View source for Estimating Odds Ratios in the presence of interaction

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 consider interaction between two variables into account by adding to the model an interaction term. Let's 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 a different combination of exposures to tiramisu and beer
{| class="wikitable" style="background-color:#FFF;"
|- style="vertical-align:middle;"
! Tiramisu
! Beer
! Equations
! Relative effect (RO)
|-
| style="vertical-align:middle;" | 0
| style="vertical-align:middle;" | 0
| style="vertical-align:middle;" | β0
| Reference
|- style="vertical-align:middle;"
| 1
| 0
| β0+β 1
| β1
|- style="vertical-align:middle;"
| 0
| 1
| β0+ β2
| β2
|- style="vertical-align:middle;"
| 1
| 1
| β0+ β1 + β2+β3
| β1 + β2+β3
|}
 

The following table shows the results of the steps in the data analysis when testing for interaction between the consumption of Tiramisu and the consumption of Beer on the occurrence of gastroenteritis in our example.

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold;"
! Model
! Constant (β0)
! Tiramisu
! Beer
! Tiramisu*beer
! LRS
! p-value
|-
| 1
| -2,9741
| style="text-align:center;" | β1 =     4,3116 OR =  74,56
| 
| 
| style="text-align:center;" | 180,3927
| style="text-align:center;" | <0,001
|- style="text-align:center;"
| style="text-align:left;" | 2
| style="text-align:left;" | -2,6740
| β1 =     4,4097 OR =  82,2419
| β2 =   -0,8895 OR =   0,41
| style="text-align:left;" | 
| 4,3210
| 0,0376
|- style="text-align:center;"
| style="text-align:left;" | 3
| style="text-align:left;" | 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 the consumption of tiramisu on the occurrence of gastroenteritis due to salmonella. Model 2 suggests that beer plays a slightly 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) suggests that there is an interaction (negative) between the consumption of tiramisu and the 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 + β2+β3) estimated in the model, including the interaction term (model 3).

The following table shows the output of the logistic regression model, including the interaction term (using a statistical package).

{| class="wikitable" style="vertical-align:middle; background-color:#FFF;"
|-
! 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 
{| class="wikitable" style="text-align:center;"
|- style="font-weight:bold;"
! style="text-align:left;" | Terms
! Coefficient
! Std.Error
! p-value
! Odds Ratio
! Lower
! Upper
|- style="vertical-align:middle;"
| style="text-align:left;" | %GM
| -2,9704
| 0,5127
| < 0.001
| 0,0513
| 0,0188
| 0,1401
|- style="vertical-align:middle;"
| style="text-align:left;" | TIRA_
| 4,8800
| 0,6374
| < 0.001
| 131,6250
| 37,7339
| 459,1393
|- style="vertical-align:middle;"
| style="text-align:left;" | BEER
| -0,0085
| 0,7830
| 0,9913
| 0,9915
| 0,2137
| 4,6006
|-
| style="vertical-align:middle; text-align:left;" | BEER* TIRA_
| style="vertical-align:middle;" | -1,2079
| style="vertical-align:middle;" | 0,9338
| style="vertical-align:middle;" | 0,1958
| style="font-weight:bold;" | 0,2988
| style="font-weight:bold;" | 0,0479
| style="font-weight:bold;" | 1,8634
|} 



<div style="display: inline-block; width: 25%; vertical-align: top; border: 1px solid #000; background-color: #d7effc; padding: 10px; margin: 5px;">
'''FEM PAGE CONTRIBUTORS 2007'''
; Editor
: Fernando Simon
; Original Author
: Alain Moren
; Contributors
: Arnold Bosman
: Lisa Lazareck
: Fernando Simon
</div>

[[Category:Fitting logistic regression models]]