Biostatistics & Epidemiology
Rothman Additive Interaction: RERI, AP, and Synergy Index (S)
Calculate Relative Excess Risk due to Interaction (RERI = RR_11 - RR_10 - RR_01 + 1), Attributable Proportion (AP), and Rothman’s Synergy Index (S) to detect public-health synergy missed by multiplicative models.
Catch biologic synergy that a multiplicative logistic/Cox product term reports as zero!
Suppose unexposed baseline risk is R_00 = 0.02 (B2), Exposure A alone raises risk to R_10 = 0.06 (RR_10 = 3.0 in B3), Exposure B alone raises risk to R_01 = 0.08 (RR_01 = 4.0 in B4), and joint exposure produces R_11 = 0.24 (RR_11 = 12.0 in B5). A multiplicative log-linear or Cox product term computes LN(12 / (3 * 4)) = LN(1) = 0.0000 and declares "no interaction"! Yet Rothman (1986) showed that independent additive excess risks predict RR_add = (3 - 1) + (4 - 1) + 1 = 6.0 (risk 0.12), meaning joint exposure doubles disease burden to 0.24!
Quantify public-health interaction across all three Rothman measures: (1) Relative Excess Risk due to Interaction RERI = RR_11 - RR_10 - RR_01 + 1 = 12 - 3 - 4 + 1 = +6.0000 (> 0); (2) Attributable Proportion AP = RERI / RR_11 = 6 / 12 = 0.5000 (50% of joint-exposure cases come purely from synergy!); and (3) Synergy Index S = (RR_11 - 1) / ((RR_10 - 1) + (RR_01 - 1)) = 11 / 5 = 2.2000 (> 1.0).
| Interaction Measure | Formula in Relative Risks | Sample Value & Null |
|---|---|---|
| Multiplicative Ratio RR_11 / (RR_10 * RR_01) | 12 / (3 * 4) | 1.0000 (Misses Additive Synergy!) |
| Relative Excess Risk (RERI / ICR) | RR_11 - RR_10 - RR_01 + 1 | +6.0000 (Null = 0) |
| Attributable Proportion (AP) | RERI / RR_11 | 0.5000 (50% of Joint Cases; Null = 0) |
| Rothman Synergy Index (S) | (RR_11 - 1) / ((RR_10 - 1) + (RR_01 - 1)) | 2.2000 (Null = 1.0) |
Relative Excess Risk due to Interaction (RERI) and Attributable Proportion (AP) Formula
Convert cell risks (R_00, R_10, R_01, R_11) into relative risks anchored at R_00 > 0 inside LET and compute RERI = RR_11 - RR_10 - RR_01 + 1.
=LET(r_00, B2, r_10, B3, r_01, B4, r_11, B5, IF(OR(r_00 <= 0, r_00 > 1, r_10 < 0, r_10 > 1, r_01 < 0, r_01 > 1, r_11 <= 0, r_11 > 1), "Invalid Risk Probability", LET(rr_10, r_10 / r_00, rr_01, r_01 / r_00, rr_11, r_11 / r_00, ROUND(rr_11 - rr_10 - rr_01 + 1, 4))))Rothman Synergy Index S = (RR_11 - 1) / ((RR_10 - 1) + (RR_01 - 1)) Formula
Guard the Synergy Index denominator (RR_10 - 1) + (RR_01 - 1): when single exposures are protective or cancel out to zero excess risk, S becomes undefined even though RERI remains well-behaved!
=LET(r_00, B2, r_10, B3, r_01, B4, r_11, B5, IF(OR(r_00 <= 0, r_00 > 1, r_10 < 0, r_10 > 1, r_01 < 0, r_01 > 1, r_11 <= 0, r_11 > 1), "Invalid Risk Probability", LET(ex_sum, (r_10 / r_00 - 1) + (r_01 / r_00 - 1), IF(ABS(ex_sum) <= 1E-9, "Undefined Synergy Denom (Use RERI)", ROUND((r_11 / r_00 - 1) / ex_sum, 4)))))Try the example
Use the original values or change them to test your understanding.