[R-sig-eco] stochastic annual survival rate

[Ben Bolker]

FrogWave <thefrenchstick at ...> writes:

> I want to simulate the trajectories over time of animal populations for
> which I know the annual survival rate, S, estimated in the field from
> several years of data. I want to look at the effect of various levels of
> environmental stochasticity, defined as independent year-to-year variations
> in the survival rate. What I did is to add a normally distributed random
> variable of mean 0 to the logit of S, and back-transformed (see below), but
> I realised than doing so leads to a lower geometric mean (i.e. the annual
> survival rate) and that the difference increases with the level of
> stochasticity . So how can I include stochasticity without changing the
> geometric mean? Any insight would be very appreciated.
> 
> logit <- function(x)
>                return( log( x / (1-x) ))
> invlogit <- function(x)
>                return( exp(x) / (1 + exp(x)))
> 
> S = 0.92  # annual survival rate
> sdS = 0.4 # standard error of S on the logit scale
> nS = 100000  # number of samples
> Svar = invlogit( logit(S) + rnorm(nS, 0, sdS))  # stochastic survival rate
> 
> gmeanS = exp( mean( log(Svar) ))  # geometric mean of S
> gmeanS    # geometric mean of stochastic survival rate less than 0.92...


  It's not possible, I think.  You can try to *correct* for it
(i.e., for a specified variance and a desired geometric mean,
increase the arithmetic mean), but you can't include stochasticity
without changing the GM -- it's a fact of life (or mathematics).

  For the log-normal equivalent, it's fairly easy to do this analytically.

For a geometric mean of exp(logmu) and a standard deviation on
the log scale of sdS, the arithmetic mean of the log-normal is
AM = exp(logmu+sdS^2/2). The mode and the median of the log-normal
are unchanged by transformation (and by noise).  Hence I think you
can get what you want in the LN case by increasing your logmu by
sdS^2/2 (this is a little fuzzy, it's late, there may be an error,
but I'm going to send it off anyway).

  For the logit-normal I'm not sure whether there's an equivalent
analytical trick.

  Ben Bolker