I'm just doing a sanity check for myself and others as it relates to constructing and interpreting confidence intervals with logistic regression. Suppose you specify a logistic regression of $y$ on a binary variable $x$ which returns a coefficient estimate $\hat{\beta_{1}}=0.05$, and a standard error $se_{\hat{\beta_{1}}}=0.001$.
Is it appropriate to claim:
1) The upper and lower bound for the 95% confidence interval for $\hat{\beta}$ is given by $\hat{\beta}{\pm}(1.96\times{se_{\hat{\beta}}})$
2) Since we can convert log odds into a probability with $\frac{e^x}{1+e^x}$, does that mean this transformation also applies to the upper and lower bounds above?
My feeling is that it's not a problem that our confidence interval is not symmetric about ${\hat{\beta}}$. (You can demonstrate this by calculating probabilities for extreme values of the log odds).
I'm also of the opinion that with these kinds of estimates even if you have an estimated coefficient that is statistically significant, it doesn't necessarily imply practical significance. For instance, if you had a coin which you knew had a long run probability of heads equal to .5000000005, a large enough number of trials would find statistical significance of unfairness, but in a practical sense you could consider the coin "fair".
My final question (and this is the hard part), is it appropriate to report these kinds of binomial probabilities in terms of the difference from a fair coin? Like, for the example in the beginning, the log odds being 0.05 instead of saying "the probability $P(y=1|x=1)=0.5125$", we instead subtract 0.5 from that value to say something like "x being equal to 1 and not 0 is associated with a change in the probability" ${\Delta{P(y=1)}=0.0125}$, just to illustrate that that the size of the effect of x on y is small even though there might be statistical significance. I apologize if my notation is screwed up. Hopefully you understand what I'm getting at though.
