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When interaction is present, the association between a risk factor and the outcome varies according to and depends upon the value of a covariate. Interaction between two variables can be positive (their joint role increases the effect) or negative (their joint role decreases the effect).
In logistic regression we will take interaction between two variables into account by adding to the model an interaction term. Let suppose we are studying the role of two exposures (tiramisu and beer) in the occurrence of gastroenteritis due to Salmonella.
The logit including an interaction between tiramisu and beer can be written as follows:
Ln (P gastroenteritis / tiramisu, beer) = β_{0 }+ β_{1} tiramisu + β_{2 }beer + β_{3} (tiramisu * beer)
The term β_{3} (tiramisu * beer) reflects the interaction.
We have therefore 2 variables and four combinations of coefficients:
Table 1: Effects of different combination of exposures to tiramisu and beer
The following table shows the results of the steps in the analysis of data when testing for interaction between consumption of Tiramisu and consumption of Beer on occurrence of gastroenteritis in our example.
Model
Constant (β_{0})
Tiramisu
Beer
Tiramisu*beer
LRS
p-value
1
-2,9741
β_{1} = 4,3116 OR = 74,56
180,3927
<0,001
2
-2,6740
β_{1} = 4,4097 OR = 82,2419
β_{2} = -0,8895 OR = 0,41
4,3210
0,0376
3
62,9704
β_{1} = 4,88 OR =131,62
β_{2} = -0,0085 OR = 0,99
β_{3 }= -1,2079 OR = 0,2988
1,6078
0,204
Model 1 tests the effect of consumption of tiramisu on the occurrence of gastroenteritis due to salmonella. Model 2 suggests that beer plays a slight confounding effect (p = 0,037, OR changing from 74 to 82) for the association found in model 1. In model 3, the introduction of the interaction term (tiramisu*beer) suggest that there is interaction (negative) between consumption of tiramisu and consumption of beer. Beer seems to decrease the risk of illness due to tiramisu consumption. However this interaction is NOT statistically significant (LRS = 1,60 and p = 0,2048).
In the presence of interaction, the effect of the different combinations of exposures should be worked out as shown in table 1, using the coefficients (β_{0}+ β_{1} + β_{2}+β_{3}) estimated in the model including the interaction term (model 3).
The following table shows output of the logistic regression model including the interaction term (using a statistical package).
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