Abstract

We often assume that an abundance estimate is log-normally distributed and use that as a basis for constructing confidence intervals. This is an aide memoire for me on that topic…

Setup

We’ve estimated the abundance of some population using some method1 and now have an estimated abundance, which we’ll call $\hat{N}$ and an associated estimate of the variance $\widehat{\text{Var}}(\hat{N})$.

So we assume: $N_\text{truth} \sim \log\mathcal{N}(\hat{N}, \text{Var}(\hat{N}))$ for R’s sake (for things like qlnorm and plnorm etc) we need the following parameters: $\mu = \mathbb{E}(\log N) \quad \text{and} \quad \sigma^2 = \text{Var}(\log N)$

Some maths

So if we know2 that: $\mathbb{E}(N) = \exp(\mu + \sigma^2/2)$ and $\text{Var}(N) = \exp(2\mu +\sigma^2)(\exp\sigma^2 -1)$ we can get back to expressions about $\mu$ and $\sigma$. For $\sigma$ first… % and for $\mu$: %

These values can then be fed to qlnorm to obtain confidence intervals, hurrah!

1. Maybe it was a density surface model? Maybe it was a Horvitz-Thompson estimate? Voodoo?

2. taken from the *lnorm help in R