I have a question about confidence interval calculations for the odds ratio $\hat{p}\over{1-\hat{p}}$ from a logistic regression model (perhaps obtained from the method of Generalized Estimating Equations, but that's of secondary importance). It's best asked after a bit of background and an example:
Let's assume we have a simple logistic regression model with a single binary independent variable ($X={0,1} $ ). For $X=1$, we can estimate the log odds by:
$\log({\hat{p}\over{1-\hat{p}}})=\hat{\beta}_0+\hat{\beta}_1$
To obtain a 95% confidence interval (CI) for the odds ratio, textbooks that I've consulted indicate that you calculate the 95% CI for the odds-ratio, ${\hat{p}\over{1-\hat{p}}}$, by:
Lower Bound= $e^{[\hat{\beta}_0+\hat{\beta}_1-1.96Var(\hat{\beta}_0+\hat{\beta}_1)]}$
Upper Bound= $e^{[\hat{\beta}_0+\hat{\beta}_1+1.96Var(\hat{\beta}_0+\hat{\beta}_1)]}$
What I am wondering is if it is appropriate to use the the Delta Method to obtain the variance of the odds ratio $\hat{p}\over{1-\hat{p}}$ $= \exp{(\hat{\beta}_0+\hat{\beta}_1)}$ instead? For example, could obtain a 95% confidence interval using the following method?:
(1) First, exponentiate to obtain the point estimate of the odds ratio: $\hat{p}\over{1-\hat{p}}$$=\exp(\hat{\beta}_0+\hat{\beta}_1)$=$f(\hat{\beta}_0,\hat{\beta}_1)$.
(2) Find the approximte large sample variance of this non-linear point estimate by the Delta methods by finding the gradient of $f$ as: $\nabla f =[\exp(\hat{\beta}_0+\hat{\beta}_1), \exp(\hat{\beta}_0+\hat{\beta}_1)]$
(3) Calculate $Var({\hat{p}\over{1-\hat{p}}})$ $\approx$ $\nabla f \hat{\Sigma} \nabla f^T$. where $\hat{\Sigma}$ is the empirical covariance matrix of $\hat{\beta}$.
(4) Find the approximate standard error, $SE$ by taking the square root of the approximate variance estimate $SE = \sqrt{\nabla f \Sigma \nabla f^T}$
(5) Finally, obtain the 95% CI by $\exp(\hat{\beta}_0+\hat{\beta}_1)\pm Z_{1-.05/2}SE$.
Is the delta method I have described appropriate to find a 95% confidence interval, and if it is, why is it never described in textbooks on logistic regression? If it isn't, why not, when it seems you can use it to linearize it for similar problems.
UPDATE: I updated the step 3 to make it clear that I'm using the estimated, empirical covariance matrix.
UPDATE 2
I still haven't received a very satisfactory answer to this question. It has even made me wonder why people don't use the Delta Method to compute confidence intervals around estimates of just $\hat{p}$ now, instead of talking the inverse of the logit link of the upper and lower confidence intervals. Maybe @FrankHarrel can shed some light as he seems especially knowledgeable about these types of calculations?