Statistical Science

Analysis of Local Decisions Using Hierarchical Modeling, Applied to Home Radon Measurement and Remediation

Andrew Gelman, David H. Krantz, Chiayu Lin, and Phillip N. Price

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This paper examines the decision problems associated with measurement and remediation of environmental hazards, using the example of indoor radon (a carcinogen) as a case study. Innovative methods developed here include (1) the use of results from a previous hierarchical statistical analysis to obtain probability distributions with local variation in both predictions and uncertainties, (2) graphical methods to display the aggregate consequences of decisions by individuals and (3) alternative parameterizations for individual variation in the dollar value of a given reduction in risk. We perform cost­benefit analyses for a variety of decision strategies, as a function of home types and geography, so that measurement and remediation can be recommended where it is most effective. We also briefly discuss the sensitivity of policy recommendations and outcomes to uncertainty in inputs. For the home radon example, we estimate that if the recommended decision rule were applied to all houses in the United States, it would be possible to save the same number of lives as with the current official recommendations for about 40% less cost.

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Statist. Sci., Volume 14, Number 3 (1999), 305-337.

First available in Project Euclid: 24 December 2001

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Bayesian decision analysis hierarchical models small area decision problems value of information


Lin, Chiayu; Gelman, Andrew; Price, Phillip N.; Krantz, David H. Analysis of Local Decisions Using Hierarchical Modeling, Applied to Home Radon Measurement and Remediation. Statist. Sci. 14 (1999), no. 3, 305--337. doi:10.1214/ss/1009212411.

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  • ance" (Environmental Protection Agency, 1997). But to Bayesians, this approach is tantamount to "sampling from the prior"; the Monte Carlo method is being used only to simulate values from assumed
  • e.g., Rai and Krewski, 1998). Still, with a few notable exceptions (Taylor, Evans and McKone, 1993; Brand and Small, 1995; Dakins, Toll, Small and Brand, 1996), the risk assessment literature seems in need of more formal Bayesian thinking, for which the present work (and the earlier work of Wolpert, Steinberg and Reckhow, 1993) may well serve as a blueprint. One way in which the risk assessment literature is ahead of that in statistics is in its willingness to discuss the value of human life on a dollar (or some other meaningful quantitative) scale. In clinical trials, for instance, real advances in decision theoretic solutions to the interim monitoring and final analysis problems have been stymied by the unwillingness of most statisticians, epidemiologists and clinicians to even contemplate such a mapping (though a few brave first attempts have been made
  • by Berry and Ho, 1988; Stangl, 1995). An important feature of the present paper is the authors' description of how already established government guidelines for what constitutes a radon exposure level worthy of remediation implicitly determines dollar values per microlife (Section 4.2). Clearly such a linear scale is not appropriate when we move far from the origin (no reasonable person would surrender one million of his own microlives for any dollar amount), but discussions of this sort may well have beneficial impact in risk assessment strategies far beyond environmental settings, if in no other way but informing decision-makers as to what implicit values their recommendations are placing on fractions of lives. Turning then to specific comments on the authors' approach, given the power of modern MCMC techniques I was surprised that the model components considered in Section 3 were essentially confined to normal distributions. The model apparently treats the variance parameters 2 and 2 (as well as a variety of tuning parameters in Section 4) as constants, instead of more plausibly assuming distributions for them. Indeed, some of the modeling is not even being shown: (2) is written as a prior (or a "predictive" in the authors' nomenclature), but in fact it must be the result of a preliminary prior-to-posterior calculation, combining some prior on the regression parameters with some preliminary data y on typical radon concentrations in U.S. homes. What is this preliminary data and model? Do its residuals suggest any evidence of lingering spatial correlation? Also, the two-stage implementation of the preliminary (Section 3.1) and house-specific (Section 3.2) models is odd, since it forfeits the usual Bayesian advantage of a single unifying model that enables all sources of variability and uncertainty to correctly propagate throughout its levels. As the authors mention, the paper's main focus is on the decision analysis in Section 4. Here there are any number of assumptions with which one could quibble (the flat $2000 to remediate any home regardless of location, the 70-year life expectancy for every occupant, etc.); one could either place distributions on these quantities as well, or simply undertake a variety of sensitivity analyses (as the authors describe in some detail in Section 6). While I don't wish to nitpick further here, I did find the approach for "discounting the value of a life," described near the beginning of Section 4.1, to be somewhat confusing. At first blush, if Dd is the amount we are willing to pay to save one microlife now, then since lives saved 20 years in the future are worth less, it seems the revised Dd should be decreased (not increased) by a factor of 1 0520. However, recall that the paper does not really specify Dd from first principles, but rather "backs it out" by viewing the $2000 remediation cost as fixed. Thus if the value of the lives saved decreases, our cost per life saved must go up. Yet even here, it seems that the appropriate increased cost must be backed out from (5) and (6) as well, discounting each future year's risk separately in the thirty-year decision period rather than applying a single inflation factor to Dd. Of course, the actual dollar amount any given person would spend per microlife saved is probably more a function of their own financial resources and aversion to risk than any governmentally recommended remediation levels. I am personally acquainted with a suburban couple with three children who, after reading an early report on the alleged dangers of living near high-voltage electrical lines like the ones near their home, immediately sold the place and moved. Because they did this at just the time when popular concern over this potential risk was at its zenith, their total financial loss in the transaction (including moving expenses and remodeling their new home) was in the neighborhood of $80-100,000: in the light of more recent data on the subject, a colossal amount spent per expected microlife saved. In the language of (6), for this couple Dd (hence Dr) was essentially infinity for this perceived risk, and thus Raction Rremed.
  • (e.g., Stokey and Zeckhauser, 1978). But the critical importance of quantifying underlying probability distributions and sources of uncertainty-indeed, the understanding that decision-makers can even use such information-has been overlooked by other health researchers. This has created the sad situation where policy analysts are waiting for probability data for use in decision analysis, while health researchers are providing only the point estimates that they apparently assume to be preferable and useful. An issue like radon testing provides a useful way to introduce the social value of data gathering into public health without the medical arena's limits of allowable practice. Medical testing is functionally equivalent to other areas of public health, such as radon testing or finding out how safe a highway is by building it and watching what happens. But the culture of medicine and the legal climate complicate things. While allowing the gathering of patient data for the social good, the current culture makes it difficult to perform tests with expected social benefits but a net expected cost to the particular patient, or to withhold tests that have negative expected value but have any chance of improving a diagnosis. Indeed, it is sometimes difficult to even discuss making more efficient decisions in the medical arena. An environmental health issue like radon allows socially optimal recommendations because most people are amenable to persuasion about the right choice, given their underlying lack of knowledge, the relatively low individual costs and risks and the noninvasiveness of most actions. At the same time, the social costs and risks are fairly high, and it is worth the effort to try to minimize them. One major policy advantage of the situation described by Lin et al. (one which should probably be given more attention in the research literature and policy process) is that it allows individuals with different tastes for risk to take different actions. Unlike public health decisions that must be made by a central authority for everyone (cases ranging from effluent regulation to airplane safety features), the decision about radon parallels the decisions about medical care and consumption. If someone is more willing to risk disease or less willing to spend money to avoid it (or does not believe that the risk is actually real), then he has the option of making a different decision than the official recommendation. This would be particularly reasonable if, as suggested by Lin et al., EPA made recommendations that ignored household composition. Single nonsmoking assistant professors could rationally choose to ignore the radon risk in their homes. An extension of these principles in a different direction (and into more controversial areas) is to assess what population-level research would be most useful given our current data and priors (Phillips
  • and Maldonado, 1999). Epidemiologic studies, along with most quantitative health research, tend to conclude by calling for more research, but very seldom assess exactly what the further research should do. Further research can be used to eliminate some of the measurement error, simply assess the level of measurement error, eliminate confounders, measure confounders or just increase the sample size. Within all these choices, there are continuous ranges of choices along multiple dimensions. Yet the decision analysis principles are still the same as those in Lin et al. By fully assessing what we know and what more we would be likely to know following future research, we can better determine when to act, when to walk away or what more we want to know.
  • ing, rebuilt and new buildings (Cole, 1993). In the United States, EPA and state health department officials have told us that when people ask them for radon advice, they don't want to have to think about a lot of different issues; they just want to know what a "safe" radon level is. Whether or not the policymakers are right about the need for simplicity, it is clear that official radon recommendations will in fact be based on quite simple monitoring and remediation criteria. So when it came to deciding what policies to analyze, we decided to restrict ourselves to fairly simple variations on the EPA's current recommendations. We make no claim that the resulting policies are the best of all possible ones; we only claim that they would be improvements to the current recommendations.
  • treatment (Dehejia, 1998). In response to Carlin's comments about the underlying statistical model: yes, we previously fit a fully Bayesian model to a large set of shortand long-term radon measurements, along with other information on houses in the dataset and counties in the United States. We used the posterior distribution of that analysis as the prior distribution for the analysis in this paper. For each county and house type, we used the posterior simulation draws from our previous analysis to compute a prior mean and standard deviation for the mean log radon level for those counties and house types. We then assumed that the standard deviations of the measurements within that county and house type were estimated to a high precision (and could thus be summarized by posterior point estimates?), which is not too bad an approximation given the large datasets used to construct that posterior distribution. If we were less confident that the posterior distribution was close to normal, then we would have worked with the simulation draws themselves, but in this case, we wanted the convenience of the normal approximation, which allowed some of the steps of the decision analysis to be performed analytically. Another approach would be to use the normal approximation, but then check it (or correct for it) at the end of the analysis, using importance sampling. Finally, we believe it is important to link the concerns of statistical modeling to those of decision analysis. Sensitivity analysis is already recognized as a crucial step in any practical decision analysis. In addition, the iterative steps of modeling, fitting and model-checking are as relevant for decision analysis as for inference. In particular, in a decision problem, it makes sense to check that the decision recommendations for the model applied to the data are consistent with what would be expected under the model; that is, decision recommendations can be used as test variables in predictive checks
  • as in Gelman, Meng and Stern (1996). In the radon example, other natural predictive checks arise from concerns expressed with the model; for example, are there pockets of high-radon homes in otherwise lowradon counties, beyond that predicted by the model? More generally, the decision analysis should guide the model-checking as well as the inference and the modeling itself.
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