normalize dependent variable with respect to one independent variable

I have a data set with 50-60 independent variables and 1 categorical (1-0) dependent variable. Out of all the independent variables say, A is one variable which has majority of the effect on the target variable. For ex

A        target variable
a              1
a              1
a              1
a              1
b              0
b              1
b              1
b              1
c              0
c              0
c              0
c              1
c              1


Clearly whatever analysis I do here, whatever probabilities I calculate for With respect to other variables, I would not be able to get the actual picture with respect to other variables.

Also I do not want to build any model at this point, I just want to see the individual effects of other variables on the dependent variable after removing the effect of variable 'A' i.e. normalizing dependent variable with respect to 'A'. How do I do? If the dependent variable was a continuous variable, I would have simply divided it by the respective averages and removed the effect. How do I do this in this case where the dependent variable is 1-0?

Though continuum regression methods appear to be significantly different from binary classification methods, the crux of the difference is usually tied up in a sigmoid function.

$$S(t)=\frac{1}{1+e^-t}$$

Sigmoid literally means sigma like or "S" like, as becomes obvious in looking at its graphical representation:

The sigmoid provides a tractable means to map from continuum analytics into discrete space while retaining differentiability and integrability needed for some theoretical derivations.

The important part is that in classification methods, one has to choose a threshold value of the sigma function for mapping to a 0 or 1. This is overwhelmingly chosen as 0.5, but there are cases of unbalanced classes where other values might work better.

That said, you should be able to proceed exactly as you have already suggested: "I would have simply divided it by the respective averages and removed the effect." Do this, but then you have to turn the results back into 0's and 1's by choosing where your cutoff threshold lies.

Hope this helps!