Statistical and biological interaction
The term interaction is frequently used alternatively with effect modification. Statistical packages for multivariable analysis offer to test for interaction. They often are based on a multiplicative assumption. Interaction is present only if the joint effect of A and B is more than the multiplication of the respective effect of A without B by the effect of B without A. This is called "statistical interaction". It is assessed after logarithmic transformation.
Epidemiologists are interested in "biological interaction". Where there is no biological interaction between two exposures (A and B), the risks related to A and B are added to each other when both A and B are present. If biological interaction occurs, we expect the joint risk to be higher that the sum of A and B risks. The difference is attributable to the joint effect.
Formulae
In cohort studies the risk due to interaction between two exposures A and B can be calculated as follows [1]:
Interaction = R_{AB}  R_{A}  R_{B} + R_{O}
In which R_{A} is the risk when only exposed to A, R_{B} is the risk when only exposed to B, R_{AB} is the risk when exposed to both, and R_{O} the risk when exposed neither to A nor B.
If there is no interaction (i.e. the exposures are independent of one another) the expected risk when exposed to both factors can be computed as:
R_{AB} = R_{A} + R_{B}  R_{O }
Cohort Studies
During an outbreak of Salmonella enteritidis gastroenteritis two risk factors were suggested by the data, consumption of undercooked chicken (exposure A) and taking antiacid medications (exposure B). The risk of illness was respectively 5/1000 among those who were not exposed to any of the 2 risk factors, 10/1000 among those who took antiacid medication but did not eat undercooked chicken, 20 among those who ate undercooked chicken but did not use antiacid medication and 100/1000 among people eating undercooked chicken and taking antiacid medication.
Exposures


Cases

Total

Risk

neither A nor B

No chicken, no antiacids

1

1000

0.001

A but not B

Chicken but not antiacids

20

1000

0.020

B but not A

Antiacids but not chicken

10

1000

0.010

A and B

Chicken and antiacids

100

1000

0.100

If there was no interaction between exposure to undercooked chicken and antiacids, the expected risk when exposed to both risk factors would be:
R_{AB} = 0.02 + 0.01  0.001 = 0.029
However, the observed risk is 0.100, suggesting a biological interaction between consumption of undercooked chicken and taking antiacid medications. The joint risk is more than the simple addition of the two risks. The additional risk linked to the exposure to both undercooked chicken and antiacids is potentially responsible for an incidence of 71 cases per 1000, potentially explaining 71% of disease occurrence.
Casecontrol studies
In case control studies biological interaction can be measured using the following formula [2]. As risk cannot be computed in a case control study, the odds ratio (OR) is used instead:
Interaction = (OR_{AB}  1)  (OR_{A}1)  (OR_{B}1)
Given no interaction (independence between the two exposures):
OR_{AB} = (OR_{A}1) + (OR_{B}1) + 1
In a national case control study looking at Salmonella enteritidis in children in France [3]
Season

Eggs

Cases

Controls

OR

Not Summer

< 2 weeks storage

32

36

ref

Not Summer

> 2 weeks storage

7

3

2.63

Summer

< 2 weeks storage

52.

64

0.91

Summer

> 2 weeks storage

12

2

6.75

The joint effect of A and B is therefore computed as:
OR_{AB} = (0,911) + (2,631) + 1 = 2,74
However, the observed value for OR_{AB} is 6,75.
Therefore, 6.75  2.74 = 4,01 of the effect is due to biological interaction when both exposures are present. The biological interaction represents 60.7% (4.01/6.75 * 100) of the effect when both exposures are present.
In 1976, an outbreak of Ebola viral haemorrhagic fever occurred in the Bumba zone of Zaire (now Democratic Republic of Congo). The disease was amplified by exposure to a large, active hospital [4].
Hospital

Case

Cases

Controls

OR

Unexposed

Unexposed

41

266

ref

Exposed

Unexposed

85

22

25.1

Unexposed

Exposed

149

26

37.2

Exposed

Exposed

43

4

69.1

If there was no interaction between "exposure to hospital" and "exposure to a case", the OR associated with contact to both exposures:
OR_{AB} = (37.2  1) + (25.11) + 1 = 61.3
This is slightly different from 69.1 as shown in the table. We could conclude for the presence of very little additive interaction between the two risk factors. In addition we would need to explain the biological meaning of such an interaction, when it seems unlikely that cases in hospital would do much mixing  being too ill to socialize in that setting.
The above represents a simplified explanation of additive and multiplicative models when testing for interaction. Further explanation can be found in major text books [1].
Using a statistical package for multivariable analysis (based on a multiplicative assumption) could have lead to different conclusions. However it is possible to assess biological interaction with such package if using an additive model. For doing so we simply need to create four level of exposures (dummy variables) exposure to A and B, exposed to A and not B, exposed to B but not A, exposed to neither. The later exposure is used as reference for the 3 others and variables and biological interaction can be measures using additive assumptions as in the above examples.
References
1. K.J.Rothman, S.Greenland, T.L.Lash. Modern Epidemiology. Third ed. Philadelphia, USA: Lipincott Williams and Wilkins; 2008.
2. Kalilani L, Atashili J. Measuring additive interaction using odds ratios. Epidemiologic Perspectives & Innovations 2006;3(1):5.
3. DelarocqueAstagneau, E., Desenclos JC, Bouvet P, Grimont PA. Risk factors for the occurrence of sporadic Salmonella enterica serotype enteritidis infections in children in France: a national casecontrol study. Epidemiol Infect 1998;121(3):5617.
4. Ebola haemorrhagic fever in Sudan, 1976. Bulletin of the World Health Organisation 1978; 56 (2): 271:93.