I believe my question is related to, but distinct from this one: Poisson Binomial Distribution - confidence intervals
I am working to estimate species richness by summing the results of 12 individual species distribution models (Maxent) and then determine whether the richness falls above or below certain thresholds.
So, for each species I estimate a probability of occurrence and I can then sum the probabilities to estimate the richness. The probabilities are unequal so the species richness estimate should follow a Poisson binomial distribution, and as discussed in this paper, estimating the variance of the species richness estiamte would be straightforward if the probabilities were known exactly: http://portal.uni-freiburg.de/biometrie/mitarbeiter/dormann/calabrese2013globalecolbiogeogr.pdf
where ps is a vector of the probabilities of occurrence for the 12 individual species.
However, each probability is estimated with uncertainty. I've used 30 fold cross-validation to get an estimate of the uncertainty. I'd now like to construct a confidence interval (or something similar)
I did have one idea, but I'm not sure if it's valid:
Because I used cross-validation, I have a measure of uncertainty for each of the 12 values. My thought was that I could use bootstrapping to come up with a confidence interval (or something similar) for the sum as follows:
- Randomly select the prediction from 1 of the 30 model-replicates (from the cross validation) for each of the 12 values.
- Sum the 12 values.
- Repeat 1000 times.
- Take the 2.5% and 97.5% quantile from the thousand replicates as the lower and upper bounds.
Is this appropriate? Is there a better way to use the information I have to do this?