raction�, the decisions of whether to moni-tor and whether to measure are made as describedin Section 4.5, and expected number of lives savedand cost spent are assessed if remediation is imple-mented.

For any of the decision strategies, in any givenhouse, we evaluate the total cost,

Expected cost = $50 Pr�measurement�

+ $2000 Pr�remediation�;(15)

where

Pr�measurement�

=

1�Mlow<M<Mhigh�; for strategy 1;

1; for strategies 2, 3 and 4,


and

Pr�remediation�

=

Pr�M>Mhigh�+ Pr

(�Mlow <M <Mhigh� and �y > y0�

)

= 1�M>Mhigh� + 1�Mlow<M<Mhigh�

·(

1−8(3−y0√V

)); for strategy 1;

Pr�y > log�Raction��

= 1−8(3− log�Raction�√

V

);

for strategy 2;

Pr�yst − log b > log�Raction��

= 1−8(Mnew − log�Raction�

Snew

);

for strategy 3,

Pr�yst > log�Raction��

= 1−8(Mnew − log�Raction + log b

Snew

);

for strategy 4

(with $50 replaced by $15 for strategies 3 and 4 inwhich short-term measurements are used), and weevaluate the expected lives saved,

Expected lives saved

= A · E�max�eθ −Rremed;0��remediation�· Pr�remediation�

= A ·RreducedPr�remediation�;

(16)

where

Rreduced = exp�3+V/2�8(3+V− log�Rremed�√

V

)

−Rremed8

(3− log�Rremed�√

V

);

and A is the expected lives lost in a thirty-year pe-riod per pCi/L of home radon exposure, given byDr/Dd from equation (5) for any home, and equalto 0.0113 for the “average household” of 0.3 maleever-smokers, 1.07 male never-smokers, 0.27 femaleever-smokers and 1.16 female never-smokers. In theabove formulas, 3 and V are given by (4), and Mnewand Snew are given by (14).

We evaluate the expectations in (15) and (16) bysimulation. First, we simulate 5000 draws of y ∼N�M;S2+σ2� (for strategies 1 and 2) or y ∼N�M+log b; S2+ = σ2

st� (for strategies 3 and 4). Second,for each draw of y, we compute A · Rreduced under

the constraints of M > Mhigh or ��Mlow < M <Mhigh� and �y > y0�� or y > log�Raction�, and thenestimate (16) and (15) as the average of these 5000draws. Simulations average over uncertainties inhome radon levels R and variability in measure-ments ey (or eyst ). For these calculations we usedthe actual model estimates of S, rather than set-ting them all equal to a single value as was donefor illustrative purposes in the previous section.

We then multiply by the total number of groundcontact houses for each �M;S�, that is, for eachhouse type and for each county, and sum them upto get expected total costs and lives saved over athirty-year period in the United States.

5.4 Results

For the present county-level radon model, withineach county monitoring is recommended for somesubset of homes: for all homes, for all homes withbasements, for all homes with living-area base-ments or for no homes. The maps in Figure 4display, for each county, the fraction of houses thatwould measure and the estimated fraction of housesthat would remediate if the recommended decisionstrategy were followed everywhere with Raction = 4pCi/L. About 26% of the 70 million ground-contacthouses in the United States would monitor. Thiswould result in detection of and remediation of2.8 million homes above 4 pCi/L (74% of all suchhomes), and 840,000 of the homes above 8 pCi/L(91% of all such homes). Some additional estimatesof the program’s effectiveness are presented in Ta-bles 3 and 4, and Figure 5 displays similar mapsfor an 8 pCi/L action level.

In order to understand the effects of the differ-ent decision strategies on aggregate outcomes, wehave developed a series of graphs. Figures 6 and7 illustrate the efficiency of the recommended re-mediation strategy by showing the overall distribu-tions of radon levels (and total radon exposures) andthe distributions of homes to be monitored and re-mediated. As is apparent in the figures, even withthe large uncertainties in individual county distri-butional parameters the recommended program isquite effective at focusing on the homes with thehighest indoor radon concentrations.

Figure 8 displays the trade-off between expectedcost and expected lives saved over a thirty-year pe-riod for the four strategies listed in Section 5.2.The numbers on the curves are action levels Raction.This figure allows us to compare the effectivenessof alternative strategies of equal expected cost orequal expected lives saved. For example, the rec-ommended strategy (the solid line on the graph) atRaction = 4 pCi/L would result in an expected 83,000


Table 3Some summary statistics on the effectiveness of various home radon measurement and remediation strategies

for the 4 pCi/L action level*

Strategy

1 2 3 4

Fraction of all U.S. homes that would measure 26% 100% 100% 100%Fraction of all U.S. homes that would remediate 5% 6% 8% 17%Fraction of all homes over 4 pCi/L that would remediate 74% 89% 74% 92%Fraction of all homes over 8 pCi/L that would remediate 91% 100% 95% 99%

Total cost ($ billion) 7.32 11.56 12.20 25.06total cost of measuring 0.97 3.50 1.06 1.06total cost of remediation 6.35 8.06 11.14 24.00

Expected lives saved 84,000 97,000 88,000 110,000ever-smokers 49,000 57,000 51,000 64,000never-smokers 35,000 40,000 37,000 46,000

Aggregate $ cost per life saved 87,000 119,000 138,000 228,000

*(1) recommended strategy based on decision analysis using the hierarchical model, (2) long-term measurements on all houses, (3) bias-corrected short-term measurements on all houses, (4) uncorrected short-term measurements on all houses. All are based on an actionlevel of Raction = 4 pCi/L. Costs and lives saved cover 30 years.

Table 4Some summary statistics on the effectiveness of various home radon measurement and remediation strategies

for the 8 pCi/L action level*

Strategy

1 2 3 4

Fraction of all U.S. homes that would measure 5% 100% 100% 100%Fraction of all U.S. homes that would remediate 0.7% 1.4% 2.5% 7.0%Fraction of all homes over 4 pCi/L that would remediate 12% 26% 37% 67%Fraction of all homes over 8 pCi/L that would remediate 44% 87% 70% 91%

Total cost ($ billion) 1.11 5.54 4.63 10.94total cost of measuring 0.19 3.50 1.06 1.06total cost of remediation 0.92 2.04 3.53 9.84

Expected lives saved 27,000 50,000 51,000 82,000ever-smokers 16,000 29,000 30,000 48,000never-smokers 11,000 21,000 21,000 34,000

Aggregate $ cost per life saved 42,000 110,000 90,000 133,000

*(1) recommended strategy based on decision analysis using the hierarchical model, (2) long-term measurements on all houses, (3) bias-corrected short-term measurements on all houses, (4) uncorrected short-term measurements on all houses. All are based on an actionlevel of Raction = 8 pCi/L. Costs and lives saved cover 30 years.

lives saved at an expected cost of $7.3 billion. Let uscompare this to the EPA’s implicitly recommendedstrategy based on uncorrected short-term measure-ments (the dashed line on the figure). For the samecost of $7.3 billion, the uncorrected short-term strat-egy is expected to save only 32,000 lives; to achievethe same expected savings of 83,000 lives, the un-corrected short-term strategy would cost about $19billion.

Figure 9 displays these results in another way,as estimated cost per life saved, as a function of ex-pected cost, for the four strategies. Finally, Figure

10 displays the estimates for both marginal andaverage cost per life saved, for the recommendeddecision strategy, as a function of the radon ac-tion level Raction. The average cost per life savedis estimated as described above, and the marginalcost per life saved is simply 106Dd (as defined inSection 4.1). Average cost per life saved is alwayslower than marginal cost because, for any actionlevel, the average includes all houses at or abovethat level, and remediations are more efficient (interms of lives saved per dollar) in the higher-radonhouses.


Fig. 6. Estimated distributions of annual living area average radon concentrations in (upper, solid line) all U.S. houses, (middle, dottedline) all houses where measurement is recommended under the optimal strategy (with Raction = 4 pCi/L), and (lower, dashed line) allhouses where remediation is recommended immediately or after measurement.

Fig. 7. Fraction of total radon exposure, as a function of indoor radon concentration. Where the previous plot shows f�θ�; this plot showsθf�θ�. Curves are shown for (upper, solid line) all U.S. houses, (middle, dotted line), all houses where measurement is recommendedunder the optimal strategy (with Raction = 4 pCi/L) and (lower, dashed line) all houses where remediation is recommended immediatelyor after measurement.

6. SENSITIVITY TO ASSUMPTIONS

Our results are subject to potential error in:

• Estimates of annual average living area radon ex-posure (and its variation) from home radon mea-surements and the hierarchical model (includingbasement information and geographic predictors).

• The magnitude of cancer risk from a given radonconcentration (including the assumed linearity ofcancer risk as a function of radon level).• The effects of remediation.

In this section, we consider each of these issues inturn and then discuss other factors, involving indi-


Fig. 8. Expected lives saved versus expected cost, for vari-ous radon measurement–remediation strategies discussed in Sec-tion 5:2. Numbers indicate values of Raction. The solid line is forthe recommended strategy of measuring only certain homes; theothers assume that all homes are measured. All results are esti-mated totals for the United States over a thirty-year period.

vidual preferences and behavior, that might affectthe decisions.

Statistical model of home radon levels. The modelhas been extensively validated (see Price, Nero andGelman, 1996; Price and Nero, 1996; Price, 1997).

Fig. 9. Estimated cost per life saved versus expected cost (over athirty-year period), for various radon measurement–remediationstrategies discussed in Section 5:2. The solid line is for the recom-mended strategy of measuring only certain homes; the others as-sume that all homes are measured. For comparison, remediatingevery house without making any measurements has an estimatedcost per life saved of $2.2 million.

Fig. 10. Estimated average and marginal costs per life savedversus action level Raction for the recommended decision strategy.Average cost per life saved is computed averaging over the distri-bution of U.S. houses, as displayed in Figure 8. Marginal cost perlife saved is 106D (as defined in Section 4.1) based on a householdwith 0:3 male and 0:27 female ever-smokers and 1:07 male and1:16 female never-smokers. Marginal cost is always higher thanaverage cost because the marginal houses are those for which itis just barely cost-effective to remediate.

In general, the model behaves well; cross-validationindicates that the uncertainty intervals are approx-imately correct, for example. However, it is likelythat the lognormality assumption (for homes in agiven county, with a given set of explanatory vari-ables) underestimates the number of homes in thehigh tail of radon concentrations for some counties.For instance, Hobbes and Maeda (1997) suggest thatsome counties in southern California might be bet-ter fit as a mixture of two lognormals, one with alow geometric mean for most of the homes and onewith a high geometric mean for the small fraction ofhomes on a particular geologic deposit. Similar high-radon pockets or exceptionally high within-countyvariability are known to occur in a few counties inFlorida, New York, Washington and elsewhere.

From the standpoint of individual decisions, anunderestimate of the size of the very high tail ofradon concentrations would generally have a smalleffect: as long as the cumulative exposure for homesexceeding the action level is not seriously in error,the recommendation of whether or not to monitorwill not be affected, so if the fraction of homes over4 pCi/L or 8 pCi/L is fairly accurately estimated us-ing the lognormal approximation, the exact distri-bution of a small number of very high homes is notcritical. The fraction of homes over 4 or 8 pCi/L isfairly well estimated under the lognormal approxi-mation for most of the counties with GM’s over 1 or2 pCi/L, respectively, and most counties with GM’slower than that have such low numbers of homes


over the action level that even a large relative errorin their prevalence would probably not change themonitoring recommendations. Given the large num-ber of counties in the United States (over 3000) itis very likely that there are at least a few for whichnonlognormality is a significant issue, but it is un-likely to seriously affect most of our results.

This whole issue becomes more important,though, if the action level is set very high (e.g.,for a female nonsmoker living alone); we would nottrust the model’s exact predictions when estimat-ing the frequency of rare cases such as homes over20 pCi/L.

On a different model-related topic, it is possi-ble that the model can be improved by includingmore spatial or geological information (see, e.g.,Boscardin, Price and Gelman, 1996; Geiger andBarnes, 1994; Mose and Mushrush, 1997; Miles andBall, 1996), which would cause predictions for indi-vidual homes to become more precise and the priorstandard deviations S to decrease. For instance,radon mapping within counties would allow recom-mendations to discriminate more precisely amonghouses and thus increase expected lives saved forany given dollar expenditure. Indeed, such targetedrecommendations may already be possible in lo-calized (subcounty) areas that can be confidentlyidentified as having disproportionate numbers ofhigh-radon homes.

Magnitude of cancer risk from radon exposure.There is disagreement as to the estimate of lungcancer risk attributable to radon exposure, despitethe efforts of the BEIR committee to thoroughly re-view the data available. The main issue is whetherthe results from the analysis of the data for min-ers could be generalized and applied to the progenyof radon in homes (Lubin and Boice, 1997). Even ifa linear no-threshold model is appropriate, the co-efficients (the risk per unit exposure) are uncertainby at least a factor of 1.4. In addition, the modelof discrete risks for smokers and nonsmokers is asimplification since smoking levels vary and manynonsmokers are exposed to secondhand smoke.

Linearity of the dose-response function. Exper-iments on animals, plus epidemiological studieswith miners and others exposed at very high doses,suggest that at high doses the dose-response is ap-proximately linear (see Nazaroff and Nero, 1988,Chapters 8–9). However, there are really no gooddata at low doses. The Environmental Protec-tion Agency assumes the function is linear all theway to zero, but others have suggested that thereare a threshold (an exposure below which thereare no effects) or even a protective effect at lowconcentrations (Cohen, 1995; Bogen, 1997).

Case-control studies suggest that, if there is a pro-tective effect at low levels, it cannot be large, butmild protective effects, or a threshold so that lev-els below 5 pCi/L or so have no effect, cannot beruled out. However, in spite of claims to the con-trary by Cohen (1995), we are confident that long-term exposure to 2 pCi/L is safer than exposure to,say, 10 pCi/L (see, e.g., Lubin and Boice, 1997).

Moreover, our results are less sensitive than onemight suppose to nonlinearities in the dose-responsefunction at low concentrations. This is because weassume that remediation reduces radon levels to 2pCi/L, so the dose-response below that concentra-tion is irrelevant. For instance, if long-term expo-sure at 2 pCi/L were actually safer than no exposureat all, that would have no effect on our analysis un-der the present assumptions.

To get some idea of the sensitivity of our resultsto the details of the dose-response relationship atlow doses, we consider the effects of a relationshipwith a threshold at 4 pCi/L, so that exposure belowthat level has no health effect. One might examinethis issue in several ways. For instance, we couldask what the optimal strategy would be under thismodified dose-response relationship and see howthe recommended actions (e.g., which homes shouldmonitor and which should remediate) would changecompared to the recommendations based on thelinear dose-response. Instead, we look at how thenumber of lives saved would change if the strategybased on the linear dose-response were imple-mented; that is, if all of the same homes monitor orremediate as for the linear dose-response, but if thedose-response actually has a threshold. This seemsto us to be the more relevant question, since ourgoal is to understand the robustness of the presentanalysis rather than to seriously propose analy-ses under alternative dose-response functions. Also,alternative recommendations would merely entailfurther restrictions on which homes are candi-dates for monitoring, so determining exactly whichhomes those are is not likely to be particularlyinstructive.

Given a threshold at 4 pCi/L, remediations inhomes close to that threshold are mostly wasted(and all remediations are less beneficial), so weexpect a reduction in lives saved. Some summarystatistics are given in the columns labeled (b)in Table 5. As expected, the resulting number oflives saved substantially changes according to thisassumption; compared to the situation with a lin-ear dose-response, 37% fewer lives are saved forRaction = 4 pCi/L, and 23% fewer are saved forRaction = 8 pCi/L. Costs per life saved are stilllowest under the recommended strategy 1.


Table 5Sensitivity analysis: expected total lives saved and cost per life saved under four strategies for a grid of Raction, under three

different models*

ActionTotal lives saved (30 years)

Total costAggregate dollars per life saved

Level Strategy (a) (b) (c) ($ billion) (a) (b) (c)

2.5 pCi/L 1 116,000 59,000 108,000 18.4 158,000 310,000 169,0002 119,000 60,000 113,000 20.6 172,000 342,000 183,0003 108,000 57,000 105,000 22.0 204,000 380,000 210,0004 119,000 60,000 126,000 39.7 332,000 664,000 317,000

3 pCi/L 1 105,000 57,000 94,000 12.9 122,000 222,000 136,0002 113,000 60,000 102,000 16.5 145,000 273,000 161,0003 102,000 56,000 95,000 17.7 175,000 314,000 185,0004 116,000 59,000 119,000 33.6 288,000 565,000 282,000

4 pCi/L 1 84,000 53,000 73,000 7.3 87,000 138,000 101,0002 97,000 59,000 84,000 11.6 119,000 195,000 136,0003 88,000 53,000 80,000 12.2 138,000 231,000 151,0004 110,000 57,000 108,000 25.1 228,000 430,000 232,000

6 pCi/L 1 48,000 37,000 40,000 2.7 57,000 74,000 66,0002 68,000 51,000 59,000 7.2 105,000 141,000 123,0003 67,000 45,000 59,000 7.0 104,000 157,000 119,0004 95,000 54,000 89,000 15.8 165,000 289,000 177,000

8 pCi/L 1 27,000 22,000 22,000 1.1 42,000 51,000 49,0002 50,000 40,000 43,000 5.5 110,000 137,000 129,0003 51,000 37,000 45,000 4.6 90,000 126,000 104,0004 82,000 49,000 73,000 10.9 133,000 219,000 148,000

12 pCi/L 1 8,000 7,000 7,000 0.2 28,000 32,000 33,0002 29,000 25,000 25,000 4.3 148,000 171,000 173,0003 32,000 25,000 28,000 2.7 83,000 107,000 97,0004 61,000 41,000 54,000 6.3 103,000 153,000 117,000

*(a) cancer risk without threshold and postremediation radon level 2 pCi/L; (b) cancer risk with threshold 4 pCi/L and postremediationradon level 2 pCi/L; (c) cancer risk without threshold and postremediation radon level with a lognormal distribution with GM equal tothe square root of the preremediation radon level and GSD of 1.3. The four strategies are (1) recommended strategy based on decisionanalysis, (2) long-term measurements on all houses, (3) short-term measurements on all houses, adjusted for bias, (4) unadjustedshort-term measurements on all houses, uncorrected.

Effect of remediation. We have assumed that re-mediation reduces a home radon level to 2 pCi/L.This cannot be accurate for several reasons. First,the postremediation radon level must, in reality,vary among houses. In the context of our lineardose-response model, we can account for variationby considering the assumed postremediation levelas an expected radon level, averaging over houses.Second, the assumed reduction level of 2 pCi/L is arough estimate from sparse data on remediation ef-fects. Raising or lowering this postremediation levelwould correspondingly raise or lower the recom-mended action level Raction and raise or lower theestimated costs per life saved. Third, the postreme-diation level must certainly, in reality, depend on theinitial radon level in a more complex way than sim-ply E�postremediation level�R� = min�R;Rremed�.In particular, we would expect that, for some houseswith initially low radon levels (below 2 pCi/L), re-mediation might still have an effect. Unfortunately,

available data on remediation effectiveness havebeen collected only for houses with fairly highpreremediation levels; see Henschel, 1993, for ex-amples.

For a sensitivity analysis, we consider a modelin which the postremediation radon level is log-normally distributed with GM equal to the squareroot of the preremediation radon level (in pCi/L)and GSD of 1.3, further constrained to not exceedthe preremediation level. This rule is arbitrary, ofcourse, but it behaves reasonably in that postreme-diation radon levels are variable and are sometimesabove 2 pCi/L for houses originally above 4 pCi/L.Under this model, high-radon houses are typicallynot remediated all the way down to 2 pCi/L, so it isnot surprising that the effects of the measurement–remediation strategy are less, with reductions of12% and 17% of estimated total lives saved forRaction = 4 and 8 pCi/L, respectively (see columns(c) of Table 5).


Additional modeling and decision issues. We havemade several simplifying assumptions and choicesregarding what parameters to calculate. These in-clude the following:

1. Examining benefits in terms of “lives saved”rather than, say, “quality-adjusted life-yearssaved.”

2. Ignoring the influence of age and latency on per-sonal risk: it takes several to many years for lungcancer to develop and to kill, once it has beeninitiated, so there is little benefit of remediationfor, say, 70-year-old persons. If a cancer has al-ready been initiated then remediation is too late,whereas if they don’t yet have cancer then theyare likely to die of another cause before a cancercan kill them.

3. Ignoring possible interactions of radon exposureand age (e.g., children may have a different dose-response from adults).

4. Assuming risk is a function of cumulative thirty-year exposure: if risk per dose is highly nonlinearthen details of the temporal variation in radonexposure become important (so, for example, theeffect of people moving from home to home mustbe considered).

5. Ignoring the distinction between remediating toreduce personal risk and remediating a housethat is to be sold (thus reducing the risk to futureoccupants).

6. Implicitly assigning zero cost to the hassle andstress of performing radon testing and remedi-ation (a simplification that could be handled byadjusting the associated dollar costs).

7. Ignoring the pattern of residential mobility: ifpeople currently living in high-radon homesremediate their houses, the majority of the re-sulting health benefits will accrue to futureoccupants of their homes (Warner, Courant andMendez, 1995; Warner, Mendez and Courant,1996).

All of these issues, and more, could in principle beaddressed by adding additional parameters to theoverall model of risks and values. We chose insteadto keep the model relatively simple, since our maingoals are to illustrate how the hierarchical radonmodel can feed into a cost-benefit analysis and tobegin to bridge the gap between modeling of haz-ards and recommendations of actions; for both ofthese goals our conceptually straightforward modelseemed appropriate. In particular, we think the EPAmight be persuaded to make geographically specificrecommendations and possibly even to make rec-ommendations that vary for smokers and nonsmok-ers, but is very unlikely to make recommendations

that vary by household size, age and so on, in partbecause more complicated recommendations mightlead to considerable confusion.

7. DISCUSSION

We have used a Bayesian hierarchical model toanalyze radon data in the United States, therebygenerating estimated distributional information,and uncertainties, for different types of homes inevery state of the conterminous United States. Weused these results, along with estimates of radonrisk taken from epidemiological data, to constructa formalism by which monitoring and remediationprograms can be evaluated, allowing for individualvariation in risk tolerance. To illustrate the use ofthis formalism, we examined the implications of apolicy derived from the current EPA recommenda-tion that sets 4 pCi/L as a remediation level, butthat takes account of the wide variation in radonlevels among counties. This sort of analysis can inprinciple be used by individuals trying to decidewhat actions to take but, more importantly, canbe used by policy-makers to decide what actions torecommend or legislate.

As for the results themselves, under the assump-tions used in this paper, radon is indeed a majorcause of lung cancer in the United States, associ-ated with thousands of extra lung cancers per year,and yet, we recommend monitoring only for 26%of the population (or less, if separate action lev-els are to be used for smokers versus nonsmokers),and remediation is recommended only for homes inthe highest few percent of all homes in the UnitedStates. Our baseline recommended strategy (basedon Dd = 0:21, equivalent to a marginal cost perlife saved of $210,000), would save only approxi-mately 3,000 lives per year out of the estimated15,000 radon-related deaths per year at an aver-age cost per life saved of $140,000. The problemis that because of the lognormality of the radondistribution, most of the total exposure (and thus,most of the expected radon deaths) is in people ex-posed at low levels of radon that cannot be substan-tially reduced by remediation (see Figure 7). Thatis unfortunate from the standpoint of cancer pre-vention but fortunate from the standpoint of ouranalysis since it renders our recommendations rel-atively insensitive to the dose-response at very lowconcentrations. However, if cancer risk is a stronglynonlinear function of radon concentration for con-centrations in the range of 2–10 pCi/L, then boththe details of the dose-response and the effects of re-mediation for low-radon homes are crucial unknownquantities in the decision. Unfortunately, we see lit-


tle hope for clarification of the dose-response issuefor many years to come.

7.1 Policy Implications

As discussed in this paper, smokers are thoughtto be at much higher risk of radon-induced lungcancer than are nonsmokers. This makes radon apeculiar issue from the standpoint of public policy,as noted by Ford et al. (1998). Under the assump-tions made in this paper, a large majority of remedi-ations should be performed by smokers, but smokersmight be willing to accept more risk for lung cancerthan are never-smokers. As Nazaroff and Teichman(1990) comment in an article that touches on manyissues of radon risk reduction, “it seems unlikelythat most smokers would make the necessary in-vestment to reduce the radon-related risk of lungcancer when the dominant cause of their risk issmoking.”

The results presented above incorporate un-certainties in the county radon distributions andexplicitly allow for estimation using different as-sumptions about risk tolerance. As illustrated inthe discussion of sensitivity analysis, it is also pos-sible to tinker with the dose-response function andthe assumptions about remediation effectiveness.An optimal decision strategy, within the frameworkof the model, can be determined for any choice ofthese parameters, but any such strategy is optimalonly in the simplified world of the model. Realitydiffers from the model in many ways: not all peoplewill act rationally or follow the recommendations ofthe model; it may be politically difficult to call forradon testing in some areas and not others, sincedoing so may lower property values; similarly, itmay be difficult to call for different action levels forsmokers and nonsmokers, though in some sense itclearly makes sense to do so; people are impatientand may vastly prefer short-term tests to long-termones and so on. In the policy world, psychologi-cal, political and economic considerations can beat least as important as the scientific and statisti-cal issues considered in this paper. Moreover, evensome scientific issues (most notably, uncertainty inthe dose-response relation) are not fully addressedin our results.

However, this is not to say that our scientificand statistical results are useless. To the contrary,some conclusions are so clear that we think thatpolicy can and should be changed to reflect them.Even considering possible nonlognormality withincounties and variation in risk tolerance, there is noplausible scenario in which it makes sense to mon-itor every house in the country with a short-termmeasurement. The fact that high-radon homes (i.e.,

over 4 pCi/L) have been found in every area of thecountry, which the EPA states when recommendinguniversal testing, is true but irrelevant. Someoneliving in a nonbasement home in Louisiana surelyhas many risk-reduction options that are vastlymore efficient uses of money and time than is per-forming a radon test (e.g., buy a smoke detector, getthe car’s brakes checked, visit a doctor, etc.). Thisis true even if we use the EPA’s recommended re-mediation level of 4 pCi/L. As we have seen, thataction level itself is not unreasonable, but it doesnot justify monitoring every home. Of course, wesay this with the luxury of having a great dealmore information on the geographical distribu-tion of radon than was available when the EPA’srecommendations were first promulgated.

7.2 Generalizations to Other Decision Problems

In the Bayesian approach to decision analysis,decision options are evaluated in terms of their ex-pected outcomes, averaging over a probability dis-tribution that is assigned jointly to all unknownquantities. The probability distribution is typicallyobtained by elicitation from experts, literature re-view, and sometimes data analysis (in which case itis identified as a posterior rather than a prior distri-bution). However, it is not yet common for decisionanalyses to use the sorts of hierarchical models thatare becoming standard in Bayesian statistics (see,e.g, Carlin and Louis, 1996; Gelman et al., 1995), aswe have done in the present paper. Indeed, we areunaware of any other case in which spatially vary-ing recommendations have been made based on theoutput of a hierarchical model, with correct incorpo-ration of spatially varying statistical uncertainties.

A nonhierarchical model that has geographicvariation would allow spatially varying recommen-dations, but, given the (inevitable) existence ofspatial variation unexplained by the model, wouldyield less accurate predictions and thus yield de-cision recommendations that were not as wellcalibrated locally. The hierarchical model, in con-trast, allows parameter estimates and uncertaintiesto vary by area, so that location-specific recom-mendations can be made, and the influence ofrecommended actions within local areas can beassessed.

More generally, we suspect that hierarchicalmodeling can be combined with decision analy-sis in a wide variety of problems, which we hopewill make the data analysis more useful and thedecision-making more individually focused. Wealso anticipate more sophisticated methods forcomputation (since, in general, the hierarchical pos-terior distributions that are input to these decision


analyses will be summarized by simulation) andgraphical display of the varying decision recom-mendations, continuing on the work developed inthis case study.

ACKNOWLEDGMENTS

We thank the Editors and reviewers for helpfulcomments, the NSF for Grants DMS-94-04305 andSBR-97-08424 and Young Investigator Award DMS-97-96129 and the Department of Energy for grantDE-AC03-76SF00098.

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CommentBradley P. Carlin

First, congratulations to the authors on a fine pa-per, which shows quite clearly how formal Bayesiandecision-theoretic tools may be combined with mod-ern hierarchical modeling techniques to produceclear and sensible guidelines in an important en-vironmental health problem setting. Of course,Bayesians have long argued that their techniquesoffer significant advantages over the traditional,more informal analytic procedures often used bydecision makers, but only with the advent of mod-ern Markov chain Monte Carlo (MCMC) computingmethods in the last decade or so can these bene-fits be fully realized. As seen in the present paper,the Bayesian engine does not obviate the needfor a variety of (potentially highly influential) as-sumptions in the analysis, but it does provide aframework in which these assumptions can becarefully structured, and their impact assessed.

Before commenting on specific aspects of theauthors’ work, it is worth mentioning a possibleconfusion in the use of the term “Monte Carlo anal-

Bradley P. Carlin is Associate Professor, Divisionof Biostatistics, School of Public Health, Univer-sity of Minnesota, Minneapolis, Minnesota 55455�e-mail: [email protected]�.

ysis” by the risk assessment and applied Bayesiancamps. As already mentioned, to the latter groupthis typically refers simply to the integration meth-ods used to evaluate the “denominator integral” inBayes’ Rule; that is,

p�u�y� = p�y�u�p�u�∫p�y�u�p�u�du

;(1)

where y denotes the observed data and u the vec-tor of unknown parameters. However, to risk as-sessors, “Monte Carlo analysis” is an approach bywhich a sample of potential risk or exposure valuesis obtained by first specifying distributions relat-ing the various observed and unobserved quantitiesin the model and then simulating values from theresulting hierarchical risk model. While my read-ing of this literature is admittedly only cursory, thisseems to be the approach taken in several risk as-sessment textbooks (e.g., Vose, 1996), and one thathas been recently codified by an EPA panel assem-bled to “promote scientific consensus on risk assess-ment issues and to ensure that this consensus isincorporated into appropriate risk assessment guid-ance” (Environmental Protection Agency, 1997). Butto Bayesians, this approach is tantamount to “sam-pling from the prior”; the Monte Carlo method isbeing used only to simulate values from assumed


distributions, not to assist in the formal prior-to-posterior updating of (1) above. To be fair, the usualapproach does encourage using observed data whendetermining distributions for unknown quantitieswhenever possible, and the literature is beginningto distinguish the variability in unknown quanti-ties (which is sometimes called “uncertainty”) fromthe variability of observed quantities given the un-knowns (which instead is called “variability”; see,e.g., Rai and Krewski, 1998). Still, with a few no-table exceptions (Taylor, Evans and McKone, 1993;Brand and Small, 1995; Dakins, Toll, Small andBrand, 1996), the risk assessment literature seemsin need of more formal Bayesian thinking, for whichthe present work (and the earlier work of Wolpert,Steinberg and Reckhow, 1993) may well serve as ablueprint.

One way in which the risk assessment literatureis ahead of that in statistics is in its willingness todiscuss the value of human life on a dollar (or someother meaningful quantitative) scale. In clinicaltrials, for instance, real advances in decision theo-retic solutions to the interim monitoring and finalanalysis problems have been stymied by the unwill-ingness of most statisticians, epidemiologists andclinicians to even contemplate such a mapping(though a few brave first attempts have been madeby Berry and Ho, 1988; Stangl, 1995). An impor-tant feature of the present paper is the authors’description of how already established governmentguidelines for what constitutes a radon exposurelevel worthy of remediation implicitly determinesdollar values per microlife (Section 4.2). Clearlysuch a linear scale is not appropriate when wemove far from the origin (no reasonable personwould surrender one million of his own microlivesfor any dollar amount), but discussions of this sortmay well have beneficial impact in risk assessmentstrategies far beyond environmental settings, if inno other way but informing decision-makers as towhat implicit values their recommendations areplacing on fractions of lives.

Turning then to specific comments on the authors’approach, given the power of modern MCMC tech-niques I was surprised that the model componentsconsidered in Section 3 were essentially confined tonormal distributions. The model apparently treatsthe variance parameters τ2 and σ2 (as well as a vari-ety of tuning parameters in Section 4) as constants,instead of more plausibly assuming distributions forthem. Indeed, some of the modeling is not even be-ing shown: (2) is written as a prior (or a “predictive”in the authors’ nomenclature), but in fact it mustbe the result of a preliminary prior-to-posterior cal-culation, combining some prior on the regression

parameters b with some preliminary data y∗ ontypical radon concentrations in U.S. homes. Whatis this preliminary data and model? Do its residu-als suggest any evidence of lingering spatial corre-lation? Also, the two-stage implementation of thepreliminary (Section 3.1) and house-specific (Sec-tion 3.2) models is odd, since it forfeits the usualBayesian advantage of a single unifying model thatenables all sources of variability and uncertainty tocorrectly propagate throughout its levels.

As the authors mention, the paper’s main focus ison the decision analysis in Section 4. Here there areany number of assumptions with which one couldquibble (the flat $2000 to remediate any home re-gardless of location, the 70-year life expectancy forevery occupant, etc.); one could either place dis-tributions on these quantities as well, or simplyundertake a variety of sensitivity analyses (as theauthors describe in some detail in Section 6). WhileI don’t wish to nitpick further here, I did find the ap-proach for “discounting the value of a life,” describednear the beginning of Section 4.1, to be somewhatconfusing. At first blush, if Dd is the amount weare willing to pay to save one microlife now, thensince lives saved 20 years in the future are worthless, it seems the revised Dd should be decreased(not increased) by a factor of 1:0520. However, recallthat the paper does not really specify Dd from firstprinciples, but rather “backs it out” by viewing the$2000 remediation cost as fixed. Thus if the valueof the lives saved decreases, our cost per life savedmust go up. Yet even here, it seems that the appro-priate increased cost must be backed out from (5)and (6) as well, discounting each future year’s riskseparately in the thirty-year decision period ratherthan applying a single inflation factor to Dd.

Of course, the actual dollar amount any givenperson would spend per microlife saved is proba-bly more a function of their own financial resourcesand aversion to risk than any governmentally rec-ommended remediation levels. I am personallyacquainted with a suburban couple with three chil-dren who, after reading an early report on thealleged dangers of living near high-voltage electri-cal lines like the ones near their home, immediatelysold the place and moved. Because they did this atjust the time when popular concern over this po-tential risk was at its zenith, their total financialloss in the transaction (including moving expensesand remodeling their new home) was in the neigh-borhood of $80–100,000: in the light of more recentdata on the subject, a colossal amount spent perexpected microlife saved. In the language of (6), forthis couple Dd (hence Dr) was essentially infinityfor this perceived risk, and thus Raction ≈ Rremed.


Looking again at Figure 3, this means the couplewould not consider taking a new risk measure-ment, and apparently the decision minimizing theirexpected loss was to remediate (i.e., move) immedi-ately. While this is obviously an extreme example,I would venture to guess that many middle- toupper-class persons would adopt a similar strategyfor radon remediation, especially given the rela-tively small cost involved here (only $2000, thoughthis does not include the headaches of getting thework done and then maintaining the fan systemonce installed).

Finally, in Section 4.3.1, the authors correctly rec-ognize that the “inner decision” of whether to reme-diate or not given the future observation y mustbe considered first; only then can we decide thebroader question of whether to take this extra mea-surement at all (versus simply deciding based onthe prior) by integrating over the possible valuesof y we might see. This approach is a special caseof the general Bayesian decision-theoretic approachto sequential analysis problems, backward induc-tion (see, e.g., DeGroot, 1970, Chapter 12), in whichmultistage decision problems are decided by “work-ing backward” through the potential future obser-vation stages, alternately minimizing expected lossand integrating over the as-yet-unseen data values.It is easy to show (and intuitively clear) that it is al-ways better to continue sampling if there is no costassociated with obtaining these new samples. Still,backward induction is seldom used in practice, dueto the explosion in analytical and bookkeeping com-plexity as stages are added to the model. Recently,

however, Carlin, Kadane and Gelfand (1998) haveproposed a “forward sampling” algorithm that sub-stantially eases the analytic and computational bur-den and can be used to identify the best member ofa plausible class of strategies when backward in-duction is infeasible. Such an approach has obviousappeal in clinical trial monitoring (where a moni-toring board wishes to check the trial’s progress atvarious intervals and stop the trial as soon as thebest decision is clear) and might also be useful inenvironmental settings where a series of measure-ments is anticipated, with remediation an availableoption at each interval.

In summary, this paper makes important method-ological contributions to the field of environmentaldecision analysis and similarly important contribu-tions to the substantive problem of radon remedia-tion (indeed, homeowners would do well to consultthis paper and its Web site rather than simply relyon any of the “one-size-fits-all” government guide-lines). I look forward to future developments in fullyBayesian decision analysis and its further incorpo-ration into the practice of risk assessment.

ACKNOWLEDGMENTS

The author is grateful to Ms. Li Zhu and Mr. CongHan for helpful discussions during the preparationof this comment. The author’s work was supportedin part by National Institute of Allergy and Infec-tious Diseases (NIAID) Grant R01-AI41966, andby National Institute of Environmental HealthSciences (NIEHS) Grant 1-R01-ES07750.

Comment: In Praise of Decision Analysis inEnvironmental HealthCarl V. Phillips

If we knew all the right decisions ex ante, thenmaking public health policy (or advising individ-uals making decisions) would be relatively easy.If we had no information to inform a certain deci-

Carl V. Phillips, Division of Environmental and Oc-cupational Health and Center for Environmentaland Health Policy, University of Minnesota Schoolof Public Health, Minneapolis, Minnesota 55455.

sion, then any new test or study could be usefuland it would be difficult to predict what new studywould be the most useful. For most public healthquestions, of course, our knowledge lies somewherebetween these extremes: we do not know enoughto make a definitive decision or recommendation,but we know (or would know if we looked carefully)what further studies or tests would help make thedecision. Yet somehow, most of the public health lit-erature fails to recognize the implications of this.


The failures include implicit assumptions that nofurther data is available (making policy recommen-dations prematurely), the next study will inevitablyclear up all uncertainty (ignoring the need to figureout what the next study should focus on), and fur-ther data gathering cannot be targeted to subpopu-lations based on existing data. (Sadly, to the extentthat targeted data collection is recommended, it isoften focused on the subpopulation that is judged tobe most “at risk.” This is often the population thatis least in need of further study, since existing datais sufficient to warrant intervention with a high de-gree of confidence.)

Lin, Gelman, Price and Krantz (1999) do a greatservice by reminding us, using a wonderfully clearexample, how ex ante knowledge helps us target fur-ther data gathering and thereby helps assess whenthe benefits exceed the costs. They observe that ourpriors about the risk from household radon exposurevary by geography and household characteristics,and that we can use this convenient data to dra-matically improve the efficiency of decisions abouttesting and remediation. The addition of previousmeasurements further improves the precision of thegeographic estimates, and thus the value of the rec-ommendations. (Decision analysis could still be usedwithout this addition, just using the predicted meancounty radon concentrations and household charac-teristics, since they vary enough to produce differ-ent recommendations. It might be, however, that theprecision would be so lowered that the benefits ofdecision analysis would be diluted.)

Social scientists and policy analysts have longdiscussed how to use existing information to decideabout gathering further information or to make dif-ferent recommendations for different individuals(e.g., Stokey and Zeckhauser, 1978). But the criticalimportance of quantifying underlying probabilitydistributions and sources of uncertainty—indeed,the understanding that decision-makers can evenuse such information—has been overlooked byother health researchers. This has created the sadsituation where policy analysts are waiting forprobability data for use in decision analysis, whilehealth researchers are providing only the point esti-mates that they apparently assume to be preferableand useful.

An issue like radon testing provides a usefulway to introduce the social value of data gather-ing into public health without the medical arena’slimits of allowable practice. Medical testing is func-tionally equivalent to other areas of public health,such as radon testing or finding out how safe ahighway is by building it and watching what hap-pens. But the culture of medicine and the legal

climate complicate things. While allowing the gath-ering of patient data for the social good, the currentculture makes it difficult to perform tests with ex-pected social benefits but a net expected cost to theparticular patient, or to withhold tests that havenegative expected value but have any chance of im-proving a diagnosis. Indeed, it is sometimes difficultto even discuss making more efficient decisions inthe medical arena. An environmental health issuelike radon allows socially optimal recommendationsbecause most people are amenable to persuasionabout the right choice, given their underlying lackof knowledge, the relatively low individual costsand risks and the noninvasiveness of most actions.At the same time, the social costs and risks arefairly high, and it is worth the effort to try tominimize them.

One major policy advantage of the situation de-scribed by Lin et al. (one which should probablybe given more attention in the research literatureand policy process) is that it allows individualswith different tastes for risk to take different ac-tions. Unlike public health decisions that must bemade by a central authority for everyone (casesranging from effluent regulation to airplane safetyfeatures), the decision about radon parallels thedecisions about medical care and consumption. Ifsomeone is more willing to risk disease or less will-ing to spend money to avoid it (or does not believethat the risk is actually real), then he has the op-tion of making a different decision than the officialrecommendation. This would be particularly rea-sonable if, as suggested by Lin et al., EPA maderecommendations that ignored household composi-tion. Single nonsmoking assistant professors couldrationally choose to ignore the radon risk in theirhomes.

An extension of these principles in a different di-rection (and into more controversial areas) is to as-sess what population-level research would be mostuseful given our current data and priors (Phillipsand Maldonado, 1999). Epidemiologic studies, alongwith most quantitative health research, tend to con-clude by calling for more research, but very seldomassess exactly what the further research should do.Further research can be used to eliminate some ofthe measurement error, simply assess the level ofmeasurement error, eliminate confounders, measureconfounders or just increase the sample size. Withinall these choices, there are continuous ranges ofchoices along multiple dimensions. Yet the decisionanalysis principles are still the same as those inLin et al. By fully assessing what we know and whatmore we would be likely to know following future re-search, we can better determine when to act, whento walk away or what more we want to know.


Given the clear policy prescriptions in an analysislike Lin et al., it is crucial that authors are carefulto identify what policy question they are answeringand get certain key social-science-based parametersright. It is here that I take some issue with the ar-ticle. (It should be emphasized that this criticismis an indication that Lin et al. are victims of theirown success. Criticism at this level would not beworthwhile if the article did not have such greatpotential for guiding policy.) The key number relat-ing to human preference, the value of a life saved,is too low. There is, of course, no clear number forthe value of a life saved, and willingness to pay fora probabilistic saved life (by a consumer or regula-tory agency) varies wildly. Nonetheless, the implicitvalues of $100,000 to $500,000 per life saved pre-ferred by Lin et al. are low by almost an order ofmagnitude compared to the typical discussion in theeconomics-of-health or economics-of-regulation liter-ature. This has a substantial impact on the recom-mended policy. The choice of discount rates also af-fects the decision through the same pathway and isalways controversial in policy discussions. The useof 5% for discounting lives saved will be criticized bysome as being high (though others have used evenhigher values).

Lin et al. allude to the difference between publicand private health impacts of remediation, partic-ularly in their acknowledgment of simplifying as-sumptions. But they understate the importance ofthis issue, which is probably the most importantchallenge to actual decision making, dwarfing thepractical implications of their other assumptions orsensitivity analyses. Most houses change residentsmany times over two or three decades. Israeli andNelson (1992) report that the mean total residencetime for U.S. households is less than five years, andfor people who own their home it is about elevenyears. The medians are considerably lower. It is alsolikely that tenure is lower among those who wouldbe most inclined to remediate (the relatively youngand affluent). If the radon exposure were perfectlycapitalized into the value of a house, then the pri-vate and public decision would be the same (set-ting aside the different impacts of radon based onthe number, age, and smoking habits of the resi-dents). The remaining benefit from a remediationexpenditure would be captured by the current ownerwhen the house was sold (or rented) and so the pri-vate optimization decision about remediation wouldachieve the social optimum.

However, there is little chance that this perfectcapitalization will occur. Currently, most consumersignore radon risk when making housing choices,while a few overreact to it. Greater attention to the

risk in the media or by the government would tendto increase the level of concern. But there is no rea-son to believe that it will get to the “correct” levelof concern (or that if it does, that it will not shootpast it into widespread overreaction). In general,the vaunted invisible hand of microeconomics canonly promise that prices will be set correctly whenthere is room for someone to make an arbitrageprofit from someone else’s miscues. If you under-price the wheat you are selling, I can make a profitby buying it. But if you underprice the value of pro-tecting yourself from radon, there is no way for meto make a profit, and thus no market pressure forthe price to rise to its proper level. Thus, the majordecision variable for individual homeowners, per-haps more important than even the concentrationof radon, is likely to be how long they plan to stayin the house, a variable which is omitted from theanalysis.

Given the failure of perfect capitalization, the op-timal public health result could still be achieved byrequiring all property owners to take the steps rec-ommended by the decision analysis, regardless ofpersonal taste or plans to leave the house. However,this would be such an implementation nightmarethat it is not even worth discussing. The analysis inLin et al. does not clearly position itself as either arecommendation to the homeowner (in which caseit should consider expected tenure in the house andlevel of capitalization from the risk) or for some na-tional public health initiative (in which case, issuesof implementation, social attitudes toward risk, andpolitics will probably dominate the rational assess-ment).

Despite addressing some of the uncertainty be-tween households through the hierarchical model,Lin et al. cannot do much to deal with the greatunderlying uncertainty of how risky low-dose radonreally is, especially for nonsmokers. In many cases,policy should be made based on the best-availableestimates of important values. However, there arelimits to this approach. On a practical level, evenwith an impeccably flawless decision analysis, itmay be difficult to get consensus on a precise ac-tion point for a decision tree when there is hugedisagreement about the value of the central param-eter. Apart from this, when certain actions haveirreversible costs (such as $1500 to install a radonremediation system), it may pay to wait for moreinformation, in case the new information mightsuggest a different optimal action. (In economics,this concept is known as option value.) This, inturn, creates another layer of optimization decisionbased on our priors about what new informationwill emerge and when.


Before paralysis sets in about the enormity of thetask of optimizing our decisions under risk and un-certainty, we should remember the value of a high-quality analysis that covers most of the importantoptions and sources of uncertainty. Such an analy-sis is much more likely to produce good decisionsthan is taking rhetorical refuge in the complexity ofthe decision and resorting to rules of thumb, politi-cal pressure or simple inertia. Lin et al. show howto conduct such an analysis, and (modulo our dis-agreement about economic parameters and the com-

plication of individual versus social decisions) carryit out. Their analysis is part of an important trendin health research toward considering all costs andbenefits of an action and seriously analyzing whatwe would learn from further data gathering. This isa huge improvement over the standard practices ofeither making policy recommendations as if a givenstudy were the last word in an area or assumingthat we will have all the answers soon and refusingto make a recommendation until then.

RejoinderChia-yu Lin, Andrew Gelman, Phillip N. Price and David H. Krantz

1. INTRODUCTION

We thank both discussants for their comments,especially for their explorations of the connectionsbetween statistical modeling, decision analysis andpublic health outcomes. Our paper has two maingoals: to illustrate the benefits of hierarchical mod-eling in probabilistic decision analysis and to de-termine an improved radon recommendation thatwould have a chance of actually influencing the gov-ernment’s radon policy. In this rejoinder, we respondto the specific comments of the discussants in thecontext of our major concerns about radon and de-cision analysis in general.

Phillips points out that we never stated exactlywhat policy question we are attempting to answer,and that it is unclear whether we are making recom-mendations to individuals or to policy-makers. Theanswer to the latter question is that we are hop-ing to influence government radon policy; we do notexpect our recommendations to reach a substantialnumber of individual homeowners. We do think thatthe government, that is, the Environmental Protec-tion Agency, or perhaps state health departments,could recommend use of some house- and occupant-specific information in making radon decisions, but,as discussed below, the complexity of the decisionswould probably be kept very low. Of course, eventhough it is the government’s radon policy that weare attempting to influence, the eventual costs andbenefits (and decisions) would still be up to indi-vidual homeowners (except for a few government-owned buildings such as schools and military base

housing, some of which have already been moni-tored and remediated).

As to our not explicitly stating the policy ques-tion that we were trying to answer, that’s certainlya valid point. Rather than claim that what we didis “right,” we will explain why we did the analysisthe way we did, since we think the same issues willapply to many other decision analyses.

2. RADON POLICY

We did not set out to answer the open-endedquestion of what the U.S. government’s radon pol-icy should be, which is such a complicated questionthat it is hard to see how to directly address itwith a decision analysis. For instance, the questionof whether radon policy should be set by the fed-eral government, state health departments or localzoning boards, is both a political question and amatter of organizational efficiency that we have noclear way to analyze. In practice, the radon policycan only be chosen from within a universe of pos-sible policies, and we don’t even know what thatuniverse encompasses.

However, we have had several years of experienceanalyzing radon data with the goal of identifyinghigh-radon homes, as part of a project sponsored bythe EPA, the Department of Energy, and the U.S.Geological Survey. During this time we grew dissat-isfied with several aspects of the current U.S. radonprogram. For instance, many people make decisionsabout radon remediation decisions (and even deci-sions about what house to buy) based on short-termmonitoring in the basement, which bothers us since


we know that this protocol leads to a very large frac-tion of “false positive” results. Also, when we foundthat some large areas of the country have nearlyno high-radon homes, it seemed silly to recommendmonitoring everywhere. Finally, there’s the risk dif-ference between smokers and nonsmokers, whichcertainly seemed as though it ought to influencedecisions.

2.1 Why Not Look for the OptimalPolicy for Individuals?

Once we begin to consider the factors that canaffect people’s decisions, it seems appropriate to an-alyze the whole problem from the perspective ofindividual homeowners, but this approach quicklybecomes extremely complicated. To fully model risk,one should consider, in addition to the predicted in-door radon concentration, at least the number ofoccupants, and the sex, age and smoking status ofeach occupant (or in the case of young nonsmokers,the probability that they will become smokers). Ofcourse, all of this is in addition to the very large un-certainty in the dose-response relation, particularlyat low doses.

Also, in contrast to our simplified model, both thecosts of remediation and the postremediation radonconcentration are variable (and unknown). As far ascosts are concerned, some unknown fraction of theremediation cost is, or ought to be, recovered uponsale of the home, which takes place after a vari-able and uncertain time period, with the recoveredcost dependent on both the risk tolerance of the newbuyers and the makeup of their household.

In short, attempting to determine the best courseof action for a particular person or home is a mess.We could, of course, create models and distribu-tions for all of the factors listed above and gener-ate individual recommendations, but to what end?There’s no chance that any sizable fraction of home-owners would actually do the work needed to de-termine what we recommend they should do, andwe also can’t picture radon policy-makers siftingthrough the resulting reams of analysis in orderto try to formulate a new radon policy. And per-haps they shouldn’t; would the benefits of having abetter-targeted, but much more complicated, radonpolicy outweigh the ill effects of added confusion andcomplexity?

2.2 What Do Governments Think a Radon PolicyShould Look Like?

Most governments that have official radon poli-cies (this includes most of the “developed” nations)have a single recommended action level, or some-times two or three separate target levels, for exist-

ing, rebuilt and new buildings (Cole, 1993). In theUnited States, EPA and state health department of-ficials have told us that when people ask them forradon advice, they don’t want to have to think abouta lot of different issues; they just want to know whata “safe” radon level is. Whether or not the policy-makers are right about the need for simplicity, itis clear that official radon recommendations will infact be based on quite simple monitoring and reme-diation criteria. So when it came to deciding whatpolicies to analyze, we decided to restrict ourselvesto fairly simple variations on the EPA’s current rec-ommendations. We make no claim that the result-ing policies are the best of all possible ones; we onlyclaim that they would be improvements to the cur-rent recommendations.

3. VALUE OF A MICROLIFE

3.1 Why We Used the Values We Used

The discussion of which simple policies shouldbe considered leads us to one of the specific issuesraised by the discussants. Phillips objects to the low“value of a microlife” that we used, and Carlin saysthat our derivation of it from the EPA’s recommen-dations is confusing.

We considered three parameterizations (value ofa microlife, or a radon action level, or a dollar valueassociated with reduction of 1 pCi/L), and it is pos-sible to perform the analysis conditional on any ofthese. One reason for allowing decisions to be ana-lyzed in terms other than “cost per life saved” wasto allow more direct treatment in, say, a regulatoryframework in which decisions are made based onan action level. Of course, a value for any of thethree parameters determines values for the others,so there is no escaping that any decision implies amarginal cost per life, but, as a practical matter,radon decision-makers might prefer to work withone of the other parameterizations.

Our analytical results were worked out with theparameter values unspecified, but when we pluggedin example values we chose them for a reason. Asa practical matter, retaining the action level is im-portant. There is probably no stone tablet in Wash-ington that reads “Thou shalt use a 4 pCi/L actionlevel,” but there might as well be: the EPA has al-ready faced significant heat from nonbelievers whothink that breathing radioactive gas is good for you,or at worst harmless, and there is no chance that theaction level will be decreased. On the other hand,there is also little chance that the EPA will com-pletely abandon its long-standing 4 pCi/L thresholdin favor of a higher threshold, particularly in view of


the fact that under conventional dose-response mod-els it is not a very protective standard; as we showin the paper, the implied marginal cost per life savedis only around $200,000, which, as Phillips notes, isvery low compared to the values people usually use.

However, interpreting this implied marginal costis a bit tricky. The radon decision is about payingnow to reduce the risk of lung cancer death between5 and 35 years in the future (under our assumptionthat remediation is effective for 30 years and as-suming 5 years between the cancer initiation eventand death). On average, ignoring the age distribu-tion of the population for the moment, performingremediation now can be thought of as saving a frac-tion of a statistical life at the middle of that period,or 20 years. This delay can also be interpreted interms of “discounting” of the future; in our paper,we suggested that a discount rate of 5% leads to a“net present value” of a life of 1:0520Dd = 2:7Dd,but that is actually slightly wrong. We should haveused

∫ 355 1:05tDd dt = 2:9Dd. However, we prefer to

avoid this discounting formulation and rather sim-ply recognize that the lives saved by radon remedi-ation will be on the order of 20 years in the future.Another way of analyzing the decision, consistentwith the medical decision analysis literature, is tolook at years of life saved. If lung cancer deathsfrom radon are approximately uniformly distributedacross the population of smokers, this gives on theorder of 25 years of life saved, so that, for example,$200,000 per life corresponds to about $8,000 perundiscounted year of life.

Getting back to Phillips’ comment that thesedollar amounts are an order of magnitude lowerthan typical values in the economics-of-health oreconomics-of-regulation literature, we note that inaddition to the discounting issue discussed above,in many health and regulatory decisions there is apotential for prescribing the action to be taken: re-quiring (by law) that insurance companies pay fora particular procedure, or that companies reducetheir emissions below a certain level. In contrast,with a few exceptions radon recommendations arejust that—recommendations—and the fact thatmost people do not now remediate, or even moni-tor for radon, suggests that there would be littlepoint to determining a radon policy based on ahigher value per microlife with its correspondinglylower recommended action level. We hope that animproved radon policy will also meet with bettercompliance with the recommendations of that pol-icy, and if so then the action level could be revisited,but for now a lower action level would probablysimply not be respected.

3.2 Variation in the Value of a Microlife

Carlin suggests that the amount people are actu-ally willing to spend is “probably more a functionof their own financial resources and aversion torisk than any governmentally recommended reme-diation level.” Actually, we think all three of thesefactors are important. In practice, based on am-ple anecdotal evidence, many people do take therecommended action level of 4 pCi/L into accountwhen deciding on whether to remediate (thoughthey do not necessarily follow the EPA’s recom-mendations; see Evdokimoff and Ozonoff, 1992),and many radon mitigation companies guaranteethat the long-term postremediation concentrationwill be below 4 pCi/L and will perform additionalremediation if that standard is not met.

However, Carlin’s point is well taken. If “risk-aversion” is measured in dollar terms, then it maybe more a measure of financial resources than ofpsychological attitude toward risk. Consider themost extreme case of avoiding certain risks. Carlinpoints out that a linear scale of value per microlifeis “not appropriate far from the origin since no rea-sonable person would surrender one million of hisown microlives for any dollar amount.” Althoughtrue, the question that is more relevant to our anal-yses is not how much someone would have to payyou in order for you to tolerate a given risk, butrather how much you would pay to avoid a givenrisk. There is clearly a finite answer to this latterquestion: you cannot pay money that you cannotraise. The theoretical “ability to pay” to save a mil-lion of your own microlives might be a couple ofmillion dollars for a reader of this journal, up tomany tens of billions of dollars for, say, Bill Gates.

4. THE ROLE OF FORMAL “DECISIONANALYSIS” IN DECISION-MAKING

Formal decision analysis requires setting up adecision–uncertainty tree, estimating the costs andprobabilities associated with the potential outcomes,setting up a value–utility function for the outcomesand evaluating the tree using averaging and max-imization. It is often said that the most importantparts of formal decision analysis are (a) explicitlysetting down the possible decisions and outcomesand (b) revealing possible incoherence in existingdecision procedures.

The key difficulty of using a decision analysis tomake an actual recommendation is that the inputsto the analysis may be more controversial than theoutputs. For example, in our analysis, any home-owner can obtain a recommendation to remediatesimply by increasing the value of a microlife past


a certain value. To put it another way, there maybe as much arbitrariness in choosing the relativevalues of money and life as there is in setting aperfect-information remediation threshold, or evenin making a measurement–remediation decision. Aswith the medical decision-making literature, we canuse decision analysis as a tool to produce best esti-mates of costs in dollars and lives, which then mustbe balanced by policymakers and individuals.

In the radon example, the clearest benefit of thedecision analysis is in allowing us to construct a spa-tially varying family of decision recommendationsthat we expect would save more lives at a lowercost than a uniform national recommendation. Inaddition, the formal analysis allows us to calibratedecision thresholds (in pCi/L) in terms of dollarsper microlife. Both these benefits require a realisticstatistical model of measurements and home radonlevels.

5. DECISION ANALYSIS ANDHIERARCHICAL MODELING

To see the connection between statistical modelsand decision analysis, we consider how decision rec-ommendations change as the underlying statisticalmodels become more complex.

The simplest decision analyses are uniform acrossthe population and are expressed as, for example,Should a patient with a certain medical conditionundergo the risk of a certain diagnostic test? Therecommendation might then be in terms of dollarsper life saved, or even simply as positive or nega-tive expected lives saved. More sophisticated analy-ses allow relevant probabilities to depend on knowncovariates, so that the question of undergoing thediagnostic test might be conditional on the age, sex,and some assessment of the health status of thepatient.

From a statistical standpoint, conditional deci-sion recommendations correspond to interactionsbetween treatment effect and covariates, and theycan have important practical considerations. Forexample, optimal recommendations for cancerscreening depend on age, with the particular agerecommendations depending on the pattern of can-cer onset (see, e.g., Eddy, 1990), and in economics,a program that has a negative effect on the popu-lation can be estimated to have a highly positiveeffect if targeted on the individuals with covari-ates that predict a highly positive interaction withtreatment (Dehejia, 1998).

In response to Carlin’s comments about the un-derlying statistical model: yes, we previously fit afully Bayesian model to a large set of short- and

long-term radon measurements, along with other in-formation on houses in the dataset and counties inthe United States. We used the posterior distribu-tion of that analysis as the prior distribution for theanalysis in this paper. For each county and housetype, we used the posterior simulation draws fromour previous analysis to compute a prior mean andstandard deviation for the mean log radon level forthose counties and house types. We then assumedthat the standard deviations of the measurementswithin that county and house type were estimatedto a high precision (and could thus be summarizedby posterior point estimates?), which is not too badan approximation given the large datasets used toconstruct that posterior distribution. If we were lessconfident that the posterior distribution was close tonormal, then we would have worked with the simu-lation draws themselves, but in this case, we wantedthe convenience of the normal approximation, whichallowed some of the steps of the decision analysis tobe performed analytically. Another approach wouldbe to use the normal approximation, but then checkit (or correct for it) at the end of the analysis, usingimportance sampling.

Finally, we believe it is important to link the con-cerns of statistical modeling to those of decisionanalysis. Sensitivity analysis is already recognizedas a crucial step in any practical decision analy-sis. In addition, the iterative steps of modeling, fit-ting and model-checking are as relevant for decisionanalysis as for inference. In particular, in a deci-sion problem, it makes sense to check that the deci-sion recommendations for the model applied to thedata are consistent with what would be expectedunder the model; that is, decision recommendationscan be used as test variables in predictive checksas in Gelman, Meng and Stern (1996). In the radonexample, other natural predictive checks arise fromconcerns expressed with the model; for example, arethere pockets of high-radon homes in otherwise low-radon counties, beyond that predicted by the model?More generally, the decision analysis should guidethe model-checking as well as the inference and themodeling itself.

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