Appendix
Proof that choosing the category with the lowest risk when both factors are considered jointly as the reference category will give consistent results among the three measures of additive interaction.
Clearly, RERI > 0 if and only if AP > 0 since \( {\text{AP}} = {\frac{\text{RERI}}{{{\text{RR}}_{{{\text{A}} + {\text{B}} + }} }}} \) and likewise RERI < 0 if and only if AP < 0. If the factors are recoded so that the category with the lowest risk when both factors are considered jointly is selected as the reference category then we will have that RRA+B− ≥ 0 and RRA−B+≥0. When RRA+B− ≥ 0 and RRA−B+≥0, we have that S > 1 if and only if \( {\frac{{{\text{RR}}_{{{\text{A}} + {\text{B}} + }} - 1}}{{({\text{RR}}_{{{\text{A}} + {\text{B}} - }} - 1) + ({\text{RR}}_{{{\text{A}} - {\text{B}} + }} - 1)}}} > 1 \) which holds if and only if \( {\text{RR}}_{{{\text{A}} + B + }} - 1 > ({\text{RR}}_{{{\text{A}} + {\text{B}} - }} - 1) + ({\text{RR}}_{{{\text{A}} - {\text{B}} + }} - 1) \)which holds if and only if \( {\text{RERI}} = {\text{RR}}_{{{\text{A}} + {\text{B}} + }} - {\text{RR}}_{{{\text{A}} + {\text{B}} - }} - {\text{RR}}_{{{\text{A}} - {\text{B}} + }} + 1 > 0 \). And similarly, with RRA+B− ≥ 0 and RRA−B+≥0, we have that S < 1 if and only if RERI < 0.
Example demonstrating that if recoding is done one factor at a time rather than jointly, the three measures of additive interaction may disagree and S may be negative.
Consider a case control study with two dichotomous factors (G and E) with 600 individuals with E = 0, G = 1, 600 with E = 0, G = 1, 200 with E = 1, G = 0 and 200 with E = 1, G = 1 with the number of cases and controls in each category reported below.
Odds ratios representing joint effects |
E = 0, G = 0 | 48 | 552 | 1.00 (reference) |
E = 0, G = 1 | 66 | 534 | 1.42 |
E = 1, G = 0 | 12 | 188 | 0.73 |
E = 1, G = 1 | 6 | 194 | 0.36 |
Odds ratios representing single effects |
E = 0 | 114 | 1,086 | 1.00 (reference) |
E = 1 | 18 | 382 | 0.45 |
G = 0 | 60 | 540 | 1.00 (reference) |
G = 1 | 72 | 528 | 1.23 |
If the factors were recoded one at a time then we would choose E = 1 as the reference category for E as the OR for E = 1 is 0.45 and we would choose G = 0 as the reference category for G since the OR for G = 1 is 1.23. If the factors are recoded jointly then we see that E = 1, G = 1 is the category with the lowest odds and so E = 1 would be chosen as the reference category for E and G = 1 would be chosen as the reference category for G.
If we proceeded by recoding the factors one at a time so that the reference category A− was E = 1 and the reference category B− was G = 0, we would obtain the following odds ratios:
Odds ratios representing joint effects |
A − B − (E = 1, G = 0) | 12 | 188 | 1.00 (reference) |
A − B + (E = 1, G = 1) | 6 | 194 | 0.48 |
A + B − (E = 0, G = 0) | 48 | 552 | 1.36 |
A + B + (E = 0, G = 1) | 66 | 534 | 1.94 |
Here we would obtain a synergy index of: \( {\frac{{{\text{RR}}_{{{\text{A}} + {\text{B}} + }} - 1}}{{({\text{RR}}_{{{\text{A}} + {\text{B}} - }} - 1) + ({\text{RR}}_{{{\text{A}} - {\text{B}} + }} - 1)}}} = {\frac{1.94 - 1}{(1.36 - 1) + (0.48 - 1)}} = - 5.86 \). The synergy index is negative. With the coding in the Table above RERI = 1.1 and AP = 0.57.
If instead we proceed by recoding the factors jointly by choosing the combined category with the lowest risk as the reference so that the reference category A− was E = 1 and the reference category B− was G = 1, we would obtain the following odds ratios:
Odds ratios representing joint effects |
A − B − (E = 1, G = 1) | 6 | 194 | 1.00 (reference) |
A − B + (E = 1, G = 0) | 12 | 188 | 2.06 |
A + B − (E = 0, G = 1) | 66 | 534 | 3.93 |
A + B + (E = 0, G = 0) | 48 | 552 | 2.81 |
Now we obtain a value of the synergy index within the range from 0 to infinity:
\( {\frac{{{\text{RR}}_{{{\text{A}} + {\text{B}} + }} - 1}}{{({\text{RR}}_{{{\text{A}} + {\text{B}} - }} - 1) + ({\text{RR}}_{{{\text{A}} - {\text{B}} + }} - 1)}}} = {\frac{2.81 - 1}{(3.93 - 1) + (2.06 - 1)}} = 0.45 \). The value of S < 1 indicates a negative interaction which is in agreement with what is indicated by RERI = −2.18 < 0 and AP = −0.76 < 0.