e., each

cow) from a beta distribution. Beta-binomial distributions are typically described with two shape parameters, a and b. The mean per-trial probability is equal to a/(a  +  b). We use an alternative parameterization presented by Morris (1997); here the beta-binomial distribution is described by a mean per trial probability (r) and an overdispersion parameter, θ, equal to a  +  b. With large values of θ (minor overdispersion), the beta-binomial converges on the binomial distribution; when θ approaches zero (large overdispersion), the distribution see more becomes U-shaped (Bolker 2008). Zero-inflated models allow for more zeros in the data than are allowed by binomial or beta-binomial distributions; they are mixture distributions whereby a binomial or beta-binomial distribution is combined with a zero density distribution. An additional parameter describes the probability that an observation of zero did not come from the binomial or beta-binomial model. Code for zero-inflated binomial and zero-inflated beta-binomial models is provided in Bolker (2008). The four distributions were fit to the entire data set with years pooled and the best distribution for the data was selected using AIC (Burnham and Anderson 2002); this distribution selleck compound was then used to estimate annual calf:cow ratios, annual estimates of dispersion, and to model sources of variation in the ratios.

We examined the following potential predictors to better understand the spatial and temporal variability in calf:cow ratios: If calf mortality occurs during the survey period, the calf:cow ratio would decline as a function of date. Date was defined as the number of days since January 1 within each survey year, minus the earliest day cows were classified in any study year. Across all survey years, cow groups medchemexpress were classified from 11 July (defined as day 1) to 12 September (defined as day 63). Time of day and longitude was recorded for each group observed. Using the algorithms of Meeus (1991), we calculated the offset between local Bering Sea Time (GMT minus

11 h) and solar noon for the longitude of each group observed. This offset, ranging from −1.1 h to +4.7 h, was added to the local time to make local noon correspond to solar noon. Solar Time was also examined with a squared term (i.e., Solar Time + [Solar Time]2) to allow for a quadratic relationship between time of day and r. The calf:cow ratio may vary as a function of group size, defined as the number of cows in a group. Understanding how the calf:cow ratio may vary as a function of the number of cows in a group is important for designing surveys but also for correctly simulating calf/cow groups in the Monte Carlo simulations (see below). Group Size was recorded for each group that was classified and the calf:cow ratio was modeled as a function of group size. Group Size was also examined with a squared term (i.e.