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) =  β0 + β1 tiramisu +  β2 beer + β3 (tiramisu * beer)

The term β3 (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 Reference
1 0 β0 β1
0 1 β0+ β2 β2
1 1   β0+ β1 + β23   β1 + β23

 

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 (β0)

Tiramisu

Beer

Tiramisu*beer

LRS

p-value

1

-2,9741

β1 =     4,3116 OR =  74,56

 

 

180,3927

<0,001

2

-2,6740

β1 =     4,4097 OR =  82,2419

β2 =   -0,8895 OR =   0,41

 

4,3210

0,0376

3

62,9704

β1 =     4,88      OR =131,62

β2 =   -0,0085 OR =   0,99

β3 =   -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 (β0+ β1 + β23) 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 p-value 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

 

 

<<Back to Logistic regression