I have fitted a model with 7-8 covariates.
Here's what I do to reduce it:
I first look at the p-values. I select all covariates with p-values > 0.05. Then I remove them one by one, get the AIC, and then see what had the lowest values. I pick that model.
Then I repeat: again, look at p-values, pick candidates, do AIC.
Is this how it's usually done?