Sample size determination for a multi-season study is complicated by several factors. First, there are now potentially a large number of parameters we wish to estimate with specified precision. To keep things manageable we will focus on the following, assuming for now that these parameters are constant over time (beyond initial occupancy, which of course is by definition) and are not heterogeneous among sites:
Initial occupancy -psi0
Colonization probability -col
Extinction probability -ext
In addition, we now have under our control (at least potentially) the duration of the study, so we will consider the impact of the following design factors
number of sites sampled
number of replicates per sample
number of seasons (years ) sampled
detection probability (may or may not be under investigator control)
Finally, we need to specify assumed values for psi0, col, and ext, which will then be used to simulate data under various designs .
I have written R code to simulate data under selected designs, estimate the model parameters of interest, compute CV for each, and summarize the results in a database. Here are a few runs of the simulation comparing various designs.
> sample_sim(psi0=0.8,col=0.1,ext=0.1,p=0.3,n_sites=100,n_reps=3,n_sim_reps=100,n_years=5)
psi0 col ext p n_sites n_reps n_years med.cv.psi0 med.cv.col med.cv.ext
1 0.8 0.1 0.1 0.3 100 3 5 0.07900938 0.6158428 0.3350781
> sample_sim(psi0=0.8,col=0.1,ext=0.1,p=0.3,n_sites=100,n_reps=5,n_sim_reps=100,n_years=5)
psi0 col ext p n_sites n_reps n_years med.cv.psi0 med.cv.col med.cv.ext
1 0.8 0.1 0.1 0.3 100 5 5 0.05904509 0.4160138 0.2425879
> sample_sim(psi0=0.8,col=0.1,ext=0.1,p=0.3,n_sites=100,n_reps=3,n_sim_reps=100,n_years=10)
psi0 col ext p n_sites n_reps n_years med.cv.psi0 med.cv.col med.cv.ext
1 0.8 0.1 0.1 0.3 100 3 10 0.06784754 0.2644118 0.189456
> sample_sim(psi0=0.8,col=0.1,ext=0.1,p=0.3,n_sites=100,n_reps=5,n_sim_reps=100,n_years=10)
psi0 col ext p n_sites n_reps n_years med.cv.psi0 med.cv.col med.cv.ext
1 0.8 0.1 0.1 0.3 100 5 10 0.05684191 0.2190034 0.1558434
>
Note that precision in estimating psi0 is quite a bit better than in estimating the transition probabilities (col and ext), which is typical for these kind of problems. This suggests that if interest focusses on turnover rates rather than simply initial occupancy, it will require a substantially more demanding design.
See the R code attached for the user-defined function to enabled these and similar analyses, which may be used if desired to export data for tabular or graphical summaries of alternative designs