Model Uncertainty
Introduction
For purposes here, a model is a mathematical statement of a theory or hypothesis, that may or may not have a verbal counterpart. However, for the most part, the discussion will also apply to theories which are formulated verbally. Thus, when talking about model uncertainty, we are largely concerned with what might also be called "theoretical" or "interpretational" uncertainty. Models are required to infer answers where facts are unavailable. Model uncertainty arises when two or more alternative interpretations give rise to different predictions.
In health risk assessment, the best known example of model uncertainty is the use of a model to extrapolate the frequency of cancer occurrence in a population from high to low doses. However, in addition to relating a dose to an effect, model uncertainty can arise at virtually every turn of an assessment. For example, in addition to relating a dose to an effect, alternative plausible interpretations may arise in judging the shape of frequency distributions used to describe a parameter in a population, the interpretation of dose-independent effects, or in the extrapolation from one species to another.
In a sense, model uncertainty may be viewed as an instance of what is known in philosophical literature as the "problem of induction" -- any generalization drawn from previous experience goes beyond experience and may prove to be incorrect in the face of new experience.[1] Theories will always be constructed which predict, or posit the occurrence of virtually any future event. To use a classic example, one can propose that an object is "grue" rather than blue, where grue is defined to be blue until the year 2000 and green thereafter. Both the "blue" theory and the "grue" theory are consistent with experience, yet yield different predictions.
Thus, without some constraint on the choice of theories to be used, virtually anything can be predicted from anything, and all data and information is rendered useless for guiding a decision. In the process of developing information which is intended to guide decisions, constraints are placed. In fact, there are several methods for doing so. The remainder of this discussion will compare those methods.
* Technocracy. The first approach to dealing with model uncertainty, which shall be referred to as technocracy, involves assigning an individual expert the entire responsibility for resolving the impact of uncertainties arising from alternative interpretations of the decision. Because there is no objective procedure, this is not a "method" in a scientific or logical sense at all. This method is "simpler" from an administrative point of view, not because the issues that become apparent with a more explicit treatment are unimportant, but because the object of the present discussion is hidden. This may be an impetus for preferring this method precisely because model uncertainty is prevented from becoming the focus of an argument. Thus, the principal difference between subjective and explicit methods of inference lies in the appearance of model uncertainty as a topic of discussion.
Often, decisions may be made by committees of experts rather than individuals. Unless model uncertainty becomes a topic for discussion, this may still be viewed as a technocratic exercise -- the process necessarily centers on discussion of the action taken, rather than any underlying belief.
* Default factors. The oldest and most widely used technique for dealing with model uncertainty in the context of regulatory toxicology involves the application of a fixed arbitrary factor whenever an instance of model uncertainty (or in some cases, statistical uncertainty) is encountered. For instance, factors of 10 are often applied when it is necessary to make statements about large populations based on the observations of a few individuals, about humans based on the observations of rodents, or about long-term exposures based on the observations from short-term exposures. Judging from appearances, the general rule is to divide by 10 upon each instance of model uncertainty. Because selection of the factor is not dependent on the state of the evidence, it is clear that the application of safety or uncertainty factors is not really a method for evaluating model uncertainty. Rather, it is a technique for avoiding the issue by not making a prediction.
Default factors were originally designed for setting regulatory levels.[2] However, the levels are also sometimes represented as factual statements, i.e. thresholds.[3] In this circumstance, the default factor approach may be thought of as a default model (see below), rather than a decision-producing policy. Similarly, default factors are sometimes represented as instruments of a technocratic exercise (see above) - in which case the factors are merely window dressing for a decision that has already been made. They cannot, however, be considered to be both procedures and products of technocratic judgment as these concepts are antithetical. However, it is possible to mix in some of each to create a hybrid.[4] For instance, a factor of 3 may be used instead of 10 because, on the basis of expert judgment, the degree of model uncertainty is smaller. The process is still semi-procedural to the extent that the factors must follow half-orders of magnitude.
* The default model. Another common technique for evaluating data in the context of public health is to employ an explicitly specified model. This technique has arisen out of the acknowledgment that the model employed may be a determinant in making the prediction. However, in spite of the fact that there may be multiple plausible interpretations of the data, a default model treatment of model uncertainty resolves the issue, by dictating one of the many models available as the basis for the prediction, irrespective of the state of the evidence. The default model approach deals with model uncertainty by suppressing it. The best known example of default model methodology in regulatory toxicology is the employment of the linear multistage model in cancer risk assessment.
In theory, one can depart from the default model if the evidence is sufficient. However, the conditions under which a departure might be justified are never specified and are rarely discussed, and in practice, almost never occur. Because the default model is not chosen on the basis of the available data, it is singled out from competing in the face of the critique to which the alternative models are subjected. It may therefore be selected even if it is contrary to the available data.
* The best model. A single model could be selected from available alternatives by explicitly formulated criteria that define the "best" model. Somewhat surprisingly, this method has not been employed in regulatory toxicology, although it appears to be the primary technique recently advocated by a National Research Council committee.[5] Perhaps the main reason that this method has not found favor is that there may often be very little to choose from between the best model and the alternatives. Thus, relatively small changes in the evidence may result in rather large consequences to assessment outcome.
* Weighting by experts. By interviewing experts in the field studied, alternative models or theories may be identified and assigned relative probabilities[7]. The relationship between data and theory may be viewed as a psychological exercise in which the scientist or expert is the subject. This technique has been employed in health risk assessment in the context of an evaluation of chloroform.[8] The major focus in this evaluation was to identify which data was important, which theories were plausible and the culmination in assignment of the probability that each theory was correct. Each of these tasks was accomplished subjectively, i.e. by consulting the opinion of a group of experts.
* Weighting by algorithm. Instead of selecting the best model, another possible use of explicit criteria for model preference is as a basis for assigning relative model probabilities.[9] The essential notion which defines this approach is that the weights assigned to the models for any given evidence is preordained by an explicit algorithm. Although models which are judged to be best are given more weight in generating a distribution of possible outcomes, any model judged a priori to have some likelihood of being correct will have an influence on the outcome of the assessment. As with weighting by expert, the generation of a weighting algorithm is a normative endeavor or process of explication. The most important difference is that an algorithm is expected to transcend particular issues.
To attain comparable results, it will also be necessary to preordain a list of plausible theories which will be used for a particular problem. Still, the constraint is loosened: Instead of one default model, the methodology is defined by a set of default models, the size of which is limited only by practical considerations. While it should be possible to revise the list of plausible theories, doing so constitutes a revision of the methodology, which may make the present evaluation incomparable to previous assessments. In such a circumstance, the value of improving the methodology by adding a new theory must be weighed against the cost of recalculating prior assessments to yield comparability.
With regard to this criteria, technocratic decision making and the default factor approach may be quickly placed at one end of the spectrum. These methods do not attempt to separate risk assessment and risk management; at the end of the process the decision has been made.[10] The two single model approaches are better in this regard, but they both dictate to some degree the impact model uncertainty will have (or not have) on the outcome of the decision.
The application of "conservative" criteria in selecting a level or model also necessarily compromises the goal of delivering information because it is clearly intended to bias the result of the decision. There is also a potential problem with selection of best or central tendency criteria for selecting a single model -- the possibility of other interpretations is not allowed to play a role in the outcome of the decision process and therefore, in some way biases the decision. If the assessment process is to be truly separate, then it is the decision-maker or policy-process which must determine the impact of the uncertainty, including model uncertainty, on the outcome of the decision, not the technical person or process. Model uncertainty cannot under any circumstances be portrayed with the results of a single model.
To make model uncertainty transparent, the development of a suitable vocabulary, whether it be words[11] or numbers, for describing model uncertainty is essential. Conversely, to the extent that any of the evaluation is subjective or requires direct participation of expert judgment, it is not transparent. Therefore, virtually the entire weight-of-the-evidence method and in part the methods employing defaults (assuming there is ever to be some deviation from the default) and decision analysis cannot be said to be transparent. Although the application of uncertainty or safety factors appears to be transparent as a policy, it is not transparent with regard to model uncertainty.
Transparency and the ability to separate risk assessment and risk management correlate strongly. Yet, as difficult as it may be to separate risk assessment from management, it may be even harder to combine them. In examining the rationale behind a putative action, discussion of intent or desire vs. belief or practicality with apparent necessity become separate issues. However, transparency may yield different decision-making paradigms. For instance, the default model approach, by seeking to resolve the issue as a matter of policy makes model uncertainty an issue of risk management, while the model weighting technique makes model uncertainty a matter of risk assessment.[12]
The evaluations conducted subjectively may be very simple or very difficult, depending on the level of effort expended by the individual(s) involved. A dictum by a single technocrat may be quite efficient. However, if the issue is to be decided by committee, the task may become very difficult. Because the need for expert opinion about model uncertainty arises from the absence of knowledge, the likelihood that an issue will be resolved may decrease as the expenditure for expert opinion increases -- it may be very difficult to get experts to agree about what they don't know.
As long as there is no deception involved, methods which rely entirely on expert judgment will presumably reflect the opinion of the experts queried. Attempting to make an analysis transparent will tend to reduce correspondence to expert opinion, because it is fraught with the difficulty of communicating the state of knowledge or degree of uncertainty to a less knowledgeable (at least with respect to the issue) person. Also, because transparency requires general measures, there is always the danger that important considerations will be excluded from the analysis. Therefore, to the extent that the goal of the assessment is to meet a standard of scientific objectivity, expert opinion is the gold standard by which other methods must be judged.
However, if the expert is entrusted with the entire decision, then there is more than belief involved, the expert also charts an action. If there are multiple experts involved (i.e. a committee), there will be no way of knowing whether any disagreement that the experts may have over an action involves belief or valuations of the beliefs, unless they are able to convey (at least among themselves) their state of belief. Therefore, one cannot determine whether or not correspondence to scientific opinion has been attained without some transparent segregation of risk assessment and risk management.
At the other end of the spectrum, employment of a default factor or model virtually guarantees that the result will not accurately reflect current scientific opinion. First, default methodologies may not meet the decision-maker's criteria. The same model or factor cannot always be the best model for all sets of evidence. Similarly, if the decision-maker wants to be "conservative", the same model cannot be equally conservative all the time. The motivations of reproducibility and simplicity which favor default-based methodology necessarily make the evaluations relatively data-insensitive. Even if departures from the defaults are permitted, rather substantial evidence must be assimilated before this result is obtained and there will inevitably be a bright line which in some circumstances can greatly accentuate small differences in the bodies of evidence for two different chemicals. Second, default methodologies do not, by design, give the decision-maker any representation of the extent of the uncertainty arising from interpretation. Therefore, the decision-maker is given an appearance of certainty which matches no one else's opinion.
Between judgment given free reign and thoughtless application of rules lie those methods which employ criteria for evaluating models. In the final analysis, objective model preference criteria must also be thoughtless rules, but because they are applied more abstractly, good rules may be expected to generate a result which is better adapted to the evidence under scrutiny, and will (by design) more closely match the judgment of an expert. Consistent variance of rules with scientific opinion may be taken as evidence that the rules are in need of revision.
The desire for an accurate portrayal of the state of current knowledge must often be balanced against the need for a method to be simple and/or transparent. Meeting the demand for objectivity tends to introduce superficiality.[14] This can be overcome by making the process more elaborate, but the cost of executing the process is then sacrificed. If the issues are at all complex, then responding to those issues in a reasonable manner will necessarily become more difficult, and communicating or explaining the results of an analysis will require more effort.
To put this issue in another light, it may take on the heading of risk comparison. Even when there is only one interest at stake, there is often a need to compare risks. If uncertainty is an issue to be decided by each expert, one has no common standard unless the same experts perform all the evaluations. There is also the matter of justice. Regardless of whether or not the method is intended to have separate steps for information development and decision-making, there is often a need to be able to demonstrate that an even-handed process is applied.
The clear example of such law-like preemption of discussions of model uncertainty is use of the default model. Default factors, best model and weighting by algorithm also have potential for the attainment of impersonality -- so long as the involvement of experts is a thing of the past. Since expert judgment and weighting involve the direct expert participation in the decision or information development, impersonality is not achieved. However, even if it doesn't explicitly say where weights come from, the weighting by expert method may be given some credit for stating which theories are employed.
Transparency is prerequisite for impersonality; thus, impersonal methods will also be transparent. This correlation is not entirely reflected in summary Figure 2.
The correlation is not reflected because the methods have been rated for impersonality based on the entire process, whereas the transparency rating refers to model uncertainty only. Yet, transparency might refer to expert judgment, clearly not impersonal. Impersonality may also affect execution cost if a process can be performed by computer.
Beyond that, Figure 2 may be used in two ways: Given motivation or criteria, a method may be selected. Or, given a method, a motivation may be gleaned. The split rating for Technocracy in terms of Cost of Execution reflects the efficiency of the method when a single expert is used versus its inefficiency when a a committee is employed.
[1] Bertrand Russell, A History of Western Philosophy 659-74 (1946).
[2] Arnold J. Lehman, O. Garth Fitzhugh, 100-Fold Margin of Safety, 18 Ass'n Food Drug Officials Q. Bull. 33 (1954).
[3] Michael L. Dourson, Jerry F. Stara, Regulatory History and Experimental Support of Uncertainty(Safety) Factors, 3 Reg. Pharm. Tox. 224 (1983).
[4] See, e.g., Robert L. Maynard et al., Setting Air Quality Standards for Carcinogens: An Alternative to Mathematical Quantitative Risk Assessment, 14 Hum. & Exper. Toxicol. 175 (1995).
[5] Committee on Risk Assessment of Hazardous Air Pollutants, National Academy of Sciences, Science and Judgment in Risk Assessment (1995).
[6] Ian Hacking, The Emergence of Probability (1975). The philosophical term for model weighting is probability logic or logical probability. Hacking presents Pascal's wager on the existence of God as the first historical employment of probability logic.
[7] Granger M. Morgan & Max Henrion, Performing Probability Assessment, in Uncertainty: A Guide to Dealing With Uncertainty in Quantitative Risk and Policy Analysis 141 (1991).
[8] John S. Evans et al, A Distributional Approach to Characterizing Low-Dose Cancer Risk,14 Risk Anal. 25 (1994).
[9] Clark D. Carrington, Logical Probability and Risk Assessment, 2 Hum. Ecol. Risk Assess. 62 (1996).
[10] For instance, it may dictate some level of a chemical above which some action should be taken, or toss a compound into the jaws of the Delaney clause. Of course, this "decision" can be, and often is, second-guessed and preempted.
[11] Paul Krause et al., An Argumentation-Based Approach toRisk Assessment, 5 IMA J. Math Appl. Bus. Ind. 249 (1994). It is argued that it is inappropriate to use numbers to describe uncertainty because the reasoning is non-quantitative.
[12] For instance, in the recently proposed Environmental Protection Agency Cancer Risk Assessment Guidelines (16 Fed. Reg. 7998), the agency declined to represent uncertainty as a series of competing models because "This would obviate the function of the policy default."
[13] The use of experts to apply default factors (as opposed to someone else) may be thought of as an administrative cost, rather than a cost of execution.
[14] Theodore M. Porter, Trust in Numbers, The Pursuit of Objectivity in Science and Public Life (1995). Exploring the tradeoff between relying on rules and relying on expert judgment.
