Derivation of confidence interval of incidence rate ratio I am trying to understand the confidence interval equation for a Incidence Rate Ratio (IRR) given several places:
$ 95\text{% CL(IRR)} = \exp(\log(\text{IRR}) \pm 1.96\times \text{SE(log(IRR))})$, with
$\text{SE(log(IRR))} = \sqrt{\frac{1}{e_1} + \frac{1}{e_2}}$, where $e_{1,2}$ is the number of events in each arm. (See here for example)
Though the equation seems to be written many places I have a hard time finding a derivation or good discussion besides lofty sayings such as "because it is a ratio we take the log" which doesn't really explain anything (to me at least).
The setup for each Incidence Rate ($\text{IR}$) is given by:

the number of incidents for an individual is $Y_i \sim \text{Pois}(\lambda \, T_i)$ where the incidence rate per unit time is $\lambda$, $T_i$ is the exposure time for individual $i$ and the $Y_i$ variates are independent. Then we define the random variable $e=\sum_i Y_i$ and the total exposure $T=\sum_i T_i$ and the Incidence $\text{IR} = \frac{e}{T}$.

I understand why the standard deviation becomes $\sigma\text{(IR)} = \frac{\sqrt{e}}{T}$ as discussed here but how I get to the IRR equation is not clear to me.
I tried approximating both IR's as Gaussian and taking the ratio assuming that to be Gaussian as well (which is discussed more thoroughly here) which leads to:
$95\text{% CL(IRR)} = \text{IRR} \pm 1.96\times \text{IRR} \times \sqrt{\frac{1}{e_1} + \frac{1}{e_2}}$ which seems to be just the lowest order expansion of the top equation.
So why are we taking the logarithm and why is that alright given that it is a ratio of 2 scaled Poisson variables? Why does $\log(\text{IRR})$ end up being Gaussian and does that imply that $\log(\text{IR}_{1,2})$ is Gaussian as well?
 A: Reading up on the background of generalized linear models (glm) may help.  The Wikipedia article may help: https://en.wikipedia.org/wiki/Generalized_linear_model
Generalized linear models work with distributions that can be factored into a certain pattern (Exponential family), part of that factoring gives what is called the "canonical link" function between the mean of the distribution and the linear combination of the predictors.  In the case of the Poisson distribution the canonical link is the natural log.  You can use other link functions, but the canonical link has some theoretical benefits and is the default for most software.  You do not have to use the log, but without good reason to use another link function, it makes the most sense.
There are multiple ways to create confidence intervals, the intervals where you take the maximum likelihood estimate and add and subtract a constant times the standard error is called a Wald Style interval.  The constant to multiply actually comes from a chi-squared distribution, but since the chi-squared with 1 degree of freedom is just a standard Gaussian squared, the constant is the same for a single estimated interval.  These intervals work well when the estimate of the standard error is stable at the ML estimate and the likelihood function is close to symmetric around the ML estimate.  This is exact in the case of a Gaussian likelihood, but is a pretty good approximation most of the time for the Poisson and can be reasonable for the Binomial and other distributions, and it is fairly quick and simple, so a good starting place.  But there are also cases where the Wald style intervals do not work well (the main example I am aware of is with the binomial/logistic regression, google for Hauk-Donner effect) and the likelihood ratio or other intervals become much preferred.
