A. Status of the Spectacled Eider - U.S. Fish and Wildlife Service

APPENDIX II
TECHNICAL DETAILS OF THE BAYSIAN ANALYSIS
CALCULATION OF POSTERIOR DISTRIBUTIONS
Estimation of the amount of risk to which Spectacled Eiders are exposed must be based
primarily on an analysis of the rate of change ofpopulation abundance where abundance is
estimated by various survey techniques. For the Spectacled Eider three time series of
abundance estimates are analyzed: the North American breeding bird survey (number of
estimates (n) = 39), aerial surveys of the YKD coast (n = 8), and ground plot surveys where
random ground plots stratified by habitat type are exhaustively surveyed for nests (n = 10).
For each survey we would like to know the probability of obtaining these data given various
hypotheses concerning the unknowns: population growth rate (r--the slope estimate of a
regression of population size against time) and variance in the estimate of population growth
rate (the standard error ofthe estimate). Given a probability distribution for population
growth rate for our particular set of data we can directly answer questions about the probability
that the population is stable or growing. The range of hypotheses tested constitute the prior
distribution and the resultant distribution, which is conditional on the data, is called the
posterior distribution.
Our primary interest is in estimating the population growth rate from observed data, which are
abundance index estimates with associated precision estimates. We assume an exponential model
ofpopulation growth (equation 1).
= N 0 e(ree) t (1) (2.)
where N = number ofbreeding pairs, t = time (years), N0 = initial number ofbreeding pairs, r =
population growth rate and E~ is Gaussian distribution symbolized as G (~, 5r2), where ~ = 0 and 5r
is the standard deviation ofr. When s~ = 0 the model is deterministic. When ~r > 0 then the
model is stochastic with annual growth rate drawn from a distribution. The parameter ofinterest.
for classification decisions is r. Parameters are estimated by fitting the model with the available
time-series ofabundance estimates using Bayesian methods. This is analogous to a weighted nonlinear
regression using classical statistical methods (essentially because the contribution ofeach
abundance estimate to the estimation is weighted by the inverse ofits estimated precision or CV).
Due to the nature ofthe available data set, several other parameters have to be defined and
estimated. The abundance estimates and their associated coefficients ofvariation, CV (which
were estimated from the sampling design for each survey), represent the observed data in this
analysis. Because the number of eiders that breed and are thus available to be counted may vary
through time (more than explained by trends in r), the estimated CV may not account for all ofthe
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