Interaction

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 than the sum of A and B risks. The difference is attributable to the joint effect.

Formulae

In cohort studies, the risk due to the interaction between two exposures A and B can be calculated as follows :

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 anti-acid 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 anti-acid medication but did not eat undercooked chicken, 20 among those who ate undercooked chicken but did not use anti-acid medication and 100/1000 among people eating undercooked chicken and taking anti-acid 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 the consumption of undercooked chicken and taking anti-acid medications. The joint risk is more than the simple addition of the two risks. The additional risk linked to exposure to undercooked chicken and antiacids is potentially responsible for 71 cases per 1000, potentially explaining 71% of disease occurrence.

Case-control studies

In case-control studies, biological interaction can be measured using the following formula . 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

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 = (0,91-1) + (2,63-1) + 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 Ebola viral haemorrhagic fever outbreak occurred in the Bumba zone of Zaire (now the Democratic Republic of Congo). The disease was amplified by exposure to a large, active hospital .

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) + (25.1-1) + 1 = 61.3

This is slightly different from 69.1, as shown in the table. We could conclude the presence of very little additive interaction between the two risk factors. In addition, we would need to explain the biological meaning of such interaction when it seems unlikely that cases in the 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 textbooks .

Using a statistical package for multivariable analysis (based on a multiplicative assumption) could have led to different conclusions. However, it is possible to assess biological interaction with such a package if using an additive model. To do so, we need to create four levels of exposure (dummy variables) exposure to A and B, exposed to A and not B, exposed to B but not A, and exposed to neither. The later exposure is used as a reference for the 3 others, and variables and biological interaction can be measured using additive assumptions as in the above examples.

Credit

FEM Editor 2007

• Naomi Boxall

Original Author

• Alain Moren

FEM Contributors

• Arnold Bosman
• Sabrina Bacci
• Lisa Lazareck
• Naomi Boxall