How to determine if independent variables in multiple logistic regression model are independent or not? I am trying a multiple logistic regression model. But I am suspicious that one of my independent variables is dependent on another. I wonder how to prove that the independent variables are truly independent? 
There are previous findings suggesting they have a positive correlation. But in my research I have found no correlation between them.  
 A: It's rare to perform a real-world logistic regression (or any regression analysis for that matter) where you don't have positive covariances between the regressors.  This is called multicollinearity.  This becomes an issue when there is a high degree of multicollinearity and really is only a problem when you are trying to make statistical inferences from the coefficients in your model. If you are simply trying to make the best predictions, multicollinearlity shouldn't really concern you.
One quick and dirty way to determine if variable $X_1$ is, at least linearly dependent on $X_2$, would be to regress $X_1$ onto $X_2$ and test if the coefficient is significantly different from zero.  There are many other tests and tutorials available for testing for correlation between two variables.
That being said, I'm suspicious why you are asking this question.  I suspect your motivation for asking this question may alter my response to your questions.  If you can clarify what you are actually trying to do, I may update my answer, if needed.  Can you tell us more about what you are doing?
A: Run logistic models with and without the predictors in question and see if model GOF changes with the addition or subtraction of each variable.   If your research does not indicate significant correlation between predictors which others have reported as being highly correlated, then just say in your report(talk) "we didn't observe correlation between these predictors and therefore we nevertheless included these predictors in our models." 
