I have a binary outcome (cesarean section - yes or no), that I model in logistic regression. I have an explanatory variable called parity which has three levels 1=no prior births, 2=prior births delivered vaginally and 3= prior births delivered by cesarean section.
I also have an explanatory variable called "high volume hospital" which is a binary indicator for whether the hospital that the woman gave birth in has > 4000 births per year. I want to find out if the effect of parity on cesarean section differs according to whether a hospital is high volume or not. So I create an interaction for this:
$X_1$ is a set of dummy variables for parity; $X_2$ is the high volume hospital indicator; $X_1X_2$ is the set of dummy indicator variables for the interaction terms.
$$Y=B_0 +B_1 X_1 + B_2X_2 + B_3(X_1X2)$$
The results I get back are as follows:
Now, these interactions mean that women in higher volume hospitals have a lower odds of CS than women in low volume hospitals. For example, the OR of CS for a a woman without a prior CS (group=3) in small hospital is 8.86. The odds for a woman in a high volume hospital is 0.33 times this so OR = 2.92 (8.86*0.33).
I would like to calculate confidence intervals around this odds of 2.92. How do I do this? Is it correct multiply the 95% CI for the interaction by the 95% CI for the main effect?
lower bound: 7.24 * 0.26 = 1.8824
upper bound: 10.85 * 0.416 = 4.5136
Giving me an OR of 2.92 (95% CI 1.88-4.51) for women in group 3 in high volume hospitals. Is this right?