Original ArticlesAveraging Attributable Fractions in the Multifactorial Situation: Assumptions and Interpretation
Introduction
In a recent article in this journal [1] we considered the epidemiologic problem of estimating attributable fractions in a multifactorial framework; that is, we developed a procedure to quantify the impact of multiple exposures on the disease load in a population while simultaneously taking account of their interrelationships potentially influencing disease risk. In particular, we introduced the concepts of sequential attributable fraction (saf) and average attributable fraction (aaf) as aids for attributing the risk of disease to different exposures.
The saf considers sequences of exposure variables of interest and quantifies the additional effect of one exposure on disease risk after the preceding variables in the sequence have already been taken into account. Thus, the saf depends on the ordering within the sequence and is not unique for an exposure. In order to remedy this ambiguity we proposed to average the safs for one exposure over all possible orderings and termed the resulting parameter average attributable fraction due to the way of its construction (alternatively, we introduced the synonym partial attributable fraction for the same measure to emphasize its epidemiologic meaning in the multifactorial setting). We referred to earlier work by Cox 2, 3 who pointed to the analogy between game–theoretical reasoning in profit allocation and the epidemiologic task of apportioning disease risk among multiple exposure factors. We used his result on the uniqueness of the aafs as the only solution to the problem of partitioning disease risk under the special assumptions stated in his article [2].
The transfer of game–theoretical results to the epidemiologic context, however, can be elegantly rephrased in a framework of assumptions that are better suited for epidemiologic needs than those given by Cox. Furthermore, these considerations lead to an improvement in understanding the epidemiologic interpretation of these parameters. In this note we try to explain these new game–theoretical insights in a nontechnical manner focusing on the implications for the practical application in epidemiologic studies. Details on the underlying mathematical line of reasoning and the comparison of Cox's axiomatics to ours are provided elsewhere 4, 5. As in our original article [1], data of the Hordaland study on obstructive lung disease [6] serve as an illustrative example for the methodologic discussion.
Section snippets
Characterization of the multifactorial situation
Typically, the multifactorial etiology of a disease does not obviously give reason to a “natural ranking” among the exposure variables. Therefore, the task of dividing joint attributable fractions into exposure-specific components requires methods that do not impose any hierarchy among the exposures to be analyzed. Conventional adjustment procedures for attributable fractions 7, 8 clearly violate this requirement and alternative approaches have been developed in order to accomplish this. All
Sequential and average attributable fractions
In the Hordaland study [6] the relationship between several exposures and the presence of asthma and respiratory disorders (including, among others, “breathlessness during exercise” and “chronic cough”) was investigated based on a random sample of 4469 persons from the general population of the Norwegian county Hordaland. For the purpose of this illustration we consider the following dichotomized explanatory variables:
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Smoking (S): ever daily smoking vs. never daily smoking;
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Occupational Exposure
The bridge to game theory
The concept of averaging safs is rooted in methods of mathematical game theory and the Shapley-solution in particular. The latter provides a classical procedure for dividing up the total profit that several players gained by acting together in coalitions. Shapley [11] as well as Young [12] proved that this method is the only one satisfying reasonable fairness-axioms. The Shapley-solution has successively been transferred to various related topics such as cost allocation [13], price calculation
Symmetry
When assessing those shares of the disease burden in the population that can be attributed to each one of several exposures of interest it should be mandatory that the method used for dividing up the combined attributable fraction among the exposures is not influenced by the enumeration of variables or by any ordering among them. A method that satisfies this condition of independence is called symmetric. The example above has illustrated that the safs are not symmetric. Note, however, that the
Uniqueness of the aaf
In the preceding section several properties have been introduced as desirable characteristics of procedures assessing exposure-specific contributions to the joint attributable fraction in a multifactorial framework. Although these properties appear to be simple and natural, they are not fulfilled in general. The safs, for instance, are neither symmetric nor internally marginally rational. On the other hand, the analysis in terms of adjusted attributable fractions does not account for all
Conclusions
In summary, our original claim in [1] that the aafs represent a well-founded solution to the problem of apportioning disease risk among several interrelated exposures is still valid. The applicability of the concept to epidemiologic problems is even improved by our recognition that only symmetry and marginal rationality have to be assumed in order to guarantee the uniqueness of the aafs.
The epidemiologic repertoire of approaches aiming at solutions to this problem of partitioning disease risk
Acknowledgements
The first author's work was partially supported by a grant from the European Union (grant no. ERB CHRX-CT 940 693) and has been completed during a stay at the Institute of Statistics at the Catholic University of Louvain, Belgium. The second author's work on this project was supported by a grant from the German Research Foundation (grant no. Ge 637/3-1).
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