Ecological Archives E091-040-A5

Steven E. McMurray, Timothy P. Henkel, and Joseph R. Pawlik. 2010. Demographics of increasing populations of the giant barrel sponge Xestospongia muta in the Florida Keys. Ecology 91:560–570.

Appendix E. Additional methods of randomization tests and bootstrap calculations.


Randomization tests were used to determine if population growth rate and the damping ratio were affected by time or location (Manly 1997, Caswell 2001). A(00-03) and A(03-06) were compared to examine temporal differences and A(CR15), A(CR20), A(CR30), and A(PR15) were compared to examine location differences. The records of state and fate of individuals were randomly permuted among the treatments being compared, keeping sample sizes fixed.  For each permutation, population matrices were constructed and λ and ρ were calculated. For temporal comparisons of population growth rate, a two-tailed test of the null hypothesis λ00-03 = λ03-06 was conducted with the statistic θ=|λ00-03–λ03-06| from a set of 10,000 permutations (Brault and Caswell 1993, Caswell 2001). Similar tests were conducted with ρ between time intervals and both population parameters between sites.


Bootstrap calculations were used to generate 95% confidence intervals for the population parameters λ and ρ and stage-specific mortality rates. Individuals and their records were resampled with replacement until a fixed sample size was drawn. For each of 10,000 resampled datasets, population matrices were constructed and λ, ρ, and stage-specific mortality rates were calculated. Confidence intervals were taken as the 0.025 and 0.975 percentiles of the distribution of each parameter. The percentile intervals were corrected for median bias of the bootstrap estimates (Efron and Tibshirani 1993; Caswell 2001). Bootstrap calculations and randomization tests were conducted using Resampling Stats software (vers. 3.2 for Excel, Resampling Stats, Arlington, Virginia, USA).


Brault, S., and H. Caswell. 1993. Pod-specific demography of killer whales (Orcinus orca). Ecology 74:1444–1454.

Caswell, H. 2001. Matrix population models. Sinauer, Sunderland, Massachusetts, USA.

Efron, B. and R. J. Tibshirani. 1993. An introduction to the bootstrap. Chapman and Hall, New York, New York, USA.

Manly, B. F. J. 1997. Randomization and Monte Carlo methods in biology. Second edition. Chapman and Hall, New York, New York, USA.

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