Logistic regression and odds ratio If the odds ratio is greater than one with an insignificant p value for a variable in logistic regression should the variable be kept in the model?
Can I select the variable with odds close to 1?
Question 2: In the validation and testing dataset do not have same mean how to handle it?
 A: It would invalidate the model to remove insignificant variables.  
What does not have the same mean?  If you are using binary logistic regression and are referring to the prevalence of $Y$, this can vary from sample to sample as long as what explains the difference is captured in the covariates.  But the bigger problem is that data splitting is a highly inefficient way to validate a logistic model.  My course notes present better ways: http://biostat.mc.vanderbilt.edu/rms .
A: (Q1)  It depends whether the two predictors are highly related (in which case they'd be strongly correlated). If they are, they'll overlap in the way they help predict the outcome.  In such a case, p-values are not the soundest way to choose betwen these predictors.  One needs to make a selection based on extra-statistical criteria if one cares about explaining the outcome.  
But if they're not highly related, and one has an odds ratio far from 1 and a small p-value while the other's odds ratio is close to 1 with a large p, then most good researchers would consider the latter variable to be noise and would consider a solution that preserved it (or many predictors like it) to be overfitted.  It would not figure to crossvalidate well; it would be capitalizing on chance and would ultimately perform more poorly than a "leaner" one.
