It is possible to have negative synergy index of interaction in logistic model? I ran logistic model to check interaction between two dichotomous variables. The interaction is statistically significant, and both variables are also significant in the model.
But when I calculated synergy index (S), the value of S was negative, as I know, value of S is 0 to infinite. How could this happen?
Thanks.
 A: As background for the few of us who care, the Synergy Index was proposed by Rothman as a general way of measuring interaction without the assumption of additive effects (that comes with linear models):
$$ S = \frac{RR(A , B) - 1}{RR(\neg A, B) + RR( A , \neg B) - 2}$$
So here are two relative risks (for an outcome) comparing one of four exposure levels in an epidemiologic study of two binary exposures and outcome. The referent group $\neg A, \neg B$ are those with neither exposure. The RR is the ratio of risks for having one or more positive exposures. 
Perhaps the most likely culprit of your confusion is the offset terms -1 and -2 in the numerator and denominator. Whereas RRs are called "positive/negative" on the basis of being >1/<1, subtracting 1 makes this convention numerically cogent: negative synergy is <0 and positive synergy is >0. If you ignore these offsets, the boundedness you allude to holds because the numerator and denominator are positive. 
Negative synergy happens when joint exposure reverses a common direction of effect for exposures A and B in people unexposed by B and A respectively.
Take, as a very stupid example, smoking and development of pre-eclampsia on perinatal death. Both are absolutely dreadfully harmful exposures for neonates. However, it just so happens that, when combined, smoking can treat pre-eclampsia. Here (hypothetically) $RR(\neg \text{smoking}, \text{preclampsia}) > 1$  and $RR(\text{smoking}, \neg \text{preclampsia}) > 1$, but $RR(\text{smoking}, \text{preclampsia}) < 1$.
Another possible example: mainline cancer therapies. Radiotherapy and surgery alone may do practically nothing to prolong survival in patients with high stage tumors, and the invasiveness and harm of those treatments come with their own risks as well. However, if performed together, it may kill the cancer completely or at least halt progression (remission) leading to longer survival.
More on abstract interaction metrics here
