# Binomial or Poisson structure in GEE

I have a dataset with two total number of trials and two number of successes for each individual in two different situations. In addition, I have two groups of individuals, e.g.:

group   ind   treat  trials success fail
1        1      A      1     0        1
1        1      B      1     1        0
1        2      A      3     2        1
1        2      B      1     1        0
1        3      A      2     0        2
1        3      B      1     0        1
2        4      A      1     0        1
2        4      B      1     1        0
2        5      A      3     2        0
2        5      B      1     1        0
2        6      A      2     2        0
2        6      B      1     0        1


I would like to test if the mean success rate is different for both groups and situations. As the same individual is exposed to situation A and B I think that the results for that individual will be correlated, so I think that a GEE is a reasonable way to test for differences between groups and between situations.

My response variable are rates of successes. I have some doubts:

1. I would use a binomial variance structure, but I think that a failure in the first trial could lead to a failure in the second trial. Is this really a problem or it's only a perception of mine?
2. In case of being a problem, I'm thinking on the use of a Poisson variance structure. I would use the logarithm of the number of trials as an offset. Would it be correct?
3. Means of the success rates are greater than variances for both groups. If this is a problem to use a Poisson structure, is there another suitable distribution?