J. Wilson White and Robert R. Warner. 2007. Safety in numbers and the spatial scaling of density-dependent mortality in a coral reef fish. Ecology 88:3044–3054.

Appendix D. A description of the microsite-scale mortality data analysis.

We assumed that each fish has a daily probability of mortality, p = exp(-λ), where λ is a linear combination of regression coefficient and covariates (λ = βx).  This is effectively the same regression model used in proportional hazards survival analysis.  The linear component of p, λ, consists of a constant (λ₀) and a term (λ_i) for each of the i covariates used in a particular model.  We analyzed 18 different models for λ, each containing parameters for various subsets of the variables group size, density (mean Delaunay spoke length), fish age, group size × density, group size × fish age, density × fish age, predator abundance, and habitat complexity.

Like Sandin and Pacala (2005), we are unable to derive an analytical expression for the likelihood of the parameters, β, given the observed pattern of mortality, so we simulated likelihoods using a Monte Carlo technique. For a given set of parameter values, we simulated 100 time series for each microsite (either a group or a solitary fish). Each time series is a set of census observations at that microsite: for a solitary fish that settled on the first day of observations and died between the second and third days, the time series would be [1 1 0 0 0 0]; a group of initial size 3 settling on the first day could have many potential time series including [3 3 3 2 2 2], [3 2 1 1 1], [3 1 0 0 0 0], and so forth. A fish has a probability p of dying on each day; this value may change during a single simulation if the model includes age or group size covariates. We assumed each fish settled on day 1 but simulated a time series that was as long as the observable lifespan of that fish (a fish that settled on the second-to-last day of observations received a two-day time series). After generating 100 time series, we calculated the proportion that matched the actual series of observations at that microsite (each time series had a small number of permutations, so 100 is a sufficient number of simulations). This is the probability of observing the data given the parameter values. After performing this calculation for each microsite, we took the product of all the probabilities as the likelihood for that parameter set.

We found maximum likelihood parameter values using a one-at-a-time Metropolis-Hastings algorithm with simulated annealing. These methods are explained in detail elsewhere (Kirkpatrick et al. 1983, Kirkpatrick 1984, Randelman and Grest 1986, Sandin and Pacala 2005); briefly, a candidate parameter value is sampled from an arbitrary distribution and used to calculate a likelihood, . This parameter value is accepted with probability q, a function of the ratio of to the likelihood of the most recently accepted parameter value, L:

(D.1)

The parameter z controls annealing. Initially z is set to a large value, so most candidate parameter values are accepted and the algorithm explores a large region of parameter space. After m iterations, z = z₀*exp(-0.05m), so the algorithm eventually accepts very few candidate values and converges on the maximum likelihood solution. As the algorithm progresses it generates a Markov chain of parameter values. After examining several preliminary runs for convergence, we used 1000 total iterations, discarded the first 250 values in the Markov chain and took the mean of the remaining, post-convergence values as Monte Carlo estimates of the maximum likelihood parameter values β. The Monte Carlo mean likelihood was also used to calculate AIC_C.

In this way we obtained parameter estimates and AIC_C for each maximum likelihood model. All programming was done in Matlab 7.0 (The MathWorks, Natick, Massachusetts, USA).

LITERATURE CITED

Kirkpatrick, S., C. D. Gelatt, and M. P. Vecchi. 1983. Optimization by simulated annealing. Science 220:671–680.

Kirkpatrick, S. 1984. Optimization by simulated annealing: quantitative studies. Journal of Statistical Physics 34:975–986.

Randelman, R. E., and G. S. Grest. 1986. N-city traveling salesman problem: optimization by simulated annealings. Journal of Statistical Physics 45:885–890.

Sandin, S.A., and S. W. Pacala. 2005. Fish aggregation results in inversely density-dependent predation on continuous coral reefs. Ecology 86:1520–1530.