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?
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 .
(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.
