Logistic regression for factorial design: interaction and main effects of treatments

#1
Hi,

This is my first post here. I'm hoping to get some help/feedback for the following hypothetical problem:

I have a factorial experimental design with 4 groups: (1) Controls, (2) Treatment A, (3) Treatment B, and (4) Both treatments A and B. A total of 100 patients were randomly allocated in a balanced manner to one of the 4 treatment combinations. My outcome variable is binary response (yes/no). My estimated response rates (in parantheses are the number of responses) are:
table1.PNG
Please ignore issues of small sample size or power for now, the numbers in the table above are made up, I’m just playing around trying to understand what the correct approach is for tackling this type of problem.
What I want are three things:
1) to perform test of the main effects of A and B.
Now, I know that before testing for the main effects, we’d first need to check and see if there is an interaction effect. Also, because my response is not continuous, I used a logistic regression model in R with term for treatment A, term for B and interaction term and I got a non-significant p-value (.72477) for the interaction term:
R1.PNG
Because the interaction was not significant, I re-fit my logistic model with the interaction term removed and got this where according to p-values for A and B, respectively, these main effects are not significant either:
R2.PNG
My question here is: are my results correct and if not, what is the correct way to do this to test for main effects of A and B?
2) If I want to get the 95% confidence interval for the difference in response rates between those taking only A and those taking neither A, nor B, can I use table below to compute relative risk RR= (10/25)/(5/15) and then calculate the conf.interval for this RR? Would that be correct?
Table2.PNG
3) Finally, I want to find probability of observing the response rates observed in this trial for the two groups not receiving drug B? And to check if these two groups are statistically different from each other.
We would use the table from part 2) again but not sure how to go about calculating that probability?
If anybody can please help answer my questions and provide further input/suggestions, I’d greatly appreciate it.

Thanks and sorry for the very long post!