To the Editor:
The conditions required to prove the presence of sufficient-cause interaction using the coefficients from a log-linear model with 2 binary exposures have previously been described by VanderWeele.1 We broaden the conditions and derive easy-to-remember criteria such as: “If the product of the separate relative risks is greater than or equal to their sum, testing for β3>0 implies sufficient-cause interaction” and “The interaction term on the relative risk scale should be larger than the sum of the separate relative risks divided by their product.” The criteria presented here are especially important for judging sufficient-cause interaction when the relative risk of the interaction term is below unity (a “negative” interaction term on the logarithmic scale).
VanderWeele1 derived conditions for sufficient-cause interactions in a saturated log-linear model for the probability of the outcome (D) as a function of 2 binary determinants (X 1 and X 2):
Based on this model and without assuming monotonicity of the effects of X 1 and X 2 on D, the most general condition for the presence of sufficient-cause interaction is:
(the latter being VanderWeele's Equation 91).
From this, minimal values of β1 and β2 can be derived for which a test for statistical interaction (β3 > 0) implies sufficient-cause interaction.
To derive the conditions for which a test for statistical interaction, β3 > 0, in the log-linear model corresponds to a test for a sufficient-cause interaction, VanderWeele rewrote the above formula as:
By using, rather than the factor (½), the factors x and (1 − x), with x between 0 and 1, more precise conditions can be obtained for which testing for β3 > 0 implies sufficient-cause interaction. We rewrite VanderWeele's Equation 9 as:
Detailed derivations are reported in eAppendix 1(http://links.lww.com/EDE/A409). Eventually, the conditions for β3 > 0 to imply sufficient-cause interaction become:
As β1 = log(RR10) and β2 = log(RR01) this leads to
Note that x = yields the conditions: RR01 ≥2 and RR10 ≥2, as obtained by VanderWeele. However, when x is chosen to be equal to 1/RR01, the second part of condition (2) becomes:
which can be rewritten as:
(see eAppendix 1, http://links.lww.com/EDE/A409). This leads to the easy-to-remember rule that: “If the product of separate relative risks is greater than or equal to their sum, testing for β3 > 0 implies sufficient-cause interaction.”
An alternative way to derive this rule is by rewriting VanderWeele's 9 as:
which is equal to:
and β3 >0, condition (4) is satisfied, leading to the same rule as from condition (3).
In general, condition (4) can be written as:
This leads to the other easy-to-remember rule that “the interaction term on the relative risk scale should be larger than the sum of the separate relative risks divided by their product.”
The minimum values for RR10 and RR01 to indicate sufficient-cause interaction for different β3 are represented in the Figure. Sufficient-cause interaction is present in the area above the curves. Our conditions cover more than VanderWeele's RR10 ≥ 2 and RR01 ≥ 2, as is shown in the Figure. If one of the relative risks is large, the other can be even smaller than 2. For instance, with RR10 = 6, and RR01 = 1.2, β3 > 0 still implies sufficient-cause interaction. In the case of monotonicity of the effects of X 1 and X 2 on D, the required adaptations will make it even easier to fulfill the conditions.
We expand our reasoning to 3-way interaction in eAppendix 2 (http://links.lww.com/EDE/A409).
We thank Saskia le Cessie, Jan Vandenbroucke, and Tyler VanderWeele for critical discussion of previous drafts.
Leo M. G. van de Watering
Sanquin Blood Bank Southwest region
Leiden, The Netherlands
Rutger A. Middelburg
Department of Epidemiology
Leiden University Medical Center
Leiden, The Netherlands