It has been noted that there is ambiguity in the expression "attributable fraction," and epidemiologic literature has drawn a distinction between "excess fraction" and "etiologic fraction." These quantities do not necessarily approximate one another, and the etiologic fraction is not generally estimable without strong biologic assumptions. In previous studies, researchers have explained the relations between excess and etiologic fractions in the potentialoutcome framework, and few authors have

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... xplained the relations between these concepts by showing the correspondence between the potential-outcome model and the sufficient-cause model. In this article, the authors thoroughly clarify the conceptual relations between excess, attributable, and etiologic fractions by explicating the correspondence between these 2 models. In so doing, the authors take into account the potential completion time of each sufficient cause, which contributes to further insight to clarify the 2 types of etiologic fraction, i.e., accelerating etiologic proportion and total etiologic proportion. These 2 measures cannot be distinguished in epidemiologic data, and the differences might be subtle. However, they are closely related to a very fundamental issue of causal inference, that is, how researchers define etiology. Further, the authors clarify the relation between 3 distinct assumptionspositive monotonicity, no preventive action (or sufficient-cause positive monotonicity), and no preventive sequence. causal inference; monotonicity; potential outcomes; sufficient causes More than 2 decades ago, Greenland and Robins (1) noted that there is ambiguity in the expression "attributable fraction" and drew a distinction between "excess fraction" and "etiologic fraction." These quantities do not necessarily approximate one another, and the etiologic fraction is not generally estimable without strong biologic assumptions (1). More detailed statistical discussions were addressed in related articles (2, 3), and other papers were meant to be educational for general readers (4-6). In these studies, researchers have explained the relations between excess and etiologic fractions in the potential-outcome framework (7). In some recent articles, investigators have discussed the concept of attributable fractions in the sufficient-component cause framework and have described how redundancy of sufficient causes impacts epidemiologic effect measures (8-12). In this article, we aim to clarify the relations between excess fractions, attributable fractions, and etiologic fractions in detail by explicating the correspondence between the potential-outcome model and the sufficient-cause model. In so doing, we take into account the potential completion time of each sufficient cause, which further clarifies how researchers should define etiology. Further, as we explain below, in most studies investigators have (sometimes implicitly) made assumptions such as positive monotonicity, no preventive action, and no preventive sequence, which might have resulted in some confusion regarding these concepts. Thus, we also aim to clarify the relation between these assumptions. To avoid technical complications, we do not discuss additional problems that can arise when exposure has multiple levels or when competing risks are being considered. POTENTIAL OUTCOMES FOR A BINARY EXPOSURE VARIABLE We let E denote a binary cause of interest (1 ¼ exposed, 0 ¼ unexposed) and Y denote a binary outcome of interest (1 ¼ outcome occurred, 0 ¼ outcome did not occur). Then, we let Y ei denote the potential outcomes for individual i if, possibly contrary to fact, there had been interventions to set E ¼ e (7). For each individual i, there would thus be 2 possible potential outcomes Y 1i and Y 0i corresponding to what would have happened to that individual when that person was 567 Am J Epidemiol. 2012;175(6):567-575 exposed and unexposed, respectively. As a result, individuals can be classified into 4 (i.e., 2 2 ) different response types, as enumerated in Table 1 (13). We let p j and q j , j ¼ 1-4, be proportions of response type j in the exposed and unexposed subcohorts, respectively. In some cases, the effect of the cause E may be in the same direction for all individuals. We say that E has a positive monotonic effect on Y if Y ei is nondecreasing in e for all individuals, that is, Y 0i