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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?

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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 .

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(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.

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