Why does hypothesis testing using coefficient and odds ratio give different conclusion? After fitting a logit glmer model in R, I got the following coefficient estimate:  
b  = 1.806649
se = 0.9899169
b + qnorm(0.025) * se
b + qnorm(0.05)  * se    # > 0, significant at 90%

I want to report odds ratio (OR) instead, so I calculate OR, its standard error, and re-do inference like so:  
or    = exp(b)
or.se = exp(b) * se      # According to delta method (i.e. one-step Taylor approximation)
or + qnorm(0.05) * or.se # < 1, so no longer significant at 90%

How should I interpret this? I suppose this is because exp(b) no longer follows a normal distribution. If this is the case, how does software (such as Stata) does hypothesis testing to show whether odds ratio $\neq 1$? Does Stata do inference using odds ratio SE or odds ratio CI (i.e. exponentiated log odds ratio CI)?
 A: An index having an asymmetric distribution (e.g., a ratio) is generally not one in which a standard error has much meaning.  Thus a hypothesis test based on a Wald statistic (of the form estimate minus hypothesized value divided by standard error) is not going to perform well, and confidence interval accuracy is especially affected.  Frequentist statisticians tend to favor the likelihood ratio test because it is invariant to transformation of the parameter being tested.  If not using a LRT, it is important to use a normality-based test on the right basis such as log odds ratio.  Even that has its own problems because even though the log odds ratio is much more normally distributed than the odds ratio, it is still not extremely close to a normal distribution.
A: I don't know how Stata works, but I doubt it is doing anything like what you are doing.  It is more common to use the likelihood as @FrankHarrell notes.  It may help you to read my answer here: Why do my p-values differ between logistic regression output, chi-squared test, and the confidence interval for the OR? 
More specific to your question, how good an approximation a Taylor expansion with a given finite number of steps is depends on (a) the function being approximated, and (b) what you are willing to consider 'good'.  For this case, looking at the values of the odds ratio at +1 SE, the delta method's approximation does not seem very good to me:  
or + or.se  # [1] 12.11861
exp(b +se)  # [1] 16.38827

