The cumulative completion risks with the 95% confidence intervals (CIs) for the example data in Table 1.

a<p>age40 years.</p>b<p>body mass index25 kg/m².</p

Excess Relative Risk as an Effect Measure in Case-Control Studies of Rare Diseases

<div><p>Epidemiologists often use ratio-type indices (rate ratio, risk ratio and odds ratio) to quantify the association between exposure and disease. By comparison, less attention has been paid to effect measures on a difference scale (excess rate or excess risk). The excess relative risk (ERR) used primarily by radiation epidemiologists is of peculiar interest here, in that it involves both difference and ratio operations. The ERR index (but not the difference-type indices) is estimable in case-control studies. Using the theory of sufficient component cause model, the author shows that when there is no mechanistic interaction (no synergism in the sufficient cause sense) between the exposure under study and the stratifying variable, the ERR index (but not the ratio-type indices) in a rare-disease case-control setting should remain constant across strata and can therefore be regarded as a common effect parameter. By exploiting this homogeneity property, the related attributable fraction indices can also be estimated with greater precision. The author demonstrates the methodology (SAS codes provided) using a case-control dataset, and shows that ERR preserves the logical properties of the ratio-type indices. In light of the many desirable properties of the ERR index, the author advocates its use as an effect measure in case-control studies of rare diseases.</p></div

Re-analysis of the coronary heart disease (CHD) occurrence data for personality type A and B persons^a.

a<p>from Table 7–24 in Selvin S (1991) Statistical Analysis of Epidemiologic Data. New York: Oxford University Press.</p>b<p>CI: confidence interval.</p

Additional file 1: of Sharp bounds on sufficient-cause interactions under the assumption of no redundancy

Derivations of the bounding formulas. (PDF 272 kb

Re-analysis of the mortality data from all causes for tolbutamide and placebo treatment groups^a.

a<p>from Table 15-1 in Rothman KJ, Greenland S, Lash TL (2008) Modern Epidemiology, 3rd ed. Philadelphia: Lippincott.</p>b<p>CI:confidence interval.</p

Additional file 3: of Sharp bounds on sufficient-cause interactions under the assumption of no redundancy

A proof that the bounds are sharp. (PDF 179 kb

Results for the analysis of the data in Table 1.

<p>^a Confidence Interval.</p><p>^b Excess Risk Ratio.</p><p>^c Population Attributable Fraction.</p><p>^d Attributable Fraction among the Exposed.</p><p>Results for the analysis of the data in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121141#pone.0121141.t001" target="_blank">Table 1</a>.</p

Thresholds for PRISM (peril ratio index of synergy based on multiplicativity) and RERI (relative excess risk due to interaction), when and (A), and when and (B).

<p>The solid lines are the no-synergy lines of , above which are the regions of positive synergisms, and below which, the regions of negative synergisms. The upper long-dash lines mark the thresholds of , above which are the regions of the ‘class’ synergy. The lower long-dash lines mark the thresholds of , below which are the regions of the ‘class’ synergy. The upper short-dash lines mark the thresholds of , above which are the regions of the ‘class’ synergy. The lower short-dash lines mark the thresholds of , below which are the regions of the ‘class’ synergy.</p

The no-synergy lines judged by the PRISM (peril ratio index of synergy based on multiplicativity), and the no-interaction lines judged by the SIM (synergistic index of multiplicativity) and RERI (relative excess risk due to interaction) criteria, when and (A), and when and (B).

<p>The solid lines are the no-synergy lines of , above which are the regions of positive synergisms, and below which, the regions of negative synergisms. The dash-and-dot lines are the no-interaction lines of in terms of relative risks. The short-dash lines are the no-interaction lines of in terms of odds ratios. The dotted lines are the no-interaction lines of in terms of relative risks. The long-dash lines are the no-interaction lines of in terms of odds ratio.</p