ranging from 1 to 5 for which the means and medians of estimates of A were within reasonable
limits. In my simulations, I fixed a value of 0.3 for r and 10 for A as constants and varied the
number of sites (100, 50, 25, 10) and the number of sampling occasions (3, 5, 10) to evaluate the
performance of the estimates.
I wrote the program in R, a free statistical programming environment (Vienna University
of Economics and Business Administration, 2006). Using "direct search" to numerically
calculate the values ofr and 2 to maximize the likelihood Equation 2-7 is very time consuming.
Instead, I used the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm and also used the
Nelder-Mead algorithm (Press, Teukolsky, Vetterling & Flannery, 1994). I used the logit
transformation on r to bind the values ofr between 0 and 1 during optimization.
From the likelihood-based estimates, I identified data sets that resulted in estimates quite
distant from the true value used in the simulation. I used these data sets to obtain posterior
distributions of the parameters r and A by running the Gibbs sampler algorithm with two
informed prior distributions. This algorithm was also programmed in R.
Results
The summary statistics for the estimated parameters r and A by the likelihood-based
inference is shown in Table 2-1. The results for all combinations of number of sites and number
of sampling occasions show a positive bias for the estimates of A. For sample sizes 100, 50 and
25 sites, the median value of A provided a better estimate of the true value of A as compared to
the mean. The Nelder-Mead algorithm and BFGS algorithms provided different estimates for the
mean and standard errors of A. For example, in the simulation with 50 sites and 5 sampling
occasions, the Nelder-Mead estimate of A(mean) was 26.181 24.476 while the BFGS estimate
was 16.813 10.747.