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
Tiramisu 
Beer 
Equations 
Relative effect (RO) 
0 
0 
β_{0 } 
_{Reference} 
1 
0 
β_{0}+β _{1 } 
β_{1} 
0 
1 
β_{0}+ β_{2} 
β_{2} 
1 
1 
β_{0}+ β_{1} + β_{2}+β_{3 } 
β_{1} + β_{2}+β_{3} 
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

pvalue

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).
Number of terms 
4 



Total Number of Observations 
245 



Rejected as Invalid 
0 



Number of valid Observations 
245 



Summary Statistics 
Value 
DF 
p=value 

Deviance 
153,3200 
241 


Likelihood ratio test 
186,3215 
4 
< 0.001 

Parameter Estimates 



95% C.I 
Terms 
Coefficient 
Std.Error 
pvalue 
Odds Ratio 
Lower 
Upper 
%GM 
2,9704 
0,5127 
< 0.001 
0,0513 
0,0188 
0,1401 
TIRA_ 
4,8800 
0,6374 
< 0.001 
131,6250 
37,7339 
459,1393 
BEER 
0,0085 
0,7830 
0,9913 
0,9915 
0,2137 
4,6006 
BEER* TIRA_ 
1,2079 
0,9338 
0,1958 
0,2988 
0,0479 
1,8634 
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